troy allan spencer thesis (pdf 18mb)
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Inverse Diffraction Propagation Applied to the Parabolic Wave Equation Model for Geolocation
Applications
Troy Allan Spencer
B. Eng Aerospace Avionics (Hons)
Cooperative Research Centre for Satellite Systems
Queensland University of Technology
THIS DISSERATION IS SUBMITTED IN PARTIAL
FULFILMENT OF THE REQUIREMENTS FOR THE
AWARD OF THE DEGREE
DOCTOR OF PHILOSOPHY
September 2006
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Statement of Authorship
The work contained in this thesis has not been previously submitted for a degree or
diploma at any other higher education institution. To the best of my knowledge and
belief, the thesis contains no material previously published or written by another
person except where due reference is made.
Signature:
Date:
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Key Words
Global Positioning System, Global Navigation Satellite System, Electromagnetic
Propagation Model, Inverse Diffraction Propagation, Parabolic Equation Model,
Huygens Principle Model, Blind Localisation, Passive Localisation, Geolocation,
Radio Frequency Interference
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Acronyms AAP Adaptive Array Processing
ADC Analogue-Digital Converter
AJ Anti-jam
APM Advanced Propagation Model
ATD Asymptotic Theory of Diffraction
C/A Course-Acquisition
CEP Circular Error of Probability
COTS Commercial Off-the-Shelf
CRPA Controlled Reception Pattern Antenna
DF Direction Finding
DFT Discrete Fourier Transform
DOA Direction of Arrival
DOD US Department of Defense
DOT US Department of Transport
DSB US Defense Science Board
DTT Discrete Trigonometric Transform
ECM Electronic Counter Measure
ECCM Electronic Counter Counter Measure
EEP Elliptical Error of Probability
EM Electromagnetic
ESM Electronic Support Measure
ESPRIT Estimate Signal Parameters via Rotational Invariant Technique
EW Electronic Warfare
FCT Fast Cosine Transform
FDM Finite Difference Method
FDOA Frequency Difference of Arrival
FEM Finite Element Model
FFT Fast Fourier Transform
FSS Fourier Split-Step
FST Fast Sine Transform
GIBC Generalised Impedance Boundary Condition
GO Geometrical Optics
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GNSS Global Navigation with Satellite Signals
GPS Global Positioning System
GSTAR GPS Spatial Temporal Anti-jam Receiver
GTD Geometrical Theory of Diffraction
HPM Huygens Principle Model
IBC Impedance Boundary Condition
IDP Inverse Diffraction Propagation
IDPELS Inverse Diffraction Parabolic Equation Localisation System
JDAM Joint-Direct Attack Munitions
IDP Inverse Diffraction Propagation
IFD Implicit Finite Difference
ION Institute of Navigation
JADE Joint Angle and Delay Estimation
JHU/APL John Hopkins University / Advanced Propulsion Laboratory
JLOC Jammer Location System
JSOW Joint Stand-Off Weapons
KLT Karhunen-Loeve Transform
LBC Leontovich Boundary Condition
LOP Line of Positions
LPE Low Probability of Exploitation
LPI Low Probability of Intercept
MFP Matched Field Processing
MFT Mixed Fourier Transform
ML Maximum Likelihood
MoM Method of Moments
MUSIC Multiple Signal Classification
NAVSTAR Navigation with Satellite Timing and Ranging
NB Narrow-Band
PEM Parabolic Equation Model
PHD Pisarenko Harmonic Decomposition
PDD US Presidential Decision Directive
PO Physical Optics
PPS Precise Positioning System
PTD Physical Theory of Diffraction
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PVT Position, Velocity, Time
RADAR Radio Detection and Ranging
RFI Radio Frequency Interference
RMS Root-Mean-Square
RSSI Received Signal Strength Indicator
SAASM Selective Availability and Anti-Spoofing Module
SAR Synthetic Aperture Radar
SPAWAR Space and Naval Warfare
SPE Standard Parabolic Equation
SPS Standard Positioning System
STAP Space-Time Adaptive Processing
SVD Singular Value Decomposition
TDOA Time Difference of Arrival
TDMA Time Division Multiple Access
TOA Time of Arrival
ULA Uniform Linear Array
UTD Uniform Theory of Diffraction
WAF Wall Attenuation Factor
WB Wide-Band
WSF Weighted Subspace Fitting
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Acknowledgement
The author expresses extended thanks to Dr Richard Hawkes from the Electronic
Warfare and Radar Division of Defence Science Technology Organisation (EWRD
DSTO) for his interest and valuable assistance. EWRD-DSTO radio technicians Mr
Christopher Pitcher and Mr Allan Padgham must also be recognised for their valuable
assistance with the field trials. A/Prof Rodney Walker is thanked for his interest and
insight, while Dr Tee Tang is acknowledged for always being available to discuss
microwave theory. This research program was performed with financial support from
the Cooperative Research Centre for Satellite Systems (CRCSS), for which the author
is thankful.
Troy Spencer – Geolocation Field Trials
Thanks must also be provided to my immediate family for they have provided
unconditional assistance and support. My sister Nickolet is deeply thanked for her
guidance, as she helped show that when the path taken is not the simplest, we learn so
much more about life.
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Abstract Localisation, which is a mechanism for discovering the spatial relationship between
objects, is an area that has received considerable research and development in recent
times. A common name given to localisation operations based on the absolute
reference frame of Earth is Geolocation. One important example of geolocation
research is E-911, where wireless carriers in the United States must provide the
location of 911 callers. The operation of E-911 can be based on either a network
configuration, or the Global Positioning System (GPS). With the importance of
localisation being acknowledged, a review concerning the vulnerability of the Global
Navigation Satellite System (GNSS) is provided as background and motivation for
this research. With the current vulnerability of GNSS, this dissertation presents the
results of a research program undertaken with the objective of developing an
electromagnetic localisation technique that can determine the relative position of GPS
Radio Frequency Interference (RFI) sources. Intended for operation in a hostile
environment, blind and passive localisation methodologies must be incorporated into
the developed model.
In performing localisation research, a background of current techniques is provided in
addition to a review of current electromagnetic propagation models. From the review
of propagation models, the Parabolic Equation Model (PEM) was chosen for
investigation concerning localisation. The selection of PEM is due to model
properties that are required for blind/passive localisation. The localisation system
developed in this research program is based on the integration of inverse diffraction
propagation (IDP) within the parabolic equation model. The title chosen for the
localisation method is Inverse Diffraction Parabolic Equation Localisation System
(IDPELS).
This thesis presents the simulation and field trial results of IDPELS. Under
simulation, the terrain or obstacle profiles were not based on any geodetic datum.
Any estimate provided by IDPELS under simulation is therefore a “Localisation”
solution. In the field trials however, IDPELS operation is referred to as
“Geolocation” as geodetic datum’s where used to determine the receiver’s position.
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Under simulation analysis, IDPELS operation was considered to provide good
promise as it could simultaneously perform localisation on multiple transmission
sources. In each investigated simulation scenario, a display of signals amplitude (dB
units) is displayed over the entire region. By determining the field convergence
regions, a localisation estimate of IDPELS is provided. By defining the convergence
regions as areas having the greatest signal amplitude values (i.e. ≥ 99%), elliptical
areas as low as 3.2m² were considered to indicate an excellent localisation capability.
With the theoretical validity of IDPELS operation in electromagnetics having been
established under simulation, further investigation into the practical feasibility of the
IDPELS was performed. The field trials positioned a continuous-wave (CW)
transmission source at a known location. By measuring signal phasors along a
straight section of road, the geodetic spatial-phase profile was used as the input signal
for IDPELS. Road sections used were cross-wise to the transmitter’s boresight.
Many data sets were recorded, each being made over a sixty second time period.
Different regions and ranges where used to continuously measure the spatial-phase
profile of the signal with fixed antennas in a moving vehicle. Such a measurement
process introduced an analogy with Synthetic Aperture Radar (SAR) processes. In
quantitating the accuracy of the IDPELS geolocation estimate in field trials, the linear
error of range and cross-range components was analysed. A free-space PEM model
was chosen for development of IDPELS and hence, data sets demonstrating properties
of a free-space environment were able to be considered suitable for testing of the
geolocation method. Data sets demonstrating free-space propagation characteristics
were measured at the base of the Mt Lofty ranges in South Australia, where the range
and cross-range error are respectively 3.14m, and 0.15m. Such low error values
clearly demonstrate the practical feasibility of IDPELS geolocation. With the
practical feasibility of IDPELS having been established in this research program, a
novel contribution to electromagnetic geolocation methodologies is provided. An
important characteristic of any geolocation technique concerns its robustness to
operate in a wide variety of possible environments. With continued development of
IDPELS, the robustness of this passive/blind geolocation technique can be enhanced.
Further assistance with geolocation of multiple transmission sources is also indicated
to be available by IDPELS, as shown in the simulation analysis.
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Table of Contents Chapter 1 - Introduction................................................................................................ 1
1.1 Background for Research – NAVSTAR GPS.............................................. 1
1.1.1 GPS Serviceability ............................................................................... 2
1.1.2 Dual Use............................................................................................... 2
1.1.3 Sample of GPS Applications................................................................ 3
1.2 GPS Susceptibility − Overview ................................................................... 3
1.2.1 Factors contributing to RFI Vulnerability............................................ 4
1.3 GPS Vulnerability Reports........................................................................... 4
1.3.1 Tactical Air Warfare ............................................................................ 5
1.3.2 The Global Positioning System: Assessing National Policies ............. 5
1.3.3 GPS Risk Assessment Study: Final Report.......................................... 6
1.3.4 Vulnerability Assessment of the Transportation Infrastructure Relying
on the Global Positioning System........................................................ 7
1.4 Interference Mitigation ................................................................................ 8
1.4.1 Implementation .................................................................................... 9
1.4.2 ECCM Comparison............................................................................ 10
1.5 Overview of Research................................................................................ 11
1.6 Research Objectives................................................................................... 11
1.7 Research Contributions .............................................................................. 13
1.8 References.................................................................................................. 15
Chapter 2 - Localisation.............................................................................................. 23
2.1 Introduction................................................................................................ 23
2.2 GPS RFI Localisation ................................................................................ 23
2.3 Localisation Taxonomy.............................................................................. 24
2.4 Localisation Parameters ............................................................................. 25
2.5 Electronic Warfare Localisation ................................................................ 27
2.5.1 Triangulation...................................................................................... 27
2.5.2 Trilateration........................................................................................ 29
2. 6 Precise Localisation Network Configurations .......................................... 30
2.6.1 Time Difference of Arrival (TDOA).................................................. 30
2.6.2 Frequency Difference of Arrival (FDOA) ......................................... 32
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2.7 Multiple Localisation Platforms................................................................. 33
2.8 Direction Finding ....................................................................................... 34
2.8.1 High Resolution Direction Finding.................................................... 36
2.8.2 Eigen-analysis .................................................................................... 38
2.8.2.1 Eigen decomposition Methods..................................................... 38
2.9 Propagation Model Localisation ................................................................ 39
2.9.1 Cellular Concept ................................................................................ 40
2.9.2 Propagation Database Correlation Model .......................................... 41
2.9.3 Matched Field Processing .................................................................. 41
2.10 References................................................................................................ 42
Chapter 3 - Inverse Diffraction Parabolic Equation Localisation System (IDPELS) . 53
3.1 Research Analogy ...................................................................................... 54
3.2 Research Objectives................................................................................... 55
3.3 Propagation Model Identification .............................................................. 56
3.4 Helmholtz Scalar Equation ........................................................................ 57
3.5 Boundary Conditions ................................................................................. 59
3.5.1 Signal Reflection................................................................................ 61
3.5.2 Brewster Angle .................................................................................. 61
3.5.3 Perfect Electric Conductor (PEC) ...................................................... 63
3.5.4 Classical and Impedance Boundary Conditions................................. 64
3.5.5 Open Boundary Requirement............................................................. 67
3.6 Multipath Distortion................................................................................... 68
3.7 Electromagnetic Propagation Models ........................................................ 70
3.7.1 Ray Tracing........................................................................................ 70
3.7.2 High Frequency Models..................................................................... 71
3.7.3 Finite Difference Model (FDM)......................................................... 71
3.7.4 Finite Element Model (FEM)............................................................. 72
3.7.5 Method of Moments Model (MoM)................................................... 72
3.7.6 Model Comparison and Selection ...................................................... 72
3.8 Parabolic Equation Model Development ................................................... 75
3.9 Standard Parabolic Equation (SPE) Approximation .................................. 77
3.9.1 Harmonic Frequency Assumption ..................................................... 77
3.9.2 Cylindrical Co-ordinate System......................................................... 77
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3.9.3 SPE Assumptions............................................................................... 79
3.9.3.1 Azimuth symmetry....................................................................... 79
3.9.3.2 Envelope function ........................................................................ 81
3.9.3.3 Far field Application .................................................................... 82
3.9.3.4 Slow envelope variation............................................................... 83
3.10 One-way Signal Propagation ................................................................... 84
3.11 Fourier Split-Step Propagation Solution .................................................. 87
3.11.1 Field Marching................................................................................. 90
3.12 Lower Boundary Condition - Signal Polarisation and Fourier
Transformations ....................................................................................... 94
3.12.1 Upper Boundary Condition - Transparency..................................... 98
3.13 Arbitrary Terrain and Obstacles............................................................. 100
3.13.1 Boundary Shift ............................................................................... 100
3.13.2 Boundary Decay............................................................................. 102
3.14 Horizontal Planar PEM .......................................................................... 103
3.14.1 Modelling Boundary Conditions.................................................... 105
3.15 Refractive Index Profile ......................................................................... 107
3.15.1 Wide-Angle Propagation Methods................................................. 109
3.16 Inverse Diffraction Propagation............................................................. 111
3.17 Conclusion ............................................................................................. 114
3.18 References.............................................................................................. 115
Chapter 4 - IDPELS Simulation................................................................................ 131
4.1 Objective .................................................................................................. 131
4.2 Simulation Procedure............................................................................... 131
4.3 Quantisation of Simulation Results.......................................................... 133
4.4 Test Cases ................................................................................................ 134
4.4.1 Block Scenario ................................................................................. 135
4.4.1.1 Window Domain of Input Signal ............................................... 137
4.4.2 Quantisation of Block Scenario ....................................................... 139
4.4.3 Wedge Scenario ............................................................................... 142
4.4.3.1 IDPELS Operation in NLOS Environment................................ 143
4.4.4 Multiple RFI Sources ....................................................................... 146
4.4.4.1 Three Interference Sources ........................................................ 146
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4.4.5 Long Range IDPELS Performance .................................................. 150
4.5 Segmented Antenna Arrays ..................................................................... 154
4.5.1 Two sensor array (10 elements in each sensor) ............................... 155
4.5.2 Nine sensor array (50 elements in each sensor) ............................... 158
4.6 Summary .................................................................................................. 159
4.7 Conclusion ............................................................................................... 162
4.8 References................................................................................................ 164
Chapter 5 - Geolocation Field Trials......................................................................... 167
5.1 Objective .................................................................................................. 167
5.2 Overview.................................................................................................. 167
5.2.1 Regional Characteristic .................................................................... 168
5.2.2 Radio Frequency Equipment............................................................ 169
5.3 Field Trial Methodology .......................................................................... 172
5.3.1 Field Trial Orientation ..................................................................... 172
5.3.2 Field Data Set Size........................................................................... 176
5.3.3 Least Square Fitting Polynomial...................................................... 176
5.3.4 Relative Doppler Shift ..................................................................... 177
5.3.5 Galilean Relativity ........................................................................... 177
5.3.6 Doppler Shift Transparency — Spatial-Phase ................................. 181
5.3.7 Input Signal Cross-range.................................................................. 185
5.4 Synthetic Aperture Radar (SAR) Analogy............................................... 188
5.4.1 SAR Development ........................................................................... 188
5.4.2 Focused SAR Array – Quadratic Phase Variation ........................... 189
5.4.3 Inverse Synthetic Aperture Radar (ISAR) ....................................... 190
5.5 Field Trial Regions................................................................................... 191
5.5.1 St Kilda Region................................................................................ 191
5.5.2 Mt Lofty Range Base Region........................................................... 194
5.6 Free-space Propagation ............................................................................ 197
5.7 Data Set Power Variation......................................................................... 199
5.8 Correlation of Signal Parameters ............................................................. 201
5.9 Test Signal Characteristics....................................................................... 203
5.9.1 Phasor Analysis................................................................................ 203
5.9.2 Signal Amplitude and Frequency..................................................... 205
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5.10 IDPELS Accuracy Analysis................................................................... 208
5.11 Field Trial Geolocation Results ............................................................. 209
5.11.1 Pine Creek Track............................................................................ 209
5.11.2 Woolshed road ............................................................................... 216
5.11.3 McEvoy Road ................................................................................ 219
5.11.4 Port Gawler Road........................................................................... 222
5.11.4.1 Rayleigh Fading ....................................................................... 222
5.11.4.2 Pt Gawler Road - Data Sets (08-09-10-11) .............................. 224
5.12 Field Trial Geolocation Error................................................................. 226
5.13 Conclusion ............................................................................................. 227
5.14 References.............................................................................................. 229
Chapter 6 - Thesis Conclusion .................................................................................. 233
6.1 References................................................................................................ 235
Chapter 7 - Recommendations.................................................................................. 237
7.1 IDPELS Precision Analysis ..................................................................... 237
7.2 Obstruction Modelling ............................................................................. 238
7.3 Wideband Propagation............................................................................. 238
7.4 Transmission Frequency .......................................................................... 239
7.5 Field Trial Procedure ............................................................................... 239
7.6 Two-Way Signal Propagation.................................................................. 240
7.7 3D Model ................................................................................................. 240
7.8 Huygens Principle Model — Wide Angle Propagation........................... 241
7.9 References................................................................................................ 242
Chapter 8 - Research Publications ............................................................................ 245
Appendix A - Huygens Principle Model................................................................... 247
A.1 References ............................................................................................... 249
Appendix B - Matlab Code – Field Trials................................................................. 251
B.1 Spatial-Phase Code.................................................................................. 251
B.2 Geolocation Code..................................................................................... 263
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B.3 NEWSINTR............................................................................................. 273
B.4 IDPELS XTICKLABELS........................................................................ 274
B.5 IDPELS Y TICKLABEL ......................................................................... 276
B.6 Load GPS field data file........................................................................... 277
B.7 Frequency Shift Code............................................................................... 280
B.8 Law of Cosine .......................................................................................... 282
Appendix C - Matlab Code - Simulation .................................................................. 291
C.1 PEM ......................................................................................................... 291
C.2 Signal Profile-to-Add............................................................................... 297
C.3 Control Test File ...................................................................................... 298
C.4 Inverse Diffraction Localisation .............................................................. 302
Appendix D - Huygens Principle Model Code ........................................................ 311
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List of Tables Table 1-1 Sample of Civil GPS Applications............................................................ 3
Table 1-2 Stand-alone Interference Mitigation Methods [42]................................... 8
Table 1-3 GPS AJ Evaluation and Comparison [52]............................................... 10
Table 2-1 Cellular Concept [66].............................................................................. 40
Table 3-1 Electric Properties of various Materials .................................................. 61
Table 3-2 Unique PDE solutions............................................................................. 67
Table 3-3 Model Comparisons ................................................................................ 73
Table 4-1 Location of Multiple Interference Sources (Figure 4-10) ..................... 146
Table 4-2 IDPELS Performance Comparison ....................................................... 159
Table 5-1 Field Trial Geolocation Error................................................................ 226
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List of Figures Figure 1-1 Locations of AJ-ECCM within GPS receiver [49] ............................... 9
Figure 2-1 Triangulation....................................................................................... 28
Figure 2-2 Trilateration........................................................................................ 29
Figure 2-3 Measuring difference in signal’s TOA................................................ 30
Figure 2-4 Eccentricity ......................................................................................... 31
Figure 2-5 TDOA hyperbolic isochrones (LOP) .................................................. 32
Figure 2-6 FDOA isofreq (LOP) .......................................................................... 33
Figure 2-7 Single Baseline Localisation ............................................................... 34
Figure 2-8 Monopulse DF system ........................................................................ 35
Figure 2-9 Phase Interferometry ........................................................................... 35
Figure 2-10 Eigen-analysis of Covariance Matrix.................................................. 37
Figure 3-1 Conic section analysis of second-order PDE ...................................... 58
Figure 3-2 Specular and Diffuse Reflection ......................................................... 60
Figure 3-3 Specular Reflection / Refraction of Horizontally Polarised Signal..... 60
Figure 3-4 Linear Reflection Coefficients for Wet Ground.................................. 62
Figure 3-5 Tangential field component variation ................................................. 65
Figure 3-6 Urban Multipath.................................................................................. 69
Figure 3-7 Mikhail Aleksandrovich Leontovich................................................... 75
Figure 3-8 Cylindrical Coordinate System ........................................................... 78
Figure 3-9 Amplitude of Azimuth Symmetric Field............................................. 80
Figure 3-10 Envelope function of diffracting field................................................. 81
Figure 3-11 Open Boundary FSS-PEM .................................................................. 87
Figure 3-12 P-domain, Z-domain Relationship ...................................................... 88
Figure 3-13 FFT Bit-reversed output [106] ............................................................ 96
Figure 3-14 Upper Boundary Condition of Vertical Planar PEM .......................... 99
Figure 3-15 Boundary Shift .................................................................................. 101
Figure 3-16 Forward PEM solution – Signal Amplitude (dB) ............................. 102
Figure.3-17 Boundary Decay [90] ........................................................................ 103
Figure 3-18 Horizontal Planar PEM Propagation domains .................................. 104
Figure 3-19 FFT and DTT basis function comparison ......................................... 106
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Figure 3-20 Horizontal Planar PEM solution – Signal Amplitude (dB)............... 107
Figure 3-21 Refractive Index Profiles and corresponding ray diagrams [24]....... 109
Figure 3-22 Mathematical Inversion of Propagator.............................................. 113
Figure 3-23 IDPELS field trial – Measurement of Input Signal........................... 114
Figure 4-1 Properties of Field Diagram (PEM) .................................................. 132
Figure 4-2 Forward propagation (i.e. PEM) – Block.......................................... 135
Figure 4-3 Inverse propagation (i.e. IDPELS) - Block....................................... 136
Figure 4-4 Inverse Propagation (block) – Input Signal (Solution Domain) ....... 138
Figure 4-5 Quantisation of localisation accuracy – Block scenario ................... 140
Figure 4-6 Magnification of Figure 4-5 .............................................................. 140
Figure 4-7 PEM - Wedge.................................................................................... 142
Figure 4-8 IDPELS – Wedge.............................................................................. 143
Figure 4-9 IDPELS – Input Signal (Solution Domain) ...................................... 145
Figure 4-10 Forward propagating field with multiple sources ............................. 147
Figure 4-11 IDPELS field with multiple sources ................................................. 148
Figure 4-12 PEM – Domain Range 6000m .......................................................... 150
Figure 4-13 IDPELS – 5000m range to Source .................................................... 151
Figure 4-14 IDPELS – 3000m range to Source .................................................... 152
Figure 4-15 IDPELS – 1000m range to Source .................................................... 153
Figure 4-16 2 Sensors (with 10 elements) ............................................................ 155
Figure 4-17 Direction Finding Analysis with two Sensor .................................... 156
Figure 4-18 9 Sensor array (with 50 elements)..................................................... 158
Figure 4-19 IDPELS Uncertainity versus Range.................................................. 159
Figure 5-1 Helix Transmission Antenna (Positioned for Mt Lofty data sets) .... 169
Figure 5-2 EB200 receiver.................................................................................. 170
Figure 5-3 Rojone Genius GPS Unit .................................................................. 171
Figure 5-4 EB200 Signal Measurements ............................................................ 172
Figure 5-5 Isotropic Transmitter Analogy .......................................................... 173
Figure 5-6 Measurement of Phase values (Symmetric Example)....................... 174
Figure 5-7 Quadratic Spatial-Phase Profile (Symmetric Example) .................... 175
Figure 5-8 Least Square Fitting Quadratic Polynomial ...................................... 176
Figure 5-9 Resultant Doppler Frequency............................................................ 178
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Figure 5-10 Relative receiver speeds governing Doppler shift ............................ 178
Figure 5-11 Receiver Range from Transmitter..................................................... 179
Figure 5-12 Frequency Shift ................................................................................. 179
Figure 5-13 Linear Phase variation for stationary Receiver ................................. 182
Figure 5-14 Motion of EB200 Receiver ............................................................... 182
Figure 5-15 Measured Phase and Linear Phase: Mt Lofty Base Data Set (03) .... 183
Figure 5-16 Measured Spatial Phase: Mt Lofty Base Data Set (03)..................... 184
Figure 5-17 Cross-range Distance: Mt Lofty Base Data Set (03)......................... 185
Figure 5-18 Spatial Phase: Mt Lofty Base data sets (02-03-04)........................... 186
Figure 5-19 Estimated Spatial Phase: Mt Lofty Base data sets (02-03-04) .......... 187
Figure 5-20 Geodetic Overview: Mt Lofty Base data sets (02-03-04) ................. 187
Figure 5-21 Circular Wavefront Phase Variation [17] ......................................... 189
Figure 5-22 DSTO Radio Research Station - St Kilda (looking South) ............... 191
Figure 5-23 St Kilda Map ..................................................................................... 192
Figure 5-24 McEvoy road..................................................................................... 192
Figure 5-25 Pt Gawler road .................................................................................. 193
Figure 5-26 Mt Lofty Range Base Region............................................................ 194
Figure 5-27 Mt Lofty Range Base, Nominal Field-of-View ................................ 195
Figure 5-28 Free-space Environment – Pine Creek Track.................................... 195
Figure 5-29 Pine Creek Track Tree Obstructions - Data set (02) ......................... 196
Figure 5-30 Tree Obstructions - Woolshed Road ................................................. 196
Figure 5-31 Free-space Loss................................................................................. 197
Figure 5-32 Fresnel Zones .................................................................................... 198
Figure 5-33 Mt Lofty base (free-space model) ..................................................... 198
Figure 5-34 Signal Power Variation: Mt Lofty Range Base................................. 199
Figure 5-35 Signal Power Variation: St Kilda Region ......................................... 200
Figure 5-36 Correlation of Parameter Variation................................................... 201
Figure 5-37 Phasor Components: McEvoy road, Data set (03) ............................ 203
Figure 5-38 Relative Speeds of EB200 Receiver ................................................. 204
Figure 5-39 McEvoy road data set (03): Range.................................................... 205
Figure 5-40 Signal Amplitude: McEvoy road, Data set (03)................................ 206
Figure 5-41 Audio Signal Spectrum: McEvoy Road - data set (03)..................... 206
Figure 5-42 Audio Signal Spectrum: Pine Creek track - data set (04) ................. 207
Figure 5-43 Consecutive Data Sets: Pine Creek Track......................................... 209
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Figure 5-44 Cross-range of Input Signal: Pine Creek Track (02-03-04) .............. 210
Figure 5-45 Pine Creek Track (02-03-04) Geolocation........................................ 211
Figure 5-46 Geodetic Overview: Pine Creek track - Data Sets (07-08-09) .......... 212
Figure 5-47 Cross-range of Input Signal: Pine Creek Track (07-08-09) .............. 213
Figure 5-48 Pine Creek Track (07-08-09) Geolocation........................................ 214
Figure 5-49 Power Variation: Pine Creek Track Data Sets (02) and (09) ............ 215
Figure 5-50 Consecutive Data Sets – Woolshed Road ......................................... 216
Figure 5-51 Woolshed Road – Power Variation................................................... 217
Figure 5-52 Geodetic Overview – Woolshed Road Data Sets (16-17)................. 218
Figure 5-53 Woolshed Road (16-17) Geolocation................................................ 218
Figure 5-54 Receiver Position – McEvoy Road Data Sets (02), (03) & (04) ....... 219
Figure 5-55 Power Variation – McEvoy Road (02)-(03)-(04).............................. 220
Figure 5-56 Geodetic Overview – McEvoy Road Data Set (07) .......................... 220
Figure 5-57 Cross-range of Input Signal – McEvoy Road (07)............................ 221
Figure 5-58 McEvoy Road (07) – Geolocation Estimate ..................................... 221
Figure 5-59 Port Gawler Road – General Power Variation.................................. 222
Figure 5-60 Port Gawler – Rayleigh Fading......................................................... 223
Figure 5-61 Geodetic Overview – Port Gawler Road (08-09-10-11) ................... 224
Figure 5-62 Cross-range of Input Signal – Port Gawler Road (08-09-10-11) ...... 225
Figure 5-63 Port Gawler Road (08-09-10-11) Geolocation.................................. 225
Figure A-1 Huygen’s Propagation Principle ....................................................... 247
Figure A-2 HPM Wide Propagation Angle ......................................................... 248
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Chapter 1 - Introduction
1.1 Background for Research – NAVSTAR GPS
“With the quiet revolution of NAVSTAR, it can be seen that these potential uses are
limited only by our imagination”
- Bradford W. Parkinson
This foresight [1] concerning the use of NAVSTAR GPS provided by Dr Bradford
Parkinson (founding NAVSTAR program director [2] ) in 1980, can currently be
considered a true reflection due to the prolific integration of GPS systems into modern
infrastructure. The current application of GPS has far out-reached the original
expectations of system designers back in the 1970s. During this time period when
GPS technology and systems were being developed, few could have anticipated how
much GPS would burgeon into a new capability with application in many different
areas. Not only have defence force operations been revolutionised with GPS [3], civil
organisations have intensely adopted GPS for employment in a wide range of
applications such as network synchronisation [4], archaeological discovery [5], and
law enforcement [6, 7]. An excellent review of GPS applications is provided in [8].
While the use of GPS continues to expand, current civil GPS receivers are vulnerable
to radio frequency interference [9]. New industrial and civil implementations that
rely on GPS have not properly addressed the issue of disruption of service in their
design [10]. While modernisation of GPS [11] will increase the robustness of the
system and reduce its susceptibility to unintentional interference, the impact of
intentional interference sources will remain substantial. This dissertation will present
research and development of a localisation technique intended for operation against
GPS radio frequency interference (RFI) sources. The methodology that is
investigated concerns the application of electromagnetic propagation models to
determine the relative position of interference transmission sources. The source
location is estimated by propagating a measured signal profile with the principle
1
identified in this thesis as Inverse Diffraction Propagation (IDP). The propagation
model that was investigated is the Parabolic Equation model (PEM). The reference
given to the application of IDP to PEM was phrased as the Inverse Diffraction
Parabolic Equation Localisation System (IDPELS). The ultimate objective or this
localisation research is to help ensure the availability of Standard Positioning System
(SPS) [12] signals for civilian users.
1.1.1 GPS Serviceability
Since the inception of a spaced-based service offering a position, velocity and time
(PVT) information to a user that was not subject to the limitations of weather, time or
availability [13], it was recognised that the proposed NAVSTAR GPS system would
provide utility for many additional users over the US military [14]. While the idea
concerning civilian use was known, it was only lightly addressed by founders of the
system and never formally incorporated into the planning process. Civil access to the
unencrypted signal was however permitted and members of the public who saw the
vast potential of GPS for use in peaceful applications began development and
commercialisation of GPS technology [15].
1.1.2 Dual Use
While having proven highly valuable for the military in both Gulf Wars [16], this
military success concerning GPS was primarily due to the advantages and cost
effectiveness offered by civilian, commercial off-the-shelf (COTS) merchandise [17].
During the first Gulf war, over 9000 COTS GPS receivers were purchased by the US
military [18]. With operational efficiency being available with GPS integration into
applications, there is a continuing trend concerning dual use of GPS equipment. With
dual use, the commercial sector contributes a substantial percentage of funding for
research and development of new technology. As a result of this process, dual use
has led to the production of many items that are now part of everyday life. Another
important example of dual use concerns the development of the internet, which
originated as networking research for the U. S. Department of Defense (DoD) in 1969
[19]. In equivalence to the internet, GPS has become an information technology that
is emerging as part of the global information infrastructure [20].
2
1.1.3 Sample of GPS Applications
To demonstrate the importance and wide spread application of GPS into modernised
society, an extended version of the listing provided by Parkinson [2] showing civilian
GPS applications, is displayed in Table 1-1.
Category Application Mining Proximity Warning System
Ore body delineation Vehicle tracking Drill positioning Real-time assistance for dozer operator
Air Navigation Non-precision approach and landing Domestic / Oceanic en-route Remote areas Helicopter operations Collision avoidance Air traffic control Unmanned Aircraft
Land Navigation Vehicle monitoring Schedule improvement Minimal routing Law enforcement
Marine Navigation Harbour approaches / departures Inland waterways Oceanic / Coastal
Static Positioning and Timing Offshore resource exploration Hydrographic surveying Time synchronisation Geographic information systems State Border Identification
Space Ionospheric Modelling In-flight / orbit determination Re-entry / landing Attitude measurement
Search and Rescue Position reporting and monitoring Rendezvous Coordinate search Collision avoidance
Environmental precision agriculture/ Agro-modelling flora / fauna mapping in wildlife reserves Air Pollution Monitoring Water level monitoring
Table 1-1 Sample of Civil GPS Applications
1.2 GPS Susceptibility − Overview
While the current technology basis for GPS can be considered to be mature [21], the
susceptibility of the Standard Positioning System (SPS) [22] to radio frequency
interference (RFI) is significant [23]. Given the wide variety of applications with
GPS as highlighted in Table 1-1, there have been numerous reports indicating the
3
vulnerability of GPS to interference. All of these reports have indicated the real
susceptibility of GPS to RFI. In the GPS Vulnerability Reports section, four major
reports that have been released to the public will be reviewed.
1.2.1 Factors contributing to RFI Vulnerability
As already indicated, GPS receivers that are intended for civil applications are highly
susceptible to the consequences of RFI. This in part is due to various factors. One
important factor is the extremely weak signal power levels that are received on Earth
[24]. With the power level of a received GPS signal being as low as -160dBW, an
interference signal with a small power level of 4.5pW will disable the receiver [25].
While numerous vulnerability reports have been made available to the public, the
additional cost associated with interference mitigation may explain why service
disruptions has not been properly addressed in the design of civil receiver systems
that rely on the coarse acquisition (C/A) code [10]. Another reason why GPS is
susceptible to malevolent organisations or people that wish to intentionally interfere
with GPS signals, concerns the unrestriction of key features concerning the SPS.
With all information concerning SPS available to everyone, an enemy will be able to
identify the interrelated purposes, parameters and processes of the system [26]. With
such knowledge, the vulnerability of SPS to Electronic Counter Measures (ECM) that
will disrupt or deny operation should be considered to be highly realistic.
It should be noted that unlike the SPS, military receivers that use the encrypted
precision satellite signals [27] are not subject to such a high susceptibility. This is
because during 1998, the chairman of the Joint Chiefs of Staff issued a Selective
Availability and Anti-Spoofing Module (SAASM) [28] mandate. This mandate
required Precise Positioning Service (PPS) users to procure SAASM only and to cease
use of non-SAASM equipment after October 1, 2002 [29].
1.3 GPS Vulnerability Reports
Numerous reports are currently available to the public that have considered or
investigated the impact of RFI sources against GPS. The following four reports will
be reviewed in this dissertation:-
1. Tactical Air Warfare (1993)
2. The Global Positioning System: Assessing National Policies (1995)
4
3. GPS Risk Assessment Study: Final Report (1997)
4. Vulnerability Assessment of the Transportation Infrastructure Relying on
the Global Positioning System (2001)
1.3.1 Tactical Air Warfare
The first indication of GPS vulnerability was made when the initial operational
capability of GPS was declared in 1993. During November of that year, the US
Defense Science Board (DSB) published the “Tactical Air Warfare” [30] report,
which was released to the public in December. In this report, it was indicated that
GPS receivers were vulnerable to jamming, particularly in acquisition mode at very
long ranges from low powered jammers. With tactical aircraft delivering GPS-aided
weapons, Electronic Counter Counter Measures (ECCM) were recommended to be
further developed so that a jammer could not break GPS tracking on short-range
missiles such as Joint Direct Attack Munition (JDAM) [31] or Joint Stand-off
Weapon (JSOW) [32]. While not originally intended for the public, this report saw
missile manufacturers expeditiously move to equip weapons with Anti-Jam (AJ)
systems.
1.3.2 The Global Positioning System: Assessing National Policies
In 1995, the RAND Corporation provided the “The Global Positioning System:
Assessing National Policies” report [33], which described the findings of a one-year
GPS Policy study. The RAND report identified major opportunities and
vulnerabilities created by GPS for the U.S. defence, commercial and foreign policy
interests.
In relation to the National Security Assessment provided by the RAND report, threats
concerning the successful use of GPS were grouped into Internal Threats and External
Threats. Internal threats were made in recognition of military use and concerned the
mismanagement of the systems, inadequate funding for operation and maintenance,
and excessive reliance on civilian GPS equipment. External threats to GPS originate
outside the direct control of the U.S. government. Evaluation of threats was based on
the direction taken by the threat, and whether the threat was unintentional or
intentional. As mentioned in the introduction a threat may be directed at the GPS
5
signal itself, or towards the space or ground segments. Unintentional threats could
include phenomena such as natural disasters and malfunctions which impact any of
the GPS segments. However the most significant threat considered is the denial of
GPS signals, which is due to sources that are intentional or unintentional. Smart
jamming and noise jamming were considered in the report, where smart jamming is
another terminology concerning Spoofing signals and noise jamming is an attempt to
overwhelm a receiver with radio noise. Both Narrowband (NB) and Wideband (WB)
jamming sources [34] were considered, together with narrow beam steering and
adaptive nulling of WB noise with a controlled radiation pattern antenna (CRPA).
Airborne jammers were also considered to be more effective than ground-based
jammers due to the greater coverage and that desired satellite signals are partially
filtered with a spatial filter. As with the Tactical Air Warfare report, the RAND
report recommended further development of anti-jam (AJ) capabilities. In such an
environment, adversaries must employ high power jammers and will therefore
become attractive targets for precision-guided munitions. In the Inteference
Mitigation section, an overview of techniques such as CRPA that reduce system
degradation is provided.
1.3.3 GPS Risk Assessment Study: Final Report
The “GPS Risk Assessment Study: Final Report” [35] was presented in January 1999.
This was a study performed by the Johns Hopkins University Applied Physics
Laboratory (JHU/APL) to assess the risks of reliance upon the Global Positioning
Satellite (GPS) navigation system. It also proposed augmentation systems to assist
GPS integrity, availability and accuracy for air navigation.
This report recognised that intentional interference is by far the largest risk area
associate with GPS. It however assumed that sufficient resources would be available
to vector aircraft away from jammed regions, hence this threat was not considered a
risk to safety-of-life. Possible disruption to traffic control and flight schedules were
however considered to be substantial. As a result, it was recommended that methods
should be developed to monitor, report and locate interference sources. This was
primarily to act as a deterrent to people who wished to intentionally interfere with
GPS signals.
6
1.3.4 Vulnerability Assessment of the Transportation
Infrastructure Relying on the Global Positioning System
On September 10, 2001 just hours before the appalling terrorist attack on the World
Trade Centre and the Pentagon [36], the “Vulnerability Assessment of the
Transportation Infrastructure Relying on the Global Positioning System” report [37]
was released by the U.S. Department of Transport (DoT) to the public. The report
was made in response the Presidential Decision Directive 63 (PDD-63) made on the
22nd May 1998 [38] which concerned vulnerability evaluation of the national
transportation infrastructure that relies on GPS.
The DoT assigned this task to the Volpe Transportation System Centre in Cambridge,
Massachusetts. This Volpe report removed the thought that GPS could be used a sole
means of navigation due to the vulnerability of GPS. The study noted that GPS is
susceptible to unintentional disruption from such causes as atmospheric effects, signal
blockage from buildings and interference from communication equipment, as well as
to potential deliberate disruption. It contained a number of recommendations to
address the possibility of disruption and ensure the safety of the national
transportation infrastructure. As with the JHU/APL report, one of the 16
recommendations made in the Volpe report indicated that systems and procedures
should be implemented or utilised to monitor, report, and locate interference in any
application where loss of GPS is not tolerable. Given the importance and repeated
recommendation to find the location of the interference source, this direction was
chosen for research.
7
1.4 Interference Mitigation
While the vulnerability of GPS is well recognised, there is also a wide variety of
interference mitigation techniques that can be incorporated within the design of GPS
receivers. The conflicting pair of actions as represented by mitigation technology,
and RFI sources describe the main operations involved with Electronic Warfare (EW)
[39]. EW concerns the use of electronic systems to control the electromagnetic (EM)
spectrum for the detection and impediment of adversarial unit, or the protection of
allied units.
The action of radio frequency interference (RFI) sources is the prevention of another
system from effectively using the section of the electromagnetic spectrum. Such an
action classifies the RFI source as an Electronic Counter Measure (ECM) [40]. To
overcome system degradation due to ECM, actions that ensure the effective use of the
electromagnetic spectrum despite the presence of ECM can be referred to as
Electronic Counter Counter Measures (ECCM) [41]. Mitigation technology that acts
to minimise the unwanted system degradation due to RFI can therefore be classified
as ECCM.
For applications based on using a stand-alone GPS receiver, Gray et al [42] groups the
mitigation technology into three categories. These categories are shown in Table 1-2.
Table 1-2 Stand-alone Interference Mitigation Methods [42]
8
As indicated in Table 1-2, complexity and therefore cost increases proportionally with
the resistance level against interference. The method offering the greatest protection
from interference is based on Space-Time Adaptive Processing (STAP) [43]. With
respect to the military environment, a new missile anti-jamming (AJ) system is based
on STAP and is referred to as the GPS spatial temporal anti-jam receiver (G-STAR)
[44]. G-STAR is an enhancement of the Controlled Reception Pattern Antenna
(CRPA) [45], which places antenna nulls in the direction of interference sources. The
G-STAR enhancement concerns the additional steering of beams towards targeted
satellites for optimal signal reception. While this example demonstrates that RFI
signals can be operationally ignored, the complexity, cost and classification of this
technology has seen that it is not commercially available to the public. Cost
efficiency is as important as operational capability to the public.
1.4.1 Implementation
With the exception of adaptive analogue-to-digital converters (ADC) [46], the
categories shown in Table 1-2 can be based on the implementation with respect to the
tracking loop [47, 48] in the receiver. A block diagram provided by Scott [49]
portrays where ECCM technology can be implemented within a receiver and is shown
in Figure 1-1.
Figure 1-1 Locations of AJ-ECCM within GPS receiver [49]
9
Post-correlation methods are implemented in the tracking loops of receivers and
improve tracking thresholds [50]. Post-correlation methods are generally a software
function and require minimal if any hardware modification. Being software based,
these methods will not significantly increase the power consumption of the receiver
allowing batteries to be operated for longer time periods.
Pre-correlation methods are applied prior to the tracking loops and provide the
greatest immunity against all forms of RFI. They however require a significantly
greater signal processing capability and are primarily based on Adaptive Array
Processing (AAP) [51].
1.4.2 ECCM Comparison
A comparison analysis of GPS anti-jam methods provided by Casabona et al [52] is
displayed in Table 1-3. This table indicates the effectiveness of different ECCM
methods against various forms of interference, while also indicating cost and size for
the anti-jam capability. As shown, the most effective methods also have the greatest
cost, size and receiver architectural alteration (as indicated in Retrofit column).
Table 1-3 GPS AJ Evaluation and Comparison [52]
10
1.5 Overview of Research
While there are many factors that can threaten the reliable operation of GPS, this
research program is solely focussed on radio frequency interference (RFI) sources.
Other threats could be space-based and involve attacks on satellites with propelled
pellets [3, 53]. Meteorites are also phenomenon that could impact operational
satellites. While the possibility of such actions occurring is realistic, the probability
of such events is significantly less than malicious organisations or people transmitting
interference signals.
While the ability to ensure successful operation with GPS despite the presence of
interference signal is possible, this is restricted to the military environment. Financial
cost plays a governing role in commercial operations. To ensure safe operation for
the entire commercial sector, interference sources must be deactivated. The ability to
determine the relative position of interference sources is therefore a necessary
requirement to ensure GPS signal availability. This requirement has seen research
and development of jammer location systems (JLOC) [54] and provides the objective
for this research program.
1.6 Research Objectives
Localisation, which is a mechanism for discovering the spatial relationship between
objects, is an area that received considerable research and development in recent
times. One such example of localisation research concerns the development of E-911
[55]. Localisation is the ultimate aim of this research program to help ensure proper
authorities are able to promptly determine the relative position of interference sources
against GPS and provide their deactivation.
In geophysical applications concerning oil or gas exploration [8], it has been widely
recognised that model propagation methods provide greater accuracy and finer
resolution in comparison to conventional searching methods [56]. Further
information concerning geophysical exploration methods can be found in references
such as [57-59].
11
With the potential benefits offered by propagation models for localisation, the chosen
research path would focus on investigating a suitable electromagnetic propagation
model. Validity for such a research option in the electromagnetic environment has
already been provided by Gingras et al [60] with Matched Field Processing (MFP).
Research by Wolfle et al [61] concerning localisation in multipath environments also
adds further weight to this investigation. An important objective for this research is
to determine if an electromagnetic propagation model can be used to provide blind
and passive geolocation of identified transmission sources in near real-time operation.
It should be noted that MFP is capable of performing blind/ passive localisation with
EM propagation models, it however has various problems. One substantial problem is
that thousands of replica fields are usually generated. This limits the potential of
MFP to perform in near real-time operation. Further discussion of MFP is provided in
Chapter 2.
In Chapter 3, a review of various electromagnetic (EM) wave propagation models is
provided. From this model review, the PEM was chosen for investigation. This is
because the Fourier Split-Step PEM (FSS-PEM) operates with an open-boundary
configuration, which is a requirement for blind localisation. Another important
reason is because Tappert [62] has investigated the capability of PEM to perform
localisation in the underwater acoustic domain. In performing acoustic localisation,
Tappert propagates the conjugate of the received continuous wave (CW), which is
referred to as “backpropagation”. The Tappert approach assumed a fixed
environment and searched for a focusing point which revealed the location of the
acoustic source. For geoacoustic inversion, Collins et al [63] incorporates
“backpropagation” with MFP to allow not only the acoustic source to be localised, but
also estimate environmental parameters of the ocean with an appropriate high-
resolution cost function. This process of incorporating “backpropagation” with MFP
is called “focalization”, where a variety of propagation models are applied.
An important objective of this research concerns evaluating if the localisation
capability of FSS-PEM as demonstrated in the acoustic domain, can be transferred to
the electromagnetic domain. The forward field propagating capability of the acoustic
FSS-PEM having been transferred to the electromagnetic domain as established by
12
Ko et al [64], Dockery [65] and Kuttler et al [66]. With this established background
concerning the transformation of PEM properties between different propagating
environments, the potential of PEM performing blind and passive electromagnetic
localisation with IDP is highly credible and demands investigation.
1.7 Research Contributions
One of the underlying important contributions of this dissertation concerns its
assistance it increasing the public awareness of GNSS vulnerability to system
degradation. This was provided be a comprehensive survey discussed in this chapter,
which also highlights the motivation for this research.
With respect to localisation\geolocation, a background is provided covering a wide
variety of different methodologies that can be operated to allow the location of
sources transmitting identified signals to be determined. The operational
requirements and limitation concerning each of the techniques is also elucidated. By
analysing different procedural characteristics and limitation, elements that define the
optimal localisation\geolocation strategy are identified. This qualitative information
contributes to information provided by Adamy [67], and knowledge ready available
concerning quantification of major existing geolocation techniques as demonstrated
by Poisel [68].
An analysis of electromagnetic propagation models in performing blind\passive
localisation is also provided. Operational characteristics and limitation associated
with a wide variety of electromagnetic propagation model is elucidated and provides
reasoning for selecting PEM as the basis for investigation in this research. The
discussion concerning propagation model characteristics contributes to the model
comparison provided by Hubing [69].
An important research contribution is based on the experiment results of field trials,
which were based on a single-sensor that was moved while measuring an
electromagnetic field profile. Analysis of the field results provide an important
13
contribution to understanding the realistic impact that various factors have on
electromagnetic signal propagation and localisation.
The end product of this research program is the operational validity of a novel
electromagnetic localisation technique that was independently developed. The
potential of this research contributing to society may be significant, particularly in
areas where safety-of-life is an important parameter in operations. One example
concerns public safety at airports, where an array of antennas will already be
established. In such an environment, the option of incorporating the Inverse
Diffraction Propagation methodology into existing networks to perform localisation
will be an efficient operation, both in terms of logistics, setup and operation. This is
however dependant on further research and field trials of IDP operation based on
network configuration. Should this ever be realised, an important contribution
concerning public safety will be established.
While the parabolic model was chosen for investigation, another important
contribution that is not discussed in this dissertation concerns the development of
another electromagnetic propagation model. This model has characteristics necessary
for blind\passive localisation and was developed in conjunction with this research
program. The title given to the novel electromagnetic propagation model is
Huygens’s Principle Model (HPM) and was initially developed by Hawkes [70]. An
overview of this model is provided in Appendix A. A synergy is considered to exist
between PEM and HPM and further geolocation research that incorporates both
propagation models will provide robust enhancement for the geolocation
methodology.
Finally, a substantial contribution to techniques that perform localisation on identified
interference signal is provided by this research. By allowing the public to be aware of
such a capability, the intention of people or organisations that wish to disrupt or cause
harm to users of GPS, should be diminished. The ultimate aim of ensuring the
availability of GPS signals is also further established.
14
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15
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16
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18
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19
[48] P. Ward, "Satellite Signal Acquisition and Tracking," in Understanding GPS
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[53] "Soviet Military Power," Union of the Soviet Socialist Republic (USSR),
Moscow 1983.
[54] A. Brown, D. Reynolds, D. Roberts, and S. Serie, "Jammer and Interference
location system - Design and Initial Test Results," presented at Institute of
Navigation, Nashville, TN, 1999.
[55] Z. Biacs, G. Marshall, M. Moeglein, and W. Riley, "The
Qualcomm/SnapTrack Wireless-Assisted GPS Hybrid Positioning System and
Results from Initial Commercial Deployments," presented at ION GPS 2002,
Oregon Convention Center, Portland, Oregon, 2002.
[56] S. Phadke, D. Bhardwaj, and S. Yerneni, "Wave equation based migration and
modelling algorithms on parallel computers," presented at Proceedings of SPG
(Society of Petroleum Geophysicists) second conference, Chennai, India, 1998.
20
[57] S. M. Hill, "Biogeochemical Sampling Media for Regional-to-Prospect-Sclar
Mineral Exploration in Regolith Dominated Terrains of the Curnamona
Province and Adjacent Areas in Western NSW and Eastern SA," Proceedings
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Exploration (CRC LEME) Regional Regolith Symposia, pp. 128 - 133, 2004.
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Analysis and Trend Exploration:A Case Study of the Mississippi Delta Survay,
Southeast Louisiana," Canadian Journal of Exploration Geophysics, 34(1 &
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[61] G. Wolfle, R. Hoppe, D. Zimmeramnn, and F. M. Landstorfer, "Enhanced
Localization Technique within Urban and Indoor Environments based on
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Florence, Italy, 2002.
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propagation in the troposphere," Radio Science, 26(2), pp. 381-393, 1991.
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House Publishers, 2001.
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Canada, 2005.
22
Chapter 2 - Localisation
2.1 Introduction
An area that has received considerable research and development in recent times is
the mechanism that discovers the spatial relationship between objects. This process is
referred to as localisation and has been extensively applied. This research program is
concerned with using passive receiver measurements to perform blind localisation of
interference transmitters. Any localisation operation that uses Earth as the absolute
frame of reference is also known as Geolocation [1]. An excellent reference that
provides a ready source of material to examine and quantify the performance of major
existing geolocation techniques is presented by Poisel [2]. Other areas where
localisation has been applied include autonomous mobile robot navigation [3], local
neural networks [4], E-911 [5] and airborne electronic warfare (EW) systems [6].
2.2 GPS RFI Localisation
GPS is absolutely critical for safety-of-flight aviation, this is particularly true in the
United States and other countries that originally intended to decommission various
land-based navigation aids for transitions to a satellite-based system. While early
Federal Aviation Administration (FAA) programs focussed on RFI prevention, it
latter became clear the GPS would require a significantly greater real-time RFI source
localisation capability in comparison with previous radio-navigation systems [7].
Development of real-time technology is in part due to the extensive application and
importance of GPS [8].
The FAA currently has a multi-faceted interference mitigation program that addresses
all aspects of RFI issues in relation to GPS [7]. This is primarily due to the many
incidents of reported GPS degradation and denial. Several examples include,
1. GPS denial in the New York / New Jersey area during December 1997 and
January 1998 [9]
2. GPS denial in during April, 1999 in Chesterfield, North Carolina [10]
3. Simultaneous receiver outages offering no redundancy protection [11]
23
The above listed examples are associated with unintentional interference. These and
other incidents were based on high-level GPS test signals being transmitted,
misbehaving pre-amplifiers in TV antennas or GPS system failure. Subsequently
there is a wide range of disturbance and interference concerning GPS that can
originate in completely unexpected sources [12]. Consequently the FAA has a
primary focus with fielding RFI localisation devices upon various platforms such as
aircraft, or with fixed or handheld platforms [7]. The process in performing
localisation of GPS RFI is important as it will assist in protecting the availability of
GPS signals by allowing the prompt deactivation of interference sources. With the
increased efficiency made possible with GPS implementation in many operations,
reliance on GPS has developed to a point that could lead to serious problems if the
service is disrupted. While a small number of high powered jamming sources may be
simple to locate, the ability to locate many low powered jammers over a wide area is
considered a serious problem [13]. Currently the loss of GPS may not pose a
significant risk to safety-of-life operations, however Baker [14] indicates that large
scales outages could have significant economic consequences.
2.3 Localisation Taxonomy
A major influence in the design of a localisation system concerns the specified
requirements of the application. Examples of operational requirements could be a
real-time solution capability, or a highly accurate localisation estimate. While there
are many applications that must perform localisation, there is also a broad spectrum of
localisation mechanisms appropriate for the various applications. Estrin et al [15]
discusses the following categories of localisation,
1. Active Localisation
2. Passive Localisation
3. Cooperative Localisation
4. Blind Localisation
Active localisation systems transmit signals to estimate the location of a target. This
involves deducing the objects location against distortions that accompany the received
24
signal. Radar and Sonar systems are classic examples of active localisation. Passive
localisation systems are contrary to Active systems and do not transmit any signals.
Passive systems will only detect signals that are emitted by targets. By correlating
data received at various locations, the location of the target can be estimated. The
Time Difference of Arrival (TDOA) [16] is a common passive location technique.
Cooperative localisation is associated with targets that cooperate with the system by
emitting signals with known characteristics. The altitude encoding transponder found
in commercial and military aircraft is an example of cooperative localisation. The
airborne transponders will receive, decode, and then interrogate pulses that have been
transmitted by secondary radar. If the interrogated code is correct, the transponder
will reply with a different series of coded pulses indicating the relative position of the
aircraft. The other localisation mechanism referred to as Blind localisation involves
estimating the location of a source without any a priori information concerning the
signal being known. Blind techniques must estimate the transmission channel
response based only on the channel output, without the use of training signals. A
survey of blind algorithms is provided by Kailath et al [17].
Further categories of localisation system are reviewed in [18], where Fine-grained, or
Course-grained systems are also considered. Fine-grained localisation provides
highly accurate location information based on trilateration or triangulation (discussed
latter), while course-grained localisation systems estimates a location by using
proximity to recognised beacons or landmarks.
2.4 Localisation Parameters
A radio frequency channel lends itself to being modelled as a parameter-dependant
function. Localisation methods can therefore be grouped according to the parameters
being analysed. Classical parameters used for localisation include the signals
direction-of-arrival (DOA), or the range of an object from the transmission source. A
discussion on various localisation techniques based on parameters such as signal
pattern matching, signal strength or timing is provided by Bulusu et al [19].
25
Several ad-hoc localisation methods have been developed, each being based on a
particular parameters. One example concerns the Pinpoint system [20], which
estimates the receivers distance from the source by using signal propagation time with
a Time Difference of Arrival (TDOA) method. Other systems such as Active Bat [21]
and Cricket [22] make explicit Time-of-Arrival (TOA) measurement based on the two
distinct modalities of propagation by radio and ultrasound, which have vastly different
speeds. The radio signal is used for synchronisation between the transmitter and
receiver, while the ultrasound signal is used for calculating the range of the receiver.
One system that uses the received signal strength indicator (RSSI) is the indoor
RADAR location system [23]. The distance of the receiver is estimated by applying a
Wall Attenuation Factor (WAF) based signal propagation model. The location of the
receiver is then estimated by using the distance measurements with trilateration. All
of the above mentioned localisation methods discussed are however not blind and
cooperation maybe required in the system. These methods are therefore not suitable
for intentional interference where there is no cooperation and blind localisation must
be performed.
With network configurations, the application of estimating signal parameters from
sensor array data is a problem that has been encountered in many engineering areas.
Respectively, there is a variety of algorithms that can be employed for parameter
estimation. In [24], parameter estimation techniques are divided into three categories
being either spectral-based, parametric subspace based, or deterministic parametric
based.
Ottersen et al [25] further discusses Maximum Likelihood (ML) and Multi
dimensional subspace methods that are referred to as Weighted Subspace Fitting
(WSF). While there are many parameters that can be estimated with various
algorithms, the geolocation method developed in this research program is based on
the DOA parameter [26]. There is a vast array of literature concerning the DOA
parameter, hence further discussion of the parameter can be found articles such as
[27-30].
26
2.5 Electronic Warfare Localisation
In electronic warfare localisation, Adamy [31] indicates there are four general
approaches that have been used to estimate the location of a hostile transmission
source. These approaches involve,
1. Measure the angle and range from one location
2. Measuring the Angle of Arrival (AoA) at multiple locations
3. Measuring multiple angles from one location
4. Measuring the distance from multiple locations
Being based in electronic warfare, all of these methods passively measure the
localisation parameters. Passive localisation reduces the probability of a hostile force
identifying the presence of the localisation system and hence, reduces the chance of a
retrospective electronic counter measure (ECM) attack against the localisation system.
These methods must therefore conduct blind localisation as belligerent organisations
will not make information on equipment freely available. It should be noted that
passive measurement of range is a challenging process and Stimson [6] provides a
review of various declassified methods.
Systems that provide high localisation accuracy are usually based on the
Triangulation or Trilateration methodologies. These two techniques have been widely
employed in many forms for geolocation and will therefore be reviewed. Further
information concerning quantification of these geolocation methodologies is provided
by Poisel [2].
2.5.1 Triangulation
The process of measuring a signals direction-of-arrival (DOA) at multiple locations is
a classical localisation method and is referred to as triangulation. It is based on using
Direction Finding (DF) at multiple positions to determine the location of a source. In
a free-space environment with no obstacles, an objects location will be the
intersection of relative directions as indicated by network sensors. It’s important to
however note that the arrival direction of a signal is not necessarily the direction to
the transmitter. In a terrestrial or urban environment, the interference signal will have
27
been reflected from terrain features or buildings and would also be subject to
diffraction.
An emitter located in a plane can be triangulated with two measured directions, while
three measured directions are required for localisation in three dimensions. A display
of planar triangulation with three relative directions is shown in Figure 2-1.
Localisation based on the DOA parameter has been extensively applied in electronic
warfare as a hostile emitter can not easily alter the DOA parameter [32].
Figure 2-1 Triangulation
Unlike the stability associated with DOA, time and frequency parameters can readily
be altered by hostile electronic counter measures. Stimson [33] provides a discussion
on various forms of airborne ECM deception methods. While DOA is the best
parameter to use for localisation, threat airborne radars can develop angle deception
by employing terrain bounce, cross-eye, crosspol or double cross methods. While an
enemy can increase the DOA error with these methods, its ECM susceptibility is
much less than other parameters and it has become an invariant sorting parameter in
the deinterleaving of radar signals for electronic support measures (ESM) [32]. This
provides a strong foundation for the novel geolocation method, which can determine
the DOA parameter when configured in a network orientation.
28
2.5.2 Trilateration
Another classical process for localisation is trilateration, which is based on measuring
the range parameter. Techniques for estimating range can be based on measured
signal strength or the transit time of the signal. By finding the intersection of three
range measurements, the location of the source is able to be unambiguously estimated
as shown in Figure 2-2.
Figure 2-2 Trilateration
Range based on transit time has been predominately employed over signal strength
due to several factors such as greater accuracy and less performance degradation due
to the sensitivity of the receiver. In a hostile environment where passive localisation
is being performed, only the interference signal is being transmitted. With one-way
signal transmission, the Time-of-Arrival (TOA) of a signal is simple to measure.
Range however requires the time period of the signal travelling between the
transmitter and receiver to be known. Subsequently the time of transmission from the
source must be known. With blind localisation there is no way of determining when
the signal was transmitted from the source. Only cooperative systems such as GPS
are able to perform trilateration with one-way transmission. This trilateration problem
29
has led to the development of the Time Difference of Arrival (TDOA) method, which
will be discussed next in the following Precise Localisation Network Configuration
section.
2.6 Precise Localisation Network Configurations
When the accuracy of localisation with respect to an emitter is required to have
resolution in units of tens of metres, two network configurations have been
extensively used. These precise network localisation methods are,
1. Time Difference of Arrival (TDOA)
2. Frequency Difference of Arrival (FDOA)
2.6.1 Time Difference of Arrival (TDOA)
The inability of trilateration to resolve the transmission time in a hostile scene has
been overcome with the TDOA technique. The TDOA method requires the difference
in a signal’s TOA between baseline sensors to be measured, which can easily be
performed as shown in Figure 2-3.
Figure 2-3 Measuring difference in signal’s TOA
30
With this measurement, a line-of-positions (LOP) indicating where the source can be
found is provided. In electronic warfare, this LOP is also referred to as an Isochrone
[16]. The isochrone is an infinite hyperbolic line containing all possible locations
where the emitter may be found [34]. A hyperbolic line has an eccentricity value
greater than one, where eccentricity describes the line’s variation from a perfect circle
and is the ratio of two distances that remains constant for the entire hyperbolic line.
As shown in Figure 2-4, the two distances used to define eccentricity are identified as
(A) and (B). The distance between the focus point and any chosen position on the
hyperbolic line is B, while the distance between the corresponding hyperbolic position
and the Directix (i.e. vertical axis) is (A). The distance represented by (A) is parallel
with the conic axis and the eccentricity value is defined as the ratio of B/A. Further
information concerning eccentricity is provided by Roddy [35].
Figure 2-4 Eccentricity
A selection of various isochrones corresponding to a different TDOA is displayed in
Figure 2-5. For localisation to be performed with TDOA, multiple baselines must be
used where location of the source will be at the intersection of isochrones. While
TDOA provides high accuracy, network sensors should be stationary.
31
Figure 2-5 TDOA hyperbolic isochrones (LOP)
2.6.2 Frequency Difference of Arrival (FDOA)
Another precise localisation technique based on a LOP intersection is the Frequency
Difference of Arrival (FDOA) method [36]. Unlike TDOA, the network sensors can
be dynamic. While TDOA measures a signals arrival time difference, FDOA requires
the frequency difference measurement between baseline sensors to be performed. The
result of FDOA is a three dimensional surface defining all possible transmitter
locations. The corresponding curve that can be viewed by taking a planar cross-
section is called an isofreq. A set of isofreq curves for a selection of various
frequency differences is shown in Figure 2-6 where the baseline sensors are moving
with the same velocity.
32
Figure 2-6 FDOA isofreq (LOP)
Just as TDOA requires multiple baselines for an emitter location to be determined,
FDOA also requires multiple baselines for localisation to be feasible. A benefit of
FDOA compared to TDOA is the account of sensors dynamics. While mobile sensors
are desirable, the computation load associated with moving interference sources has a
tendency to be too large. In an airborne environment FDOA is therefore generally
used only on stationary or slowing moving targets.
2.7 Multiple Localisation Platforms
In practise, localisation systems will typically use multiple platforms. This allows
multiple solutions to be considered for localisation. A system that combines TDOA
and FDOA measurements can find the precise location of a transmitter with a single
baseline. A diagram with a single baseline that has a combined TDOA / FDOA
solution is provided in Figure 2-7. The multiplicity of solutions provides more
accurate results over a wider range of operational conditions. The localisation method
discussed in this thesis will be able to provide multiple solutions for localisation
systems already being operated without any substantial financial cost. With all the
33
required microwave equipment already operational, a software implementation for
localisation software is the only requirement [36].
Figure 2-7 Single Baseline Localisation
2.8 Direction Finding
With the IDPELS localisation method being based on the DOA parameter, several
common DF techniques will be reviewed. The simplest DF method uses amplitude
comparison and a mechanically rotated narrow-beam antenna. While highly accurate
DF can be yielded, the probability of desired signal interception is relatively low [37].
While a low probability of interception (LPI) for an adversary is required in electronic
warfare to ensure a corresponding low probability of exploitation (LPE) [38], this DF
method can overcome the LPI to a certain extent by configuring an array to provide
complete azimuth coverage (i.e. 360 °). This antenna coverage is displayed with a
four-quadrant amplitude DF system in Figure 2-8.
34
Figure 2-8 Monopulse DF system
By identifying the greatest (P1) and second greatest (P2) received power levels, the
DOA can be determined. While amplitude comparison systems are frequency
independent and able to cover wide bandwidths, the DOA estimate has a high
probability of being contaminated by multiple signals simultaneously received. These
systems also require calibration with signals that have known DOA information.
Another common DF technique employed in EW is Phase interferometry, which is
shown in Figure 2-9.
Figure 2-9 Phase Interferometry
35
While the effective accuracy of localisation is often stated as a circular-error of
probability (CEP) or elliptical-error of probability (EEP), the accuracy of DF systems
is usually stated as the root-mean-square (RMS) angular error [39]. When precise DF
is desired with an accuracy of 1° RMS, phase interferometry should be employed.
While very accurate, the application of interferometry is however restricted to narrow
band signals. By measuring the phase difference between baseline sensors, the DOA
can be determined via trigonometry. In most interferometric systems, the baseline is
between 0.1 and 0.5 λ. A baseline less than 0.1 λ will not provide enough accuracy,
and if the baseline is greater than 0.5 λ, ambiguous results will be returned. These
baseline restrictions arise because the wavefront is assumed to be a planar wave, when
in reality the wavefront is circular.
While there are various other DF finding techniques that could have been reviewed in
this dissertation, a tutorial of many existing DOA estimation methods is provided by
Godara [40]. A special class of spectral estimation that will however be reviewed is
association with blind deconvolution [41], where both the system and input signal are
required to be estimated from only the measured output signal. These methods are
subspace based and are highly accurate with a high-resolution capability. Discussion
of subspace methods is provided in the following High Resolution Direction Finding
section.
2.8.1 High Resolution Direction Finding
A class of high-resolution DF methods are the subspace class of spectral estimation
techniques that determine a signal’s DOA by computing the spatial spectrum and
finding the local maxima of the spectrum. The subspace DF methods can surpass the
limiting behaviour of classical Fourier-based methods (e.g. Periodogram, Welsh [42])
in estimating frequency or DOA.
Subspace techniques require the noise and signal subspace to be extracted from the
covariance matrix [43] of signal observations. Such decomposition of a stationary
process can be referred to as Wold’s Decomposition [44]. The covariance matrix is
the function that is used to characterise a random process in the signal domain and
36
provides a measure of how much the noise and signal subspace vary together
according to some parameter. A diagram showing the noise and signal decomposition
according to the DOA parameter is provided in Figure 2-10.
Figure 2-10 Eigen-analysis of Covariance Matrix
The noise component does not vary with the desired signal component and hence has
zero covariance. This is shown by the circle in Figure 2-10 and indicates the noise
data set to be uncorrelated with DOA. The elliptic signal component has its greatest
value inclined to the direction in which it was received.
To perform received signal decomposition, the Karhunen-Loẽve transform (KLT) [45,
46] can be used on symmetric matrices, or Singular value decomposition (SVD) [47,
48] can be applied with asymmetric matrices. Both of these methods can arrange the
signal components into orthogonal data sets, where the observation interval is short
and the random signal is non-stationary. The most familiar orthogonal basis set used
in signal processing is the Discrete Fourier Transform (DFT). The DFT however
37
requires a long observation interval compared to the duration of the correlation
function. Computational efficiency is another important consideration due to the
complexities associated with matrix inversion [49]. Matrix inversion is a process
required in the decomposition methods and involves the rotation and scaling of the
coordinate system to provide data for analysis as shown in Figure 2-10. The greatest
efficiency is provided by SVD as the products involved in forming the correlation
matrix never have to be computed.
2.8.2 Eigen-analysis
In performing decomposition with these subspace methods, eigen-analysis has been
extensively used as it enhances the bearing resolution capabilities of adaptive
processing methods [50]. As shown in Figure 2-10, eigen-analysis elliptically fits the
observed covariance matrix, where eigenvalues correspond to the variance of the
subspaces [48]. The DOA is determined by the eigenvector corresponding to the
greatest eigen-value. The use of eigenvectors to characterise the desired signal and
noise portions of received signal observations has characterised one of the distinctive
properties of modern signal processing. Many DOA estimation methods utilise the
principles of eigen-decomposition and some of the popular schemes are reviewed in
the following subsection.
2.8.2.1 Eigen decomposition Methods
The first subspace method was developed by Pisarenko [51] in 1973, which is is
referred to as Pisarenko Harmonic Decomposition (PHD). It should be noted that
PHD does not directly estimate the DOA parameter. Instead PHD determines the
frequency and power of real sinusoids in additive white noise. PHD is based on
Caratheodory`s theorem which is an indication of the required data-set size for
dynamics of desired parameters to be captured [52]. The extension of PHD to DOA
estimation was made by Schmidt [53] in 1981 with Multiple Signal Classification
(MUSIC) method. While MUSIC has been extensively used for estimation, it
requires full knowledge of the antenna array manifold that must be precisely
calibrated. Another limitations associated with MUSIC is that the number of sensors
must be greater than the number of signals present. Another subspace methods with
38
the same limitation is the Estimate Signal Parameters via Rotational Invariance
Techniques (ESPRIT) [54]. Unlike MUSIC that uses the intersection between the
array manifold and signal subspace to estimate directions, ESPRIT exploits the
underlying rotational invariance among signal subspaces. A paper that compares the
direction estimation of MUSIC and ESPRIT is provided by Kangas etal [55]
The Joint Angle and Delay Estimation (JADE) method presented by Vanderveen [56]
in 1997 can overcome the limitations associated with the number of sources, provided
that signal fading is constant. JADE is further research based on the work performed
by Spielman etal [57], where a MUSIC based algorithm was used to solve a two
dimensional radar problem with the estimation of TOA and DOA. JADE is based on
multiple channel estimates and is best suited for Time Division Multiple Access
(TDMA) systems where training signals are available for channel characterisation.
One of the limitations associated with JADE is the requirement of sufficient time
delay between multiple signals. A scenario where this delay will not be provided is in
microcell or picocell models where multipath has a dominant effect on signal
propagation. Respectively JADE may not be readily applied in these environments.
2.9 Propagation Model Localisation
To conclude discussion on localisation, propagations model will be reviewed as the
IDPELS localisation method is based on inverse diffraction signal propagation.
Detailed information concerning IDPELS is discussed in the following chapter.
While the accuracy or efficiency offered by various propagation models is provided in
the next chapter, this section will demonstrate that propagation models hold potential
for conducting localisation. In [58], Phadke et al state that propagation methods
provide greater accuracy and finer resolution is comparison to conventional methods
in geophysical exploration for petroleum where time reversal [59] and back
propagation [60] is applied to acoustic or pressure signals. With respect to radio
propagation, it’s important that information concerning the dimensions of propagation
regions is reviewed.
39
2.9.1 Cellular Concept
After the birth of the wireless era in 1899 when Marconi [61, 62] demonstrated the
use of radio waves for communication over large distances, there was rapid
advancement in wireless technology immediately after World War II. During this
time period, the first mobile telephone service was made commercially available in
Saint Louis, Missouri, USA in 1946 [63]. One year latter the cellular concept was
articulated by D. H. Ring [64] in an unpublished paper at Bell laboratories [65].
Three distinct models were defined for mobile phone operation concerning outdoor
and indoor environments. Macrocell and microcell concern outdoor modelling, while
picocells represent indoor models. In Macrocells, the propagation path is dominated
by the unobstructed path over the rooftops. The typical cell radius of macrocells can
vary between 1 – 30 kilometres as indicated in [66]. Microcell will account for
reflections and diffraction from buildings and streets that often dominant the
propagation environment. Ray-tracing type methods have proven justifiable for
propagation in microcell models. Picocell are associated with cell sizes are reduced
to less than approximately 100 m and cover areas such as large rooms, corridors,
underground stations or large shopping centres, etc. Indoor areas have different
propagation conditions than those covered by macrocell and microcell systems and
thus require different considerations for developing channel models. A summary of
cell characteristics is shown in Table 2-1. Further information concerning cells is
provided by Godara [67].
Table 2-1 Cellular Concept [66]
40
2.9.2 Propagation Database Correlation Model
An example that highlights the potential of propagation models in assisting or
performing localisation is provided by Wolfe etal [68], which compares a signal’s
path-loss with a look-up table defined by a highly accurate propagation model. The
look-up table is the database and its correlation with received signals will localise
mobile objects in multipath scenarios. The application of this database correlation
method is therefore intended for use in picocell or microcell scenes. Depending on
the urban layout associated with a microcell, the workload for adequate resolution in
the look-up tables could be considerable. While any technique that contributes to
interference signal localisation in an urban environment should be considered
valuable, this database correlation method is not feasible in a hostile scenario. In
urban EW, there is no method to determine the hostile interference transmission
power level. As a result, no path-loss calculations can be made or corresponding
database correlation. This renders the database method unsuitable for RFI localisation
in a blind urban EW scene.
2.9.3 Matched Field Processing
Another application of propagation models that has previously been researched and
developed to provide passive / blind localisation is a methodology known as Matched
Field Processing (MFP). A review of MFP for underwater acoustics is provided by
Tolstoy [69], where statistical hypothesis are tested based on the results of forward
propagating models that estimate the source location parameter. The predicted and
observed fields are then correlated for a range of source locations and the
hypothesised location that produces the greatest correlation is then taken as the
estimate of source location.
With the statistical processing associated with MFP, the possibility of the modelled
field not accurately representing the measure field is very realistic. An incorrect
transmitter location being estimated is a form of “mismatch” that is not meaningful to
the localisation operation. Another problem with MFP concerns is ability to account
for scattering and coupling between steered signals [70]. It should be noted that MFP
is a computationally expensive process as it is usually necessary to calculate
41
thousands of replica fields with a variety of forward propagation models. Ray tracing
[71], parabolic equation [72], or normal mode [73] models are some of the possible
propagation models that can be used. A comparison of electromagnetic propagation
models is provided in chapter 3.
While having been extensively investigated in the acoustic domain, simulation of
MFP in electromagnetic domains (EM-MFP) has also been provided by Gingras et al
[74]. Part of their conclusion was the need for further field trials of the methodology.
It should be noted that while localisation is possible with this technique, MFP
however has not been used beyond the research community [70].
2.10 References
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46
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techniques using an array of antennas: A mobile communication perspective,"
presented at IEEE International Symposium on Phased Array Systems and
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Jersey: Prentice Hall Signal Processing Series, 1992, pp. 140 - 222.
47
[47] A.-J. v. d. Veen, E. F. Deprettere, and A. L. Swindlehurst, "Subspace Based
Signal Analysis using Singular Value Decomposition," Proceedings of the
IEEE, 81(9), pp. 1277 - 1308, 1993.
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Statistical Signal Processing, A. V. Oppenheim, Ed. Englewood Cliffs, New
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201.
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effects on ESPRIT and MUSIC direction estimators," Radar, Sonar and
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51
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52
Chapter 3 - Inverse Diffraction Parabolic Equation
Localisation System (IDPELS)
While the Global Positioning System (GPS) is a relatively mature technology, its
susceptibility to radio frequency interference (RFI) is substantial. The Volpe Report
[1] conducted under US Presidential Decision Directive (PDD-63) recommended that
methods should be developed to monitor, report and locate interference sources for
applications where loss of GPS is not tolerable. With GPS becoming an integral
utility for developed society, the significance of research projects that enhance and
expand the capabilities of GPS RFI localisation is highly important.
In response to this recommendation, a technique called “Inverse Diffraction Parabolic
Equation Localisation System” (IDPELS) was independently developed. This
technique applies knowledge of the geometrical terrain profile with an inverse
diffraction propagation model based on the Parabolic Equation Model (PEM).
Extensive research of PEM has been performed by many people, including Lee [2],
Hannah [3], Walker [4], Levy [5], Jenson [6] or Tappert [7]. A propagation-based
localisation method was chosen to be investigated, as it is recognised that propagation
methods provide greater accuracy and increased resolution in comparison to
conventional localisation methods [8].
In wave-propagation theory, an inverse problem involves determining characteristics
of the transmission source, from measured signal values in the far field region. PEM
is an electromagnetic propagation modelling method that has been extensively used
for many applications. This is in-part demonstrated by users of the Advanced
Propagation Model (APM) [9] provided by the Space and Naval Warfare (SPAWAR)
Systems Centre. Originally developed by Amalia Barrios [10], APM is currently a
Hybrid model incorporating ray-optic and PEM techniques. It should be noted that
PEM applications are not restricted to electromagnetic signal propagation. A
significant proportion of PEM development was based on underwater acoustic
propagation [7]. Application of PEM is also found in Quantum physics, where a
modified parabolic equation is referred to as the Schroedinger wave equation [11].
The Schroedinger equation defines the propagation of electrons and not
53
electromagnetic signals. Modification of the PEM is made to account for the presence
of potential energy that has the effect of accelerating electrons.
This chapter provides the theoretical background and independent development of the
inverse diffraction propagation localisation method. Initially presented to the public
at the Institute of Navigation – Global Navigation Satellite Systems (ION-GNSS)
conference at Long Beach, California, USA during 2004, this IDPELS research was
awarded the “Best Presentation” award for making an important contribution towards
determining the location of GPS interference sources [12].
3.1 Research Analogy
As acknowledged in the Research Objective section of Chapter 1, previous research
concerning blind/passive localisation with propagation models has been performed in
the underwater acoustic environment. Section 2.9.3 of Chapter 2 reviewed Matched
Field Processing (MFP) as presented by Tolstoy [13], where statistical hypothesis are
tested on the results of forward propagating acoustic models. MFP has also been
evaluated in the electromagnetic domain by Gingras et al [14]. Being based on
hypothesis testing, MFP however can incorrectly estimate a transmitter’s location.
Because of possible “mismatch”, MFP and has never been used beyond the research
community due to the “mismatch” limitation. Research into MFP however shows that
propagation models have been previously been considered for localisation.
Another area of acoustic research that forms an operational analogy with this research
is based on the inverse propagation modelling as presented by Zhu [15]. The
objective of Zhu’s research is directed towards the reconstruction of high-quality
sonar images of underwater targets. This is performed by reversing the phase of a
measured acoustic field and backward marching the field to the place of the target. In
determining the target’s image, the range parameter to the target is known in this
imaging procedure. While imaging has a different objective and uses different
environmental parameters compared to localisation, this operational imaging
procedure forms an analogy with research presented in this dissertation.
54
While a target’s range parameter is prior known with imaging, it is unknown in a
blind/passive localisation operation. In the underwater acoustic environment, the
procedure performed by Zhu’s imaging was originally considered and investigated by
Tappert [16] to perform blind/passive localisation. The process of propagating the
conjugate of the received acoustic field was discussed in the Research Objective
section of Chapter 1. Being based on acoustic PEM propagation, the fundamental
background for this GPS-RFI research program has been established. As mentioned
in the Research Objective section, the process of transforming the localisation
capability of PEM to operate in the electromagnetic environment is highly credible
and demands investigation. The acoustic investigation performed by Tappert [16]
with PEM forms the strongest analogy with the independently developed
methodology in this dissertation.
3.2 Research Objectives
From the localisation methods discussed in Chapter 2, different limitations associated
with each of the reviewed techniques were highlighted. These limitations range from
the jammer / sensor dynamics to heavy computational loads. As noted with the
combined TDOA/ FDOA method, a robust localisation solution is the ultimate goal of
any localisation method. The primary objectives that were pursued during this
research program for development of IDPELS are listed below;
1. Investigate if inversion theory can be applied to electromagnetic propagation
models to provide an accurate localisation solution
2. Investigate real-time feasibility of localisation methodology based on
propagation models
3. Determine if robust localisation is available with propagation modelling,
where operational limitations associated with conventional techniques does
not impede application
4. Determine if improved localisation can be made if detailed knowledge of the
local terrain is known
55
In pursuit of these objectives, a review of suitable propagation model characteristics is
provided. This chapter will show the suitability of PEM for localisation and then
discuss required model adaptations for the localisation operation to be available.
3.3 Propagation Model Identification
With respect to physical domains (i.e. astronomy, mechanics, geophysics, wave
propagation, etc), a forward problem is defined as a process that is oriented along a
cause – effect sequence [17]. A corresponding inverse problem is associated with the
reciprocal, effect – cause sequence. A forward problem therefore involves
determining what observations can be made from a system where the input parameters
are known. An inverse problem will determine the unknown input parameters, from
observations made of the system output.
The definition of a forward-inverse pair indicates that inversion theory must be based
on well-established physical and scientific laws, which specify the cause-effect
relationship. While many researchers, such as Keller [18] or Claerbout [19], only
consider forward and inverse problems being related, it’s important to also
incorporate the model identification problem for proper consideration of generalised
inversion theory. The relationship between model identification and theory inversion
is further discussed by Aster et al [20] and Tarantola [21].
While it has been acknowledged that the PEM was chosen as the model for
investigation in this localisation research program, there are various other propagation
models that could have been considered. Ultimately PEM was chosen due to its
numerical efficiency, open-boundary configuration, and its extensive previous
research and developed [2] for wave propagation. Further discussion of these PEM
characteristics and other qualities of PEM will be further discussed in the Parabolic
Equation Model section.
While PEM was chosen to form the basis for wave propagation localisation, a review
of other possible wave propagation models is provided. This comparison will indicate
how electromagnetic field propagation is performed in each of the models. After
56
comparing each of the models and considering their suitability for localisation,
justification for choosing PEM as the basis for GPS-RFI localisation is provided.
Before this comparison of propagation models is provided, the background
concerning electromagnetic propagation modelling is reviewed.
3.4 Helmholtz Scalar Equation
To generate a numerical solution for an electromagnetic propagation problem, a
reformulation of the vector based Maxwell’s equations is required. By combining the
reformulation with the assumption of a harmonic signal, EM signal propagation is
based on a set of equations referred to as Helmholtz’s equations that are shown in
Equation 3-1. With these equations equalling zero, they model a source-free
environment. A background providing the derivation of Helmholtz’s equation from
Maxwell’s equation is provided by Sadiku [22].
2 2 2 2 2 2 Equation 3-1E k E = 0 and ∇ H + η H∇ + η k = 0
where
∇2 − divergence of gradient (i.e. Laplacian operator [22, 23])
E – electric field
H – magnetic field
k – signal wave-number (i.e. 2 / , where λ is field wavelength)π λ
η − refractive index of propagation medium [24]
While both electric (E) and magnetic (H) fields constitute the propagating signal,
further simplification in modelling Helmholtz’s equation can be performed by only
considering an individual component of the field. This indicates only the electric field,
or the magnetic field is modelled. By analysing only one of the field components,
Helmholtz’s equations can be considered as a scalar model.
Second-order Partial Differential Equations (PDE) are used in formulating a solution
for scalar Helmholtz’s equations, which are also grouped into three categories. With
57
the polynomial representation of a second-order PDE being quadratic, each category
is defined based on the value of the quadratic discriminant. If the discriminant value
is positive, the PDE is defined to be hyperbolic. A negative discriminant value
defines an elliptic PDE, while a zero discriminant value represents a Parabolic PDE.
In addition to PDE definition based on the discriminant value, another definition that
shows the graphical properties of each PDE groups is provided with conic sections
analysis [25]. A display of the conic sections that define each PDE groups is shown
in Figure 3-1.
Figure 3-1 Conic section analysis of second-order PDE
As shown on the right-hand side of Figure 3-1, a hyperbolic PDE is defined by a
vertical plane section of the cone. In the lower left-hand side of Figure 3-1, an ellipse
is displayed and is defined by the conic section having an angle above the base of the
cone. The angle of the plane defining the ellipse must also be less than the slope of
the cone. When the angle of the plane equals the slope of the cone (as shown in the
upper left-hand corner of Figure 3-1), a parabolic equation defines the cross-section of
the cone. A plane section having no angle with respect to the base of the cone defines
the PDE of a circle.
58
3.5 Boundary Conditions
The development of boundary conditions has greatly simplified the process of
determining the field external to medium bodies. In principle, an accurate external
field should be found by knowing the material properties of the medium body to
determine the internal field behaviour. The material properties of a body are
generally classified in terms of its electrical properties represented by conductivity – σ
(s/m), permittivity – ε (F/m) and permeability – μ (H/m). Conductivity is modelled as
a constant value, it is however dependant on temperature and frequency. Material that
has low conductivity (σ << 1) is referred to as an insulator or dielectric [26]. A
dielectric is measured in terms of its ability to store an electric field without
conduction current (i.e. J = σE) flow. This measure is expressed in terms of
permittivity and material that permits conduction current flow is considered a lossy
medium. A list displaying the electrical properties of different medium types in
which signal propagation occurs is provided below [27]. Relative permittivity ( ε r )
and relative permeability ( μ r ) show the material’s properties compared to free space
values.
• Free space = = 0 , μ μ0(σ 0, ε ε = )
• Lossless Dielectric = = ε = μ )(σ 0, ε εr 0 , μ μ r 0
• Lossy Dielectric ≠ = ε = μ )(σ 0, ε ε r 0 , μ μ r 0
• Good Conductor (σ ≈ ∞, ε ε ε , μ μ μ )= = r 0 r 0
By simulating only the external field imposed on the outer surface, boundary
conditions simplify the procedure for determining the reflected/diffracted field.
Boundary condition reduce a complex, multiple media scenario into a single medium
problem [28].
All PDE groups require boundary conditions to be specified for a unique solution to
exist. Boundary conditions provide representation of electrical properties and the
terrain geometry in the propagation environment. When a terrain profile demonstrates
a smooth surface, specular reflection [29-31] will exist, thereby allowing Snell’s law
of reflection/refraction and Fresnel reflection and transmission coefficients to be used
[32, 33]. If the boundary surface is however rough, diffuse reflection will instead
59
exist. A ray diagram showing specular and diffuse reflection characteristics of a
propagating signal is provided in Figure 3-2.
Figure 3-2 Specular and Diffuse Reflection
To perform further analysis of specular reflection, an incident signal composed of a
horizontally polarised electric field and a vertically polarised magnetic field is shown
in Figure 3-3 [34]. Note should be made of the 180º phase shift associated with the
vertical and horizontal field components in the reflected field, while the transmitted
(i.e. refracted) field has the same polarisation as the incident field. Further discussion
of the reflected signal is provided in the Brewster angle section.
Figure 3-3 Specular Reflection / Refraction of Horizontally Polarised Signal
60
3.5.1 Signal Reflection
With any operation involving signal propagation, a wide variety of possible materials
and terrain profiles will affect signal reflection and scattering. A list showing the
variation in electrical properties of different material and terrain is displayed in Table
3-1. This listing is provided in ITU-R documentation [35] .
Table 3-1 Electric Properties of various Materials
Knowledge concerning the dielectric and conduction values of a reflecting surface
medium allows horizontal ( ΓH ) and vertical ( ΓV ) reflection coefficients to be
determined with Equation 3-2 and Equation 3-3 [34].
sin - - cos 2 ψ Equation 3-2ψ ε Γ =H
sinψ + ε - cos 2 ψ
ε ψ ε sin - - cos 2 ψ Equation 3-3 Γ =V 2sin + ε ε ψ cos ψ
The grazing angle (ψ) is the right-angle triangle compliment to the angle of incidence
(θ) [36] (refer to Figure 3-3), and the complex dielectric constant is represented by (ε).
3.5.2 Brewster Angle
By considering linearly polarized waves [37], Hannah [38] investigated the magnitude
and phase of horizontal and vertically polarised signals. The variation in signal
magnitude and phase was found to be different for each of the linear polarisations, but
61
a typical pattern was noticed. The magnitude and phase results for the wet ground
environment are shown in Figure 3-4. All other materials in Table 3-1 except for the
sea water have a similar phase and magnitude behaviour with different Brewster
angles.
Figure 3-4 Linear Reflection Coefficients for Wet Ground
As shown in Figure 3-4, the magnitude of the horizontal field component smoothly
decreases with increasing grazing angle, with an almost constant 180º phase shift for
all grazing angles. As also displayed, the vertical field magnitude experiences a
greater rate of decrease for grazing angles increasing up to what is known as the
Brewster angle [39-44]. The Brewster angle exists when there is 90º between the
reflected and refracted signal resulting in no reflection of the vertical field component.
Only a horizontal field component is reflected at the Brewster angle as it maintains a
transversal orientation to the propagation direction. At the Brewster angle, the
orientation of the vertical field component is collinear with the propagation direction.
Because E, H and k, as shown in Equation 3-1 are mutually orthogonal, the vertical
field component of a transverse electromagnetic (TEM) signal has not maintained
Maxwell’s conditions. As the grazing angle increases above the Brewster angle, the
magnitude of the vertical field component increases in a non-linear fashion until it
matches the magnitude of the horizontal component at a 90º grazing angle. The phase
of the vertical component also maintains a constant 180º phase shift up to the
62
Brewster angle. As the grazing angle increases above the Brewster angle, an almost
constant 0º phase shift accompanies the vertical component’s phase. With this
knowledge concerning the vertical component of signal reflection, further information
concerning the incident grazing angle in Figure 3-3 can be determined. By analysing
the vertically polarised magnetic field, the phase of the reflected signal in Figure 3-3
has experienced a 180º phase shift. This reflection behaviour of the vertical field
component indicates the incident grazing angle is below the Brewster angle threshold.
Recognition of the Brewster angle is important, as any modelling of a vertically
polarised field should account for this angle.
3.5.3 Perfect Electric Conductor (PEC)
Different materials and terrain environments will impose different signal losses with
their corresponding boundary interfaces. Propagating signal loss results from the
transfer of signal energy into the new medium. When the new medium is a good
conductor (σ >> 1), the signal experiences exponential attenuation in the skin depth
region [45] and is confined to a very thin layer of the conductor’s surface [27].
Signal penetration is further reduced as the conductivity of the new medium, or
frequency of the signal increase. A perfect conductor will have no surface current and
all signal energy is reflected.
While realistic boundary conditions will have signal losses, convenient boundary
conditions exist for scalar field models where the Earth is considered a Perfect
Electric Conductor (PEC). Initial investigation concerning the conducting properties
of Earth was made by Steinheil in 1837, who introduced Earth plates to operate the
telegraph line with a single wire [46]. By modelling the Earth as a PEC, the following
field properties will exist,
• No electric field can exist within the perfect conductor
• The external electric field must be normal to the PEC surface
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Electromagnetic theory also requires the tangential components of the field to be
continuous across the boundary interface. This field property exists when the
conditions listed below are observed.
• the incident, reflected and refracted signal have the same frequency
• phase matching exists between the mediums, where the tangential component
of respective propagation vectors is continuous
Further discussion concerning the continuous tangential components is provided in
many references such as [27, 47, 48].
3.5.4 Classical and Impedance Boundary Conditions
An overview of classical boundary conditions that provide the basis for realistic and
accurate boundary conditions is provided in the list below. While an overview is
provided in this thesis, further detailed information concerning electromagnetic
boundary conditions is provided by Senior et al [28].
• Dirichlet condition − field value on the surface of the boundary
• Neumann condition − normal gradient of the field solution on the boundary
• Cauchy condition − combination of Dirichlet and Neumann boundary
conditions, with initial value specification
When a PEC medium being modelled with a scalar propagation model, Dirichlet and
Neumann boundary conditions are commonly applied and are shown in Table 3-2.
Their combination forming the first-order Cauchy boundary condition provides the
basis for higher-order impedance boundary conditions, which are referred to as
Generalised Impedance boundary conditions (GIBC) [28]. The application of
Impedance boundary conditions (IBC) permits modelling of complex material
properties in scattering and propagation problems compared to the infinite
conductivity of a PEC. IBC are based on determining the normal impedance of the
lower medium from field characteristics in the upper medium. Brekhovskikh [49]
shows this relationship in governed by Equation 3-4.
64
E μ Equation 3-4Z = T1 = 2
N2 HT1 ε2
Only the tangential components of the electric (ET1) and magnetic (HT1) field in the
upper medium are required to compute the normal impedance (ZN2 ) of the lower
medium, which is equal to the square-root of the ratio between permeability (μ2 ) and
permittivity (ε2 ) of the lower medium. While based on a simple equation, the normal
impedance is however governed by the signal’s angle of incidence (θ). This is
highlighted in Figure 3-5 where two cases are shown corresponding to different
grazing angles, and an angle equivalent to respective angle of incidences. The signals
are also positioned on the boundary interface between the two mediums, with
respective propagation vectors k1 and k2 being displayed. The left-hand side of
Figure 3-5 has a lower grazing angle (ψ1) compared to the right-hand side grazing
angle (ψ2 ) . The magnitude of each electric field is equal and horizontally polarised.
Being horizontally polarised, the entire E field is tangential on the boundary interface
in each case. The magnetic field however has a tangential component on the
boundary interface, which is found by determining the product of the magnetic field
magnitude with the angle of incidence (θ) cosine. This equation is shown at the
bottom of Figure 3-5 for each case. With the magnitude of both H fields being equal,
variation in the tangential component is shown to be inversely proportional to
respective angle of incidences (θ).
Figure 3-5 Tangential field component variation
65
While the propagation angle of signals can greatly vary, particularly in irregular
terrain environments, original development of IBC theory performed by Schukin [50]
during World War II permitted a first-order IBC to be modelled with independence
from the propagation angle. While originally developed by Schukin [51], the first-
order IBC is commonly referred to as the Leontovich boundary condition (LBC) [52].
While mathematically simple, the LBC has operational limitations and is only valid
with highly conducting surfaces. This arises due to definition of a surface impedance
coefficient (α) of the lower medium. Equations showing the surface impedance
coefficients for vertically and horizontally polarised signals are respectively shown in
Equation 3-5 and Equation 3-6 [53]. Discussion concerning the application of α
within IBC equations is provided in section 3.12, and shown in Equation 3-47 and
Equation 3-48.
sin2 θ Equation 3-5α = jk εr −
εr
α = jk ε r − sin2 θ Equation 3-6
From inspection of Equation 3-5 and Equation 3-6, it can be seen that the surface
impedance coefficient can only be independent from the angle of incidence when
relative complex permittivity (ε r ) of the lower medium is much greater than one, i.e.
ε >> 1 . A smooth ocean surface is one environment that conforms to this principler
that validates the LBC application.
66
A chart showing what boundary conditions are required for a unique solution to exist
with respect to the three PDE groups is displayed in Table 3-2 [54].
Condition Boundary Hyperbolic
Equation
Elliptic
Equation
Parabolic Equation
Dirichlet or
Neumann
Open Insufficient Insufficient Unique, stable
solution in positive
direction, unstable in
negative direction
Closed Solution not
unique
Unique, stable
solution for
Neumann
conditions
Solution over
specified
Cauchy Open Unique, stable
solution
Solution unstable Solution over
specified
Close Solution over
specified
Solution over
specified
Solution over
specified
Table 3-2 Unique PDE solutions
As shown in Table 3-2, the Hyperbolic PDE requires an open boundary with a
Cauchy boundary condition for a unique, stable solution to exist. The Elliptic PDE
must use a closed boundary with Neumann conditions being specified, while
Parabolic PDE requires an open boundary condition with signal propagation in the
forward direction with either a Dirichlet or Neumann boundary condition being
specified.
3.5.5 Open Boundary Requirement
At any time when RFI source localisation is required to be performed, the process of
obtaining a closed set of boundary conditions for an elliptic PDE solution is not
feasible. This is because the actual location of the interference source is unknown, so
no prior knowledge of domain range is known. An open boundary domain is a
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requirement for blind localisation. While the hyperbolic PDE solution was also
shown to be stable with an open boundary, the computational-load of the hyperbolic
PDE compared to the parabolic PDE solution is significant. In any hostile GPS-RFI
environment, the range to interference equipment will cover a distance that is many
multiples of the carrier signal wavelength (i.e. >> 100 λ). The possibility of analysing
localisation data in real-time will not be feasible with the hyperbolic PDE. Real-time
localisation analysis is important, particularly in environments such as at international
airports. In such environments where real-time analysis is required, blind localisation
in far-fields can only be performed with a parabolic PDE solution.
3.6 Multipath Distortion
Since the GPS localisation process may be required in an urban or terrestrial
environment, any chosen complex model must accurately account for multipath
propagation. An extensive review of multipath affects with respect to GPS is provided
by Walker [4], while methods of modelling its affects on GPS signals are provided by
Hannah [3]. The presence of multipath arises from the combined result of signal
reflection, scattering and diffraction.
There are two types of multipath propagation that can exist, one if referred to as
specular multipath, while the other is diffuse multipath. Diffuse multipath arises
when the signal is incident upon irregular surfaces and is randomly scattered.
Scattering is a form of reflection, but it involves obstruction objects that have
dimensions small compared to the wavelength of the signal [55]. The presence of
diffuse multipath introduces fast fading and will increase the level of background
noise.
Specular multipath is of major concern and arises when good conductors that are
physically large with respect to the first Fresnel reflection zone obstruct signal
propagation (refer to figure 5.32 in Chapter 5). Not only is the transmitted signal
subject to reflection, it will also experience diffraction where objects have sharp edges.
The zone construction was first proposed by the French engineer Augustin Jean
Fresnel during 1818 in an attempt to explain the diffraction phenomena using
68
Huygen’s principle [56]. The Fresnel region is an ellipsoid defined about the line-of
sight path between the transmission source and location of the receiver. Figure 5.32
in Chapter 5 illustrates the concept of Fresnel zones.
While multipath affects will also fade interference signals, it will also create greater
confusion for localisation systems. As shown in Figure 3-6, a receiver in an urban
environment will have reception of numerous versions of the transmitted signal, each
with a different delay, amplitude, and arriving from a different direction. A direction
finding localisation system will have multiple solutions, each of which will require
further evaluation before a final localisation estimate can be provided. Multipath
affects create great problems for localisation systems as shown in Figure 3-6, where
there is no direct line-of-sight (LOS) between the transmitter and receiver.
Figure 3-6 Urban Multipath
69
3.7 Electromagnetic Propagation Models
The objective of this research was to explore how an improved localisation method
could be made through the use of an EM propagation model. This section provides a
review of different modelling techniques that could potentially be applied for
localisation.
The limitations associated with these popular propagation models, and their
advantages will be briefly discussed in the following sections. At the end of this
review of models, a comparison is made between them that justifies why PEM was
chosen for the GPS – RFI localisation problem. An excellent reference for analysing
and modelling sources of electromagnetic interference is provided by Hubing [57].
3.7.1 Ray Tracing
Geometric Optics (GO) is a ray tracing method that has been used for centuries for the
design of lenses at optical frequencies. At optical frequencies, GO modelling is
accurate as the dimensions of lenses are much greater than the signal wavelength and
reflected rays appear to originate from a signal point. Specular reflection is
performed during optical signal propagation according to Snell’s Law [32]. At
microwave frequencies, problems with GO however arise with signal propagation.
When the signal is reflected from a scattered surface, GO only returns non-zero field
elements in the specular directions. GO also discontinues the field at shadow
boundaries and doesn’t account for the field in the penumbra region. Edge diffraction
must be accurately modelled to agree with field decay according to Huygen’s
principle. GO is therefore not considered suitable for application in this geolocation
research program.
Another ray tracing method that overcomes the GO edge diffraction problem is the
Geometrical Theory of Diffraction (GTD) model [58]. While edge diffraction is
modelled with GTD, it however has shortcomings with regard to singularities in the
field. Several modified versions of GTD have however eliminated this problem by
introducing correction factors. These GTD versions include the Uniform Theory of
Diffraction (UTD) [59] and Asymptotic Theory of Diffraction (ATD) [60].
70
These ray tracing methods are commonly used for signal propagation. The total field
at any observation point is the vector sum of all the reflected and diffracted fields
arriving at that point. Multiple reflection and diffraction will be needed for accurate
predicting of fields. Ray tracing therefore has long run times as there is no effective
processing algorithm that can be uniformly applied to the variety of possible terrain
profiles. While several schemes exist to reduce ray tracing computational load, they
are only suitable for ad-hoc applications [61]. Further research is required for
development of efficient ray tracing methods to cope with complex scenarios, while
also maintaining suitable accuracy in propagation prediction results.
3.7.2 High Frequency Models
A method that is not a ray tracing technique, but is also a high-frequency method for
predicting diffraction is the Physical Theory of Diffraction (PTD). PTD provides the
same function for Physical Optics (PO) as GTD does for GO. Physical Optics (PO)
estimates surface currents induced on an arbitrary body. By applying the surface
currents to the radiation integral, the scattered field can be determined. In analogy
with GO, PO sets the current to zero at a shadow boundary hence diffracted fields can
not be estimated. PTD however accounts for diffraction by providing fringe currents
that flow along boundary edges. With PTD the total field is estimated by adding the
standard PO scatter field, with the edge-scattered field. While PTD is simple in
principle, it can however be inconvenient in practice. Fringe currents can extend
considerable distance from the edges and therefore requiring a two-dimensional
integration. Further discussion of PTD is provided by [62] and [63].
3.7.3 Finite Difference Model (FDM)
One of the most popular numerical solutions for Maxwell’s propagation equations is
the Finite Difference method (FDM). The FDM is based on a Taylor series
representation of the propagation field and replaces differential operators with finite-
difference terms [64]. The required computer storage and run time is proportional to
the size of the volume being modelled and the required grid resolution. To ensure
accuracy of computed EM field spatial derivatives, the grid spacing size (δd) is
71
required to be small (i.e. δd < λ/10). Where modelled areas are not symmetric or have
curved edges, stair-casing can be performed. While FDM can account for irregular
terrain with stair-casing, if sharp edges are present the computational load will
become significant [65]. This operational characteristic renders the FDM unsuitable
to model irregular terrain or urban environments. Such environments are a realistic
expectation for hostile jamming scenarios.
3.7.4 Finite Element Model (FEM)
Another popular propagation model is the Finite elements model (FEM). With FEM,
the domain is divided into elements where field quantities are found at each grid [66].
Initially FEM was subject to spurious solutions known as vector parasites. While this
problem was solved in 1991 by Paulsen et al [67], FEM has difficulty in modelling
open configurations where all boundary conditions are not known. An open
configuration is a requirement for RFI localisation where direction and range to the
source are unknown parameters. This therefore makes FEM unsuitable for blind
localisation.
3.7.5 Method of Moments Model (MoM)
While FD and FEM are based on differential equation solution, the Method of
Moments (MoM) is an integral equation method. It was first applied to
electromagnetic scattering problems by Harrington [68]. MoM reduces a set of
complicated integral equations to a simpler system of linear equations. A trial
solution is considered and optimised based on a method of weighted residuals.
Residuals are the difference between the trial solution and true solution and the best
solution is considered to exist when residuals have a minimum value. A limitation of
the MoM technique concerns its difficulty in dealing with arbitrary terrain, complex
physical geometries or inhomogeneous dielectrics.
3.7.6 Model Comparison and Selection
The accuracy and computational complexity of any propagation model is dependant
upon parameters associated with each specific application. The selection of an
72
appropriate propagation model for localisation should provide high accuracy, with
low computational load. A parameter that is important for consideration is the size of
the domain with respect to the signal wavelength. A comparison of various
propagation models based on domain size was made by Umashankar et al [69].
Consideration of domain ranges varying from distances less than a wavelength to
ranges greater than 100 wavelengths was made and is shown in Table 3-3.
Table 3-3 Model Comparisons
Table 3-3 shows that high frequency techniques are not suitable for small domain
sizes. High-frequency methods provide an approximation of signal propagation by
considering the signal wavelength to be small compared to the overall size of the
computational domain. Geometrical optics (GO) is a standard representation of the
high-frequency approximation where only the propagation direction of the wave front
is modelled. While high-frequency methods appear adequate for far-field propagation,
which is also a requirement of this research, such models have a high computational
loading with an irregular terrain or urban environment. Further typical difficulties
concerning the linear superstition of waves is also demonstrated with high-frequency
models.
One solution developed to overcome the above discussed limitations associated with
high-frequency methods was the formulation of Hybrid propagation models. Hybrid
methods are a combination of different propagation models to increase the robustness
73
of the field solution provided by the model. Hybid techniques ensure improved
accuracy and practicality in terms of compuatational resources. A combination of ray
tracing with fintite difference models (FDM) is typically found in many hybrid
models. In these hybrid models, ray tracing is used to analyse the wide area, while
FDM is used to study areas close to complex discontinuities where ray-based
solutions are not sufficiently accurate. While robustness is increased with Hybrid
methods, Table 3-3 however shows these models have been found to be unsuitable for
domains ranges greater than 100λ.
The same domain range limitation is also shown by Table 3-3 to exist when field
propagation is being represented with either integral and differential equation.
Method of Moments as discussed in Section 3.7.5 Method of Moments Model
(MoM) represent signal propagation with integral equations, while FDM as discussed
in Section 3.7.3 Finite Difference Model (FDM) represent signal propagation
with differential equations.
Only PEM was found to be suitable for all domain dimensions, regardless of
frequencies being modelled. This PEM characteristic allows model application
without being restricted to model domain size. Such a model characteristic is an
important requirement when performing blind localisation as interference signals will
be operated over a wide range of frequencies and transmission distances.
The importance of model accuracy for localisation was highlight by Wolfle et al [70].
An evaluation of PEM values with respect to measured field results was conducted by
Geng et al [71]. From these trials it was shown that PEM provides very accurate
field-strength predictions. Due to the accuracy and extent of possible applications
offered by PEM, it has become the benchmark tool for radio propagation. With
model benefits of accuracy and efficiency contributing to the benchmark status, it
provides further validation in choosing PEM as the propagation mechanism in this
research project.
74
3.8 Parabolic Equation Model Development
The research methodology developed in this thesis is called the Inverse Diffraction
Parabolic Equation Localisation System (IDPELS). This methodology uses the
Parabolic Equation Model (PEM), which was originally proposed by Mikhail
Aleksandrovich Leontovich in 1944 [72] for long range radio propagation. A
photograph of Leontovich [51] is shown in Figure 3-7 below.
Figure 3-7 Mikhail Aleksandrovich Leontovich
In 1946, Leontovich and Fock [73] were able to provide planar and spherical
electromagnetic PEM solutions. The PEM involves approximating the elliptic scalar
Helmholtz wave equation with a parabolic partial differential equation to reduce the
difficulties experienced in obtaining a Helmholtz solution. After the original
development of PEM, application of PEM remained significantly restricted till the
75
1970s when computer technology had advanced to allow numerical solutions to be
developed.
In 1973, Frederick D. Tappert and R. H. Hardin [74] introduced the parabolic
approximation to oceanic acoustic propagation with the powerful Split-Step method,
which performs efficient propagation based on the numerically efficient Fast Fourier
Transform (FFT) [75-77]. Claerbout [78] latter derived a finite-difference [79] PEM
version for geophysical applications. Eventually PEM returned to radio propagation
where propagation over a littoral environment (i.e. sea or flat terrain) was initially
considered. With the development of faster algorithms, Kuttler and Dockery [53]
were able to adapt the split-step method (developed by Tappert) for radio propagation.
Further application of PEM was made possible with researchers such as Barrios [10],
who evaluated the Tappert approach on a variety of irregular terrain profiles. Walker
[4] extended PEM for use in GPS propagation studies, while Hannah [3] investigated
two-way PEM propagating for GPS multipath studies. Further information
concerning the two-way PEM is provided by Levy [80] for electromagnetic
propagation over terrain, and by Collins [81] for the analogous problems in
underwater acoustics.
The historical development of the PEM discussed above has highlighted milestones
that have contributed to the application of PEM with electromagnetic signals. With
the historical overview of PEM development discussed above, the following section
will elucidate the mathematical framework of model.
76
3.9 Standard Parabolic Equation (SPE) Approximation
Development of classical radio propagation models is based on approximating the
Helmholtz equation [82]. This is because radio propagation domains are large with
respect to the signals wavelength. The computational loads will therefore be
significant if no approximation of Maxwell’s equations has been made. While the
solution of a classical propagation model is not exact, the solution will be within
acceptable and defined error limits.
The approximation made in each propagation model will distinguish its operational
characteristic. The defining characteristic of the parabolic equation model is signal
propagation within a cone that is centred on a preferred direction. The preferred
direction is referred to as the paraxial direction. It should be noted there are several
methods that can be followed to derive a parabolic approximation of Helmholtz scalar
wave equation. The parabolic derivation in this dissertation is however based on the
procedure presented by Tappert [7], which is based on a Hankel function [83]
substitution. The parabolic approximation resulting from the Tappert’s derivation is
referred to as the Standard Parabolic Equation (SPE). The following sections will
discuss the mathematical procedure and assumptions that are made in deriving the
SPE.
3.9.1 Harmonic Frequency Assumption
One of the fundamental assumptions made in many models concerns the time
dependence of the signal. By assuming the signal to have a single frequency
component, a highly desirable simplification is introduced for the propagation model
[84]. Mathematical representation of harmonic excitation is provided by the phasor j te ω [85], where j = −1 , ω = frequency (rads) and t = time (sec). This harmonic
assumption is performed in the SPE derivation procedure.
3.9.2 Cylindrical Co-ordinate System
A cylindrical or spherical coordinate system must be used in the development of the
SPE, which is due to substitution of the Hankel function. A cylindrical system is
chosen for discussion in this dissertation. With GPS jamming ranges approximately
77
200km as demonstrated with the Aviaconversia GPS / GLONASS jammer [86], such
range values are small compared to the dimensions of Earth. Such a physical
environment allows SPE development to be simplified by considering the earth to be
flat. With the earth flattening formulation, the field accuracy provided by PEM is not
reduced by using the cylindrical coordinate system.
A representation of the cylindrical coordinate system is shown in Figure 3-8, where
the base of the cylinder is a flat earth and axial height (z) represents height above
earth’s surface. Each field component is thus represented by a height (z) and range (r)
from the origin. With the cylindrical coordinate system assuming a flatten earth’s
surface, the refractive index profile [5] must be modified to account for the curvature
of the earth’s surface, when non-free-space signal propagation is being modelled.
Figure 3-8 Cylindrical Coordinate System
With the chosen coordinate system, the cylindrical Helmholtz scalar wave equation
provides the basis for SPE derivation and is shown in Equation 3-7.
2 2 2∂ Ψ 1 ∂Ψ ∂ Ψ ∂ Ψ 2 2 2 + + 2 + 2 + k η (r, z, ) Equation 3-7θ Ψ = 0
∂r r ∂r ∂θ ∂z
where
78
Ψ − magnetic or electric field component
r − range from origin
z − axial height
θ − azimuth
k − wavenumber
η − refractive index
3.9.3 SPE Assumptions
With the cylindrical Helmholtz scalar equation, four specific assumptions are made
for derivation of the Standard Parabolic Equation (SPE). These assumptions are listed
below and will be further reviewed in the following sections.
1. azimuth symmetry
2. envelope function assumption
3. far field application
4. slow envelope variation
3.9.3.1 Azimuth symmetry
By assuming azimuthal symmetry, dependence on azimuth is removed from Equation 2∂ Ψ3-7 by ignoring the 2 term. The corresponding Helmholtz scalar equation
∂θ
becomes two dimensional (2D) with the field solution being determined for each grid
point according to range (r) and height (z) values. The simplified 2D Helmholtz
scalar equation is shown in Equation 3-8.
2 2∂ Ψ 1 ∂Ψ ∂ Ψ 2 2+ + + k η (r, z) Ψ = 0 Equation 3-8∂r2 r ∂r ∂z2
With model orientation being based on a cylindrical coordinate system and the
assumption of azimuth symmetry, PEM operates with circular wave propagation. A
canonical function that represents forward circular wave propagation is the Hankel (1) function of the first kind of zero order, H (kr) [87], which is otherwise known as0
Bessel’s function of the third kind [88]. The ‘kr’ term shown in the Hankel function
is the electromagnetic representation of range. In deriving the SPE, the Hankel
function is incorporated to accurately represent the signal [6]. A figure showing
79
0
amplitude of an azimuth symmetric field is presented in Figure 3-9. Signal amplitude
is determined by the calculating Bessel’s function of the first kind of zero order,
J (kr) . As PEM provides a full wave solution where signal amplitude and phase can
be determined at each grid point, the Hankel function is analogous to the complex
exponential being expressed with real and imaginary components [89], i.e. ± θe j = cos θ ± jsin θ . The Hankel function representing forward signal propagation of
the full wave is shown in Equation 3-9 [83]. Bessel’s function of the second kind is
represented by the Y (x) term, which is also known as the Neumann function. Then
order of functions in Equation 3-9 is represented by ‘n’, while ‘x’ is the input
parameter. In this thesis, the input parameter is the range (r) of field elements from
the source and the order is set to ‘0’ for PEM development. Further detailed
information concerning Hankel functions can found in [87, 88].
(1) Equation 3-9H (x) = J (x) + jY (x) n n n
Figure 3-9 Amplitude of Azimuth Symmetric Field
80
3.9.3.2 Envelope function
In a hostile GPS jamming scenario, primary interest will be concerned with field
variations that are large compared to the signal wavelength. It is therefore convenient
to remove the rapid phase variation and only consider the envelope function u(r, z) .
A diagrammatic representation of an envelope function is shown in Figure 3-10,
where the slower varying trend remains after the rapid phase variation is removed.
Figure 3-10 Envelope function of diffracting field
By only considering the envelope function, the substitution of Equation 3-10 into the
cylindrical Helmholtz scalar equation (Equation 3-8) is performed in the derivation of
the SPE. The process of performing this substitution was first made by Tappert [7],
who devised an efficient numerical solution scheme based on the Fast Fourier
Transform (FFT). Further discussion of this substitution is provided by Walker [90]
and Levy [91].
81
ψ(r, z) = μ (r, z) ejkr Equation 3-10
kr
where
u(r, z) − Signal envelope according to range (r) and height (z)
e jkr − Asymptotic Hankel function
1 − Cylindrical spreading path losskr
The process of substituting the envelope function (Equation 3-10) into the azimuth
symmetric Helmholtz equation (Equation 3-8) is proven by Walker [92] to produce
the representation shown in Equation 3-11.
∂2u2 + ∂2u
2 + 2jk ∂u + k2 ⎡⎢η
2 (r, z) - 1+ 12
⎤⎥ u = 0 Equation 3-11∂z ∂r ∂r ⎣ (2kr) ⎦
3.9.3.3 Far field Application
An important assumption concerning PEM application is that it is only used to
provide a far-field solution. The far-field is defined by the domain region
demonstrating the k.r >> 1 property. Many radio-wave propagation applications do
not normally require near-field (i.e. k.r < 1 ) analysis, therefore the far-field
assumption does not generate any restrictions concerning PEM application in GPS
RFI localisation.
1Because the far-field is defined by k.r >> 1, the 2 term in Equation 3-11 will2kr
become negligible in calculations and is removed. The simplified equation is shown
in Equation 3-12.
∂2u ∂2u ∂u 2 2 Equation 3-12+ + 2jk + k ⎡η (r, z) - 1 u = 0 ⎤2 2∂z ∂r ∂r ⎣ ⎦
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3.9.3.4 Slow envelope variation
The last assumption for complete derivation of the SPE concerns a limited rate of
envelope variation. By assuming the presence of a slow envelope variation, the
∂2uLaplacian operator denoted by the term [22], will be much smaller than the
the changing envelope, while the changing envelope is represented by the term.
∂r2
∂u ∂r
term as shown in Equation 3-13. The ∂2u ∂z2 term represents the variation rate of
∂u ∂r
Equation 3-13∂u ∂2u 2jk >> ∂r ∂r2
∂uWith the envelope change (i.e. ) over a distance corresponding to the signal’s∂r
wavelength being small, the corresponding value of the envelope variation will be
∂usignificantly less than one, i.e. 1 >> . This envelop characteristics permits the∂r
∂2u term to be ignored in Equation 3-12. The resulting equation shown in Equation∂r2
3-14 is the Standard Parabolic Equation (SPE), which provides the functional
framework for the PEM.
∂2u ∂u 2 2 Equation 3-14 ∂z2 + 2jk
∂r + k ⎣⎡η - 1⎦⎤ u = 0
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3.10 One-way Signal Propagation
The SPE displayed in Equation 3-14 is a quadratic function and represents two-way
signal propagation. Research concerning PEM operation with two-way propagation
has been investigated by Collins [81] and Hannah [3]. This research program has
assumed that any backward propagation field components is insignificant and was not
incorporated into the propagation model. The assumption of insignificant backscatter
is based on field trials being intended for operation in a free-space environment.
When modelling only the forward propagating field, a simplified equation compared
to the SPE can be used for propagation. Derivation of the parabolic equation
governing forward signal propagation is shown in this section.
In finding the forward propagating parabolic equation, factorisation of the free-space
elliptic equation (Equation 3-12) is performed. This equation is repeated in Equation
3-15 for clarity. In performing this factorisation, coupling between the forward
propagating and backward propagating field is removed.
∂2u ∂2u ∂u 2 2 Equation 3-15 2 2 + 2jk + k ⎡ ⎦⎤+ ⎣η (r, z) - 1 u = 0
∂z ∂r ∂r
To derive the factorising of Equation 3-15, the operator definition provided by Jensen
et al [6] will be followed. The denoted ‘P’ and ‘Q’ operators are shown in Equation
3-16.
∂ 2 1 ∂2 Equation 3-16P = and Q = η + 2 2∂r k ∂z
Substitution of the ‘P’ and ‘Q’ operators into the Equation 3-15 allows Equation 3-17
to represent the elliptic far-field equation.
84
2 2 2 Equation 3-17⎡P + 2jkP + k (Q −1) ⎤ u = 0⎣ ⎦
Factorisation of Equation 3-17 is shown in Equation 3-18. Hannah [93] indicates the
commutator [94] of the P and Q operators (i.e. [PQ - QP]) can be neglected for free-
space propagation, or where there is a weak range dependence of the refractive index.
[ + − Q) ] P + jk(1 + ] − jk[PQ − QP]u = 0 Equation 3-18P jk(1 [ Q) u
By setting the commutator value to zero, the resulting equation representing the
factorisation of Equation 3-17 is shown in Equation 3-19, where coupling between the
forward and backward propagating field is lost.
[ + − Q) ] P + jk(1 + ] = 0 Equation 3-19P jk(1 [ Q) u
The backward propagating field is shown in Equation 3-20, while the forward
propagating field is shown in Equation 3-21. As previously stated, only forward
propagation is considered, therefore the backward propagation equation will no longer
be discussed in this thesis.
[ + + Q) u ] Equation 3-20P jk(1 = 0
P jk(1 = 0 Equation 3-21[ + − Q) u ]
∂uWith the envelope function ( ) for the field being modelled by PEM, Equation∂r
3-21 is re-arranged according to the form shown in Equation 3-22.
85
Pu = − jk(1 − Q)u Equation 3-22
After replacing the ‘P’ and ‘Q’ operator notations shown in Equation 3-16 and
assigning a unity value to the refractive index profile (i.e. η² = 1), free-space signal
propagation in the far-field is accurately represented by Equation 3-23. Note that
further discussion concerning the assumed value for the refractive index is provided in
the Refractive Index Profile section of this chapter.
∂u(r,z) = −
⎛⎜
1 ∂2 ⎞ Equation 3-23jk 1 − 1+ ⎟ u(r,z)
∂r ⎜⎝ k2 ∂z2 ⎟
⎠
With Equation 3-23 providing a mathematical representation of the electromagnetic
field, the following section will show how the field is propagated with the Fourier
Split-step (FSS) method. While the Fourier Split-Step method was chosen due to its
unmatched model efficiency for signal propagation, it should be noted there are
various other propagation methods that can be applied. An overview of other PEM
propagation methods is provided in the Refractive Index Profile section.
86
3.11 Fourier Split-Step Propagation Solution
As indicated by the title of the propagation solution developed by Tappert [7], a
Fourier Transform based solution is used to propagate an electromagnetic field with
the Fourier Split-Step Parabolic Equation Model (FSS-PEM). Being based on the
Fourier transformation, FSS-PEM domains having properties similar to the
time↔frequency relationship exist. The FSS-PEM domains are called the spatial
domain (z-domain) and the angular domain (p-domain). In finding a solution for
Equation 3-23, a Fourier Transform is applied with the FSS-PEM to simplify
mathematical operations in propagating the field. The relationship between the z-
domain and p-domain is generalised in Equation 3-24.
u(r,z) ⇔ U(r,p) Equation 3-24FT
As demonstrated in Equation 3-24, the z ⇔ p relationship is similar to the common
t ⇔ ω relationship associated with Fourier analysis. Before discussing
characteristics of the z ⇔ p relationship, a graphical display of FSS-PEM
propagation is provided. A two dimensional model showing propagation of the field
elements in the vertical plane is shown in Figure 3-11. A vertically polarized signal is
also considered in this discussion.
Figure 3-11 Open Boundary FSS-PEM
87
With any forward propagation simulation, the input field profile μ(0, z) on the left-
hand side of Figure 3-11 must be initially specified. This input field entered on the
left-hand side (r = 0) is a vertical distribution of field elements in the spatial domain
(z-domain). The vertical distance between each element (Δz) in the input field is
determined according to the Nyquist requirement [95], where 2N samples must be
large enough to ensure anti-aliasing. The field solution at the incremented range (r +
Δr) is then determined by incorporating all field values at range (r), with the upper
( z zmax ) and lower (z = 0) boundary conditions into parabolic model.= The open
boundary configuration on the right-hand side of Figure 3-11 can be extended to any
range required for the modelling process. The open boundary condition associated
with the PEM is a required model characteristic for blind localisation.
Part of the FSS-PEM process in marching the input field profile with a range step (Δr)
involves transforming of the z-domain into the p-domain. The p-domain represents
the vertical spatial frequency spectrum of the z-domain field and as such is the
vertical component of the z-domain wavenumber [90]. A diagram showing a
graphical relationship between signals in the z-domain and p-domain is provided in
Figure 3-12.
Figure 3-12 P-domain, Z-domain Relationship
88
The maximum propagation angle of the spatial frequency in the z-domain ( θmax ) that
is modelled with the free-space FSS-PEM has a direct relationship with the maximum
vertical spatial frequency in the p-domain ( pmax ). This is shown in Equation 3-25,
where the sin function is used in calculating the angular spectrum.
⋅ Equation 3-25p = kvertical = k sin(θ) (rad/m)
As a sufficient number of z-domain samples must be used to ensure anti-aliasing, the
analogy existing between the Fourier analysis domains and FSS-PEM domains
requires Equation 3-26 to be satisfied. The units of Δz are metres, therefore the 2π
value is included in the numerator to adjust the rads/metre units of pmax .
Δ ≤ 2πz2p Equation 3-26
max
As can be determined from Equation 3-25, as θmax increases, so to does pmax . By
considering the inverse relationship existing between the z-domain and p-domain
shown in Equation 3-26, it can be seen a limit with respect to the maximum spatial
frequency must be established. As θmax and pmax both increase, Δz must become
smaller. As Δz becomes smaller, the number of samples defined by 2N increases.
Model efficiency is one the drawcards of PEM and as such a limit must therefore be
established on how many samples can be used in the Fourier transformation. A limit
with respect the maximum propagation angle is therefore specified in the PEM model.
With the N point value for the FFT and Δz being known, the distance between p-
domain samples (Δp) is found with Equation 3-27.
Δ = 2π Equation 3-27p N zΔ
Having established the requirement to specify the maximum propagation angle with
the FSS-PEM and introduced the transformation domains, discussion of the field
marching technique is provided in the following section.
89
3.11.1 Field Marching
To discuss the marching process performed with the Fourier Split-Step method, the
forward propagating parabolic equation shown in Equation 3-23 is repeated for clarity
in Equation 3-28.
∂u(r,z) ⎛ 1 ∂2 ⎞ Equation 3-28= − jk 1
∂r⎜⎜
− 1+ k2 ∂z2 ⎟⎟ u(r,z)
⎝ ⎠
∂2
As shown by the 2 term in Equation 3-28, there is a second-order derivative that∂z
must be performed. Second-order derivatives increase the required computational
time in any propagation model. To overcome this difficult derivative, the angular
spectrum transform is employed to allow efficient signal propagation in the p-domain.
In the p-domain, a simple and efficient multiplication operation is performed in place
of the second-order derivative. With the FSS being based on the FFT methodology,
an efficient solution can be made with the transformation property shown in Equation
3-30 [96].
⎛ ∂n ⎞ n Equation 3-29
F ⎜ n u(z)⎟ ⇔ (jp) U(p) ⎝ ∂z ⎠
With the Fourier transformation being applied with FSS-PEM to provide model
efficiency, the application of the Fourier Transform to both sides of Equation 3-28
allows Equation 3-30 to represent the p-domain spectrum of the field. As can be seen
in this equation, a simplified first-order partial differential equation represents the
angular spectrum of the field.
∂U(r, p) = − ⎜⎜
⎛ 1 2 2 2 ⎞ Equation 3-30jk 1− 1+ j 4π p ⎟⎟ U(r, p)
∂r ⎝ k2 ⎠
90
To determine the propagation solution of the angular spectrum, a separation of
variables method [97] is applied. The separation of variables is a valuable tool that
can be used to solve differential equations represented by Equation 3-31.
∂y = −by Equation 3-31
∂r
The angular spectrum shown in Equation 3-30, can be modelled with Equation 3-31
⎛ 1 2 2 2 ⎞by allowing y U(r, p= and b jk 1= ⎜⎜ − +1 2 j 4π p ⎟⎟ . In finding the separation of ⎝ k ⎠
variables solution, the natural logarithm property extensively used in calculus [98] is
applied. This calculus property is the derivative of a natural logarithm and is shown
in Equation 3-32.
∂ ln y = 1 Equation 3-32
∂y y
From the above given information, Equation 3-31 is rearranged to separate the ‘b’
term as shown in Equation 3-33. With the natural logarithm property shown in
Equation 3-32, a further substitution of the 1/y term can be made and is shown in
Equation 3-34. The simplified expression is then shown in Equation 3-35.
∂y 1 Equation 3-33−b = ⋅
∂r y
−b = y ⋅ ln y Equation 3-34∂ ∂
r y∂ ∂
b ∂ ln y Equation 3-35− =
∂r
The next step in the separation of variables process is the integration of Equation 3-35.
This is shown in Equation 3-36, where B = ∫ b ∂r and ‘C’ is an arbitrary constant used
to represent the initial field conditions.
91
ln(y) = −B + C Equation 3-36
The exponential of Equation 3-36 is then performed and shown in Equation 3-37.
eln(y) = y = e−B + C = e−B ⋅ eC Equation 3-37
By using the integration constant ‘C’ to account for the input field, the assignment of
y0 = eC is made for the input field. The resulting separation of variables solution is
shown in Equation 3-38.
y y0 ⋅e−B = eC Equation 3-38= where y0
In applying the separation of variables solution to the PEM angular spectrum, the
expression for ‘B’ as defined by B = ∫ b ∂r must be known and is shown in Equation
3-39.
⎛ 1 ⎞ ⎛ 1 ⎞ Equation 3-392 2 2 2 2B jk 1= ⎜⎜ − 1+
k2 j 4π p r = jk 1− 1− k2 4π p ⎟⎟ r⎟ ⎜⎟ ⎜
⎝ ⎠ ⎝ ⎠
Before applying the ‘B’ expression shown in Equation 3-39 to the separation of
variables solution of Equation 3-38, it should be noted the angular spectrum is
marched with a stepping distance (Δr). As the marched angular spectrum U(r + Δr, p)
is provided by the separation of variables solution, the correct expression for ‘B’ is
shown in Equation 3-40. The initial field conditions are therefore represented by the
angular spectrum defined by U(r, p) . Substitution of ‘B’ and the initial field
conditions into the separation of variables solution is shown in Equation 3-41.
92
⎛ 1 2 2 ⎞ Equation 3-40B jk 1⎜⎜ − 1− 2 4π p ⎟⎟ Δr=
⎝ k ⎠
⎛ 2 2 ⎞4π p ⎟ Equation 3-41⎜
⎝ − jk 1− 1− 2 Δr
⎠U(r + Δr, p) = U(r, p) e⋅⎜ k ⎟
With u(r,z) and U(r, p) being a Fourier Transform pair, the propagated field in the
spatial field is then found by applying the inverse transform to the marched angular
spectrum. Mathematical representation of this marching process is shown in Equation
3-42.
2 2⎡ −⎛⎜ 4π p ⎞ ⎤ Equation 3-42
jk 1− 1− ⎟Δr
u(r + Δr, z) = F-1 ⎢⎢F u(r, z) ⋅e
⎜⎝ k2 ⎟
⎠ ⎥⎥
⎢ ⎥
[ ] ⎣ ⎦
⎛ 2 2 ⎞ ⎜ 4π p ⎟− jk 1− 1− 2 Δr ⎝The exponential term e ⎜ k ⎠
⎟ seen in Equation 3-42 is referred as the
Diffraction function or Propagator of the FSS-PEM. The actual propagator and
equations employed in FSS-PEM may slightly vary depending of factors that are
considered in the model. The above presented equation concern the signal
propagation in free-space. Any FSS-PEM that accounts for variation in the
atmospheric refractive index profile will be different.
By inspecting Equation 3-42, the process of marching the field to any specified range
with the FSS-PEM can be summarised with the following statements,
1. transform the field distribution to the angular spectrum
2. multiple the angular spectrum with the Propagator
3. inverse transform angular spectrum into spatial domain
By repeating the field transformation and applying the Propagator for each range step,
the efficient marching technique can be continuously stepped forward in any
simulated domain being only limited by the memory capability of computers.
93
3.12 Lower Boundary Condition - Signal Polarisation
and Fourier Transformations
With electromagnetic boundary conditions defining the electric (E) and magnetic (H)
field behaviour on a chosen surface in any propagation model, specification of
boundary conditions is an important process. The vertically planar PEM shown in
Figure 3-11, has the specified input field represented on the left-hand side, while the
open boundary of the FSS-PEM corresponds to the right-hand side. For a unique field
solution to exist with any propagation model, including the FSS-PEM, the upper and
lower boundary conditions must be specified. With discussion based on the vertically
planar field, the upper boundary condition must be transparent, while the lower
boundary condition will represent the terrain profile and therefore provide signal
reflection. The following discussion is based on flat terrain profile that provides
specular reflection.
The parabolic model developed in the simulation investigation of this research
program defined the lower boundary condition as a perfect conductor. Any signal that
is incident on a perfect conductor will have all energy reflected as the field is not
absorbed. The chosen mathematical representation of the field incident on the lower
boundary condition is shown in Equation 3-43, which is a Dirichlet condition.
u(r, 0) = 0 Equation 3-43
As previously reviewed in the Perfect Electric Conductor section, an external electric
field must have a perpendicular orientation on the PEC surface (i.e. normal), which is
represented by a vertically polarised field. There can be no horizontally orientated
field on the PEC surface. Because a horizontally polarised field is represented by the
tangential field on the PEC surface, Equation 3-43 conforms to PEC field theory. As
there is no tangential field component on the PEC surface, there is no signal transfer
into the PEC medium. Without signal transfer into a PEC, this provides reasoning for
no field within the PEC medium.
94
If a vertically polarised signal is chosen to be propagated with the scalar model,
specification of field magnitude (as performed with the horizontally polarised signal)
does not uniquely represent the PEC boundary condition. Any non-zero field
magnitude on the PEC surface can easily be replicated within the propagating field,
based on the principles of constructive interference [99]. Instead, a unique boundary
condition that can be used for propagation of a vertically polarised field requires
location specification where the vertical variation of the field is zero. Because no
vertical field component can penetrate the PEC medium, the vertical field gradient is
zero and provides a unique and uniform specification of the boundary condition.
Representation of the PEC boundary condition for a vertically polarised field is
provided by the Neumann boundary condition, which is shown in Equation 3-44.
∂u Equation 3-44= 0
∂z
Image transforms applied within digital imaging theory [100-102] provide a
background for convenient modelling of Dirichlet and Neumann boundary conditions
with discrete trigonometric transforms (DTT). To model the Dirichlet boundary
condition as used in the simulation investigation, the Fast Sine Transformation (FST)
[103-105] is applied and shown in Equation 3-45. All displayed field elements in the
following discussion are as they have been previously defined.
Fsin [f (x, z) ] = ∫∞
f (x, z) sin(p z) dz Equation 3-45 0
Use of the FST permits the propagator shown in Equation 3-42 to remain unadjusted.
The FST is developed by extending the number of data samples to twice its length and
rearranging data to be represented by an odd function, i.e. ( ) = − f (−x)f x . This
rearrangement allows the numerically efficient, single-sided FFT methodology [53]
to be applied. FFT algorithms provides efficient computation of the Discrete Fourier
Transform (DFT) by considering the harmonic relationship between data samples and
significantly reducing the number of required mathematical operations [106]. With
95
4
an N length data set, a DFT requires N² complex operations. A FFT however
provides the same result with N log N operations where the data length is equal to an2
integral power of 2. With N² data elements, a basic phase increment corresponding to
the lowest filter frequency is initially determined (i.e Δθ = 2π / N ). Only integral
multiples of the basic phase increment can be applied within FFT algorithms, which
are conventionally represented by the exponent of the W complex operator, e.g.
W = 4Δθ . Application of the FFT butterfly is then performed, which represents W
operation combined with complex summation/subtraction of data samples. By
successively applying the FFT butterfly, the transformed data is finally presented in a
bit-reversed sequence as shown in Figure 3-13. Bit reversal reordering is a necessary
part of any FFT algorithm. Original development of FFT is based on the principles of
Danielson and Lanczos [107], and Cooley and Tukey [75]. With continued
improvements of FFT operation, the efficiency of many early versions of the FFT
based on methods such as the mixed-radix [108] have been superseded by principles
such as the split-radix method [109, 110].
Figure 3-13 FFT Bit-reversed output [106]
A similar procedure for modelling the Neumann boundary condition also exist when
the Fast Cosine Transformation (FCT) [111, 112] is applied in the model. The
extended data set with the FCT requires representation of an even function, i.e.
f x( ) = f (−x) . With the extended field data being adjusted to represent an even
function, the FCT provides a simple option for the scalar PEM to model a vertically
polarised signal and is shown in Equation 3-46.
96
F [f (x, z) ] = ∫∞
f (x, z) cos (p z) dz Equation 3-46 cos 0
In addition to the DTT providing PEM propagation with boundary conditions based
on the PEC, DTT are also applied to account for IBC. As discussed in the Classical
and Impedance Boundary Conditions section, IBC are modelled based on the Cauchy
boundary condition, which is a combination of the Dirichlet and Neumann conditions.
This combination of PEC boundary conditions in forming an IBC for a horizontally
polarised field is shown in Equation 3-47, while vertical polarisation is represented in
Equation 3-48 [113].
Equation 3-47∂u + α u(z = 0) = 0 ∂z z = 0
⎡ 1 ∂η ⎤ Equation 3-48∂u
z = 0
+ ⎢⎣ η ∂z
+ α⎥⎦
u(z = 0) = 0 ∂z
In similarity to the combination of PEC boundary conditions to model the finite-
conducting boundary condition, Kuttler et al [53, 113] show the combination of the
FST and FCT forming the Mixed Fourier Transform (MFT) can be used to account
for signal propagation above an IBC. The single-sided integral equation representing
the MFT is displayed in Equation 3-49.
∫∞
u(r, z) [ ] Equation 3-49U(r, p) = α sin pz - p cos pz dz 0
The procedure for determining the inverse of the MFT is provided by Titchmarsh
[114] and is shown in Equation 3-50. By combining the Sine and Cosine Transforms
for the IBC, MFT performs approximately twice the number of computations required
97
for a PEC. Further information concerning the formulation of MFT in the discrete
domain is provided by Kuttler and Dockery [113].
-αz 2 ∞ α sin(pz) − p cos(pz) Equation 3-50u(r, z) = Ke + π ∫0
U(r, p) α2 + p2 dp
where
⎧ ∞ -αz⎪2α∫ u(r, z) e ; Re(α) > 0K = ⎨ 0
⎪⎩0 ; Re(α) ≤ 0
3.12.1 Upper Boundary Condition - Transparency
While the lower boundary condition modelled as a perfect conductor reflects incident
signals without absorption loss, the upper boundary condition must not reflect any
signal component and is therefore required to be transparent. There are various
methods that can be applied to ensure this transparent boundary transparency [115],
however a Hanning windowing function [95] was chosen for application within the
developed model. By implementing the Hanning window, fractional coefficient
values gradually force field values above Zmax to smoothly approach zero at a height
of 2 Z . Z else undesired ⋅ max A smooth decay in field values is required above max
reflection of the field will result. A display of gradual signal attenuation due to a
Hanning window is shown in Figure 3-14 [90].
98
Figure 3-14 Upper Boundary Condition of Vertical Planar PEM
While the transparent boundary condition shown in Figure 3-14 corresponds to the
upper boundary condition for the vertically planar PEM, the horizontally planar PEM
used in the field trials also required the lower boundary condition to be transparent.
Without any signal reflection being required with the horizontally planar PEM,
application of FFT and not DTT was initially considered for localisation. While the
FFT methodology can be operational with forward signal propagation, problems were
however observed with inverse diffraction propagation. Further discussion of
problems arising with the FFT methodology and localisation is provided in the
Horizontal Planar PEM section.
99
3.13 Arbitrary Terrain and Obstacles
An important advantage offered by PEM is demonstrated by the variety of methods
allowing simulated signal propagation over arbitrary terrain surfaces. Isolated
obstacles can also be readily simulated with PEM. Isolated obstacles that are not
attached to the terrain profile will exist in PEM when field propagation in a horizontal
plane is being simulated. Up until now, the vertically planar PEM has provided most
of the background for discussion of the model. In chapter 5 where the field trials of
IDPELS are discussed, a horizontal plane is instead used. The framework for the field
trials of IDPELS is therefore based on the horizontally planar PEM. A diagram
showing field trial methodology based on the horizontal PEM is shown in Figure 3-23.
Further discussion of the horizontally orientated PEM is provided is the following
section.
An important aspect in the development of this localisation research concerns the
ability of the method to account for arbitrary terrain or isolated obstacles. A hostile
urban environment where the interference signal will be reflected, diffracted and
refracted as shown in Figure 3-6 will present a major challenge to any localisation
methodology. This section will discuss two methods developed by Barrios [116] for
the Fourier Split-Step PEM. These two methods are referred to as,
• Boundary Decay
• Boundary Shift
It should be noted that Boundary Decay and Boundary Shift methods are not derived
by rigorous mathematical or physical formulation. Their development was instead
based on intuitive concepts concerning signal propagation.
3.13.1 Boundary Shift
While investigating the localisation feasibility of IDPELS under simulation, the
boundary shift methodology was applied with PEM propagation orientated within a
vertical plane. This PEM orientation has formed the basis of discussion in this
chapter. The vertical plane allows the field profile to be evaluated from a range
versus height perspective. In analysing signal loss in the vertical signal plane, the
100
propagation solution provided by PEM is referred by Barrios as the Coverage
Diagram [9].
To account for irregular terrain with PEM and IDPELS, the boundary shift technique
will shift the field array either up or down in accordance with the height variation of
the lower boundary condition. A graphical presentation of PEM boundary shifting is
provided by Walker [90] and shown in Figure 3-15.
Figure 3-15 Boundary Shift
The low field elements that will propagate into terrain are discarded with a null field
being inserted at the top of array. Conversely, the highest field elements are discarded
with a null field being inserted at the bottom of the array as terrain height descends.
101
The number of array elements associated with signal propagation remains constant.
The boundary shift method restructures the field propagating over a plane earth, while
also accounting for diffractive effects of terrain. An example of a coverage diagram
provided by the vertically processed PEM is shown in Figure 3-16. Detailed analysis
and discussion of coverage diagrams is provided in chapter 4.
Figure 3-16 Forward PEM solution – Signal Amplitude (dB)
3.13.2 Boundary Decay
The Boundary decay method offers another option for the parabolic model to account
for arbitrary terrain. A graphical display of parabolic signal propagation over terrain
being modelled with boundary decay is shown in Figure.3-17. With Boundary decay,
a signal component that is located immediately prior to a cell representing terrain will
be set to zero during the range step. The number of grid points set to zero is
determined by dividing the increase in terrain height by the z-domain spatial sampling
period. Conversely for field components that will emerge above the descending
102
terrain height, they will assume the value computed by the split-step propagator based
on adjacent field grid points.
Figure.3-17 Boundary Decay [90]
3.14 Horizontal Planar PEM
While Barrios [116] indicates the boundary decay method to be the least rigorous
method, its application is especially useful for horizontally planar PEM. When the
spatial profile on an electromagnetic signal being propagated is horizontal, isolated
obstacles may have no interaction with boundary conditions. When this situation
arises, particularly with respect to the lower boundary condition mentioned in
previous discussion, a non-reflecting boundary condition should be specified. Unlike
the reflecting boundary condition used with the vertically planar PEM (Figure 3-14),
the lower boundary condition with the horizontally planar PEM requires transparency.
The field trials performed to evaluate the practical operation of IDPELS (as discussed
in Chapter 5) required a transparent boundary condition on either side of the
propagation solution.
103
With the horizontal planar PEM propagating the horizontal field plane, the lower
boundary condition in Figure 3-14 will correspond to the right-hand side of the
propagation field with the horizontal planar PEM, while the upper boundary condition
will correspond to the left-hand side. A diagram showing the propagation solution
and window domains for horizontally planar PEM used in the field trials is shown in
Figure 3-18. The orientation of right-hand-side window domain in relation to the
image domain was found to provide adequate decay of any signal that would
otherwise be reflected from the image domain and distort the horizontal plane field
solution. It should also be noted this domain orientation is different to that applied
when the horizontal plane solution is based on the combination of the FST with the
FCT to negate signal reflection as further discussed in the following Modelling
Boundary Condition section . The window arrangement shown in Figure 3-18
therefore allows the FST, or FCT to be solely used for analysis of either the horizontly,
or vertically polarised field.
Figure 3-18 Horizontal Planar PEM Propagation domains
104
Due to no signal reflection being required at either boundary in the field trials, initial
investigations were performed with the more numerically efficient FFT, as opposed to
the DTT. While a forward propagating field based on the FFT as discussed by Eibert
[117] could be estimated, no localisation result was obtainable during the simulation
investigation.
To ensure a transparent boundary condition with the horizontal planar PEM, it’s
important to note that any of the possible Fast Fourier Transformation (FFT) methods
should not be solely used for signal propagation. While the FFT offers the best
processing efficiency, there are two reasons why a DTT should be applied, as opposed
to the FFT for signal propagation. One reason concerns accurate modelling of
boundary conditions, while is other is related to the spatial profile of the signal during
an inverse transformation. This section will discuss boundary modelling, while the
inverse transformation issue is discussed in the Inverse Diffraction Propagaton section.
3.14.1 Modelling Boundary Conditions
The reason why an efficient FFT method should not be used to account for boundary
conditions is based on the FFT basis function definition. By analysing the basis
functions of a sine transformation, it will show a complete set of sine functions. In
analogy with the sine transformation, the basis functions of a cosine transformation
are a full set of cosine functions. A display comparing the first 5 basis functions
between the DTT and FFT [103] is provided in Figure 3-19. The Fourier basis set is
located at the top (a) of Figure 3-19, while the sin basis set is in the middle (b) and the
cosine basis set is at the bottom (c)
105
Figure 3-19 FFT and DTT basis function comparison
From inspection of each distinct basis set in Figure 3-19, it can be noted that while the
Fourier transform is a combination of sine and cosine functions, it however does not
include sine functions (1), (3), (5), or cosine functions (2) and (4). This arises
because the FFT methodology is based on rewriting an ‘N’ length data set into the
sum of two N/2 length data sets. One of the reduced data sets constitutes the even-
indexed samples from the original data set, while the other data set is the odd-indexed
samples. As the FFT basis set does not constitute all of the sine and cosine function,
this provide explanation as to why the FST and FCT are the best matching functions
for the Dirichlet and Neumann boundary conditions.
Given the framework behind FFT development and that FST and FCT require odd
and even function representation of field data, it can be seen that the non-reflection
properties of the FFT can be formulated by combing the FST and FCT. To propagate
the field without a reflection boundary, a similar procedure that is applied in FFT
formulation can be followed. By splitting the number of field data samples into two
groups, the sine and inverse sine transform can be applied to the group that provide
106
odd signal reflection, while the cosine and inverse cosine transform will propagate the
group with even field reflection. The combination of these two propagation groups
effectively cancels signal reflection and therefore eliminates the knife-edge boundary
that would otherwise be present with either of the DTT being solely used.
A graphical display of a horizontally planar FSS-PEM solution based on the FST and
FCT combination is shown in Figure 3-20. The signal is propagated into an open
building, which is based on a rectangular configuration. While the isolated obstacle is
based on a rectangle, any shape could have been modelled with FSS propagation.
Figure 3-20 Horizontal Planar PEM solution – Signal Amplitude (dB)
3.15 Refractive Index Profile
While this thesis has so far been based on free-space propagation, it is important to
recognise that further research will require the inclusion of the refractive index profile
(η) for tropospheric modelling. The atmospheric refractive index governs the signal’s
speed of propagation within the medium. Given that the Earth’s surface is curved, it
107
is convenient to modify the refractive index profile so that the surface of the Earth is
mapped onto a flat plane, while maintaining representation of the curved surface. The
modified refractive index profile (M) is derived from knowledge of Earth’s radius (a)
and field height (z) is provided in Equation 3-51.
M(z) = η(z) + z Equation 3-51 a
Various other modified refractive profiles (M) are also displayed in Figure 3-21 where
the ducting effect is highlighted. Part (a) corresponds to the standard atmosphere,
where the index profile increases linearly with height. Case (b) shows an earth-
attached waveguide extending from the surface of earth to a height ( h0 ), where most
of the signal is captured within the waveguide. In part (c), similar signal propagation
is shown within an elevated waveguide extending from h1 to h2 . Case (d) is a
combination of cases (b) and (c), however the greatest amount of radiated energy is
captured within both ducts due to greater variation of M over height.
It should be noted that while the refractive profiles in Figure 3-21 are a function of
field height, range dependent refractive profiles can also be modelled [118]. The
inhomogeneous or horizontally varying refractive structures have however been found
by Barrios to provide little improvement over the homogeneous estimates [119]. This
analysis showing little model improvement is based the vertically planar PEM. It
however also provides validation for modelling refractive index profile in the field
trials, to be independent of cross-range. The field trials are based on the horizontally
planar PEM, where the refractive value over the model cross-range was assigned a
constant value based on the standard atmosphere. Further information concerning the
field trials is provided in chapter 5.
108
Figure 3-21 Refractive Index Profiles and corresponding ray diagrams [24]
3.15.1 Wide-Angle Propagation Methods
Traditionally, problems have arisen with Fourier methods of tropospheric signal
propagation due to the necessity of separating the refractive index effects, from the
diffraction part in the parabolic propagator [120]. While the solution of the free-space
PEM is an exact solution and is not limited in either propagation angle or range, this
is however not true for parabolic models that must account for the refractive index
profile [93]. Levy shows field phase error of the SPE is proportional to sin4 θ , where
θ is the propagation angle above the horizon [91]. Significant accuracy errors will
be associated with the SPE field solution provided outside the propagation angle of
±15º. While this angle limitation is considered undesirable, if the terrain profile is not
irregular and the transmission source is terrestrially based, there will be no operational
limitations associated with the SPE. An area covered by ±15º about the paraxial will
accurately model any signal that is propagating in close proximity to the horizon.
109
While such a model limitation reduces the robustness of the PEM, sophisticated
schemes have been devised to allow accurate representation of the field over greater
propagation angles about the paraxial. Many of these wide-angle schemes are based
on a different representation of the square-root operator Q shown in Equation 3-16.
One methodology is based on the Padé approximant, which is a rational function
representing a series expansion of the Q operator [121]. The Padé approximant was
first introduced to PEM by Claerbout [78] with geophysical applications. Further
information concerning the accuracy of Padé approximants is provided by Jensen et al
[6] and Hannah [93]. Another successful method developed by Collins [122] is based
on approximating the exponential operator within the parabolic marching solution.
Wide propagation angles can also be modelled by extending the parabolic equation
methodology to account for elliptic wave propagation. Fishman et al [123] show the
two approaches allowing this. One concerns the exact reconstruction of elliptical
Helmholtz operators [124, 125], while the other involves application of the Bremmer
coupling series [126, 127].
While there are a variety of methods offering wide-angle propagation, the above
mentioned schemes are not based on the efficient Fourier split-step methodology.
While wide-angle FSS problems exist due to the separation of the refractive index,
this model limitation can be overcome by applying a correction to the initial field.
Kuttler [128] has analysed the correction and found excellent agreement exists
between the wide-angle FSS solution with the exact Bragg scatter [129] solution for
propagation angles up to ±30o . This agreement has seen more common use the FSS
wide-angle method. While FSS wide-angle can be modelled, it should however be
1noted that ±90o is not possible due to the correction factor of 3 [128]. With this cos 2 θ
wide-angle correction, the source will be an infinite value when the propagation angle
is 90º.
110
3.16 Inverse Diffraction Propagation
As previously shown in this chapter, FSS-PEM propagation is based on multiplying
the Diffraction function to the angular spectrum. There is a wide variety of
Diffraction functions that can be employed with PEM, each being governed by the
parameters incorporated within the model. Model efficiency was a central
consideration in this research program, therefore a model that directly employed the
FFT methodology as opposed to the DTT was initially investigated. Such a Fourier
domain method has been developed by Eibert [117] and offers greater efficiency than
the surface impedance approach presented by Dockery and Kuttler [113]. As the
Eibert method was initially evaluated with the field trials of IDPELS, a very important
PEM property was noted. While detailed discussion of the field trials is provided in
chapter 5, an overview of the field trials is provided for discussion of the important
PEM characteristic that provides localisation feasibility.
The field trials were based on a stationary transmitter, where a receiver measured the
phase of the continuous–wave (CW) test signal. The receiver was moved with a
vehicle along a straight road-section that was perpendicular to the transmitter
boresight. A least-square quadratic estimate of the spatial-phase, based on the
measured phase was generated and used as the input signal for IDPELS in the field
trials.
During analysis of field data with the Eibert method, it was known the measured
spatial-phase profile of the CW test signal should have been quadratic in nature. To
ensure geolocation accuracy with the field data, the quadratic phase of the signal is
required to be maintained during any transform, or inverse transform algorithm. A
consequence of this requirement found the Eibert method not suitable for localisation
due to the quadratic nature not being maintained with the FFT inverse transformation.
Correct spatial-phase is important to ensure accurate geolocation with the field data.
To ensure the quadratic characteristic of the spatial-phase, the propagator according to
the wide-angle sin transform algorithm as shown in Equation 3-52 and discussed by
Levy [120] should be used in any further evaluation.
111
1−
2 2 Equation 3-52⎧⎪ π p ⎪⎫ D(p) = exp jkΔx 1 - 2 −1⎬⎨
⎪ k ⎪⎭⎩
where
D(p) − diffraction function
k − spatial frequency spectrum (z-domain)
p − vertical spatial frequency spectrum (p-domain)
Δx − paraxial range step
j −
As previously discussed in the Fourier Split-Step Propagation section, the diffraction
function must be multiplied with the angular spectrum. A high level equation
representing this forward propagation is provided by Equation 3-53.
-1u(x + Δx) = T (U×D) Equation 3-53
where
u (x + Δx) − envelope function of propagated field (z-domain)
U − angular spectrum of the signal (p-domain)
T−1 − Inverse transformation
D − Diffraction function
As IDPELS applies inverse diffraction with back propagation in order to estimate the
location of the source, it divides the diffraction function with the angular spectrum. A
high level equation representing inverse propagation is provided in Equation 3-54.
u(x − Δx) = T−1(U ÷ D) Equation 3-54
The only difference compared with forward PEM propagation is the division of the
diffraction function to provide the signal profile at the previous stepping position.
112
Inverse Diffraction propagation is a direct mathematical inversion of forward
propagation. A diagram showing the inverse relationship of the forward and inverse
propagators in provided in Figure 3-22. As can be seen in the top section of Figure
3-22, field divergence will occur during forward propagation in PEM. This is shown
by the unwrapped phase value, which increases with propagation angle. The inverse
model characteristic is shown in the bottom section of Figure 3-22. As can be clearly
seen, the unwrapped phase value decreases as propagation angle increases.
Figure 3-22 Mathematical Inversion of Propagator
The development of IDPELS is conceptually simplistic, but offer much potential for
interference localisation. A diagram showing the geometry of IDPELS in relation to
the field trials is provided in Figure 3-23. The direction of inverse propagation is
shown together with the height of the receiver during signal reception and
measurement. The principles associated with Synthetic Aperture Radar (SAR)
correspond to the reception of the input field profile for IPDELS. Further discussion
of SAR theory is provided in the Field Trial Chapter.
113
Figure 3-23 IDPELS field trial – Measurement of Input Signal
3.17 Conclusion
With localisation of radio interference sources being considered very important,
especially with GNSS, an independently developed inverse propagation model was
designed for localisation. The significance of this research program was recognised
with a “Best Presentation” award at the Institute of Navigation’s GNSS conference in
September, 2004 [12].
Being based on software models, various propagation models were reviewed. From
the model comparison, the Parabolic Equation Model (PEM), which is also the
benchmark model for radio propagation was chosen as the model for localisation.
The benchmark status of PEM arises due to the accuracy, processing efficiency and
suitability for all domain sizes [2]. A review of PEM history together with its
operating procedure was provided.
With the efficient Fourier Split-Step (FSS) propagation mechanism being chosen to
model signal propagation within both the horizontal and vertical planar PEM, model
adjustments allowing localisation were explained. The horizontal planar PEM
provides the basis for the field trials, where the input signal was terrestrially measured.
With the framework for blind localisation having been provided in this chapter,
chapter 4 will demonstrated the theoretical feasibility of the localisation method with
simulation results.
114
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Chapter 4 - IDPELS Simulation
In this thesis chapter, simulation results of the inverse diffraction propagation
localisation (IDPELS) methodology based upon the parabolic wave equation are
presented. The theoretical feasibility of IDPELS is established in this chapter and
provides reasoning for the “Best Presentation” award at the Institute of Navigation’s
GNSS conference in 2004, at Long Beach, California [1]. A variety of scenarios were
chosen for investigation to allow robustness and performance evaluation of IDPELS.
A comparison of all localisation results in provided in the summary section of this
chapter. This work leads into the following chapter, which demonstrates the practical
feasibility of using the IDPELS framework. Note should be made be that localisation
results are provided in this chapter, while geolocation results are analysed in chapter 5.
This terminology shows that geodetic datum’s were only incorporated with the field
trials, and not with the simulation investigation.
4.1 Objective
The primary objective of the simulation investigation was to determine if the IDPELS
framework is a theoretically valid methodology. In addition to testing the theoretical
feasibility, modelling and simulation reduce the time, resources and risks associated
with the emergence of the localisation / geolocation methodology. Before the more
realistic and expensive practical testing of the geolocation method could be
considered, IDPELS results serving as the best case performance benchmark were
required to be known. The best case benchmarks allow the methodology to be
analysed with the field trials discussed in chapter 5. Simulation allows the best case
benchmarks to be determined, where system operation under a noise-free environment
can be provided.
4.2 Simulation Procedure
In performing the simulation investigation, it is necessary to first calculate the field
profile that will be used as the starting point for the inverse diffraction propagation
systems (IDPELS). A Gaussian transmission source is assumed to be the source of
energy to be located with the IDPELS methodology. A forward propagation standard
131
parabolic equation model is used to propagate the interference source across a chosen
terrain profile to a distance where the field is then “virtually” measured by the
IDPELS sensor array. An example of the forward propagation model is shown in
Figure 4-1, where the Gaussian transmission source is located 80m above the terrain
profile on the left-hand side of the figure. The terrain profile incorporated a block
obstacle that is characterised by a height of 50m being demonstrated between ranges
of 200m and 400m from the Gaussian source. The colour bar on the right-hand side
of the image represents signal attenuation due to free-space loss and the impact of the
terrain. The split-step PEM is used to step the input field (at x = 0m) across the model
domain to a range value so the field at the right-hand side of the figure (x = 1000m)
can be determined. It is the vertical signal profile at chosen range (in this case at x =
1000m) that is used as input for the IDPELS process.
Figure 4-1 Properties of Field Diagram (PEM)
The forward propagation parabolic model (PEM) that has been developed is based on
the sin transform [2] and therefore has a lower boundary condition that assumes a
perfect conductor (no signal attenuation). Diffraction of the propagated signal can be
132
seen on the right-hand side of the terrain block in Figure 4-1, where the shadow
boundary and the diffracted and reflected zone are marked on the diagram. It is
important to note that the displayed field (as shown graphically in Figure 4-1) is only
a subsection of the entire field that is propagated through the parabolic equation
process. The Hanning window (shown in figure 3-9) is applied from heights of 200m
to 400m in the simulation of Figure 4-1. The point where the Hanning window is
applied is also marked on Figure 4-1. Since the sin transform is used, the other
section of the field is the mirror image of the propagation domain and the Hanning
window in the negative z direction. It should also be noted that due to the manner in
which the forward propagation and inverse propagation (IDPELS) plots have been
made, the simulated IDPELS figures are mirror image of the forward propagation
scenes. This indicates that all localisation diagrams are directed in the reverse
paraxial direction.
4.3 Quantisation of Simulation Results
To quantify simulated system results, conventional indicators of solution uncertainty
such as 2drms or SEP (Spherical Error of Probability), etc [3] are based on statistics
and are not suitable for use. Simulation trials were performed in a perfect free-space
environment, with the ground boundary being represented as a perfect conductor.
With signal propagation not being subject to noise, any repeated measurements of
signal amplitude and phase at any specific grid point will be exactly the same. Such
an environment introduces equivalence between the accuracy and precision
terminology [4], where precision describes the similarity between repeated
measurements of the same quantity. In these simulation investigations, the quantity
being evaluated is the signal phase and amplitude.
It should be noted that any further investigation of system degradation due to noise
during the simulation trials was not considered essential to analyse as,
• field trials were conducted and could provide an indication of how the system
deviates from the best capability
• the best possible accuracy was considered suitable to act as a reference to
compare field trial results, and any possible further research
133
With the application of statistics not being suitable in providing confidence units for
the localisation solution uncertainty, quantisation of the simulation results is provided
by using an analogy of the elliptical error of probability (EEP) [5]. The analogy
involves measuring the area of field convergent regions according to a specified
amplitude level. The chosen amplitude threshold for convergent regions was
specified at the 99% level of the greatest field value. In all scenarios investigated, the
greatest amplitude values are located in the convergence regions. The variation in the
range and cross-range dimension of the convergent regions were not similar, hence
the use of an elliptical area as opposed to a circle. A detailed examination of the
range and cross-range dimensions of the field convergence region will be provided
with the first simulation scenario shown in Figure 4-3. A table displaying the range
(semi-major axis) and cross-range (semi-minor axis) dimensions of the localisation
solution for all scenarios is provided in the summary section, together with the elliptic
area and inverse propagation range of the transmitter. A comparison of all the
presented scenarios is discussed in the same section.
4.4 Test Cases
In the following sub-sections, analysis and evaluation of IDPELS operating under
simulation is provided. In addition to the flat terrain profile, both block and wedge
terrain profiles were modelled for analysis of IDPELS, where the propagated signal is
subjected to obstructions. These terrain profiles allow evaluation of IDPELS when a
Non-Line-of-Sight environment exists. After evaluating localisation performance in
NLOS environments, investigation concerning system operation when multiple
signals exist in field is provided. As noted in chapter 2, the performance of all
localisation systems degrades when multiple signals are present. Different
configurations concerning the input signal are also investigated. The configuration
analysed in this investigation are:-
• a continuous input signal profile
• a segmented array configuration of the input signal profile
Another characteristic analysed concerns the dependence of system accuracy
according to the range of the input signal from the transmitter. In this investigation,
the same terrain profile is applied in each scenario, but the range of the input signal is
134
varied between 2000m to 6000m. These ranges correspond to ranges associated with
field trials discussed in chapter 5.
In each of the investigated scenarios, the forward propagation field is shown along
with the IDPELS estimate of the interference transmission source. For each scenario,
an estimate of the accuracy in localising the interference source is provided. Results
of all of these scenarios are summarised in the summary section.
4.4.1 Block Scenario
The first scene for evaluation by IDPELS involves a simple terrain (or obstacle)
configuration. The terrain is essentially flat, but has a single block obstacle with a
height and width of 20m. A Gaussian transmission source is chosen to be on the left-
hand side of the block, and 20m above the block. This places the height of source
above the floor of the propagation solution domain at 40m. A display of for this
scenario can be seen in Figure 4-2 below.
Figure 4-2 Forward propagation (i.e. PEM) – Block
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As described in the simulation procedure section, the first step in the analysis process
is to determine the forward propagation field over the range of interest. Figure 4-2
shows the forward propagation field using boundary shift to account for terrain with
the Fourier Split-Step Parabolic Equation Model. The vertical field profile located at
x = 100m is denoted as “measured field” and is representative of the field measured
by the IDPELS sensor array. This “measured field” is used as the input field to the
IDPELS process and the results of this is shown in Figure 4-3 below.
Figure 4-3 Inverse propagation (i.e. IDPELS) - Block
Figure 4-3 shows the measured field on the left-hand side of the image. The terrain
block from Figure 4-2 can be seen in Figure 4-3 between ranges of 80m and 100m
(thus, this terrain representation is a mirrored diagram of that shown in Figure 4-2 to
represent the localisation problem. The domain range has been deliberately increased
out to 200m to represent the uncertainty in the location of the interference source from
the IDPELS perspective. By inspection of Figure 4-3 and applying the IDPELS
process, it can be seen that there is obvious field convergence region and the peak of
this is marked as the estimate of source location in Figure 4-3.
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By considering the range and height section of the convergence region, an elliptical
area that contains the estimated location of the transmitter can be calculated. By
inspection, a semi-major axis (i.e. range section) is 2.55m, and a semi-minor axis (i.e.
height section) is 0.4m. The corresponding elliptical area that contains the estimated
transmitter location is 3.2m². These values were determined by using the top 1% of
amplitude values from the IDPELS analysis. A detailed discussion of this analysis is
provided in the following Quantisation of Block Scenario section. It’s important to
note that while this outdoor localisation solution can be considered to be highly
accurate, this solution has been provided under an idealised simulation with no noise
impediment.
4.4.1.1 Window Domain of Input Signal
An important characteristic of the input field that should be noted in the IDPELS
analysis of Figure 4-3, is the inclusion of additional field elements outside of the
displayed area shown in the respective diagram. This is highlighted by the labelling
in the upper left corner in Figure 4-3. The sensor array used in this analysis was twice
the height of that displayed in Figure 4-3. The window domain is applied with PEM
propagation as it mitigates undesired signal reflections from the upper boundary of the
propagation solution domain. A discussion of window domains was provided in the
PEM propagation section of chapter 3. All localisation diagrams shown within this
thesis do not display the field information within the window domain. The window
domain is not displayed because the process of measuring the window-weighted field
values is not realistic.
While the process of measuring window-weighted field values is not realistic, the
process could however be approximated by measuring a longer signal profile and
applying window weights from a suitable distance threshold. As discussed in the field
trials of chapter 5, a receiver measures the signal profile while being moved with a
vehicle along a road section. If for example a signal profile was measured over a two
kilometre distance, windows values could be applied to all distances greater than one
kilometre. It should however be noted that this procedure was not performed in the
field trials as discussed in chapter 5. To consider how the extra field values that are
137
window weighted affect the localisation estimate, the same localisation environment
displayed in Figure 4-3 is repeated with the input field profile having only field values
from the solution domain. The subsequent IDPELS localisation estimate is displayed
in Figure 4-4.
Figure 4-4 Inverse Propagation (block) – Input Signal (Solution Domain)
Inspection of Figure 4-4 reveals the field convergence region having the same
properties as demonstrated with the IDPELS analysis of Figure 4-3, where window
weighted field values were included within the input field profile. While the analysis
shown in Figure 4-4 suggests that window field values do not affect localisation
estimation accuracy, the analysis performed in the Wedge Scenario section shows a
reduction in estimation accuracy without the extra field values in the window domain.
A discussion concerning analysis of all presented scenario is provided in the summary
of this chapter, where a table of solution uncertainty for all test cases is presented.
From the summary section of this chapter, a performance analysis of IDPELS is
provided.
138
4.4.2 Quantisation of Block Scenario
As previously stated in the Block Scenario section, accuracy of the localisation
method was quantified by finding the dimensions of the elliptical area corresponding
to the top 1% of field amplitude values. The reason for quantifying the system with
an area unit was provided in the Quantisation of Simulation Results section. This
section will provide detailed discussion concerning quantisation of the IDPELS
localisation estimate shown in Figure 4-4, where a block terrain profile was used.
Quantisation of the remaining scenarios will not be discussed to the extent of this first
example.
The transmission source generating the field for forward propagation in Figure 4-2
was specified with a linear vertical aperture of 1m, beginning at 40m in height. With
the height of propagation solution domain being 100m, the total field height which
includes the window domain is therefore 200m. To ensure Nyquist sampling, 2048
samples applied in the z-domain correspond to a vertical grid separation distance of
0.098m. The transmission source was therefore modelled with 10 vertical elements.
In investigating the accuracy of the IDPELS methodology, the convergence region
defined by the top 1% of field amplitude values is shown in isolation from the
remaining field in Figure 4-5. The same propagation range used for localisation
analysis in Figure 4-4 is shown in Figure 4-5, which corresponds to 200m. To
increase the visual detail of the IDPELS localisation estimate in Figure 4-5, a
magnified diagram is shown in Figure 4-6. Figure 4-6 is a subsection of Figure 4-5,
with ranges limits corresponding to 70m and 110m and propagation domain height
being reduced to 45m.
In both Figure 4-5 and Figure 4-6, an indication of the correct location of the
transmission source is provided by highlighted lines that are segmented. As discussed
with the block scenario of Figure 4-2, the correct range to the transmitter is 100m,
while the height is between 40m and 41m. If the IDPELS localisation methodology
can be considered to be accurate, the convergence region must be superimposed upon
the intersection of the highlighted lines.
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Figure 4-5 Quantisation of localisation accuracy – Block scenario
Figure 4-6 Magnification of Figure 4-5
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From visual inspection of Figure 4-5 and Figure 4-6, the elliptical convergent region
can be seen to accurately reflect the correct location of the transmitter. While the
baseline showing 40m is just below the field convergence region, this should be
considered to be accurate as the height specification of the transmission source is
between 40m and 41m. In addition to the accurate height estimate of the transmitter,
the 100m range baseline is also clearly superimposed with the convergence region.
This example clearly validates the IDPELS methodology in performing localisation.
In defining the area of the elliptical convergence region as shown in Figure 4-6,
measurement of the semi-major axis (A) and semi-minor axis (B) is required to be
performed. From inspection of Figure 4-6, the range dimension of the elliptical
convergence regions is 5.1m, while the corresponding height dimension is 0.8m. In
calculating the area of an ellipse, the semi-major axis (A) is defined as half the range
dimensional value (i.e. 5.1 / 2 = 2.55), while the semi-minor axis (B) is defined as half
the height dimensional value (i.e. 0.8 / 2 = 0.4). By applying these values the area of
an ellipse equation as shown in Equation 4-1, the resulting area of the ellipse is 3.2m².
2 ⋅ ⋅Area of Ellipse (m ) = π A B Equation 4-1
If the semi-major or semi-minor axis were observed to possess a greater value, the
elliptical area would therefore have been greater and introduced greater uncertainty in
the localisation estimate. The desired objective of any localisation procedure is to
provide the solution estimate with minimal uncertainty. With the propagation
solution domain having a range of 200m and a height of 100m, the domain area is
20000 m². With the localisation solution being 3.2m², the percentage value of this
solution over the entire solution domain is 0.00016%. Such a minute percentage
value in uncertainty shows the IDPELS is capable of providing highly accurate
estimate.
This section has visually demonstrated the process used to quantify the localisation
solution uncertainty that was discussed in the Quantisation of Simulation Results
section. All other presented scenarios are quantified based on the same procedure
discussed in this section.
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4.4.3 Wedge Scenario
The next demonstration of IDPELS is with respect to an obstacle possessing a wedge
configuration. In this analysis of IDPELS, the Gaussian source was specified to cover
a linear vertical aperture of 1m, from a height of 20m above the floor of the
propagation domain. The Gaussian source is positioned 40m to the left-hand side of
the wedge obstruction. The wedge obstruction has a vertical profile according to an
isosceles triangle where the base length of the wedge is 25m, and the perpendicular
height is 50m. A graphical display of the forward propagated field is provided in
Figure 4-7, where the “virtually measured” input signal for IDPELS is the vertical
field profile on the right-hand side of the respective figure. The range of the IDPELS
input signal from the Gaussian source is 100m.
Figure 4-7 PEM - Wedge
This scenario was investigated to consider what effect a non-line-of-sight (NLOS)
environment has on IDPELS operation. With the height of the Gaussian source being
positioned between 20m and 21m at a range of 40m to the left of the wedge, the
marked shadow boundary in Figure 4-7 defines the geometrical distinction for
142
measured signals that were either propagated with a direct LOS, or subjected to
diffraction. By defining the shadow boundary, all measured field elements below
79m in height were subjected to diffraction, while all field elements measured above
this height propagated with a direct LOS. This NLOS environment initially considers
the “virtually measured” to include field elements from the window domain. After
analysing IDPELS operation with the input signal including the window domain,
analysis is performed again where input signal does not include field elements from
the window domain.
4.4.3.1 IDPELS Operation in NLOS Environment
A graphical display of the field propagated with inverse diffraction by IDPELS is
shown in Figure 4-8. The input field on the left-hand side of the display is marked as
the “measured field” in Figure 4-7 and range has been extended to 200m.
Figure 4-8 IDPELS – Wedge
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Unlike the localisation estimate provided by IDPELS for the block obstacle scenario,
an intersection of Line-of-Positions (LOP) provides a localisation estimate of the
Gaussian source. The LOP intersection is marked as the “estimate of source location”
in Figure 4-8, where field beams are seen centred upon two LOPs marked in the
respective diagram. The two LOPs are identified as LOP (A) and LOP (B). From
inspection of Figure 4-8, both LOPs are shown to be directed in a downward motion
from the left-hand side of Figure 4-8 until each intersects the ground at different
locations. The beams are than directed in an upward motion from the ground. LOP
(A) is reflected from the ground identified as “Ground Location (A)”, while LOP (B)
is reflected from the ground identified as “Ground Location (B)”. Ground Location
(A) covers range values on the terrain profile between 80m and 90m, while Ground
Location (B) covers range values on the terrain profile between 120m and 135m.
By analysing the LOP intersection marked in Figure 4-8 and measuring the semi
major and semi-minor axis of the intersection region, the resulting elliptic region
covers an area of 10.36 m². In comparison to the block obstacle analysed in Figure
4-4 to Figure 4-6, the uncertainty of the localisation estimate has been increased with
the greater area of the intersection region. While solution uncertainty has increased, it
is important to note that the correct location of the Gaussian transmission source in
Figure 4-8 is between heights of 20m and 21m, at a range of 100m from the input
field profile on the left-hand side. From inspecting of the beam intersection region, it
is recognised to be superimposed over the correct location of transmission source.
Feasibility in the IDPELS localisation estimate is again validated in this analysis.
Please note that detailed discussion concerning Quantisation of Block Scenario is not
repeated with the wedge obstruction.
While the uncertainty has increased with the localisation estimate, it is important to
note that a substantial region of the input field was subject to the wedge creating a
NLOS environment during the forward signal propagation shown in Figure 4-7. With
the NLOS environment, field elements from the window domain were included in the
IDPELS input signal profile. As can be seen in upper left-hand corner of Figure 4-8,
the marking shows field elements that have originate from the window domain. As a
result of including the window domain, the field beam associated with LOP (A) can
therefore be considered to exist only due to field elements from the window domain.
144
The field beam associated with LOP (B) is shown to originate completely from the
propagation solution domain. If the window domain was excluded from the IDPELS
inputs field, the identified LOP (A) would therefore not exist in the IDPELS analysis.
If there was no LOP (A) in Figure 4-8 and only LOP (B) existed, there would be no
beam intersection region. Such an environment where only one beam existed would
reduce the IDPELS estimation from localisation to direction finding (DF) in the
NLOS environment.
To consider the proposed IDPELS characteristic mentioned above, the IDPELS
analysis performed in Figure 4-8 is repeated, but the input field profile was chosen to
exclude field elements from the window domain. The field generated with inverse
diffraction propagation via IDPELS is shown in Figure 4-9.
Figure 4-9 IDPELS – Input Signal (Solution Domain)
By analysing Figure 4-9, IDPELS capability is reduced from localisation to direct
finding, where a significant proportion of the input field profile has not received the
measured signal with a direct LOS. The correct location of the source is shown to be
superimposed by the marked LOP. While the operational capability of IDPELS is
reduced, is still provides a valuable DOA estimation for localisation operations.
145
4.4.4 Multiple RFI Sources
The operational feasibility of IDPELS has been clearly demonstrated in the previous
test cases concerning both the block and wedge obstructions. Analysis of IDPELS in
these cases was initially performed as the terrain profile is a non-complex
environment. With the theoretical feasibility of IDPELS having been validated in
simple environments, the next investigation into IDPELS operation concerned the
models ability to provide localisation of multiple interference sources that are
simultaneously transmitting. The potential of a microwave system being subject to
multiple interference sources in a hostile environment should be considered to be
significant. The impact of multiple interference sources can subject the microwave
system to large scale outages. If the ability of the IDPELS methodology can provide a
localisation estimate for multiple interference sources, a highly important and desired
localisation capability will have been developed. It will also demonstrate that
IDPELS is capable of operating in an environment where convention localisation
methods have had limited success.
4.4.4.1 Three Interference Sources
To investigate the operational capability of IDPELS to perform localisation on
multiple transmitting sources, the input field profile for IDPELS is taken as the right-
hand side of the forward propagation scenario displayed in Figure 4-10. The paraxial
range to the input signal profile is therefore 500m. The Gaussians sources with equal
transmission power levels are marked as Interference Sources (A), (B) and (C) and
their relative positions are highlighted in Figure 4-10. With the origin of the forward
propagation scene being the lower left-hand corner, the range and height values to
each interference source in Figure 4-10 is shown in Table 4-1.
Source Range (m) Height (m)
A 250 30
B 50 10
C 10 80
Table 4-1 Location of Multiple Interference Sources (Figure 4-10)
146
All three sources have the same dimensions and each has the same configuration as
specified in both the Block and Wedge test cases. The height of each source is
therefore 1m and constitutes 10 vertical field elements. Each of the three sources also
has equal transmission power.
Continued analysis concerning the impact of line-of-sight reduction to the IDPELS
input field is also considered in this example. The NLOS environment is created by
including a wedge terrain profile that is specified as an isosceles triangle. The peak
height of the wedge is 52m that is centred about the 150m range. The base length of
the wedge is 26m, which is specified between ranges 137m and 163m.
Figure 4-10 Forward propagating field with multiple sources
Two of the sources (A and C) were positioned to have an unobstructed line-on-sight
to the IDPELS input signal profile. The remaining source (B) was positioned to be
obstructed by the wedge to “virtually measured” input signal profile. As indicated in
147
chapter 3, the height of the propagation solution domain was double for inclusion of
the Hanning window. Therefore the maximum height of model field elements is
200m. This height should be noted, as it will be used to determine if source (B) is
obstructed from all possible field elements, including the window domain. Analysis
of the LOS gradient between source (B) and the wedge apex shows the LOS height
limit on the right-hand side of Figure 4-10 is 195.5m. While not completely blocked
from the IDPELS input field profile, localisation analysis was chosen to be performed
with the input field profile being restricted to field elements in the propagation
solution domain. Therefore source (B) propagates no signal to the input field profile.
Since source (B) provides no input field elements for IDPELS, it should therefore not
be expected to be estimated in the localisation solution. A graphical evaluation of this
proposal can be made with the IDPELS field in Figure 4-11.
Figure 4-11 IDPELS field with multiple sources
148
As anticipated, inspection of Figure 4-11 reveals that no localisation estimate is
provided for source (B). While no estimate of source (B) is provided by IDPELS, it
should however be noted that any interference sources that is completely obstructed
from a receiving device will also have no impact on the receiver.
Further inspection of Figure 4-11 also reveals that a clear localisation estimate is
provided for all sources that were unobstructed from the input field. The
corresponding convergence region shown as (A) and (C) are also superimposed upon
the correct location of the respective transmission sources. The elliptical area of
convergence region (A) was calculated as 5.89m², while convergence region (C) was
found to be 13.32m².
As discussed in the Quantisation of Simulation Results section, localisation
uncertainty increases in proportion to the area of convergence regions. In Figure 4-11,
the paraxial range of source (A) from the IDPELS input field is 250m, while source
(C) is 490m. With convergence region (A) being 5.89m² and convergence region (C)
being 13.32m², Figure 4-11 suggest that uncertainty with IDPELS estimation will
increase as paraxial range to the source also increases. Further investigation into this
IDPELS property is made in the following Long Range IDPELS Performance section.
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4.4.5 Long Range IDPELS Performance
The maximum propagation range that has been considered with previous results is
500m, as analysed in Figure 4-11. In the field trials that are discussed in the
following chapter, the actual range between the transmitter and receiver vary between
3.5Km and 11Km. Thus it was therefore decided to form a series of simple
simulations to determine the potential of IDPELS to operate over longer ranges than
has been previously shown. These longer ranges are similar to the ranges used in the
Geolocation investigation discussed in chapter 5.
To perform this evaluation, a simple scenario was considered that comprised of a flat
terrain profile with a single Gaussian transmission source. The height of the
transmitter was 40m, at a range of 1000m from the left-hand side of Figure 4-12. As
with previous test cases, the forward field was propagated with the split-step PEM,
with the terrain profile being modelled as a perfect reflector. The input signal for
IDPELS propagation corresponds to the vertical signal profile at the paraxial range of
6000m, which is 5000m from the source.
Figure 4-12 PEM – Domain Range 6000m
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The corresponding inverse diffraction propagation field is shown in Figure 4-13.
From inspection it can be observed from this figure that the IDPELS system does
provide a clear field convergence region, which represents the localisation estimate to
the transmission source. As with all previous test cases, the convergence region is
superimposed on the correct location of the source as shown by the regions
positioning over the height value of 40m, which is 5000m to the right of the IDPELS
input signal.
Figure 4-13 IDPELS – 5000m range to Source
While the range of 5000m is larger than all other previous test cases, the area of the
convergence is also greater. Analysis of the convergence region in Figure 4-13 found
the area of the elliptical region to be 312.6 m². While the elliptic area is much greater
than that experienced in all previous test cases, it was noted in the Multiple RFI
Sources section that IDPELS uncertainty increased as propagation range also
increased. Further investigation into this IDPELS property is therefore taken with this
Long Range analysis scenario where there are no obstructions in the signal
propagation path.
151
To analyse the relationship between estimation uncertainty and propagation range, the
range of the IDPELS input signal from the left-hand side of Figure 4-12 is reduced
from 6000m, to values of 4000m and 1000m. The model dimensions used in Figure
4-13 are maintained in these two examples, thereby allowing a greater comparative
view in the analysis. This analysis will initially investigate use of the input signal
profile corresponding to the vertical field profile at 4000m in Figure 4-12. A
graphical display of the IDPELS analysis is shown in Figure 4-14. As can be seen by
maintaining the same domain dimension of Figure 4-13, there is no field propagated
in the initial 2000m. This region is shown by the dark blue colour assignment.
Figure 4-14 IDPELS – 3000m range to Source
As graphically shown in Figure 4-14, the convergence region appears to be smaller
compared to the convergence region generated by IDPELS in Figure 4-13. By
measuring the dimensions of the convergence region in Figure 4-14, the
corresponding area of the region was calculated to be 91.1m². The shorter paraxial
range propagated by IDPELS continues to suggest that solution uncertainty is
152
proportional to paraxial range. To confirm this IDPELS characteristic, the same scene
is repeated where the input signal profile for IDPELS was specified as the vertical
field profile provided at the 2000m range in the forward propagation scene displayed
in Figure 4-12. This means the distance of the IDPELS input signal profile is 1000m
from the source. The corresponding IDPELS field is shown in Figure 4-15
Figure 4-15 IDPELS – 1000m range to Source
Inspection of Figure 4-15 again shows a smaller convergence region superimposed on
the correct location of the source. Measurement of the convergence area found the
elliptic area to be 33.5m². This analysis confirms the relationship between estimation
uncertainty and paraxial range in IDPELS operation. While estimation uncertainty
increases with ranges, this analysis has also shown the methodology to be feasible in
long range environments. Before concluding the feasibility investigation of IDPELS
operation under simulation, the last test case to be analysed concerns a segmented
array configuration for the input signal profile for IDPELS. All previous test cases
have used a continuous array configuration.
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4.5 Segmented Antenna Arrays
All previous test cases of IDPELS have used an input signal that was the entire
vertical field profile according to a specified range in the respective forward
propagation scene. As the feasibility of IDPELS has been validated in all previous
test cases, it was then decided to evaluate IDPELS operation when the input profile
was configured as a segmented array. This evaluation was performed as practical
testing of IDPELS could involve the use of segmented arrays to measure the field
profile. At this point, note should be made that the field trials discussed in chapter 5
are based on a test signal that has extreme stability in frequency. As most jamming
signals are unlikely to have this sort of frequency stability, an operational limitation of
IDPELS can be perceived to exist. While this may be true where the input signal is
measured based on SAR principles as discussed in chapter 5, the segmented array
approach provides a ready option to overcome this possible operational problem by
simultaneously measuring the same input signal at all locations. In the following
simulation analysis, each segment of the array was configured as a uniform linear
array (ULA). The segmented array analysis also allows a comparison of IDPELS
operation with direction finding, which is a standard localisation method. This
comparison is made in the first scenario examined.
The objective of this segmented array analysis is the evaluation of uncertainty and
accuracy associated with the IDPELS estimate when the input field is configured as a
segmented array. It is well known that field values measured with a multi-element
antenna arrays are affected by the reception profile of the array. The reception profile
of the array can however be adjusted and array characteristics that alter the reception
profile include [6],
1. number of arrays sensors
2. number of elements in each sensors
This investigation will therefore analyse two cases of the segmented array
configuration, which are listed below.
1. Two sensor array with a 10 element in each sensor
2. Nine sensor array with 50 elements in each sensor
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The forward propagation scenario used for providing the IDPELS input field
configured as a segmented array was chosen to be the same as that provided with the
block obstacle scenario shown in Figure 4-2. While this figure is not repeated here,
the paraxial range to the vertical signal profile is 100m. This equates to the right-hand
side of Figure 4-2 and therefore the above listed segmented array configuration are
applied to the same IDPELS input signal used in Figure 4-4.
4.5.1 Two sensor array (10 elements in each sensor)
A graphical display of the IDPELS field generated with the 2 sensor array
configuration is shown below in Figure 4-16. This test case has a base height of 10m
for sensor (1), and 70m for sensor (2). With 10 elements in each sensor, the aperture
of each sensor is therefore 0.98m (refer to Quantisation of Block Scenario section)
Figure 4-16 2 Sensors (with 10 elements)
155
Inspection of Figure 4-16 reveals that field convergence is superimposed on the
correct location of the transmission source. The correct location of the source is
highlighted by the intersection of the LOP associated with both sensors. While
IDPELS accuracy is maintained with a segmented input signal, the uncertainty
associated with the convergence region is much greater compared to the same scene
propagated with a continuous input signal profile. The area of the elliptic region
associated with the continuous input signal profile was 3.2m². The estimate provided
by IDPELS with the input field being configured with 2 sensors, returned an elliptical
area of 365.68m². Solution uncertainty has significantly increased in this test case.
Having shown the solution uncertainty increased with this scenario where only two
sensors have been employed, it is convenient at this point to conduct a comparison
with the direction finding (DF) method. In performing this comparison, an accurate
DF system is considered, where the 3dB beamwidth of antenna sensors is considered
to be narrow [7]. Use of a narrow beamwidth lends itself well to accuracy and ease of
DF operation. The 3dB beamwidth was therefore assigned a 5 degree value [8]. A
display of the DF system with 5 degree beamwidths is shown in Figure 4-17.
Figure 4-17 Direction Finding Analysis with two Sensor
The uncertainty associated with the DF localisation estimate is highlight by the yellow
region surrounding the correct location of the target. The area of this DF estimate was
156
found to be 308.57m². In comparison to the IDPELS estimate for the same scenario
where the convergence region was 365.68m², this suggests the DF method has less
uncertainty with its localisation estimated. A 5º beamwidth is however a narrow
specification. If a wider DF sensor beamwidth of 15º was specified, the area of the
DF estimate would then be 3405.88m². This is a substantial uncertainty that is
significantly greater than any of the previous IDPELS results (refer to Table 4-2).
While the accuracy of a DF estimate is governed by both sensor geometry and
measurement error [9], this examples has shown that DF based on narrow beamwidth
can provide a similar operation compared to IDPELS.
157
4.5.2 Nine sensor array (50 elements in each sensor) With the remaining configuration applied to the same field profile used for IDPELS in
Figure 4-16, the number of sensors is increased to 9. Each sensor also has more
elements. In this case, the number of elements in each of the 9 sensors is also
increased to 50 elements, so the aperture of each sensor is 4.9m. The field generated
by IDPELS having this configuration for the input field profile is shown below in
Figure 4-18.
Figure 4-18 9 Sensor array (with 50 elements)
As can be clearly seen by inspecting Figure 4-18, the area of the field convergence
region is smaller than that provide in the IDPELS estimate shown in Figure 4-16.
With more sensors and more elements in each sensor, the solution uncertainty has
been reduced to 40.53m². This segmented array investigation has therefore shown
that better IDPELS performance is provided with more elements measuring the in
input signal profile.
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4.6 Summary
In summarising IDPELS performance under simulation, Table 4-2 is provided to
allow a comparative view of all test cases. Table 4-2 also permits accuracy of the
IDPELS localisation estimate to be discussed, as the table includes the quantitated
area of convergence regions. It should be noted that in all presented test cases, the
IDPELS localisation estimate was superimposed on the correct location of sources
that were not obstructed from the input field profile. Such an operational
characteristic of IDPELS has therefore validated the theoretical electromagnetic
feasibility of IDPELS to perform localisation.
Test Scenario Semi- Semi- Elliptical Range to
Case Major Minor Area Source (m)
Axis (m) Axis (m) (m²)
1 Block (Figure 4-4) 2.55 0.40 3.20 100
2 Wedge (Figure 4-8) 2.35 1.40 10.36 100
3 Multiple (A) 2.80 0.67 5.89 250
Sources (B) Not not not 450
(Figure 4-11) available available available
(C) 5.30 0.80 13.32 490
4 Range 2000m
(Figure 4-15)
19.40 0.55 33.50 1000
5 Range 4000m
(Figure 4-14)
26.30 1.11 91.70 3000
6 Range 6000m
(Figure 4-13)
62.20 1.60 312.65 5000
7 Array – 2 segments
(Figure 4-16)
14.55 8.00 365.68 100
8 Array – 9 segments
(Figure 4-18)
6.00 2.15 40.53 100
Table 4-2 IDPELS Performance Comparison
159
In Table 4-2, the block and multiple RFI scenarios are referred to as test cases (1) and
(3). By analysing the area of these convergence regions, it suggested that solution
uncertainty increases according to a function of propagation range. In test case (1),
the paraxial range to source was 100m and the convergence region had an area of
3.2m². In test case (3), the range associated with source (A) was the next greatest
value of 250m. The area of the corresponding convergence region increased to
5.89m². The next greatest range value of 490m is associated with source (C). The
localisation estimation of source (C) again displayed a greater area of 13.32m². It is
important to note that terrain profile used in test case (1) and (3) is different, therefore
this analysis only suggests the relationship between localisation uncertainty and
paraxial range.
To determine if IDPELS uncertainty is related to a function of paraxial range, a
forward propagation scenario was generated with a flat terrain profile covering a
distance of 6000m (Figure 4-12). From this PEM scenario, the input signal profile for
IDPELS was selected as vertical field elements corresponding to range values of
2000m, 4000m and 6000m. In Figure 4-12, the range of the Gaussian source from the
left-hand side of the domain was 1000m. The height of the transmission source was
also specified as 40m. Because the forward propagation range to the source was
1000m, the paraxial range to respective input signal profiles is 1000m, 3000m and
5000m. Each of these scenarios are respectively labelled as test cases (4), (5) and (6)
in Table 4-2. Analysis of tabled information shows that convergence region area for
test case (4) was 33.5m², test case (5) was 91.7 m² and test case (6) was 312.65m².
With localisation uncertainity being a measure of the convergence region`s area, a
graphical display of IDPELS uncertainity as a function of range is shown in Figure
4-19. This display clearly shows that under simulation the localisation uncertainity
increases accoding to some quadratic function of paraxial range. By taking a least-
square fit of a quadratic function to represent the simulation results, localisation
uncertainity at any desired range can be determined. The least-square fit is
highlighted by the blue line in Figure 4-19 and was determined according to
coefficients provided by the matlab “polyfit” function.
160
Figure 4-19 IDPELS Uncertainity versus Range
This simulation analysis has shown IDPELS uncertainty increases according to a
quadratic function, where in the independent variable is the paraxial range. In
addition to establishing the uncertainty property of IDPELS, test cases (5) and (6) also
validated the feasibility of using IDPELS at larger ranges as performed in the field
trials. Discussion of field trials is provided in the following chapter.
The segmented array scenarios labelled as test cases (7) and (8) were analysed so that
optimal design of sensing arrays could be determined. The same scenario and range
was used in both test cases, however the number of segments and the number of
elements in each segment was adjusted. In test case (7), two segments with 10
elements were analysed, while test case (8) had nine segments with 50 elements in
each segment. As shown in Table 4-2, the convergence region for test case (7) was
365.68m², while the convergence region for test case (8) was 40.53m². This analysis
has shown that improved IDPELS performance is provided by using apertures with
more field elements in the signal measurement process.
161
With the segmented array test case (7), a convenient comparison with the
conventional direction finding method was also performed. In the same scenario,
uncertainty in the direction finding estimate method was measured based on the
intersection region of sensor beams having a 3dB beamwidth of 5² and 15º. The
respective areas were 308.57m² and 3405.88m². This comparison indicated that
IDPELS can operate with similar performance compared to the narrow beamwidth DF
method. Such a comparison shows that IDPELS is capable of providing standard
localisation estimates that are based on the input signal being measured with a simple
array configuration.
4.7 Conclusion The simulation investigation clearly demonstrated the theoretical feasibility of inverse
diffraction propagation to perform localisation. Recognition of this localisation
method was made with the “Best Presentation” award at the international Institute of
Navigation’s GNSS-2004 conference at Long Beach, California, USA [1].
All scenarios were based on a noise-free environment introducing equivalence
between the accuracy and precision [10] to quantitate the localisation system. System
accuracy was therefore quantitated based on the localisation uncertainty. Localisation
uncertainty was a direct measurement of the elliptic area corresponding to the
convergence region in each scenario. In all investigated scenarios, the correct
location of the transmitter was positioned within the field convergence regions. A
display of all quantitated results was provided in Table 4-2.
The quantitated results indicated greater localisation certainty is provided when the
input signal was based on a continuous field profile, instead of a segmented array
configuration. While an array configuration was initially considered for measurement
of an input field profile in field trials, a SAR analogy was chosen to the measure a
continuous signal phase. Further information concerning the field trials is provided
the following chapter.
It was also noted that a requirement for IDPELS to provide an accurate localisation
estimate was that a significant proportion of the measured input signal experienced
free-space propagation. Analysis of obstructions was performed with a wedge in
162
Figure 4-9. Another analysis of the obstruction was made with source (B) in Figure
4-11. In Figure 4-9, only 21% of the input field propagated in free-space. The
remaining field percentage had been blocked by the wedge and any measured field
was subject to diffraction. The IDPELS result in Figure 4-9 was reduced from a
localisation estimate, to a Direction Finding (DF) estimate. Only a Line-of-Position
(LOP) estimate was provided in Figure 4-9.
The investigation of Source (B) in Figure 4-11 further evaluated the obstruction effect
by completely blocking Source (B) from the measured input field with a wedge. In
addition to testing signal obstruction, localisation of multiple sources was also
investigated in this scenario. There were another two sources identified as Sources (A)
and (C) in Figure 4-11 that could propagate in free-space to the input field. With
source (B) being completely obstructed from the input field, it was shown that no
localisation estimate could be provided to the source in Figure 4-11. While the
location of source (B) could not be provided by IDPELS, the localisation estimate
however provided a clear location estimate for sources (A) and (C). As discussed in
chapter 1, research by Casabona et al [11] showed the operational capability of all
GPS interference mitigation and localisation techniques to reduce as the number of
interference sources increased. This undesired characteristic is not demonstrated by
IDPELS and indicates an important possible future use of the localisation method.
Similar accuracy between DF and IDPELS also indicated the ability of IDPELS to
match the operation performance of conventional localisation methods.
This chapter has clearly demonstrated the theoretical feasibility of IDPELS and
provided a review of observed operational characteristics in a variety of different
scenarios. Given the highly credible evidence behind the localisation method, the
final investigation in this research program was to test the practical application of
IDPELS. A detailed discussion of the field trials for IDPELS is provided in the
following chapter 5.
163
4.8 References
[1] T. A. Spencer, R. A. Walker, and R. M. Hawkes, "GNSS Interference
Localisation Method Employing Inverse Diffraction Integration with Parabolic
Wave Equation Propagation," presented at ION GNSS 2004, Long Beach
Convention Centre, Long Beach, California, 2004.
[2] E. Kreyszig, "Fourier Series, Integrals and Transforms," in Advanced
Engineering Mathematics, 7th ed. Columbus, Ohio: John Wiley & Sons, INC,
1993, pp. 566-625.
[3] F. v. Diggelen, "GPS Accuracy: Lies, Damn Lies, and Statistics," in GPS
World, vol. 9, 1998, pp. 41-45.
[4] E. M. Mikhail and F. E. Ackermann, Observations and least squares. New
York: IEP - A Dun-Donnelley Publisher, 1976.
[5] D. Adamy, "Emitter Location - Reporting Location Accuracy," Journal of
Electronic Defense, 2002.
[6] C. A. Balanis, "Arrays: Linear, Planar, and Circular," in Antenna Theory :
Analysis and Design, Second ed. New York: John Wiley and Sons, 1997, pp.
249 - 338.
[7] E. McCann and H. Hibbs, "Electrically small D. F. antenna," presented at IRE
International Convention Record, 1959.
[8] R. Schmidt and R. Franks, "Multiple source DF signal processing: An
experimental system," Antennas and Propagation, IEEE Transactions on
[legacy, pre - 1988], 34(3), pp. 281-290, 1986.
[9] X. Jian-juan, X. Jian-hua, and H. You, "Location error analysis of direction
finding location system," presented at Microwave, Antenna, Propagation and
164
EMC Technologies for Wireless Communications, 2005. MAPE 2005. IEEE
International Symposium on, 2005.
[10] A. El-Rabbany, "Appendix A GPS Accuracy and Precision Measures," in
Introduction to GPS: The Global Positioning System. Boston: Artech House,
2002, pp. 161 - 162.
[11] M. M. Casabona and M. W. Rosen, "Discussion of GPS Anti-Jam
Technology," GPS Solutions, 2(3), pp. 18-23, 1999.
165
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166
Chapter 5 - Geolocation Field Trials The previous chapter demonstrated the theoretical concept of inverse diffraction as
applied to the localisation problem. A numerical electromagnetic propagation model
was developed and a series of simulations were conducted that showed under ideal
conditions that the concept of inverse diffraction propagation could be used to locate
the relative position of the source transmitting an identified interference signal. It was
observed that there were limitations with the performance of the approach, namely
that knowledge of terrain was required and that a line-of-sight path to the interference
source provided the best results. Overall it was considered that the results were
promising enough to take to the next stage of evaluation, where the performance of
this process using actual field measurements would be tested. A series of field trials
were conducted in collaboration with the Electronic Warfare and Radar Division of
the DSTO, Edinburgh, South Australia. The signal measurement process
incorporated geodetic datum information and therefore the localisation process is
referred to as geolocation in the field trials. These field trials were conducted over a
four day period in June 2004 and were designed to evaluate the practical application
of the IDPELS methodology.
5.1 Objective
The objective of the field trials was to perform measurements of signal characteristics
that would allow the spatial-phase profile to be used as the input signal for IDPELS.
The spatial-phase profile represents the phase of the received signal according to the
geodetic location of the receiver. The receiver’s geodetic location is determined with
GPS data files, while measurements of in-phase (I) and quadrature (Q) phasor
samples allow the corresponding phase to be calculated with Equation 5-1. Further
information concerning signal phasor components is provided by Stimson [1].
phase = tan−1(Q / I) Equation 5-1
5.2 Overview
With the objective of field trials being to measure the received signal profile over a
know region with adequate dimensions to determine if the IDPELS methodology
167
could locate the transmission source of energy. Data was collected between the 8th
and 11th June 2004, using measurements made in two different South Australian
regions that were chosen for the field trials.
The geodetic location of the transmitter in each chosen South Australia region is listed
below in combination with the name of the respective region.
1. St Kilda radio research station − (34° 43’ 26.2” S, 138° 32’ 15.6” E)
2. Base of Mt Lofty Ranges (Baldon rd) − (34° 25’ 2.85” S, 139° 14’ 10” E)
In both regions, the transmitter boresight was directed along an approximate Northern
orientation. The chosen direction of receiver movement is based on an orthogonal
orientation to ensure the limitation associated with propagation angle did not degrade
the field trials as the free-space model was used in the trials. The receiver was
therefore moved on road sections that approximated either a western or eastern
direction. Further discussion and diagrams that provide greater perspective of the
field trials is given the subsequent sections.
Also as was shown in Chapter 4, greater accuracy and certainty could be achieved for
the localisation solution when the input signal profile was chosen to be a continuous
signal profile and not based on a segmented array configuration. The signal profile
was therefore chosen to have a continuous profile and was measured with a helix
antenna in a moving vehicle. The decision to use a horizontally measured continuous
signal profile introduces an analogy of the field trials with Synthetic Aperture Radar
(SAR) principles. Further detailed information of the input field profile is provided in
the following sections.
5.2.1 Regional Characteristic
During the 8th of June 2004, the transmission site was at the St Kilda radio research
station, while on the 11th of June the transmission site was at the base of the Mt Lofty
Ranges on Baldon road. No experimental data was measured on the 9th and 10th of
June. Field data recorded in the St Kilda region was analysed on the 9th, while rain
and showering weather conditions prevented field measurements on the 10th of June.
These sites were chosen because the regions over which the test signal was
transmitted approximated a flat terrain profile. With such a terrain profile, a free
168
space propagation path to the receiving antenna could be provided for the experiment.
While the free-space environment was desired due to its incorporation into
propagation model, the presence of trees, vegetation and buildings however degraded
the validity of various data sets in ascertaining the practical feasibility of IDPELS.
Further discussion of the trial regions is provided in the Field Trial Regions section,
while data set validity has been incorporated with the respective data sets in the Field
Trail Geolocation Results section.
5.2.2 Radio Frequency Equipment
In performing the field trials, the transmission source was a 1.399GHz tone signal that
was transmitted by a right-hand circular polarised (RHCP) axial helix antenna, with a
gain of approximately 15dB. A photograph showing the helix antenna, rubidium
oscillator, signal generator and RF power amplifier is displayed in Figure 5-1. It can
be seen that it is a mobile installation within a trailer, which is attached to an
automobile. The RF power supply is a petrol generator at the font of the trailer. It
should be noted that the transmitter remained stationary at each previously listed
location while field data was being measured.
Figure 5-1 Helix Transmission Antenna (Positioned for Mt Lofty data sets)
169
It should be noted that the field trials should not be regarded as a practical geolocation
technique, as the evaluation procedure is only valid with a stable continuous wave
(CW) signal. Since the stability of the tone signal was an important consideration [2]
for these trials due to the measurement of a frequency offset, rubidium reference
oscillators were chosen as signal references for the transmitter and receiver. A
photograph showing rubidium incorporation with the EB200 receiver is displayed in
Figure 5-2.
Figure 5-2 EB200 receiver
A Rohde & Schwarz EB200 Miniport receiver [3] was chosen for this investigation as
it’s internal design permits signal mixing and filtering for generation of an audio
frequency offset. A trigonometric identity showing generation of a frequency offset
term ‘ cos(f − f ) ’ by signal mixing is shown in Equation 5-2.1 2
1 ⋅ 2 12
[ 1 2 1 − f ) 2 ] Equation 5-2cos(f ) cos(f ) = cos(f + f ) + cos(f
170
With the EB200 receivers IF bandwidth being selected as 600Hz, a frequency offset
of 300Hz was chosen for measurement as it could easily be recorded using a laptop
computer sound card. The mixing frequency was selected by tuning the EB200 to
1.399GHz + 300Hz. In addition to the internal mixing and low-pass filtering that
provided the 300Hz offset signal, the EB200 also provided in-phase (I) and
quadrature (Q) signal samples. The I and Q samples of the offset frequency signal
were recorded into a stereo WAV file [4].
The remaining radio frequency equipment used in the field trials was the GPS
receiver. The chosen receiver was a Rojone Genius unit [5] that was serially
connected to a RS-232 port of the laptop computer. The output format of GPS data
from the Rojone unit was NMEA GGA (National Marine Electronic Association -
Global Positioning System Fix Data) [6]. NMEA GGA data was recorded with a
sampling rate of 1 Hz. A display of the Rojone Genius unit is shown in Figure 5-3.
Figure 5-3 Rojone Genius GPS Unit
171
5.3 Field Trial Methodology
As previously discussed, the primary objective of the field trials was to determine the
spatial-phase profile of the received signal and use it as the input signal for IDPELS.
To calculate the spatial-phase profile of the test signal, the geodetic location of the
receiver was also required to be measured in addition to measurements of In-phase (I)
and Quadrature (Q) phasor components of the received offset frequency. The phasor
components were measured by the EB200 receiver, while the geodetic location of the
receiver was provided by the Rojone Genius unit.
5.3.1 Field Trial Orientation
From the simulation results, it was shown that accurate localisation could be provided
when a continuous signal profile was chosen as the input signal for inverse diffraction
propagation. The field trial methodology was therefore concerned with measuring a
continuous field profile. The offset frequency signal was therefore measured by the
EB200 receiver in a moving vehicle that was driven along a chosen section of road.
The road sections used in the field trials were straight, thereby allowing a linear input
signal. A diagram showing the orientation of the receiver motion with respect to the
transmitter boresight is provided in Figure 5-4.
Figure 5-4 EB200 Signal Measurements
172
To provide further assistance for the readers understanding, Figure 5-5 shows an
analogy of the field trial methodology with an isotropic transmission source. It is
however important to note the actual transmitter is not isotropic. In addition to
increasing the reader’s understanding, Figure 5-5 also provides an indication of how
the spatial-phase should appear with field trial measurements. While information
concerning spatial-phase calculation is provided in the Doppler Shift Transparency −
Spatial Phase section, an introduction is provided below.
Figure 5-5 Isotropic Transmitter Analogy
As shown in Figure 5-5, the location of the EB200 receiver at the start of the
measurement procedure is highlighted. This is shown as the process for determining
the spatial-phase requires all measured phase values to be subtracted from the first
phase measurement. A diagram showing eleven selected phase samples and
respective range values from the Isotropic transmitter is shown in Figure 5-6, where
the EB200 is moving in the right-hand direction. Samples 2 to 10 are labelled, while
the first and eleventh samples are at respective ends of the measurement path. It
should be noted that the displayed movement of the EB200 receiver in Figure 5-6 is
geometrically symmetric with respect to the transmitter. During the field trials,
measured data sets may not have been geometrically symmetric with respect to the
transmitter.
173
As the EB200 begins movement in the right-hand direction after measuring the first
phase value, Figure 5-6 shows the phase values vary according to some function that
would represent the arc of the circle. The greatest unwrapped phase value compared
to the first measured phase values concurs with the EB200 receiver being positioned
at its nearest location to the transmitter. The nearest location of the EB200 receiver to
the transmitter was highlighted in Figure 5-5 and can be determined by finding where
spatial-phase gradient is zero as highlighted in Figure 5-7.
Figure 5-6 Measurement of Phase values (Symmetric Example)
As the phase variation can be expressed by an equation representing the arc of a circle,
a quadratic polynomial function will mathematically represent the spatial-phase
profile [7]. A general representation of a quadratic polynomial is shown in Equation
5-3, where ‘a’, ‘b’ and ‘c’ are the quadratic coefficients [8] and ‘x’ is the input
variable. For a quadratic polynomial to exist, the ‘a’ coefficient can not be zero.
f (x) = ax 2 + bx + c Equation 5-3
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The spatial-phase profile corresponding to the field measurements of Figure 5-6 is
shown in Figure 5-7. With the measurement path being geometrically symmetric
relative to the transmitter, the phase measured at the start of the data set will be the
same as the phase value at the end of the data set. All other phase values highlighted
in Figure 5-6 will have a greater unwrapped phase value. The substraction of
measured phases from the first phase value will therefore result in lower spatial-phase
value as described by the quadratic function.
Figure 5-7 Quadratic Spatial-Phase Profile (Symmetric Example)
In applications where finite-valued ranges are associated with system operation, the
use of quadratic functions to represent signal phase is very important for accurate
signal processing. It should however be noted that in many applications signal phase
has also been considered to be linear. A linear consideration is valid where phase
variation between samples is negligible. A linear phase variation is therefore
associated in astronomy where distances to objects can be considered to be infinite.
While this section has introduced the spatial-phase concept, further discussion of the
quadratic phase variation is provided in the Synthetic Aperture Radar (SAR) section.
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5.3.2 Field Data Set Size
In recording phasors of the frequency offset signal, the time period for measurements
was limited to 60 seconds. This time limit corresponds to the default setting of the
freely distributed Microsoft sound recorder [9]. With a 60 second time period and a
sampling rate of 44.1 kHz, the size of data sets approximates 10MB. Such a data set
size was considered prudent for preliminary analysis in these field trials. It should be
noted that displayed parameters in this thesis are taken from different data sets in both
test regions.
5.3.3 Least Square Fitting Polynomial
For IDPELS to provide an accurate geolocation solution, the input spatial-phase
profile must correctly represent the measured signal. It should be noted that direct
application of measured spatial-phase was not performed. Instead the spatial-phase
was modelled with a least square fitting (LSF) quadratic polynomial. A diagram
showing error between the LSF polynomial and field data is shown in Figure 5-8.
Additionally, if the terrain profile was not flat and obstructions existed in the
propagation path of the test signal, multipath and diffraction will degrade the validity
of field data sets in proving the practical feasibility of IDPELS. This is because a
free-space propagation model was chosen as the basis for the model. Field
characteristics that do not conform to free-space principles will be discussed before
the results section.
Figure 5-8 Least Square Fitting Quadratic Polynomial
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5.3.4 Relative Doppler Shift
As previously stated in the Field Trial Orientation section, a continuous signal profile
was chosen for measurement in the field trials. This was because greater localisation
accuracy was demonstrated under simulation when the input signal was continuous.
By placing the EB200 receiver in a moving vehicle, the continuous input signal
profile could be measured with greater efficiency in terms of cost and equipment
compared to a fixed array configuration. The EB200 receiver was therefore moved in
a vehicle along a straight section of road. With the EB200 receiver being moved
during the measurement period, this introduced a Doppler shift [10] in the measured
frequency offset signal. This section will discuss the relative Doppler shift
experienced in the field measurements. The operations that were performed to
overcome the relative Doppler shifts are discussed in the following sections.
5.3.5 Galilean Relativity
As the velocity magnitude of the vehicle carrying the EB200 receiver is negligible
compared to the speed of light, the Galilean Principle of Relativity [11] governs the
field trials. With the field trials being configured with a stationary transmitter and a
moving receiver, the frame of reference is based on the stationary receiver at the start
of measurement for each data set. As the EB200 receiver will be in motion during
signal phase measurement, a Doppler shift with respect to the transmitter will be
incurred in respective field measurements. In addition to the Doppler shift relative to
the transmitter, another Doppler shift with respect to the repeater will also be
experienced in the offset frequency measurement. Both of these Doppler shifts can be
determined by knowing the relative speed with which the EB200 receiver was moved.
A diagram showing the resultant Doppler frequency shift experienced while recording
data set (03) on Pine Creek track is shown in Figure 5-9.
The difference between the two relative Doppler shifts provides the resultant Doppler
shift shown in Figure 5-9. The speed at which the vehicle carrying the EB200
receiver moved along the road section was recorded with the Rojone Genius GPS unit.
By measuring the vehicles speed along the road section, the Doppler shift with respect
to the repeater can be directly determined. By knowing the vehicles speed and the
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corresponding distance moved, the relative speed of the EB200 with respect to the
transmitter could than be determined by incorporating the NMEA GPS location data.
The relative speed of the EB200 with respect to the transmitter varied in direct
proportion to the range variation between the transmitter and receiver. The resultant
Doppler frequency shown in Figure 5-9 was calculated based on the relative EB200
receiver speeds shown in Figure 5-10.
Figure 5-9 Resultant Doppler Frequency
Figure 5-10 Relative receiver speeds governing Doppler shift
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Due to the road orientation, the receiver experiences an increase in range from the
transmitter during the measurement period, as shown in Figure 5-11. This was
interpreted as a negative receiver speed with respect to the transmitter. A positive
speed along the road was interpreted if an East heading was observed. The respective
frequency shift of the measured offset signal is shown in Figure 5-12.
Figure 5-11 Receiver Range from Transmitter
Figure 5-12 Frequency Shift
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Analysing of Figure 5-12 shows the measured offset frequency to be nearly constant
at 301Hz during the initial 4 seconds of that data set. During this initial time period,
the EB200 was stationary as shown by the zero velocity magnitude in Figure 5-10.
The section of data shown in Figure 5-12 between the time limits of 4 and 15 seconds
shows an almost linear rate of reduction in the frequency offset. This time period
viewed in Figure 5-10 shows that an almost constant acceleration rate was maintained
while increasing the speed of the vehicle. After accelerating the vehicle, the time
period defined by the limits of 15 and 46 seconds in Figure 5-12 shows a lower
reduction in the offset frequency. This time period in Figure 5-10 shows that an
almost constant speed was maintained. Between 46 and 57 seconds in Figure 5-12,
the offset frequency returns to its initial value of 301Hz. Analysis of Figure 5-10
shows the vehicle and EB200 receiver was slowed down, and completely stopped just
prior to the 60 second time limit. By terminating receiver movement near the end of
the data set, an analysis of the signal stability between the receiver and transmitter can
be performed. With the same frequency offset being shown at the beginning and end
of the measurement period, this indicates the test signal has remained stable during
measurement of the data set.
By also comparing the frequency shift of Figure 5-12 with the range variation in
Figure 5-11, it is shown that as the receiver is moving away from the transmitter, the
offset tone signal experiences a decrease in frequency. This frequency behaviour
corresponds with the Doppler principle, which will be further discussed in the Test
Signal Characteristics section.
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5.3.6 Doppler Shift Transparency — Spatial-Phase
To determine the spatial-phase profile of the measured signal, there is a requirement
to find the phase variation as a function of the EB200 receiver’s motion with respect
to the transmitter. As mentioned in the previous section, the speed of the receiver
with respect to the transmitter was calculated by incorporating GPS data with the
speed of the receiver as indicated by the speedometer of the carrying vehicle.
By knowing the speed of the receiver relative to the transmitter, the change in
distance between the transmitter and receiver can also be determined with GPS
information. It should be noted that the change in distance between the transmitter
and receiver in this thesis is also referred to as the range variation. With the range
variation being known, the spatial-phase is calculated by incorporating the
corresponding phase variation. To determine the phase variation in the field trials,
there is a requirement to find the difference between the directly measured moving
phase profile and phase profile that would be measured by a stationary receiver. The
stationary receiver would also remain at the location decided as the starting point for
each data set.
If the EB200 receiver remains stationary, the received frequency will be constant.
With the received signal having a constant frequency, the corresponding offset
frequency will also be constant. By sampling the constant offset frequency at a
constant rate of 44.1kHz, the corresponding phase profile will be linear. An example
of a linear phase profile can be shown with the data set (1) measured on Pine Creek
Track at the base of the Mt Lofty ranges. In this data set, the receiver remained
stationary for the entire 60 second measurement period. A display of the measured
phase for data set (1) on Pine Creek Track is shown in Figure 5-13. As can be seen in
the figure, the phase profile is linear. Further discussion of linear phase profiles is
provided by Smith [12].
For a data set to provide evaluation of IDPELS, the EB200 receiver was moved
during the measurement period of one minute. The linear phase model was therefore
determined by extrapolating the phase gradient according to the initial time period
where the receiver was stationary. The driving pattern for each data set is similar to
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the motion shown in shown Figure 5-14 that corresponds to data set (04) recorded on
Pt Gawler road in St Kilda region.
Figure 5-13 Linear Phase variation for stationary Receiver
Figure 5-14 Motion of EB200 Receiver
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As shown in Figure 5-14 the receiver initially remained stationary for several seconds
in each data set. Acceleration along the road section was then intended to be
increased in a linear fashion until a desired maximum speed had been reached. The
maximum speed was maintained until sufficient time remained for a linear de-
acceleration, where the receiver would be stationary just prior to the 60 second limit.
Different maximum speeds were reached in each region, according to the road quality.
With the above description of EB200 receiver movement, the difference between the
stationary linear phase model and the observed phase allows the phase variation to be
determined by the range variation between the transmitter and EB200 receiver. A
diagram showing the extrapolated linear phase and measured phase for data set (03) at
the base of Mt Lofty Ranges is presented in Figure 5-15
Figure 5-15 Measured Phase and Linear Phase: Mt Lofty Base Data Set (03)
By incorporation the range variation as provided by GPS information with the phase
variation, the spatial-phase profile can be correctly determined. The spatial-phase for
Mt Lofty data set (03) is shown in Figure 5-16, where wavelength units instead of
degrees are presented.
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Figure 5-16 Measured Spatial Phase: Mt Lofty Base Data Set (03)
The spatial-phase as can be viewed in Figure 5-16 does not have a linear variation.
The spatial-phase instead has a variation that can be modelled as a quadratic function.
The quadratic nature arises because in a realistic three dimensional environment, the
wavefront conforms to a spherical wavefront and should not be considered a plane
wave. Further discussion of the quadratic spatial-phase variation is provided in the
Focused SAR Array – Quadratic Phase Variation section
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5.3.7 Input Signal Cross-range
With the signal spatial-phase demonstrating the characteristic of a quadratic function,
a Least Squares (LS) estimate [13] for the quadratic polynomial was determined.
With such a polynomial being known, the signal phase according to any cross-range
distance on the road could be determined. However to ensure the polynomial function
accurately represented the spatial phase, the required distance to be covered while
actually recording the signal was found be approximately one kilometre. The cross-
range covered in many data sets was not sufficient for an accurate quadratic
polynomial to be determined. The cross-range distance covered in Mt Lofty data set
(03), is shown as a function of time in Figure 5-17.
Figure 5-17 Cross-range Distance: Mt Lofty Base Data Set (03)
The cross-range distance covered was 389 metres. While this data set in isolation will
not allow an accurate polynomial to be estimated, numerous data sets that were
consecutively measured could be used. One example can be shown with data sets
(02), (03), and (04) that were measured at the base of the Mt Lofty Ranges on Pine
Creek track. The corresponding cross-range distance for this combined set of data is
1104.6 metres. A display of the measured spatial phase for these consecutive sets of
data is shown in Figure 5-18. The spatial-phase as determined by the least-square
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fitting polynomial is displayed in Figure 5-19. The quadratic polynomial allows
spatial-phase over any cross-range distance to be specified.
Figure 5-18 Spatial Phase: Mt Lofty Base data sets (02-03-04)
An important note that should be considered with spatial-phase estimation concerns
the extent of cross-range values. The curvature of the Earth was chosen to be ignored
in model processing. For this assumption to be valid, the cross-range values chosen
for the input spatial-phase should be kept small in relation to the radius of Earth.
In Figure 5-19, the initial phase calculated from EB200 data is located at the (0, 0)
coordinate. The coordinate corresponding to the final EB200 measurement is (1105,
1040). The receiver was moved in an Eastern direction. To provide a spatial
overview, Figure 5-20 shows the actual measurement path that is linear, and its
extension. The transmitter location is also shown. With an accurate estimate of
spatial-phase, the minimum spatial-phase value will have a cross-range corresponding
to the boresight of the transmission shown in Figure 5-20. To perform this spatial-
phase test, the Cosine Rule used when angles are unknown in the triangle is applied.
According to the Cosine Rule, the transmitter boresight intercepts the extended
measure path 410m to the west of the first phase measurement. By plotting the
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spatial-phase, it was also found that the minimum spatial-phase value is located 410m
to the west of the first measured phase value. This agreement indicates the error of
spatial-phase polynomial to be acceptable with cross-range values in close proximity
to the boresight intercept.
Figure 5-19 Estimated Spatial Phase: Mt Lofty Base data sets (02-03-04)
Figure 5-20 Geodetic Overview: Mt Lofty Base data sets (02-03-04)
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5.4 Synthetic Aperture Radar (SAR) Analogy
The chosen method to continuously move in a straight direction with a receiving
antenna pointing in a fixed direction to the side of the van introduces an analogy
between the Synthetic Aperture Radar (SAR) concept, and the field trials that measure
a continuous wave (CW) test signal. The quadratic variation of the phase can also be
explained based on the principles associated with a focused SAR system.
Correspondingly, a review of SAR is provided. It should however be noted that the
primary application of SAR concerns imaging, mapping or target detection [14].
With geolocation and imaging conforming to different objectives, not all SAR
characteristics are required to be considered and so only a brief overview will be
provided. One of the primary differences that should be noted concerns the signal
propagation path. SAR requires two-way propagation of the transmitted signals for
its fine azimuth resolution capability. This research program is based on blind
geolocation, where only the interference signal will be propagating. Signal
propagation with respect to inverse diffraction geolocation is therefore one-way.
5.4.1 SAR Development
Original development of the SAR principle was established by Carl Wiley in 1951
[15]. The forward motion of a fixed side-looking antenna was found to be
advantageous in overcoming the problems associated with obtaining adequate
azimuth resolution to recognise objects at long ranges.
With a real array, azimuth resolution is governed by the horizontal width of the beam
at the corresponding range of a target. To determine the azimuth resolution of an
aperture, both the half-power beamwidth ( β3dB ), and the targets range (R) are required
to be known. With these parameters, the linear azimuth resolution distance ( ΔAzires )
can be determined by Equation 5-4, [16].
ΔAzires = β3dB ⋅ R = λ
⋅ R Equation 5-4L
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The main-lobe ( β3dB ) can be determined by the division of the signals wavelength (λ),
by the physical length of the antenna (L). Therefore to increase the azimuth
resolution, signals with small wavelengths together with large antenna apertures
should be used. However in airborne radar mapping the problem arose where
impractically long antennas apertures was required, or signal wavelengths were so
short that severe atmospheric attenuation impacted the system. SAR overcame the
array length problem by taking advantage of an aperture forward motion to integrate
the signals for synthesize of a very long antenna array.
5.4.2 Focused SAR Array – Quadratic Phase Variation
Further explanation of why the spatial-phase of the CW test signal has a quadratic
variation is provided by the principles associated with the development of a Focused
SAR array. The Focused SAR array is therefore reviewed in this section.
Beginning with Equation 5-4, it is indicated that azimuth resolution will become finer
with increased array length. With an unfocussed array, a length is however ultimately
reached where both the antenna gain will begin to decrease and beamwidth will widen,
both of which are undesirable properties for imaging or mapping. This problem arises
because the distance travelled by the signal continuously increases to elements in the
uniform linear array that are further from the boresight. In a 2D scene this indicates
that the signal is not propagating with a planar wavefront, but a cylindrical wavefront.
The Hankel function as previously discussed and used in conjunction with the wave
equation, is based on either cylindrical, or spherical wave propagation. A cylindrical
wavefront with increasing range distances to array elements is shown in Figure 5-21.
Figure 5-21 Circular Wavefront Phase Variation [17]
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With the wavelength of the test signal being 21.43cm, considerable phase differences
will result with the increasing path distance to each array element. As already
demonstrated, it has been shown the phase difference at each array element will vary
according to a quadratic function.
To overcome this unfocused SAR problem, the focussed SAR can be used where fine
azimuth resolution is achieved by using a small aperture [18], a characteristic that is
not demonstrated in Equation 5-4. Finest azimuth resolution is achieved by ensuring
all spatial elements of the synthetic array during the measurement process receive the
returned signal from the region being imaged. The width of the main-lobe must
therefore entirely cover the region or object. With a higher sampling rate of the signal,
the gain of the synthetic antenna (which is the sum of phasors) will increase. With
this increase in gain, there will be a narrowing of the synthetic beamwidth. A small
aperture is therefore required to cover large regions being mapped with fine resolution.
While the aim of SAR mapping is not the same as geolocation, its application
involves the use of many similar parameters found in the geolocation field trials.
There are also various SAR operating modes that can be applied, which provide the
basis for an alternative approach in measuring the spatial-phase for further field trials.
A review of other SAR modes is provided by Stimson [19].
5.4.3 Inverse Synthetic Aperture Radar (ISAR)
To conclude the SAR analogy section, the Inverse Synthetic Aperture Radar (ISAR)
will be briefly reviewed. While the inverse term in its title may indicate an analogy, it
should be noted that Sullivan [20] states it has no mathematical inversion with respect
to SAR imaging and can be considered a misnomer. Inverse Diffraction propagation
concerns mathematical inversion of the Diffraction term associated with PEM.
The principle application of ISAR involves imaging a target that has rotational motion
with respect to a stationary radar system. The targets rotational motion requires
Doppler processing, where angular resolution improves with the angle through which
the target rotates. With ISAR using stationary radar, it has no association with the
field trials and will not be further discussed.
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5.5 Field Trial Regions
With the methodology of field trials having been discussed, maps and pictures of the
trial regions will be provided to enhance the orientation perspective. The St Kilda
region will initially be presented, juxtaposed with the Mt Lofty Range base region.
5.5.1 St Kilda Region
The DSTO St Kilda radio research station was a chosen transmission site to allow a
received signal profile to be measured on chosen roads with close proximity. A photo
of the radio research station is shown in Figure 5-22. The transmission boresight
approximated a northern direction during field trials.
Figure 5-22 DSTO Radio Research Station - St Kilda (looking South)
Roads that were used to measure the test signal are McEvoy road, and Pt Gawler road.
Eight (8) data sets were recorded on McEvoy road, where transmission power was
10W. On Port Gawler road, eleven (11) data sets were recorded with transmission
power set to 30W. A map displaying their orientation with respect to the St Kilda
radio research station is shown in Figure 5-23.
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Figure 5-23 St Kilda Map
A photo of the McEvoy road section used to measure the test signal is shown in
Figure 5-24, while the Pt Gawler road can be viewed in Figure 5-25.
Figure 5-24 McEvoy road
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Figure 5-25 Pt Gawler road
A repeater can be seen on both McEvoy road and Pt Gawler road. The direction
towards the transmitter is shown on McEvoy road (Figure 5-24), where the helix
antenna is pointing in a southern direction. Early in the field trials, the use of
repeaters was considered to account for Doppler shift. It was however noted that
signal strength from the repeater was not sufficient and had little effect on results.
The repeater was therefore not employed with measurements at the Mt Lofty Range
base. A possible option that would have enhanced the use of repeater to account for
Doppler shift is a high gain amplifier being applied to the transmitter. This option
was however not taken in the preliminary field trials.
It can also be seen from both photographs (Figure 5-24 and Figure 5-25) that
obstructions such as trees or building exist in the signal propagation path between the
receiver and transmitter. This indicates that the use of raw field data may not be
suitable for analysing IDPELS feasibility as the model was developed for a free-space
environment. Further discussion of raw data set suitability is provided in the Free-
space Progagation section.
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5.5.2 Mt Lofty Range Base Region
The base of the Mt Lofty ranges, approximately 10km east of Truro along the Sturt
Highway was the other region chosen for field trials. The road sections used to
measure the test signal were along Pine Creek track, and Woolshed road. The
transmission power setting for the region was 1W. Nine (9) data sets were measured
on Pine Creek track, and eight (8) on Woolshed road. A regional map is shown in
Figure 5-26. The transmitter was positioned on Baldon road, with the roads chosen
for receiver measurement approximating a perpendicular orientation to the transmitter
boresight. With the beamwidth on the transmission antenna approximating 35
degrees, the nominal field-of-view is highlighted in Figure 5-27
Figure 5-26 Mt Lofty Range Base Region
A photograph showing the general environment at the base of the My Lofty ranges is
displayed with Figure 5-28. The scenario shows the propagation path of the signal is
not subject to any major obstructions such as buildings or hills, and the terrain profile
can be considered to be flat. While such a scenario will be feasible with free-space
modelling, not all data sets taken in the region can be considered obstruction free.
Signal obstruction by vegetation was occasionally experienced as shown in Figure
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5-29. This figure displays trees that obstructed signal propagation while measuring
data set (02) on Pine Creek track.
Figure 5-27 Mt Lofty Range Base, Nominal Field-of-View
Figure 5-28 Free-space Environment – Pine Creek Track
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Figure 5-29 Pine Creek Track Tree Obstructions - Data set (02)
Woolshed road (Figure 5-30) on the northern side of Pine Creek was the other road
chosen for phase measurement. The receiver is on an elevated plateau with respect to
Pine Creek track and as shown, large trees will impact signal propagation.
Figure 5-30 Tree Obstructions - Woolshed Road
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5.6 Free-space Propagation
A free-space propagation model was chosen to analyse the spatial-phase being
calculated from field trials. The free-space model provides the benefits of simple
model implementation, and more efficient analysis. A definition of free-space for
radar applications has been defined by United States Patent Office [21] as, “Space
where the movement of energy is any direction is substantially unimpeded, such as
interplanetary space, the atmosphere, the ocean and other large bodies of water or the
earth”
In a free-space model, path-loss is represent by the only the propagation loss. It does
not account for other loses such as absorption or diffraction losses. Propagation loss
results from an increase in the surface of the sphere as the signal propagates. With
conservation of energy, there is a corresponding reduction is signal density. The one-
way free-space path loss can be determined with Equation 5-5, [22]. Range between
the transmitter and receiver is represented by (R), while signal wavelength is
represented by (λ). In assessing the free-space path-loss for each of the data sets,
application of Equation 5-5 is displayed in Figure 5-31 for each of the roads used in
the field trials. The transmitter power in all examples in Figure 5-31 was assumed to
be 10W.
Free-space Loss (dB) = 20log ⎢⎡ 4π R
⎥⎤ Equation 5-5
⎣ λ ⎦
Figure 5-31 Free-space Loss
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For a test region to be considered feasible for free-space modelling, any possible
obstructions should not significant impede the first Fresnel zone as shown in Figure
5-32. The Mt Lofty base region is shown to approximate a free-space region, as
viewed in the mid-section of Figure 5-33. The terrain profile is flat and obstacle free.
Figure 5-32 Fresnel Zones
Figure 5-33 Mt Lofty base (free-space model)
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The formula to approximate the radius of the elliptic Fresnel zone of order ‘n’ at an
obstruction is shown in Equation 5-6, [23]. The obstructions range from each antenna
is shown with d1 , and d2 , while the signal wavelength is represented by ‘λ’. By
estimating the Fresnel zone radius, corresponding heights that objects should not
significantly exceed can be determined.
λ 1 2 Equation 5-6n d d rfres = 2
d1 + d2
5.7 Data Set Power Variation
To determine if a data set is feasible for free-space modelling, its power variation
should be uniform. Where power variation was seen to have a variation less than 2dB,
a good line-of-sight existed between the receiver and transmitter. If the power
variation of a data set exceeds 6dB, reflection modelling would then be required to be
incorporated in the IDPELS model. The following diagrams show the most stable
data set measured on each road, where the power variation being shown is in
logarithmic units.
Figure 5-34 Signal Power Variation: Mt Lofty Range Base
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Figure 5-35 Signal Power Variation: St Kilda Region
As shown in Figure 5-34, the least power variation is shown with Pine Creek track
data set (04). It should be noted that Pine Creek track data set (03) is almost similar
to data set (04). From Figure 5-28 and Figure 5-33, the region closely resembles a flat
surface with no obstructions. With such conditions, raw field data can be used to
directly determine the spatial-phase input signal for Inverse Diffraction propagation.
Data set (10) recorded on Woolshed road is shown in Figure 5-34 to have a time
period of at least 10seconds where power variation exceeded 6dB. This corresponds
to approximately 17% of data not being suitable for free-space propagation. A similar
situation exists with McEvoy road data set (07) shown in Figure 5-35, which however
has a lower variation level of 4dB with unsuitable data.
The data set presented with the greatest power variation is data set (11) on Port
Gawler road (Figure 5-35), where over 50% of data exceeds the 6dB variation. This
indicates that modelling of obstructions and accounting for reflection should be
incorporated with all data sets measured on Port Gawler road. It’s important to note
that all other data sets measured on each road, have a greater power variation.
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5.8 Correlation of Signal Parameters
Pine Creek Track has been shown to have the greatest resemblance of a free-space
environment according to power variation. With the receiver being subject to motion
in free-space, there should therefore be a high correlation in the variation of the
following received signal parameter,
1. spatial frequency
2. spatial phase
3. GPS position
Each of the data sets with the least power variation on each road has their signal
variation for each of the above listed parameter graphically shown in Figure 5-36.
Figure 5-36 Correlation of Parameter Variation
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As shown with the Pine Creek track data set, Woolshed road and Port Gawler road
data set show an increasing value of signal parameters. The increasing pattern is due
to the range between the receiver and transmitter also increasing during measurement.
The opposite pattern is demonstrated with McEvoy data set (07). In this data set, the
range between the transmitter and receiver was reduced during signal measurement.
The difference between parameters for Pine Creek track is minimal. Phase and
frequency variation is identical, with GPS variation being identical except for the
initial 13 seconds of signal measurement. This analysis confirms that raw data
measured on Pine Creek track is suitable for free-space modelling.
Data set (10) measured on Woolshed road has a continuous increase in the variation
between signal parameters. This difference in variation is attributed to large trees that
exist in the signals propagation path, and the increase plateau elevation as shown in
Figure 5-30.
In the St Kilda region, data set (07) measured on McEvoy road has an identical
variation concerning signal phase, and frequency for the initial 25 seconds of
measurement. The difference between these two parameters then remains less than
the difference with respect to GPS. With approximately 80% of data having a power
variation less than 4dB (Figure 5-35), a free-space scenario could be considered to
exist. The inverse diffraction propagation results via IDPELS will provide an
indication of how geolocation accuracy is reduced with such correlation and power
variation. Another important point that should be considered with McEvoy road is the
fact that consecutive data sets were not measured. Cross-range distances of the
measured signal were between 350 – 450 metres and will not provide sufficient
accuracy for spatial-phase estimation.
With respect to Port Gawler road, data set (11) has a considerable difference between
all parameters. As shown in Figure 5-25, there are many trees and other obstructing
that has impacted the test signal. The use of raw data from Port Gawler road is
therefore not considered suitable for the free-space IDPELS model.
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5.9 Test Signal Characteristics
Before reviewing the field trial results of the free-space IDPELS model, further
suitability of data sets was evaluated by analysing other test signal characteristics such
as amplitude and frequency. Data sets measured on McEvoy Road are used in this
section as they clearly demonstrate multipath in the spectral analysis. While signal
amplitude and frequency are analysed in this section, it should be noted that only
phasors of the offset frequency were measured. In determining the amplitude and
frequency characteristics of the received signal, EB200 data sets were divided into
bins for data processing. Frequency and amplitude analysis was performed with
Fourier processing and frequency interpolation on the data bins. The signal
processing procedures used in this analysis can be viewed in Appendix B.
5.9.1 Phasor Analysis
As signal phasors of the offset frequency were directly measured by the EB200
receiver, a review of In-phase (I) and Quadrature (Q) phasor components is initially
provided. A plot of I and Q phasor components from data set (03) measured on
McEvoy road is displayed in Figure 5-37.
Figure 5-37 Phasor Components: McEvoy road, Data set (03)
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The Time Samples axis shown in Figure 5-37 corresponds to a time-lag as the signal
phase was being measured for the data set. From radar principles, it is known that
phasor analysis allows positive and negative Doppler frequencies to be differentiated
[1]. Because the ‘I’ phase component has the same variation for positive or negative
Doppler shifts, the relative position of the ‘Q’ phasor is stated. As time-delay
increases directly with the time sample value in Figure 5-37, the ‘Q’ phasor
component leads the ‘I’ phasor component. When ‘Q’ leads ‘I’, the Doppler shift is
negative and indicates the receiver is moving away from the transmitter. The EB200
receiver was moved in the Eastern direction while recording data set (03) on McEvoy
road and a display of the relative speeds of the receiver is shown in Figure 5-38. The
negative speed with respect to the transmitter shows the EB200 moved away during
measurement of data set (03). The repeater was also positioned to the East of all data
sets measured on McEvoy road. As the EB200 was moved towards the repeater
during this data set, the relative speed to the Repeater is shown to be positive.
Figure 5-38 Relative Speeds of EB200 Receiver
As the relative speed of the EB200 to the transmitter is negative, the range between
the transmitter and receiver will therefore have increased. A display of the range
between the transmitter and EB200 while measuring data set (03) on McEvoy road is
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shown in Figure 5-39. As the range value has increased, this clearly shows the EB200
moved away from the transmitter.
Figure 5-39 McEvoy road data set (03): Range
While phasor analysis correctly identified relative EB200 motion in this data set, it is
important to notes that this phasor analysis is not valid. By conducting a phasor
analysis on all data sets, it was shown that ‘Q’ leads ‘I’, regardless of the relative
EB200 motion. Phasor analysis is not valid because a frequency difference was
measured by the EB200 receiver. The only valid parameter that can be provided with
phasor analysis in all of the field trial data sets is the relative offset of the Local
Oscillator (LO) in the EB200 receiver to the input frequency. Because ‘Q’ leads ‘I’,
the LO is greater than then received signal. Knowledge of this is not required,
therefore phasor analysis is no longer discussed.
5.9.2 Signal Amplitude and Frequency
With a 10W test signal being transmitted to the receiver on McEvoy road, the
indicated power level was shown to approximate 60dBm. While not directly
measured, the absolute signal amplitude is shown in Figure 5-40 where the significant
variation in signal intensity demonstrates multipath and diffraction effects.
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Figure 5-40 Signal Amplitude: McEvoy road, Data set (03)
A spectral analysis of the audio signal measured with the EB200 receiver is displayed
in Figure 5-41. As shown, the tone signal has been subject to multipath. Multiple
signals are represented by the various spectral peaks. The sheds shown on the left-
hand side of Figure 5-24 may have contributed to the multipath characteristic.
Figure 5-41 Audio Signal Spectrum: McEvoy Road - data set (03)
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With the amplitude variation in Figure 5-40 coupled with the multipath feature of
Figure 5-41, data set (03) from McEvoy road is not suitable for free-space
propagation. In a free-space environment, only a single spectral peak should be seen
in the spectral analysis. Data set (04) measured on Pine Creek track has such a
spectral characteristic for the tone signal, as shown in Figure 5-42.
Figure 5-42 Audio Signal Spectrum: Pine Creek track - data set (04)
The displayed spectral peak is centred at 292 Hz and therefore represents a Doppler
shift of 8Hz. In comparison with the frequency shift diagram of Figure 5-12, the
spectral analysis of Figure 5-42 was performed for the thirtieth second in the
measurement period. This spectral characteristic provides further evidence that data
sets measured on Pine Creek track are suitable for the free-space IDPELS model used
in the preliminary field trials.
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5.10 IDPELS Accuracy Analysis
The most important issue concerning the application of an emitter-geolocation system
is the accuracy with which it can locate the transmitter [24]. In analysing the
accuracy of the location estimated by IDPELS, it’s important to note that only the
linear error will be considered. As will be shown in the following geolocation results,
IDPELS provides a range and cross-range estimate to the emitter. An angle-of-arrival
(AOA) is not analysed or provided by IDPELS in this field trial investigation. The
RMS error associated with direction-finding systems [25] is therefore not required for
analysis in this investigation. Only the linear error associated with the range, and
cross-range estimate will provide an indication of the geolocation accuracy provided
by IDPELS.
The primary objective of these field trials was to determine the practical feasibility of
IDPELS to geolocate a radio frequency transmitter. With the frequency of the tone
test signal being known, the process of identifying the interference signal does not
form part of this research program. With only a limited number of data sets being
measured, the use of describing system accuracy with the Circular Error of
Probability (CEP) [26] or Elliptical Error of Probability (EEP) [25] is therefore not
feasible. Any statistical results returned by CEP or EEP can not properly describe
system accuracy, due to an insufficient number of data sets. The limited number of
data sets adds further weight to system validation with analysis of linear error.
After presenting the most accurate geolocation results from each of the regions in the
Field Trial Geolocation Results section, a comparison of linear error in each test case
is provided in the following Field Trial Geolocation Error section. By analysing the
range error in each of the optimal test cases, the practical feasibility of IDPELS can be
established.
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5.11 Field Trial Geolocation Results
With the developed IDPELS model being based on a free-space environment, data
sets that provided free-space propagation will initially be analysed. From the Free-
space Propagation section, it was shown that data sets measured on Pine Creek track
provided the greatest indication of free-space signal propagation. Pine Creek track
data sets are therefore initially be analysed to determine the practical feasibility and
accuracy of the geolocation technique. Investigation of IDPELS operation using other
data sets will then be analysed to provide an indication of system operation when
factors such as diffraction and multipath affected free-space signal propagation.
5.11.1 Pine Creek Track
As stated in the Input Signal Cross-range section, the cross-range of the input spatial-
phase was found be one kilometre. Such a cross-range distance was reached with data
sets that were consecutively measured on Pine Creek track. These consecutive data
sets were sets (02-03-04), and (07-08-09). The geodetic location of the receiver in
both combined data sets is shown in Figure 5-43.
Figure 5-43 Consecutive Data Sets: Pine Creek Track
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While Figure 5-43 shows the EB200 location appears to be almost identical in each
combined sets of data, the EB200 motion was different in each set. The EB200 was
moved in an Eastern direction while measuring phase for data sets (02-03-04). The
opposite, Western direction was moved while measuring data sets (07-08-09).
Analysis of data sets (02-03-04) will be provided before data sets (07-08-09).
5.11.1.1 Data Sets (02-03-04)
A diagram providing an overview of receiver motion with respect to the transmitter
was shown in Figure 5-20. As the EB200 was moved in the Eastern direction, the
range between the transmitter and the receiver also continually increased. The
measured spatial-phase profile was shown in Figure 5-18. The quadratic coefficients
for the least-square fit polynomial are highlighted in Figure 5-44, which also shows
the section for the measured (green), estimated (blue) and specified spatial phase used
as input for the IDPELS operation (red).
Figure 5-44 Cross-range of Input Signal: Pine Creek Track (02-03-04)
In Figure 5-44, the position of the EB200 at the beginning of signal measurement for
the combined data set is indicated by the index corresponding to where the cross-
range and wavelength are both zero. The specified limits of the spatial phase were
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defined as -1300 and 915m in this analysis. The length of the input signal was
therefore 2215m. By using the spatial-phase profile indicated by the red highlight in
Figure 5-44, the IDPELS geolocation estimated is shown in Figure 5-45.
Figure 5-45 Pine Creek Track (02-03-04) Geolocation
By analysing the geolocation estimate shown in Figure 5-45, the IDPELS method
indicates that at a cross-range of 410m to the west of the first field measurement, the
range to the transmitter is 4.8Km. The geolocation estimate is indicated by the peak
field value. The cross-range and range estimate are highlighted in Figure 5-45.
By using the Cosine Rule, the correct cross-range and range to the transmitter are
calculated to be -410.15m and 4803.14m, respectively. With the IDPELS range
estimate being 4800 metres, the range error is less than 3.2m. While the range
estimate is within 3.2 metres of the actual range, it’s important to note that model
stepping size governs the accuracy resolution of the IDPELS range estimate. A
specified stepping size of 100m was applied to IDPELS while analysing all field data
sets. With the FCC indicating required system accuracy between 50 – 100m for E
911 [27], 100m was considered to offer high accuracy for outdoor geolocation. A
smaller stepping size such as 1m was not chosen, as system efficient is one of the
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reasons why PEM was chosen to be investigated with inverse diffraction propagation
geolocation.
The IDPELS cross-range estimate shown in Figure 5-45 is within one (1) metre of the
correct transmitter cross-range. This geolocation result has indicated the practical
feasibility of inverse diffraction propagation to be valid. While this result has
provided high accuracy, it should however be noted the cross-range estimated did not
remain invariant with a changing input cross-range specification. Additionally, as
these trials are concerned with feasibility testing of the system, repeatability is another
factor that should be considered. A repeatability test for Pine Creek track data sets
(02-03-04) is potentially offered with data sets (07-08-09) on Pine Creek track.
5.11.1.2 Data Sets (07-08-09)
Repeatability is an important model characteristic, particularly in a testing
environment. While accuracy is highly important, one single accurate measurement
out of many is usually worthless [28]. As shown in Figure 5-43, the highly similar
EB200 location in both combined sets allows data sets (07-08-09) to act as a
repeatability test for IDPELS results based on data sets (02-03-04). The geodetic
overview for Pine Creek track data sets (07-08-09) is shown Figure 5-46.
Figure 5-46 Geodetic Overview: Pine Creek track - Data Sets (07-08-09)
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The only significant difference between data sets (02-03-04) and (07-08-09) is the
direction of motion. The EB200 was moved in a Western direction during data sets
(07-08-09). The cross-range distance moved during data sets (07-08-09) is 1137.57m.
This cross-range distance is 32.9m greater than the cross-range distance with data sets
(02-03-04).
A display of the least square fitting polynomial for Pine Creek track data sets (07-08
09) is shown in Figure 5-47. As with the spatial-phase estimate of data sets (02-03-04)
in Figure 5-44, the same colour legend indicates the estimated (blue), measured (green)
and specified (red) spatial-phase profiles.
The position of the EB200 receiver at the start of signal measurement in Figure 5-47
is indicated by the index corresponding to where the cross-range and wavelength are
both zero. The specified cross-range of the polynomial spatial-phase was chosen to
be between 590m and 2600m. The IDPELS geolocation estimate for this input
spatial-phase profile is shown in Figure 5-48.
Figure 5-47 Cross-range of Input Signal: Pine Creek Track (07-08-09)
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Figure 5-48 Pine Creek Track (07-08-09) Geolocation
From Figure 5-46, the cosine rule shows the correct range and cross-range to the
transmitter to be 4801.69m and 1537.52m, respectively. In this example the range
error has increased to 99m, where the IDPELS boresight range estimate of 4900m was
provided. While this IDPELS range estimate does not demonstrate the same accuracy
provided with data sets (02-03-04), the cross-range estimate remained similar with the
linear error being less than one metre. The input spatial-phase profile was also more
symmetric. The reason why the range error in comparison to data sets (02-03-04) has
significantly increased is not exactly known. It could however be attributed to
obstructions in the signals propagation path.
In Figure 5-29 where the EB200 receiver was at the starting position of data set (02),
tree obstructions are highlighted. With the receiver moving in a western direction
with data sets (07-08-09), the same obstruction will have impacted data recorded
towards the end of data set (09). A plot the power variation for data sets (02) and (09)
is shown in Figure 5-49
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Figure 5-49 Power Variation: Pine Creek Track Data Sets (02) and (09)
While the power variation of data set (02) exceeded the 6dB limit defined for free-
space modelling, this was only for approximately 17% of the data. It did not have a
significant impact with the geolocation results provided in Figure 5-45, but could
explain why the input spatial-phase was not as symmetric with data sets (07-08-09).
While the variation remained below 20dB with data set (02), it has increased to 32dB
with data set (09). The only operational difference while measuring each data set was
the direction of receiver motion. General traffic on the Sturt Highway may therefore
have attributed to this increased power variation in data set (09).
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5.11.2 Woolshed road
On Woolshed road at the base of the Mt Lofty Ranges, there were two cases of data
being consecutively measured. As previously discussed, consecutive data sets are
required where the cross-range motion of the receiver covered a distance less than
approximately 1000m. This distance is necessary to provide sufficient accuracy in
estimating the signal’s spatial-phase profile along the road section. The maximum
speed reached on Woolshed road was approximately 30km/hr. Corresponding cross-
range distances covered in each separate data set varied between 350m – 450m. Data
sets that were sequentially measured on Woolshed road are identified below,
• Data sets(10-11) – receiver moved in eastern direction
• Data sets(16-17) – receiver moved in western direction
These two groups of data again permit a repeatability test of geolocation via inverse
diffraction propagation. A diagram showing similarity between the receiver’s
geodetic positions while recording combined data sets is shown in Figure 5-50. A
latitude separation of approximately 4m is maintained between the data sets, as the
van was driven on opposite sides of Woolshed road in either direction.
Figure 5-50 Consecutive Data Sets – Woolshed Road
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While future feasibility and repeatability testing was intended with these combined
sets, it was shown in Figure 5-30 the receiver is on an elevated plateau compared to
Pine Creek Track. This indicates that diffraction will contribute to signal loss. In
addition, the signal is also obstruction by tree cultures. The combined effect of these
two signal impediments is demonstrated by further analysing the power variation in
each separate data set. The geodetic receiver positions in data sets (10) and (17) is
similar, hence the relative placing and colour scheme of the power variations in
Figure 5-51. The same analogy exists with data sets (11) and (16).
Figure 5-51 Woolshed Road – Power Variation
As can be clearly seen with data sets (11) and (16), either combined sets of data can
not be applied to the free-space model. Any geolocation result based on these
combined data sets will not have any reputable accuracy. The alternative of using
single data sets does not assist any geolocation purpose. From numerous trials of the
single data sets, the estimated boresight range varied between 2.5km and 3.9km. For
data sets (16-17), the correct geolocation result is a boresight range of 5.921km,
corresponding to a cross-range of 1697m from the first signal measurement by the
EB200. These values calculated according to the Cosine rule are shown in Figure
5-52, while the inaccurate IDPELS geolocation result is shown in Figure 5-53. While
the cross-range error is only 1.1m, the boresight range error is a substantial 2578m.
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Figure 5-52 Geodetic Overview – Woolshed Road Data Sets (16-17)
Figure 5-53 Woolshed Road (16-17) Geolocation
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5.11.3 McEvoy Road
As previously shown in Figure 5-35, the most stable data set measured on McEvoy
road is set (07) where approximately 85% of data has a power variation less than 4dB.
While this was considered to be suitable for free-space propagation, the cross-range of
data sets is not sufficient to allow an accuracy estimate of the spatial-phase. The
largest cross-range distance was made during data (02) where 493.9m was travelled.
The remaining 7 sets of data covered distances between 360m and 386m. While data
sets where not consecutively measured on McEvoy road, sets (03) and (04) could
provide a longer cross-range, when their fractionally extension was combined with set
(02). A display of the respective measurement paths is shown in Figure 5-54.
Figure 5-54 Receiver Position – McEvoy Road Data Sets (02), (03) & (04)
While the cross-range distance had been extended, it did not exceed the 1000m
threshold established with Pine Creek track data sets. The greatest cross-range was
590.78m with data sets (02-04). With data set (02) being the initial set in both cases,
the boresight range according to the Cosine rule is 3874.99m. The corresponding
cross-range intersection of the boresight is 1059.37m west of the first measurement in
data set (02). With combined data set (02-04), the boresight range was estimated to
be 4.3km. This estimate has a linear range error of 425m. By combining data sets to
increase the cross-range distance, there was no geolocation accuracy improvement
compared to the use of data set (07) by itself. This could only be attributed to the
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combined data sets not experiencing free-space propagation of the test signal. The
power variation in data sets (02), (03) and (04) is shown in Figure 5-55.
Figure 5-55 Power Variation – McEvoy Road (02)-(03)-(04)
As shown, data set (02) has approximately 30% of data exceeding the 6dB threshold,
while sets (03) and (04) exceed the limit with approximately 45% and 85%
respectively. All three data sets have a greater power variation in comparison to data
sets (07), which was shown in Figure 5-35.
While a range error of 425 might be considered suitable for coarse geolocation, this
error is reduced to 223.75m with data set (07). A boresight range of 4.1km is
estimated, with a cross-range of 881m. This geolocation cross-range estimate
corresponds to an error less than 3m. A display of the correct boresight range and
cross-range, according to the Cosine rule is shown in Figure 5-56.
Figure 5-56 Geodetic Overview – McEvoy Road Data Set (07)
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The input spatial-phase profile for data set (07) is displayed in Figure 5-57, while the
IDPELS geolocation estimate is shown in Figure 5-58.
Figure 5-57 Cross-range of Input Signal – McEvoy Road (07)
Figure 5-58 McEvoy Road (07) – Geolocation Estimate
The McEvoy road geolocation estimate, in combination with Woolshed road has
demonstrated the importance of data sets being suitable for free-space modelling in
the preliminary trials. Raw field data was applied in determining the least-square
fitting spatial-phase profile for IDPELS operation.
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5.11.4 Port Gawler Road
The IDPELS geolocation estimate based on data sets from Woolshed road and
McEvoy road have demonstrated the importance of the test signal propagating in a
free-space environment. The most stable data set observed on Port Gawler road was
data set (11), with its power variation being shown in Figure 5-35. As shown in the
respective diagram, approximately half of the data set experienced a power variation
in excess of 6dB. A similar power variation was demonstrated on Woolshed road
with the combined data set (16-17). With this combined data set from Woolshed road
being used as the input spatial-phase profile, the estimated boresight range had a
significant range error of 2578m, as shown in Figure 5-53. A similar range error in is
therefore expected with the most stable data set measured on Port Gawler road.
5.11.4.1 Rayleigh Fading
With 11 data sets being recorded on Port Gawler road, the power variation in the
remaining sets is similar to that shown for sets (07) and (09) in Figure 5-59. In each
case, at least 85% of observed data has a power variation in excess of 6dB and with
data set (09), a 40dB variation was experienced. Such power variation can be
explained based on the dense vegetation of region as shown in Figure 5-25. The
propagated test signal will have been subject to possible shadowing, scattering and
reflection giving rise to the multipath effect. In such a environment, Rayleigh fading
[29] must therefore be considered on Port Gawler road.
Figure 5-59 Port Gawler Road – General Power Variation
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In the development of mobile radio networks, the statistical properties of the Rayleigh
and Rice processes have been extensively used to model fast fading [30]. The model
behind Ricean fading involves the presence of a dominant spectral component in the
spectral analysis of a narrowband signal. Such a situation means there is clear line-of
sight (LOS) between the transmitter and moving receiver. Rayleigh fading is used
when the scene has multiple indirect paths between transmitter and receiver, with no
distinct dominant path. In the extreme situation, there is no clear desired signal and in
analysing the frequency spectrum there will be multiple spectral peaks, without any
being the dominant one. This situation is approximated with Port Gawler data sets
(07) and (09) as demonstrated in Figure 5-60. In each data set there is no distinctive
dominant component in the spectrum, which was demonstrated in Figure 5-42 on Pine
Creek track.
Figure 5-60 Port Gawler – Rayleigh Fading
Any inverse diffraction geolocation using Port Gawler road data sets will obviously
display a similar error shown with Woolshed Road. This means that both Port Gawler
roads and Woolshed road data sets provide no credible evidence towards the
feasibility of this geolocation method. This situation arises because the software was
developed for experimentation and hence, was developed upon the free-space model.
While this is the position that should be considered with Port Gawler data sets,
geolocation results corresponding to the use of combined data sets (08-09-10-11) will
be provided to act as an error reference for further research.
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5.11.4.2 Pt Gawler Road - Data Sets (08-09-10-11)
As previously discussed, as cross-range of the measured signal profile is increased,
the accuracy of the polynomial will also increase. The largest cross-range of
2605.81m is available by combining data sets (08-09-10-11), where the EB200
receiver was moved in an Eastern direction along Port Gawler road. A geodetic
overview of the respective data sets, together with the correct boresight range and
cross-range intersection is shown in Figure 5-61
Figure 5-61 Geodetic Overview – Port Gawler Road (08-09-10-11)
The input spatial-phase profile is shown in Figure 5-62, with the respective
geolocation via IDPELS provided in Figure 5-63. As shown, the boresight range
estimate has a substantial error of 1364m, while the cross-range estimate error is less
than 1m. While the range error is substantial, it is not as large as the error returned on
Woolshed Road. This is attributed to the two following factors,
1. Greater cross-range distance of measured signal on Pt Gawler road.
2. Shadow and diffraction effects associated with the increased plateau elevation
concerning Woolshed road.
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Figure 5-62 Cross-range of Input Signal – Port Gawler Road (08-09-10-11)
Figure 5-63 Port Gawler Road (08-09-10-11) Geolocation
225
5.12 Field Trial Geolocation Error
This section provides a comparison of the IDPELS geolocation errors in the field trial
investigations. As sub-metre accuracy or resolution is not required for geolocation
over large distances, distance units are also rounded to metre values. The linear cross-
range error and range error is shown together with its percentage error value. The
percentage for all cross-range errors are less than 0.01% and hence, not shown.
Region Data Sets Linear Geolocation
Error (m)
Range Error
Percentage
Cross-range Range
Mt Lofty
Range Base
Pine Creek Track
(02-03-04)
1 3 0.06%
Pine Creek Track
(07-08-09)
1 99 2.06%
Woolshed Road
(16-17)
1 2578 43.53%
St Kilda McEvoy Road (07) 3 224 5.78%
Port Gawler Road
(08-09-10-11)
1 1364 12.6%
Table 5-1 Field Trial Geolocation Error
Range error values were governed by the data sets suitability to be modelled as a free-
space environment. Pine Creek Track data sets displayed a signal power variation
remaining below 6dB (i.e. Figure 5-34) and provided an accurate geolocation estimate
as the range error percentage is less than 2.06%. Woolshed road data sets were
subject to diffraction and obstructions with a substantial range error of 43.53%. A
display of the Woolshed environment is shown in Figure 5-30. The largest cross
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range error of 3m is shown with McEvoy Road. This is attributed to the distance of
379m being much less than the 1km distance associated with the input spatial-phase
associated with Pine Creek Track. Port Gawler road was also subject to dense
vegetation and was not considered suitable for free-space modelling.
5.13 Conclusion
Simulation results of IDPELS presented in Chapter 4 indicated the potential of inverse
diffraction propagation to be highly feasible and accurate. Subsequently field trials
were conducted in collaboration with the Navigation Warfare group of the Electronic
Warfare and Radar Division at the Defence Science Technology Organisation
(DSTO), Edinburgh, South Australia. The terrestrial field trials were conducted in
June 2004 with 1.399GHz tone transmitter. The objective of the field trials was to
record a 300Hz offset frequency into a WAV file. The spatial-phase profile of the
measured signal was then calculated by incorporating GPS data and was used as the
input signal for inverse diffraction propagation. Signal phase was simultaneously
measured as the EB200 receiver was moved with a van that was equipped with
relevant instrumentations.
With vehicle (and therefore receiver) motion being perpendicular to the transmitter
boresight, a horizontally planar free-space propagation model was applied to the field
data. With such a measurement process, an analogy between the field trials and
Synthetic Aperture Radar (SAR) was considered.
The software developed for geolocation was based on a free-space model for the field
trials, which would permit efficient analysis of the system. Such software
configuration however placed a restriction on data samples acquired in the field trials.
With various roads in different regions being considered for investigation of system
feasibility, the only sets of data that could be considered to have signal propagation in
free-space were those recorded on Pine Creek Track. A threshold for free-space
signal propagation was based on 6dB variation in signal power.
227
The spatial-phase variation of the test signal along the road section was modelled with
a quadratic polynomial. To ensure an accurate geolocation estimate, a sufficient
cross-range distance of approximately one kilometre was necessary with the measured
field data. With data being measured in 60 second time blocks, such a cross-range
distance was not always covered in each data set. Data sets were however
consecutively measured and could be joined together to ensure a sufficient cross-
range distance of the input spatial-phase.
The geolocation results developed from field data measured on Pine Creek Track
demonstrated the practical feasibility of the IDPELS methodology. This is because
data being recorded on Pine Creek Track could be modelled for free-space signal
propagation. A table showing the geolocation errors for each of the optimal data sets
was provided in Table 5-1. From this table, Pine Creek Track data sets were shown to
have range error value of 3m (0.06%) for data sets (02-03-04), while a range error of
99m (2.06%) was shown for data sets (07-08-09). A larger range error for data sets
(07-08-09) is attributed to the time period (approximately 1 second) when signal
power variation exceeded 30dB (Figure 5-49). Factors that may have contributed to
such an undesired property are vehicles travelling along the Sturt highway and
vegetation shown in Figure 5-29.
With geolocation accuracy being dependant of the accuracy of the input parameter (i.e.
spatial-phase), further research should consider other possible methods of acquiring
field data. Factors such as implementation and environmental conditions should be
analysed. A simple implementation was undertaken in the preliminary field trials,
where fixed helix antennas were employed. With an analogy existing between SAR
and the receiver motion, there are various SAR operating modes [19] that could
provide the basis for further field trials.
Further research and development is required to improve the operational capability of
geolocation with inverse diffraction modelling. By enhancing system software to
account for factors such as signal obstructions and terrain elevation, the application of
geolocation based on propagation modelling in combination with other methods such
as interferometry, will improve geolocation accuracy and capability.
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5.14 References
[1] G. W. Stimson, "Key to a Nonmathematical Understanding of Radar", in
Introduction to Airborne Radar. New Jersey: SciTech Publishing, Inc, pp. 59 -
70, 1998
[2] J. R. Vig, "Introduction to Quartz Frequency Standards", Army Research
Laboratory, Electronics and Power Sources Directorate, Fort Monmouth, NJ,
USA SLCET-TR-92-1 (Rev. 1), October, 1992.
[3] "Miniport receiver EB200 Datasheet", Rohde & Schwarz, Munich, Germany
2005.
[4] T. Mock, "Music everywhere", Spectrum, IEEE, vol. 41(9), pp. 42-47, 2004.
[5] C. Rizos, "How can my position on the paddock help my future direction?"
presented at Geospatial Information & Agriculture Conference, Sydney,
Australia, July 16-19, 2001.
[6] Novatel, "GPSCard - Command Description Manual", Novatel
Communications Ltd OM-20000008 Rev 2.0, 31 March, 1995.
[7] C. Budd and C. Sangwin, "101 uses of a quadratic equation: Part II", Plus, vol.
(30), pp. 2004
[8] C. Budd and C. Sangwin, "101 uses of a quadratic equation", Plus, vol. (29),
pp. 2004
[9] T. Nelson and M. O'Connor, "Make Your Own Sounds", PC Today, vol. 2(4),
pp. 56-57, 2004
[10] G. W. Stimson, "Doppler Effect", in Introduction to Airborne Radar. New
Jersey: SciTech Publishing, Inc, pp. 189 - 199, 1998
229
[11] I. M. Yaglom, A Simple Non-Euclidean Geometry and Its Physical Basis: An
elementary account of Galilean Geometry and the Galilean Principle of
Relativity. New York: Springer-Verlag, 1979.
[12] S. W. Smith, The Scientist & Engineer's Guide to Digital Signal Processing,
2nd ed. San Diego: California Technical Publishing, 1999.
[13] C. W. Therrien, "Linear Models", in Discrete Random Signals and Statistical
Signal Processing, A. V. Oppenheim, Ed. Englewood Cliffs, New Jersey:
Prentice Hall Signal Processing Series, pp. 503 - 584, 1992
[14] G. W. Stimson, "Meeting High Resolution Ground Mapping Requirements",
in Introduction to Airborne Radar. New Jersey: SciTech Publishing, Inc, pp.
393 - 401, 1998
[15] C. Wiley, "Synthetic aperture radars–a paradigm for technology evolution",
IEEE Transaction on Aerospace and Electronic Systems, vol. AES-21, pp.
440-443, 1985.
[16] B. R. Mahafza, "Chapter 12 - Synthetic Aperture Radar", in Radar Systems
Analysis and Design Using MATLAB. New York: Chapman & Hall /CRC, pp.
12.1 - 12.12, 2000
[17] G. W. Stimson, "Principles of Synthetic Array (Aperture) Radar", in
Introduction to Airborne Radar. New Jersey: SciTech Publishing, Inc, pp. 403
- 424, 1998
[18] L. J. Cutrona, "Synthetic Aperture Radar", in Radar Handbook, M. I. Skolnik,
Ed., 2nd ed: McGraw-Hill, Inc., pp. 21.1 - 21.21, 1990
[19] G. W. Stimson, "SAR Operating Modes", in Introduction to Airborne Radar.
New Jersey: SciTech Publishing, Inc, pp. 431 - 437, 1998
230
[20] R. J. Sullivan, "Introduction to Imaging Radar", in Radar Foundations for
Imaging and Advanced Concepts: SciTech Publisher, pp. 159 - 189, 2004
[21] "US Patent Class 343 Class Notes", United States Patent Office 1999.
[22] "One-Way RADAR Equation and RF Propagation", in Electronic Warfare and
RADAR Systems Engineering Handbook. Point Mugu, CA, USA: Naval Air
Warfare Center Weapons Division, pp. 4.3.1 - 4.3.8, 2002
[23] T. S. Rappaport, Wireless Communications - Principles & Practice, 2nd ed.
Upper Saddle River, N.J: Prentice Hall Professional Technical Reference,
2001.
[24] D. Adamy, "Emitter Location - Conversion of AOA Errors to Location Errors",
Journal of Electronic Defense, 2003.
[25] D. Adamy, "Emitter Location - Reporting Location Accuracy", Journal of
Electronic Defense, 2002.
[26] L. P. Harter, "Circular Error Probabilities", Journal of the American Statistical
Association, vol. 55, 1960.
[27] J. R. Beal, "Contextual Geolocation: A Specialized Application for Improving
Indoor Location Awarenewss in Wireless local Area Networks", presented at
35th Annual Midwest Instruction and Computing Symposium (MICS)
Proceedings, College of St. Scholastica, Duluth, Minnesota, USA, April 11 -
12, 2003.
[28] G. D. Rash, "GPS Jamming in a Laboratory Environment", Naval Air Warfare
Centre Weapons Division (NAWCWPNS), China Lake, California, USA
November 5, 1997.
231
[29] M. Pätzold, "Rayleigh and Rice Processes as Reference Models", in Mobile
fading channels. Chichester, England: John Wiley & Sons, Ltd, pp. 33 - 54,
2002
[30] B. Sklar, "Rayleigh Fading Channels in Mobile Digital Communication
Systems, Part 1: Characterization", IEEE Communications Magazine, pp. 90 -
100, 1997
[31] R. M. Hawkes and C. P. Baker, "Tropospheric Propagation Model for Land
Warfare", presented at Land Warfare Conference 2003, Adelaide Convention
Centre, Adelaide, Australia, 28 - 30 October, 2003.
[32] R. M. Hawkes, T. A. Spencer, and R. A. Walker, "Tropospheric Propagation
Model using Huygens’ Principle", presented at The Second International
Association of Science and Technology for Development (IASTED)
Conference on Antennas, Radar and Wave Propagation (ARP), Banff, Alberta,
Canada, 19-21 July, 2005.
232
Chapter 6 - Thesis Conclusion
Importance is placed on localisation research due to its recommendation being made
in the Volpe report [1] to ensure GPS signal availability. Currently there is a wide
variety of interference mitigation methods available to reduce a systems susceptibility
to interference as discussed in chapter 1. However, since mitigation technology is not
solely associated with localisation, a review of classical and recently developed
localisation techniques was made in Chapter 2. Limitations associated with these
methods were highlighted, helping to determine what path to follow for development
of a new electromagnetic localisation technique. Subsequently a backward
propagating model coupled with inverse diffraction, also referred to as inverse
propagation was considered to be a feasible path to follow. Validity for such a
research option in the electromagnetic environment has already been established by
Gingras et al [2] with Matched Field Processing (MFP). The MFP methodology
however has substantial problems and has never been used outside the research
community. One problem is that thousands of replica fields are usually generated, so
that near real-time operation is not feasible. Another problem is that incorrect
transmitter locations can be returned. This incorrect estimated is a form of
“mismatch” that is not meaningful for any localisation operation.
Various propagation models were reviewed in Chapter 3, however the Parabolic
Equation Model (PEM) was chosen for localisation research due to its methodology
incorporating an open boundary on the paraxial. An open boundary is a requirement
for blind localisation. Undesired requirements associated with other models were also
noted. Feasibility of performing localisation with the PEM has also been previously
established by Tappert [3] in the underwater acoustic environment. This research
propagated the conjugate of the received continuous wave (CW) signal, which is
referred to as “backpropagation”. This approach assumed a fixed environment and
searched for a focusing point which revealed the location of the acoustic source. Zhu
[4] has also used the focus-marching procedure with phase reversal to provide fine-
resolution in target imaging, where operational procedures bear an analogy to the
localisation method investigated in this research program.
233
IDPELS was initially developed under simulated terrain conditions. Corresponding
results were shown in Chapter 4 and were considered to be accurate while also
accounting for diffraction and reflection. Notably, the “Best Presentation” award was
received for this work when presented at the Institute of Navigation Global
Navigation Satellite Systems (ION GNSS) International conference in Long Beach,
California, 2004 [5] for the paper showing simulation results of IDPELS. With the
theoretical validity having been established with the electromagnetic simulation
investigation, the practical feasibility of the geolocation method was investigated.
Discussion concerning the practical evaluation of IDPELS was made in chapter 5.
The practical feasibility testing was performed in collaboration with the Navigation
Warfare group in the Electronic Warfare and Radar Division of DSTO, Edinburgh,
South Australia.
The objective of the field trials was to measure a test signal on road sections that were
approximately perpendicular to the transmitter’s boresight. Various geodetic regions,
together with roads sections at different ranges from the transmitter were used in the
trials. The input signal from the trials was the spatial-phase of the measured signal,
where the geodetic position of the EB200 receiver had been incorporated with GPS –
NMEA files. A quadratic polynomial was estimated based on the least-square
principle from the measured spatial-phase of the signal. The polynomial estimate
thereby allows the localisation operator to specify any cross-range section as the
system input. As the selected input phase became more symmetric, the geolocation
accuracy also improved. The ultimate accuracy of the system was dependant on the
accuracy of the input spatial-phase, and it was noted that a cross-range of
approximately 1000m with respect to measured field data was required for sufficient
accuracy to be obtained in the quadratic polynomial estimate.
Development of software was based on the free-space parabolic equation model and
field regions that approximated this environment provided range and cross-range
estimation to the source with small errors. The practical feasibility of IDPELS
geolocation was demonstrated with data being measured on Pine Creek Track, where
the power variation of signal on the respective roads did not exceed 2dB. Data sets
measured on McEvoy road (St Kilda region) demonstrated power variation below
4dB, while Woolshed road and Port Gawler road where not suitable for free-space
234
modelling. From the 36 measured data sets from both trial regions, only 6 data sets
demonstrated characteristics of free-space propagation.
While not all data sets demonstrated an accurate geolocation estimate, these undesired
field trial results arose because the free-space environment did not accurately reflect
signal propagation. The range error percentage with these data sets varied between
12.6% and 43.53%. The data sets providing the 43.53% range error experienced
obstructions and diffraction in the signal measurement process.
In analysing the error percentage of the geolocation estimate, the range error
percentage varied between 0.06% and 2.06% with data sets demonstrating free-space
propagation. The cross-range error percentage in all cases was less than 0.01%.
Given the small errors associated with the IDPELS geolocation estimate where free-
space signal propagation was performed in the field trials, the authors considers an
important contribution to electromagnetic geolocation methods has been provided.
A list of research papers developed during this research program is listed in chapter 8.
6.1 References
[1] "Vulnerability Assessment of the Transportation Infrastructure Relying on the
Global Positioning System," John A. Volpe National Transportation Systems
Center for the Office of the Assistant Secretary for Transportation Policy, U.S.
Department of Transport, 29 August 2001.
[2] D. F. Gingras, P. Gerstoft, and N. L. Gerr, "Electromagnetic matched-field
processing: basic concepts and tropospheric simulations," Antennas and
Propagation, IEEE Transactions on, 45(10), pp. 1536-1545, 1997.
[3] F. D. Tappert, L. Nghiem-Phu, and S. C. Daubin, "Source localization using
the PE method," The Journal of the Acoustical Society of America, 78(S1), pp.
S30, 1985.
235
[4] D. Zhu, "Application of a Three-Dimensional Two-way Parabolic Equation
Model for Reconstructing Images of Underwater Targets," Journal of
Computational Acoustics, 9(3), pp. 1067-1078, 2001.
[5] T. A. Spencer, R. A. Walker, and R. M. Hawkes, "GNSS Interference
Localisation Method Employing Inverse Diffraction Integration with Parabolic
Wave Equation Propagation," presented at ION GNSS 2004, Long Beach
Convention Centre, Long Beach, California, 2004.
236
Chapter 7 - Recommendations
PEM is a powerful benchmark for signal propagation and there has been extensive
research of PEM under simulation [1]. By adapting its application to localisation, the
efficiency provided by PEM can provide the basis for real-time localisation. The
ultimate aim of this research program was to test the practical feasibility of the
electromagnetic geolocation method based on inverse diffraction propagation.
With this research program being the preliminary investigation of electromagnetic
geolocation with inverse diffraction propagation, field trials were intended to be kept
simple. This was to allow prompt analysis and evaluation of experimental procedures
in the field trials. There were no known guidelines to provide assistance in the trials
and these field trials should therefore not be considered the ‘best’ option to be
employed if ever adapted in a real geolocation operation.
The following sections indicate areas where both field operation and system
performance should be further investigated for improvement in system operation.
7.1 IDPELS Precision Analysis
The primary objective of simulation analysis in this research program was to
fundamentally determine if the IDPELS localisation method is feasible, which in turn
decides if the system can possibly be realised. Absolute accuracy of the system
provides a definite indication of feasibility and provides benchmark performance
indications of the system. System quantification showing “best” operation was
considered highly important for this research program. Accuracy variation as
provided by precision analysis does not provide a definite indication for realisation of
system feasibility.
It is however realised that for a more convenient comparison of IDPELS with other
conventional localisation methods such as triangulation, precision analysis accounting
for variation in system accuracy should be investigated. In such an environment, the
presence of noise is a primary factor contributing to the system accuracy variation. In
237
any further investigation of IDPELS operation, field noise should be included in the
simulation analysis.
7.2 Obstruction Modelling
Free-space modelling was chosen for preliminary investigation of the inverse
diffraction geolocation methodology. While its choice was intended for simplified
implementation, this however placed a severe restriction on suitability of field data.
To enhance the operational capability of inverse diffraction geolocation, modelling of
obstructions, foliage, refractive index and other parameters that affect signal
propagation should be incorporated with any further development of functional
software.
7.3 Wideband Propagation
An important recommendation concerns the bandwidth of test signals. The field trials
were performed with a continuous-wave (CW) signal. There are various other
categories of interference and the definition adopted in this thesis is that provided by
Rash [2]. The primary motivation for this research was based on GPS interference.
Further research should therefore consider incorporation of a wideband (WB) channel
impulse response with the PEM providing inverse diffraction propagation. The
impulse response provides the frequency response of all specified frequencies within
the specified bandwidth. Wideband modelling with respect to PEM has been
previously investigated by Gerstoft et al [3] and Ekkelkamp et al [4]. Evaluation of
the wideband impulse response is performed in the frequency domain, where signal
propagation over the scene is repeated with each frequency in the bandwidth. The
repeat propagation will involve uniformly spaced samples of frequencies in the
bandwidth. With the channel response of each single frequency component being
known at each grid point within the domain, the channel impulse response at each
respective grid point is found by inverse transforming all of the frequency
components. The complex envelope of the signal is then calculated by translating the
channel frequency response to baseband. By being able to account for wideband
238
signals, geolocation with inverse diffraction propagation will not have any application
restrictions.
7.4 Transmission Frequency
While this research program has been based on a GPS background and modelled
signals where within the L-band, further analysis into system operation at other
frequencies is an area that demands further investigation. In particular, the
relationship between the uncertainty of the estimate of the source location and
transmission frequency should be determined. Duplicates of simulations shown in
this thesis are recommended to be performed with various other frequency bands such
as VHF, C-band, X-band, Ku-band, etc. Such an investigation will allow a more
encompassing analysis of the localisation methodology and indicate other areas where
the system could be used.
7.5 Field Trial Procedure
Another important consideration concerns the signal measurement process in the field
trials. From the performed field trials, geolocation accuracy was shown to be
dependant on the accuracy of the quadratic polynomial representing the input spatial-
phase profile. There are numerous other methods of acquiring field data and with an
analogy existing between the field trials and SAR, different SAR operating modes
could provide the basis for greater accuracy in estimating the spatial-phase profile.
One SAR example is the Multilook Mapping method [5], where a region is scanned
several times with a fixed antenna. By superimposing the scans, the effects of
scintillation are reduced thereby providing greater resolution and accuracy. A
sensitivity analysis of various data acquisition method is an area that is suggested for
further investigation, where implementation and environmental conditions are
considered.
239
7.6 Two-Way Signal Propagation
The parabolic propagation model adapted for localisation / geolocation in this
research program has been based only one-way signal propagation. Two-way signal
propagation that accounts for back-scattering has been chosen to be omitted in this
research program due to the intention of performing field trials in a free-space
environment. Simplification in the field trial implementation was considered
important to ensure proper validation of the localisation methodology. However, by
not accounting for two-way signal propagation in the model, operational limitations
will arise in complex urban environments. Any further investigation of inverse
diffraction propagation should incorporate two-way signal propagation to account for
objects that provide significant signal reflection. Two-way signal propagation is
based on the one-way model, where an account of the location concerning reflecting
obstacles is recorded. By storing the forward propagating field value at these
locations, reflection coefficients and surface roughness factors are then applied to the
back-propagating field. Further investigation and information of two-way signal
propagation is provided by Collins [6, 7], Levy [8] and Hannah [9].
7.7 3D Model
Another recommendation concerns the development of a 3D simulation model. The
actual physical process of acquiring a two dimension plane profile was not considered
simple to achieve, and hence a 3D model was only considered in simulation analysis
during this research program.
3D simulation trials of IDPELS were performed based on the Eibert imaging method
[10]. However, as noted in chapter 5, the Eibert method fails to maintain the
quadratic phase profile during an inverse Fourier transformation. No 3D results of
IDPELS could be provided in this research program. If any further research is based
on a 3D approach, the localisation model must be based on the surface impedance
approach discussed by Dockery and Kuttler [11], where discrete trigonometric
transforms are instead applied.
240
7.8 Huygens Principle Model — Wide Angle
Propagation
The inverse diffraction propagation methodology investigated and proven in this
research program has been based on the parabolic equation model (PEM). This
propagation model was chosen for investigation as it has the necessary model
characteristics for blind/passive localisation, and it has been extensively researched
and developed. While PEM has set benchmarks for modelling signal propagation, in
a non-free-space environment where the refractive index profile must be modelled,
the PEM field is only accurate within a defined angular region about the paraxial.
With unmatched numerical efficiency provided by Fourier-Split-Step (FSS)
propagation, the Standard Parabolic Equation (SPE) model is only accurate within a
±15º region about the paraxial. While this could restrict model operation in certain
situations, the FSS-PEM can also provide a wider propagation angle limit of ±30º
about the paraxial [12], where a correction to the starter field must be applied.
Discussion of this PEM adjustment was made in the Wide-Angle Propagation
Methods section of chapter 3.
While PEM was chosen for investigation, it is important to realise that the underlying
localisation methodology is based inverse diffraction propagation (IDP). Note should
be made that the IDP principle can also be applied with different propagation models
that have characteristics permitting blind/passive localisation. Another propagation
model with such characteristics and was analysed for IDP localisation is the Huygen`s
Principle Model (HPM). Initially developed by Hawkes [13, 14] , the IDP principle
applied to HPM is able to accurately model signal propagation within a ±90º angle
limit with the efficient Fourier-split-step propagation method [15]. Accurate field
propagation within a ±90º angular region about the paraxial is important, particularly
in the advent of a hostile jamming environment where there is no prior indication of
relevant direction to the transmitter. While the field trials were able to demonstrate
the accurate geolocation capability of IDPELS, the relative position of the transmitter
and receiver allowed the receiver to be moved in an orthogonal orientation to the
transmitter boresight. This orientation of the receiver ensured the measured field was
within the ±15º angle limit for the standard parabolic equation. Accurate signal
241
modelling to a ±90º angle limit is not possible with the PEM due to necessary
assumptions made in its development.
Any further research of localisation based on inverse diffraction propagation should
include analysis with HPM as it is considered to form an excellent synergy with PEM.
Any overview of HPM operation is provided in Appendix A.
7.9 References
[1] D. Lee, A. D. Pierce, and E.-C. Shang, "Parabolic Equation Development in
the Twentieth Centuary," Journal of Computational Acoustics, 8(4), pp. 527
637, 2000.
[2] G. D. Rash, "GPS Jamming in a Laboratory Environment," Naval Air Warfare
Centre Weapons Division (NAWCWPNS), China Lake, California, USA
November 5, 1997.
[3] D. F. Gingras and P. Gerstoft, "The Effect of Propagation on Wideband DS
CDMA Systems in the Suburban Environment," presented at IEEE Signal
Processing Workshop on Signal Processing Advances in Wireless
Communications, Paris, 1997.
[4] M. P. H. Ekkelkamp, P. van Genderen, and J. S. van Sinttruijen, "On some
wide band radar propagation effects over sea," presented at Radar, 1996.
Proceedings., CIE International Conference of, 1996.
[5] G. W. Stimson, "SAR Operating Modes," in Introduction to Airborne Radar.
New Jersey: SciTech Publishing, Inc, 1998, pp. 431 - 437.
[6] M. D. Collins and R. B. Evans, "A Two-way Parabolic Equation for Acoustic
Backscattering in the Ocean," Journal of the Acoustic Society of America, 9(1),
pp. 1357 - 1368, 1992.
242
[7] M. D. Collins, "A Two-way Parabolic Equation for Elastic Media," Journal of
the Acoustic Society of America, 9(3), pp. 1815-1825, 1993.
[8] M. F. Levy, "Parabolic Equation Modelling of Backscatter from the Rough
Sea Surface," presented at Target and Clutter Scattering and their Effects on
Military Radar Performance (AGARD-CP-501), Ottawa, Ontaria, Canada,
1991.
[9] B. M. Hannah, "Chapter 4 GPS Parabolic Equation Model," in Modelling and
Simulation of GPS Multipath Propagation. Brisbane: Electrical and Electronic
Systems, Queensland University of Technology, 2001, pp. 115 - 148.
[10] T. F. Eibert, "Irregular terrain wave propagation by a Fourier split-step wide-
angle parabolic wave equation technique for linearly bridged knife-edges,"
Radio Science, 37(1), 2002.
[11] J. R. Kuttler, "An improved Impedance-Boundary Algorithm for Fourier Split-
Step Solutions of the Parabolic Wave Equation," Antennas and Propagation,
IEEE Transactions on, 44(12), pp. 1592 - 1599, 1996.
[12] J. R. Kuttler, "Differences between the narrow-angle and wide-angle
propagators in the split-step Fourier solution of the parabolic wave equation,"
Antennas and Propagation, IEEE Transactions on, 47(7), pp. 1131-1140,
1999.
[13] R. M. Hawkes and C. P. Baker, "Tropospheric Propagation Model for Land
Warfare," presented at Land Warfare Conference 2003, Adelaide Convention
Centre, Adelaide, Australia, 2003.
[14] R. M. Hawkes and C. P. Baker, "Propagation Model for Littoral
Environments," Defence Science Technology Organisation, Electronic
Warfare and Radar Division System Sciences laboratory, Salisbury, South
Australia DSTO-RR-XXXX, 2003.
243
[15] R. M. Hawkes, T. A. Spencer, and R. A. Walker, "Tropospheric Propagation
Model using Huygens’ Principle," presented at The Second International
Association of Science and Technology for Development (IASTED)
Conference on Antennas, Radar and Wave Propagation (ARP), Banff, Alberta,
Canada, 2005.
244
Chapter 8 - Research Publications
T. A. Spencer and R. A. Walker, "A Case Study of GPS Susceptibility to Multipath
and Spoofing Interference," presented at Australian International Aerospace Congress
incorporating the 14th National Space Engineering Symposium 2003, Brisbane,
Queensland, Australia, 2003.
T. A. Spencer and R. A. Walker, "Prediction and analysis of GPS susceptibility to
multipath and spoofing interference for land and space," presented at The 6th
International Symposium on Satellite Navigation Technology Including Mobile
Positioning & Location Services, Melbourne, Victoria, Australia, 2003.
T. A. Spencer, R. A. Walker, and R. M. Hawkes, "GNSS Interference Localisation
Method Employing Inverse Diffraction Integration with Parabolic Wave Equation
Propagation," presented at ION GNSS 2004, Long Beach Convention Centre, Long
Beach, California, 2004.
T. A. Spencer, R. A. Walker, and R. M. Hawkes, "Inverse Diffraction Parabolic Wave
Equation Localisation System," presented at GNSS 2004, University of NSW, Sydney,
2004.
T. A. Spencer, R. A. Walker, and R. M. Hawkes, "Inverse Diffraction Parabolic Wave
Equation Localisation System," Journal of Global Positioning Systems (JPGS), vol. 4,
2005.
R. M. Hawkes, T. A. Spencer, and R. A. Walker, "Tropospheric Propagation Model
using Huygens’ Principle," presented at Submitted for review of The Second
International Association of Science and Technology for Development (IASTED)
Conference on Antennas, Radar and Wave Propagation (ARP), Banff, Alberta,
Canada, 2005.
245
R. M. Hawkes, T. A. Spencer, and R. A. Walker, "Three Dimensional Model for
Propagation in the Troposphere and Inverse Diffraction," presented at Workshop on
Applications of Radio Science (WARS), Leura, New South Wales, Australia, 2006.
246
Appendix A - Huygens Principle Model
In the review of electromagnetic propagation models provided in section 3.7 of
Chapter 3, limitations associated with models provided a reason for investigating the
localisation capability of PEM. One propagation models that was not however
discussed was the Huygens Principle Model (HPM), initially developed by Richard
Hawkes [1]. HPM is based on Huygens` principle [2] where each point on the
primary wave front at time ‘t+Δt’, is the result of phase summation due to secondary
spherical wavelet sources at time ‘t’. The primary wavefront is considered as a
function of these wavelets and is shown in Figure A-1.
Figure A-1 Huygen’s Propagation Principle
HPM operation is similar to PEM, but offers greater flexibility as a propagation tools
for localisation. Unlike FSS-PEM that operates signal marching in the angular
domain, HPM can operate in either Time or Frequency domains. The stepping size
used in HPM can also be significantly greater for propagation of the signal, while
maintaining required accuracy and resolution. PEM requires stepping distance to be
reduced to ensure required accuracy. Another benefit offered by HPM is the wide
angle propagation capability, where up to ±90° is possible. While FSS-PEM can
provide wide-angle propagation up to ±30º with source corrections [3] , it can not
247
accurately account for a ±90° propagation angle about the boresight. PEM
propagation angles are usually limited by the required paraxial assumption where
efficiency is desired. A ±90° propagation angle is however considered important in
the advent of hostile jamming, where no relative direction to the interference source is
prior known.
This same benefit of the HPM can also be advantageous in the field trials of inverse
diffraction propagation (IDP). In the field trials of IDPELS, the relative direction to
the transmitter was known and spatial phase was measured in close approximation to
the transmitter boresight. With the ±90° propagation angle of HPM, prior knowledge
of where the transmitter is not required. The signal phase as determined according to
chapter 5 in this thesis can be found with the measurement path orientated so the
paraxial direction is not in close approximation to the transmitter boresight. This
orientation principle is demonstrated in the lower section of Figure A-2, which
permits a much greater degree of freedom for input parameter measurement.
Figure A-2 HPM Wide Propagation Angle
248
The HPM propagation method concerns the circular convolution shown in Equation
A-1. The input field at range x, (i.e. u(x, z ') ), is stepped to provide the solution at
range x + Δx (i.e. u(x + Δx, z) ). With this account of diffraction concerning HPM,
zero padding is required with the N elements of the wavefront. This will however
unfortunately increase the transform time by a factor of 4.
u(x + Δx, z) = ∫ u(x, z ') h(z − z ')dz ' Equation A-1
In two dimensions, the axial height is represented by z and the h(z) term provides the
correct amplitude and phase for each possible path between the N points of the
primary and secondary wavefronts. The larger variation in stepping size is provided
by the h(z) term, which is further expanded in Equation A-2 [4].
h(z) = (Δz/δ) exp(jkδ) / δ Equation A-2⋅
The ( x / ) term in Equation A-2 is the obliquity factor and it describes theΔ δ
directionality of the secondary emissions and therefore negates back propagation [5].
The δ term used in defining the obliquity factor is expanded in Equation A-3.
Equation A-3
A combination of HPM and PEM will allow wide propagation angles as required with
HPM, while also allowing the faster and efficient PEM. A listing of HPM code
conducting localisation on four jammers under simulation is provided in Appendix D.
A.1 References
[1] R. M. Hawkes and C. P. Baker, "Tropospheric Propagation Model for Land
Warfare," presented at Land Warfare Conference 2003, Adelaide Convention
Centre, Adelaide, Australia, 2003.
2 2x zδ = Δ +
249
[2] B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens'
Principle, 2nd ed. Oxford: Clarendon Press, 1950.
[3] J. R. Kuttler, "Differences between the narrow-angle and wide-angle
propagators in the split-step Fourier solution of the parabolic wave equation,"
Antennas and Propagation, IEEE Transactions on, 47(7), pp. 1131-1140,
1999.
[4] R. M. Hawkes, T. A. Spencer, and R. A. Walker, "Tropospheric Propagation
Model using Huygen's Principle," presented at The Second International
Association of Science and Technology for Development (IASTED)
Conference on Antennas, Radar and Wave Propagation (ARP), Banff, Alberta,
Canada, 2005.
[5] F. A. Jenkins and H. E. White, "Fresnel Diffraction," in Fundamentals of
Optics, 4th ed. Singapore: McGraw Hill, 1976, pp. 378 - 402.
250
Appendix B Matlab Code – Field Trials
B.1 Spatial-Phase Code %************************************************************************* %* SPATIALPHASE - analyse GPS and WAV field data files %* and determine least square fitting quadratic polynomial %* %* In addition to finding spatial-phase, it will also analyse the %* field data looking at parameters such as phase or frequency %* %* Troy Spencer – November 2005 %************************************************************************ close all clear all
% store directory location with inverse diffraction code % field data is located in following folders % ....\FIELD_DATA\gps or .....\FIELD_DATA\eb200 code_dir = cd;
%************************************************************* %* user specified parameters (with default values) %************************************************************* imprompt = {}; inprompt{1} = 'Specify frequency of signal (MHz)'; inprompt{2} = 'Specify the audio sampling frequency (Hz)'; inprompt{3} = 'Define time from start where van remained stationary (sec)'; inprompt{4} = 'Specify number of consecutive file to be opened'; inprompt{5} = 'Specify stepping size to reduce computational load of all WAV data'; inprompt{6} = 'Estimate percentage of data that is repeated at start'; inprompt{7} = 'Estimate percentage of data that is repeated at end'; inline = 1; indef = {'1399','44100 ','0.74304', '3', '1000', '2', '98'}; intitle = 'Frequency, Audio sampling frequency, and Linearity'; str_inputs = inputdlg(inprompt, intitle, inline, indef); rfreq = str2double(str_inputs(1)); fs = str2double(str_inputs(2)); stationarygrid = str2double(str_inputs(3))*fs; stationarygrid = round(stationarygrid); filenos = str2double(str_inputs(4)); wavstep = str2double(str_inputs(5)); lowtax = str2double(str_inputs(6)); lowtax = lowtax / 100; hightax = str2double(str_inputs(7)); hightax = hightax / 100; c = 288792458; % m/s wavelength = c / (rfreq * 1e6);
%******************************************************************************************************** % initalise CONTINUEPHASE that ensures phase at beginning of next data set continue from % the end of previous data set, and define arrays %******************************************************************************************************** continuephase = 0;
totaldist = []; totalphase = []; eachset = {}; % stores each seperate set of data
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setnumbers = {}; % stores each data sets number for use in filename
%*************************************************************************************************** % when multiple files are specified, ensure consecutive WAV and GPS files are opened %**************************************************************************************************** for fileloop = 1 : filenos
%************************************************************ % load wav data and find unwrapped signal phase %************************************************************ fclose('all');
% display last data file opened (for correct sequencing) wavtitletext = 'Select WAV data file'; if fileloop == 1
wavtitle = strcat(wavtitletext); else
wavtitle = strcat(wavtitletext, ' (Last file - ', wavname, ')'); % after first file selection, can change directory to WAV data cd(firstwavpath)
end
% save the first PATH to data files (allows quicker selection) if fileloop == 1
[wavname, wavpath] = uigetfile('*.wav', wavtitle); wavfile = strcat(wavpath, wavname); disp(sprintf('uigetfile `%s`, please wait ...', wavfile)); firstwavpath = wavpath;
% Based on field data being in folder with name FIELD_DATA fieldataindex = strfind(firstwavpath, 'FIELD_DATA'); gpspath = firstwavpath(1 : fieldataindex + 10); firstgpspath = cat(2, gpspath, 'gps\');
else cd(firstwavpath) [wavname, wavpath] = uigetfile('*.wav', wavtitle); wavfile = strcat(wavpath, wavname); disp(sprintf('uigetfile `%s`, please wait ...', wavfile));
end
% store the data set number (for filename) setnumbers{fileloop} = wavname(1:2);
% load the wav data wavdata = wavread(wavfile);
% find length of wav data file wavlen = length(wavdata);
% create vector with indexs of wav file wavindexs = 1 : wavlen;
% phase calculation takes time, notify user msgtext = 'Calculating PHASE for each WAV sample in '; msg = sprintf('%s %s, please wait ...', msgtext, wavname); msgtitle = 'WAV data phase calculation'; msghandle = msgbox(msg, msgtitle); pause(1)
% calculate sample phases
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wavphase = 180 .* unwrap(atan2(wavdata(wavindexs, 1), ... wavdata(wavindexs, 2))) ./ pi;
% close gui close(msghandle)
clear wavdata wavindexs
%************************************************************ %* load GPS data %************************************************************ cd(firstgpspath) gpsread; % sub-program to load gps data file
% must ensure that distance is reference to inital data set if fileloop == 1
first_lat = latitude(1); first_lon = longitude(1);
%***************************************************** %* determine where source was positioned %*****************************************************
% select first 5 characters from gps filename to allow code % to know where transmission source is positioned
firstpart = gpsname(1:5); underscoreindex = find(firstpart == '_'); setnumber = firstpart(1 : underscoreindex - 1); region = firstpart(underscoreindex + 1);
% St Kilda region if (region == 'M') | (region == 'P')
% St Kilda Tx site tx_lat = 34 + 43./60 + 26.2./3600; tx_lon = 138 + 32./60 + 15.6./3600;
end
% Mt Lofty ranges (Truro) at base if region == 'B'
% Baldon Rd, Truro Tx site tx_lat = 34 + 25./60 + 2.85./3600; tx_lon = 139 + 14./60 + 10./3600;
end
end
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%************************************************************ % find distance moved along road while recording %************************************************************ a_earth = 6378.137; b_earth = 6356.752; N_earth = (a_earth.^2)./ sqrt((a_earth .* cos(pi * first_lat ./ 180)) .^2 + ...
(b_earth .* sin(pi * first_lat ./ 180)) .^ 2);
r_ns = abs(first_lat - latitude) .* N_earth .* pi ./ 180; r_ew = abs(first_lon - longitude) .* N_earth .* pi .* cos(first_lat .* pi ./ 180) ./ 180;
distance = 1000 .* sqrt(r_ns .^ 2 + r_ew .^ 2);
figure plot(time_sec, distance, 'm') title('EB200 Track - road')
xlabel('Time (sec)') ylabel('Distance (m) along road')
grid on
%************************************************************************* % find distance moved by receiver from transmitter (RANGE)
%************************************************************************** r_ns_range = abs(tx_lat - latitude) .* N_earth .* pi ./ 180; r_ew_range = abs(tx_lon - longitude) .* N_earth .* pi .* cos(first_lat .* pi ./ 180) ./ 180; range = 1000.*sqrt(r_ns_range.^2 + r_ew_range.^2);
figure plot(time_sec, range, 'linewidth',2)
titlepart = 'Range of EB200 receiver from transmitter'; titletext = cat(2, gpsname4title, ' - ', titlepart); title(titletext)
xlabel('Time (sec)') ylabel('Range (m) from Transmitter ')
grid on
%**************************************************************** % Calculate range * crossrange to TX via COSINE RULE
%**************************************************************** a = range(1); b = range(length(range)); c = distance(length(distance)) - distance(1);
ang_rads = acos( (b .^ 2 + c .^ 2 - a .^ 2) ./ (2 .* b .* c) ); dist_ew = b .* cos(ang_rads) - c; dist_ns = b .* sin(ang_rads);
msg1 = gpsname4title; msg2 = 'Dimension of RX is relation to TX at start of signal measurement'; msg3 = ' '; msg4 = sprintf('In an E-W orientation, Rx was %g (m) from Tx', dist_ew); msg5 = sprintf('In a N-S orientation, Rx was %g (m) from Tx', dist_ns); msg = strvcat(msg1, msg2, msg3, msg4, msg5);
% limitation on amount of characters in title msgtitle = 'Law of Cosine - Geodetic Spatial Relationship'; icondata = 1 : 64; icondata = (icondata'*icondata)/64; msgbox(msg, msgtitle, 'custom', icondata, hot(64));
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%**************************************************************** %* Calculate relative velocities, v = delta range / delta time *
%**************************************************************** % as based on 1Hz sampling rate, distance per second will be velocity % want a time lagged set of the data (i.e. time and distance) time_sec2 = time_sec(2: gga_len); time_sec2(gga_len) = 0; distance2 = distance(2: gga_len); distance2(gga_len) = 0;
% must know if van has moved E to W, or W to E % data files have the direction specified in their titles % will have either `e2w`, or `w2e` in title, so look for last '2' and % consider letter on its right. NB there could be multple '2'
two_points = []; two_points = find(gpsname == '2'); two_to_use = two_points(length(two_points)); going_to = gpsname(two_to_use + 1);
%********* find velocity of van in relation to repeater ********* % van moved in e -> w fashion if going_to == 'w'
vel2repeater = (distance2 - distance) ./ (time_sec2 - time_sec)'; % can use below since sampling at 1Hz % vel2repeater = (distance2 - distance);
end
% van moved in w -> e fashion if going_to == 'e'
vel2repeater = -(distance2 - distance) ./ (time_sec2 - time_sec)'; % can use below since sampling at 1Hz % vel2repeater = -(distance2 - distance);
end vel2repeater(gga_len) = 0;
%********* find velocity of van in relation to Tx source ********* distance2 = range(2 : gga_len); distance2(gga_len) = 0; vel2txsource = (distance2 - range) ./ (time_sec2 - time_sec)'; % can use below since sampling at 1Hz % vel2txsource = (distance2 - range); vel2txsource(gga_len) = 0;
figure plot(time_sec, vel2repeater, 'r', time_sec, vel2txsource, 'g', 'linewidth',2) titlepart = 'Velocity Magnitude of EB200 Reciever'; titletext = cat(2, gpsname4title, ' - ', titlepart); title(titletext)
xlabel('Time (sec)') ylabel('Velocity Magnitude (m/s)') legend('Repeater', 'Tx Source', 3)
grid on pause(1)
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%**************************************************************** %* Calculate Doppler frequencies f = v / lambda *
%**************************************************************** rfreqmhz = 1399; lambda = 3e8/(rfreqmhz*1e6); dopfrq = abs(vel2txsource - vel2repeater)./lambda;
figure plot(time_sec, dopfrq, 'c','linewidth',2)
titlepart = 'EB200 Receiver Doppler Shift'; titletext = cat(2, gpsname4title, ' - ', titlepart); title(titletext)
xlabel('Time (sec)') ylabel('Doppler Frequency (Hz)')
grid on pause(1)
%**************************************************************** %* show receiver position while on used roads *
%**************************************************************** gps_param(:,1) = time_sec'; gps_param(:,2) = distance;
xposn(:,1) = 1000 .* r_ew_dist; yposn(:,1) = 1000 .* r_ns_dist;
figure plot(xposn, yposn, 'linewidth',2) titlepart = 'Location of Rx during signal measurement'; titletext = cat(2, gpsname4title, ' - ', titlepart); title(titletext) xlabel('Longitudinal distance (m)') ylabel('Latitudinal distance (m)')
grid on pause(1)
%********************************************************************************** % compare measured phase with phase if van is stationary at each point
%********************************************************************************** gradient = abs(wavphase(1)) + abs(wavphase(stationarygrid) - wavphase(1)) ...
* wavlen / stationarygrid; linearphaseend = sign(wavphase(wavlen)) * gradient;
% generate a vector with linear phase variation stationaryphase = linspace(wavphase(1) , linearphaseend, wavlen)';
% find the difference between linear and actual data phase_dif = (stationaryphase - wavphase)/360;
% reduce the size of wav data uspec_samples = 1 : wavstep : wavlen; time_samples = uspec_samples ./ fs;
figure plot(time_samples, wavphase(uspec_samples) ./ 360, ...
time_samples, stationaryphase(uspec_samples) ./ 360, 'r') title('Observed Phase and Linear Phase(i.e. stationary RX)')
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xlabel('Time (sec)') ylabel('Wavelengths') legend('Observed', 'Linear Model', 2) grid on
%***************************************************************** %* join phase from previous set to next data set & update
%***************************************************************** spatial_phase(1:length(uspec_samples)) = phase_dif(uspec_samples)+…
continuephase; continuephase = spatial_phase(end);
%********************************************************************************** %* View FIELD PHASE vs DISTANCE along road (ie. SPATIAL PHASE)
%********************************************************************************** % find when van starts to move (check against van speed) distance2 = distance(2: length(distance)); distance2(length(distance)) = 0;
absvel = abs(distance2 - distance); absvel(length(distance)) = 0;
finish_flag = 0; deadstart_index = 1; for i = 1 : length(distance)
if absvel(i) > 1 & finish_flag ~= 1 deadstart_index = i; finish_flag = 1;
end end deadstart_index = deadstart_index - 2; sixty_sec = 1 : 60;
% adjust samples from 'distance' vector to match number of wav phases distanceinterp = interp1(sixty_sec, distance(deadstart_index : deadstart_index + …
60 - 1 ), time_samples);
figure plot(distanceinterp, spatial_phase, 'r' )
% Display the name of the data file in the title wavnamedot = find(wavname == '.'); wavtitlename = wavname(1 : wavnamedot - 1); wavunderscore = find(wavtitlename == '_'); wavtitlename(wavunderscore) = '-';
titletext = cat(2, 'Spatial Field Phase -- ',wavtitlename); title(titletext)
xlabel('Distance (m)') ylabel('Wavelengths')
grid on
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%***************************************************** %* find FREQUENCY for each bin (coarse size) %***************************************************** coarse_timebin_size = 2.^15; no_of_coarse_bins = fix(wavlen/coarse_timebin_size); coarsebins_per_wavcycle = coarse_timebin_size./fs;
if no_of_coarse_bins >= 1
% initalise vector for signal FREQUENCY and AMPLITUDE freq_center = zeros(1, no_of_coarse_bins); amp_sig = zeros(1, no_of_coarse_bins);
for block_loop = 1 : no_of_coarse_bins % specify WAV indexs for each bin bin_start = 1 + (block_loop - 1) * coarse_timebin_size; bin_stop = block_loop * coarse_timebin_size; bin_wavindexs = bin_start : bin_stop;
%---- below removes the spectral mirror -----fft4bin = fft((wavdata(bin_wavindexs, 1)+...
i.*wavdata(bin_wavindexs, 2)).*… hanning(coarse_timebin_size));
absfft4bin = abs(fft4bin); [fftmax, fftmax_index] = max(absfft4bin);
% Frequency interpolation y0 = abs(absfft4bin(fftmax_index - 1));
y1 = fftmax; y2 = abs(absfft4bin(fftmax_index + 1)); interp_freq_index = fftmax_index + (y0-y2)/(2*(y0+y2-2*y1));
% Add new value to FREQ and AMP vector freq_center(1, block_loop) = (coarse_timebin_size ...
- interp_freq_index + 1).* fs./coarse_timebin_size; amp_sig(1, block_loop) = fftmax;
end end
alltimesamples = no_of_coarse_bins * coarsebins_per_wavcycle; coarsetimesamples = linspace(1, alltimesamples, no_of_coarse_bins ); figure plot(coarsetimesamples, freq_center, 'r') titlepart = 'Max Frequency per Time Bin'; titletext = cat(2, gpsname4title, ' - ', titlepart); title(titletext)
xlabel('Time (sec)') ylabel('Frequency (Hz)') grid on
%******************************************************* %* integrate frequency difference to get phase %******************************************************* if no_of_coarse_bins > 1
freq_diff = freq_center - freq_center(1);
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freq_bytime = - freq_diff .* coarsebins_per_wavcycle; freq_phase = zeros(1, no_of_coarse_bins);
for phase_loop = 1:no_of_coarse_bins freq_phase(1,phase_loop) = sum(freq_bytime(1:phase_loop));
end end
figure plot(sixty_seconds, rangechange_lambda, 'b', ...
reducedwavindexs_perwavcycle, phase(reduced_wavindexs), 'r', ... coarsetimesamples, freq_phase, 'g')
titlepart = 'Change in Range Between TX and RX'; titletext = cat(2, gpsname4title, ' - ', titlepart); title(titletext)
xlabel('Time (sec)') ylabel('Wavelengths')
grid on legend('GPS','EB200 - phase','EB200 - freq', 0)
%************************************************************ %* Signal Amplitude Analysis (Coarse Resolution) %************************************************************ figure plot(coarsetimesamples, amp_sig, 'b') titlepart = 'Signal Amplitude'; titletext = cat(2, gpsname4title, ' - ', titlepart); title(titletext)
xlabel('Time (sec)') ylabel('Abs Value') grid on
% Log plot figure semilogy(coarsetimesamples, amp_sig, 'm') titlepart = 'Signal Amplitude (log)'; titletext = cat(2, gpsname4title, ' - ', titlepart); title(titletext)
xlabel('Time (sec)') ylabel('Abs Value')
axis([ 0 60 10^3 10^5 ]) grid on
%*************************************************** %* Use fine time resolution for power plots %*************************************************** % Amplitude variation with more time points fine_timebin_size = 2.^8; no_of_fine_bins = fix(wavlen / fine_timebin_size); finebins_per_wavcycle = fine_timebin_size ./ fs;
if no_of_fine_bins >= 1 % predefine no of blocks fine_amp_block = zeros(1, no_of_fine_bins);
for fine_block_loop = 1 : no_of_fine_bins finebin_start = 1 + (fine_block_loop - 1) * fine_timebin_size;
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finebin_stop = fine_block_loop * fine_timebin_size; finebin_wavindexs = finebin_start : finebin_stop; fft4finebin = fft(wavdata(finebin_wavindexs, 1) + ...
i .* wavdata(finebin_wavindexs, 2)); absfft4finebin = abs(fft4finebin);
[fine_max_field, fine_max_index ] = max(absfft4finebin);
% Amplitude fine_amp_block(1, fine_block_loop) = fine_max_field;
end end
finetime_samples = linspace(1, no_of_fine_bins * finebins_per_wavcycle, … no_of_fine_bins);
max_amp = max(fine_amp_block); pwr_variation = 20 .* log10(fine_amp_block ./ max_amp); plot(finetime_samples, pwr_variation, 'r') titlepart = 'Signal Power Variation'; titletext = cat(2, gpsname4title, ' - ', titlepart); title(titletext)
xlabel('Time (sec)') ylabel('dB')
grid on
%************************************************* %* Signal Amplitude Analysis (fine resolution) %************************************************* figure plot(finetime_samples, fine_amp_block, 'b') titlepart = 'Signal Amplitude'; titletext = cat(2, gpsname4title, ' - ', titlepart); title(titletext)
xlabel('Time (sec)') ylabel('Abs Value') grid on
% Log plot figure semilogy(finetime_samples, fine_amp_block, 'm') titlepart = 'Signal Amplitude (log)'; titletext = cat(2, gpsname4title, ' - ', titlepart); title(titletext)
xlabel('Time (sec)') ylabel('Abs Value') grid on
axis([ 0 60 10^1 10^3 ])
%***************************************************************************************** %* eliminate repeated data at flanks of data to allow quadratic approximation
%***************************************************************************************** dist_len = length(distanceinterp); tax = round(dist_len * lowtax : dist_len * hightax); distancetax = distanceinterp(tax); phasetax = spatial_phase(tax);
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%************************************************************ %* store continual (& each) set of distance and phase %************************************************************ totaldist = [totaldist ; distancetax']; totalphase = [totalphase ; phasetax']; eachset{fileloop, 1} = distancetax; eachset{fileloop, 2} = phasetax;
%******************************************************************* %* clear gps variables to allow next gps file be processed %******************************************************************* clear g* lat* lon* dist* spat* phase* observed* disp(' ');
end %fileloop
% change directory back to original setting cd(code_dir)
%*************************************************************** %* determine quadratic coefficents %*************************************************************** quad_coefs = polyfit(totaldist, totalphase, 2);
% determine polynomial values fn_phase = polyval(quad_coefs, totaldist);
%****************************************************************** %* save the quadratic coefficients (need file name for titles) %****************************************************************** % want potential filename part for plot titles fn_date = date; fn_type = '.mat';
% make filename from WAVNAME wavnamedot = find(wavname == '.'); wavpart = wavname(4 : wavnamedot - 1);
% add data set numbers used switch filenos case 1
setnos = cat(2, '_', setnumbers{1}); case 2
setnos = cat(2, '_', setnumbers{1}, '_', setnumbers{2}); case 3
setnos = cat(2, '_', setnumbers{1}, '_', setnumbers{2}, '_', setnumbers{3}); case 4
setnos = cat(2, '_', setnumbers{1}, '_', setnumbers{2}, '_', setnumbers{3}, '_', setnumbers{4});
end
phasetext = '_PHASE_direct'; filename = strcat(wavpart, setnos, phasetext, fn_type);
questext = 'Do you want to save Quadratic coefficents ?'; save_quadcoefs = questdlg(questext); switch save_quadcoefs case 'Yes'
% user gui selection
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ui_title = 'Select directory to save workspace'; [guifilename, pathname] = uiputfile(filename, ui_title);
% check if user has changed filename changetest = isequal(filename, guifilename); if changetest == 0
filename = strcat(guifilename, fn_type); end
filepathname = strcat(pathname, filename); disp(sprintf('Saving `%s`',filepathname));
save(filepathname, 'quad_coefs', 'totaldist', 'totalphase') end
%**************************************************************************** %* plot each set of spatial WAVLENGTHS with a different colour %* (max of 4 consecutive sets - Pt Gawler 8-9-10-11) %**************************************************************************** figure switch filenos case 1
plot(eachset{1,1}, eachset{1,2}, 'r', 'linewidth', 2) case 2
plot(eachset{1,1}, eachset{1,2}, 'r', eachset{2,1}, eachset{2,2}, 'g', 'linewidth', 2) case 3
plot(eachset{1,1}, eachset{1,2}, 'r', eachset{2,1}, eachset{2,2}, 'g', ... eachset{3,1}, eachset{3,2}, 'm', 'linewidth', 2)
case 4 plot(eachset{1,1}, eachset{1,2}, 'r', eachset{2,1}, eachset{2,2}, 'g', ...
eachset{3,1}, eachset{3,2}, 'm', eachset{4,1}, eachset{4,2}, 'b', 'linewidth', 2) end
title('Signal Spatial Phase') xlabel('Distance (m)') ylabel('Wavelengths') grid on
%*************************************************************** %* provide plot of function %*************************************************************** figure plot(totaldist, totalphase, 'linewidth', 2) title(‘'Measured Phase and Least Squares Fit') hold on plot(totaldist, fn_phase, 'r', 'linewidth', 2) legend('Measured Phase', 'Estimated Phase') grid on xlabel('Distance (m)') ylabel('Wavelengths')
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B.2 Geolocation Code %********************************************************************************* %* IDPELS – perform geolocation with inverse diffraction propagation of input %* signal calculated from field data %* displays WATERFALL plot of propagated spatial-phase %* %* Troy Spencer – November 2005 %*********************************************************************************
close all clear all
% user specification of geolocation parameters (with default values) inprompt = {}; inprompt{1} = 'Transmitter frequency (MHz)'; inprompt{2} = 'Domain width (m)'; inprompt{3} = 'Domain range (km)'; inprompt{4} = 'Range step (km)'; inprompt{5} = 'FFT power (e.g. 32768 = 2 .^ 15)'; inprompt{6} = 'Spatial-phase spectrum extension factor'; inprompt{7} = 'Flag to view data plots';
inpdef = {}; inpdef{1} = '1399'; inpdef{2} = '1000'; inpdef{3} = '7'; inpdef{4} = '0.1'; inpdef{5} = '14'; inpdef{6} = '3'; inpdef{7} = '0'; inptitle = 'IDPELS Parameters'; pemstr = inputdlg(inprompt, inptitle, 1, inpdef);
rfreq = str2double(pemstr(1))*1e6; crmax = str2double(pemstr(2)); rgmax = str2double(pemstr(3)); rginc = str2double(pemstr(4)); nfft_pwr = str2double(pemstr(5)); nfft = power(2, nfft_pwr); expansion = str2double(pemstr(6)); viewplots = str2double(pemstr(7));
% Calculate wavelength in meters wavl = 3.0e+8 / rfreq; kwavn = 2*pi / wavl;
% Small constants epsm = 0.000000000001; epsc = epsm + i*epsm;
% Calculate Cross Range increment in m crinc = 2*crmax/nfft;
% Set number of points in calculation: i.e. Cross Range and range points nrg = round(rgmax / rginc); windfrac = 1.0 - crmax /(nfft*crinc);
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% Set up Cross Range domain z = (0:crinc:nfft*crinc);
% set up domain for spatial frequency in horizontal direction - calculate max propagation % angle also maximum propagation angle % need to also specify p domain window via pwfac pwfac = 0.25; pangmax = 180*asin((1-pwfac)*wavl/(2*crinc))/pi; if (pangmax > 45)
warn = 'Max Propagation Angle too big - reduce nfft or increase crmax'; warntitle = 'IDPELS Sampling is NOT suitable'; warndlg(warn, warntitle) disp(warn); break pause % ensure program is halted
end
psamp = pi / (2*crmax); pangmax2 = 180*asin((1-pwfac)*psamp*nfft/kwavn)/pi; p = (0:psamp:psamp*nfft) + i*zeros(1,nfft+1);
% Equivalent in degrees for angle domain plots pa_deg = real(180.*asin(p./kwavn)./pi);
% Window for p-domain windowp = ones(1,nfft+1) + i*zeros(1,nfft+1); windowp(nfft+1) = 0.0 + i*0.0; for jw = 0:nfft*pwfac
arg = 0.5*pi*(-1+2*jw/(nfft*pwfac)); win = (0.5 + 0.5*sin(arg)); windowp(nfft-jw) = win + i*0.0;
end
if viewplots == 1 figure
plot(pa_deg, abs(windowp)) title('FFT window for p domain') xlabel('Angle in degrees')
ylabel('Abs value') grid on end
% NOTE WINDOW function below has been modified to mask reflections that would % otherwise be obtained from BOTH outer boundaries
% Set up window for fft calculation and plot window = ones(1,nfft+1) + i*zeros(1,nfft+1); window(1) = 0.0 + i*0.0; window(nfft+1) = 0.0 + i*0.0; windfrac = 0.3; for jw = 0:nfft*windfrac
arg = 0.5*pi*(-1+2*jw/(nfft*windfrac)); win = (0.5 + 0.5*sin(arg)); window(2+jw) = win + i*0.0; window(nfft-jw) = win + i*0.0;
end
if viewplots == 1 figure clf
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plot(z, abs(window)) title('FFT window - absolute value') xlabel('Cross Range (m)')
grid on end
% Set up diffraction term and plot aksq = (p./kwavn).^2; chok = find(aksq > 1); aksq(chok) = 1;
% KD 96 version term = 1000.*rginc.*kwavn.*(1-sqrt(1-aksq)); pdif = exp(i.*term).*windowp; pdifi = exp(-i.*term).*windowp;
if viewplots == 1 figure clf % display diffraction term subplot(2,1,1)
plot(pa_deg, unwrap(angle(pdif))) title('Diffraction term') xlabel('Angle (deg)')
ylabel('Unwrapped phase (rad)') grid on
% display INVERSE DIFFRACTION term subplot(2,1,2) plot(pa_deg, unwrap(angle(pdifi)), 'r') title('Inverse Diffraction term')
xlabel('Angle (deg)') ylabel('Unwrapped phase (rad)')
grid on end
% Initialise antenna OMNI y = ones(1,nfft+1) + i.*ones(1,nfft+1).*epsm;
%*************************************************************************** %* Load estimated and measured PHASE found via FINDPHASE %*************************************************************************** fclose('all'); uigettext = ' Select the file with quadratic coefficients'; [quadcoefname, quadcoefpath] = uigetfile('*.mat', uigettext);
% check file is valid, else quit if isequal(quadcoefname, 0) | isequal(quadcoefpath, 0)
disp('File not found') break
end
quadpathfile = strcat(quadcoefpath, quadcoefname); load(quadpathfile) quad_coefs;
% Analyse the filename and find direction travelled while measuring signal
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going2index = min(find(quadcoefname == '2')); going_to = quadcoefname(going2index + 1); switch going_to case 'e'
obsheading = '(Eastern Direction ->)'; case 'w'
obsheading = '(Western Direction ->)'; end
% find distance between roots and display phase spectrum over a multiple of this distance quad_roots = roots(quad_coefs); dist2add = abs(round(quad_roots(1) - quad_roots(2)))*expansion;
% account for direction of measurement if quad_roots(1) < quad_roots(2)
extradist = round(quad_roots(1) - dist2add/2) : round(quad_roots(2) + dist2add/2); else
extradist = round(quad_roots(2) - dist2add/2) : round(quad_roots(1) + dist2add/2); end
extra_phase = polyval(quad_coefs, extradist);
figure plot(extradist, extra_phase, 'linewidth', 2) h_index = find(quadcoefname == 'H'); titletext = quadcoefname(1 : h_index - 3); underscore = find(titletext == '_'); titletext(underscore) = '-'; titletext = cat(2, titletext, ' : Signal Spatial Phase'); title(titletext) grid on
%***** Highlight the spatial-phase that was actually measured ***** obsphase = polyval(quad_coefs, totaldist); hold on plot(totaldist, obsphase, 'r', 'linewidth', 2) legend('Estimated', 'Measured') xlabeltext = {}; xlabeltext{1} = 'Crossrange (m)'; xlabeltext{2} = obsheading; xlabel(xlabeltext) ylabel('Wavelengths')
% want non-exponential X and Y labels defxticks = get(gca, 'xtick'); set(gca, 'xticklabel', defxticks); defyticks = get(gca, 'ytick'); set(gca, 'yticklabel', defyticks); pause(1)
%****** want separate plots with measured and specified cross-range ****** disp('require a symetrical setting about the minimum value'); figure plot(extradist, extra_phase, 'linewidth', 2) titletext1 = 'Specifed cross-range section : '; h_index = find(quadcoefname == 'H'); titletext2 = quadcoefname(1 : h_index - 3); underscore = find(titletext2 == '_');
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titletext2(underscore) = '-'; % want to use this filename latter in the phase movie, so reassign pmfilename = titletext2; titletext = cat(2, titletext1, titletext2); title(titletext) xlabel(xlabeltext) ylabel('Wavelengths') grid on
% want non-exponential X and Y labels defxticks = get(gca, 'xtick'); set(gca, 'xticklabel', defxticks); defyticks = get(gca, 'ytick'); set(gca, 'yticklabel', defyticks);
% need to determine how far spectrum start is shifted wrt LHS for gui specification of % spatial-phase limits phase_lhs = min(extradist); phase_lhs = abs(phase_lhs);
%***** user to gui specify limits of spatial phase ***** % specified region will be highlight in RED limitvec = []; hold on for phaselimits_loop = 1 : 2
phaselimit = []; while isempty(phaselimit)
% must use DRAWNOW to avoid `Segmentation Violation` % www.mathworks.com/support/solutions/data/25049.shtml drawnow phaselimit = round(ginput(1));
end
% find the index value corresponding to user input and highlight boundaries plot(round(phaselimit(1)), extra_phase(round(phaselimit(1)) + …
phase_lhs),'*','MarkerEdgeColor','r', 'MarkerFaceColor', 'r', 'MarkerSize',10)
% store the element locations limitvec = [limitvec ; round(phaselimit(1))];
end
%****** make vector with xrange grids ***** lhlim = min(limitvec); rhlim = max(limitvec); xrange_values = lhlim : rhlim;
%***** highlight the specified phase region in RED ***** hold on phaseindex2show = xrange_values + phase_lhs; plot(xrange_values, extra_phase(phaseindex2show), 'r', 'linewidth', 2) legend('Estimated', 'Specified')
%************ update the title, with the limits shown in it **************** switch going_to case 'e'
lhlim2show = sprintf('West limit %g(m)', lhlim); rhlim2show = sprintf('East limit %g(m)', rhlim);
case 'w' lhlim2show = sprintf('East limit %g(m)', lhlim);
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rhlim2show = sprintf('West limit %g(m)', rhlim); end
titlelimits = sprintf('%s : %s', lhlim2show, rhlim2show); titletext = strvcat(titletext, titlelimits); title(titletext)
%*****GENERATE SPATIAL PHASE (radians) ****** yvalues = -2 .* pi .* polyval(quad_coefs, xrange_values); yvaluespos = yvalues - min(yvalues);
if viewplots == 1 % display specified signal spatial-phase figure plot(xrange_values, yvaluespos, 'r', 'linewidth', 2) titletext = sprintf('Selected SPATIAL-PHASE (%s)', pmfilename); titletext = strvcat(titletext, titlelimits); title(titletext) xlabel(xlabeltext)
ylabel('Phase (rad)') grid on end
% normalise the cross-range and display xtestphase = 0 : length(xrange_values) - 1;
if viewplots == 1 figure plot(xtestphase, yvaluespos, 'g', 'linewidth', 2) title('Specified Spatial-phase (normalised crossrange)')
xlabel('Crossrange (m)') ylabel('Phase (rad)') grid on end
% interpolate the phase in accordance with PEM grids and plot yangle = interp1( xtestphase, yvaluespos, z ); % yangle = interp1( xtestphase, yvalues, z );
if viewplots == 1 figure plot(yangle, 'm', 'linewidth', 2)
grid on title('Interpolated Spatial-phase for domain width')
xlabel('IDPELS grids') ylabel('Phase (rads)') end
% apply spatial-phase to the uniform signal (i.e constant power) y1 = abs(y(1+nfft/2)).*exp(i.*yangle); y = y1.* window; input_y = y;
if viewplots == 1 figure plot(abs(input_y), 'linewidth', 2)
grid on
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title('Input signal (amplitude) - Ensure full window is visible ') end
% find how many elements are numbers nan_size = []; % ensure this is EMPTY nantext = isnan(input_y); nan_yes = find(nantext == 1); nan_size = length(nan_yes);
if nan_size ~= 0 figure(4) msgtitle = 'Input Signal Size Test'; msgtext1 = 'You MUST halt the program and use a different crossrange'; msgtext2 = sprintf('Limits were (%g) and (%g)', lhlim, rhlim); msgtext = strvcat(msgtext1, msgtext2); msgbox(msgtext, msgtitle) break
end
%********************************************************************* %* define dimensions of WATERFALL plot %********************************************************************* xrangeindex2show = fix(nfft/4):fix(3*nfft/4); index2showtally = length(xrangeindex2show); display = zeros(1 + nrg, index2showtally); display(1,:) = y(xrangeindex2show);
%* sample the cross range, to show during movie crossrangenfft = round(linspace(lhlim, rhlim, index2showtally));
%********************************************************************* %* Inverse Diffraction propagation loop and display plot %********************************************************************* % wbh = waitbar(0, 'IDPELS progress, please wait ...'); % phase_movie = {}; % record of movie of propagated phase figure % for phase propagation clf for id_loop = 1:nrg
% Forward FFT & Sine option y(2:nfft) = newsintr( y(2:nfft) ); y(1) = 0; y(nfft+1) = 0;
% Apply diffraction term with window y = y.*pdifi;
% Inverse FFT & Sine option y(2:nfft) = newsintr( y(2:nfft) )./(nfft/2); y(1) = 0; y(nfft+1) = 0;
% Window y = y.*window; display(1 + id_loop,:) = y(xrangeindex2show);
% plot the input field y2view = y(xrangeindex2show); plot(crossrangenfft, abs(y2view), 'm')
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grid on xlabel('crossrange (m)') ylabel('Signal Phase')
axis([lhlim rhlim 0 20])
current_range = id_loop * rginc;
titletext1 = sprintf('%s -> Spatial-phase at range %g (km)', … pmfilename,current_range);
% show user (in title) at what range the maximum value is found help to decide % if movie should be saved % initialise with first value If id_loop == 1
max_signal = max(abs(y2view)); tx_range = current_range;
% now find crossrange of peak crossindex = find(abs(y2view) == max_signal);
end
% compare current value against previous maximum value current_max = max(abs(y2view)); if current_max > max_signal
max_signal = current_max; tx_range = current_range;
% now find crossrange of peak crossindex = find(abs(y2view) == max_signal);
end
hold on cross4max = crossrangenfft(crossindex); plot(crossrangenfft(crossindex), abs(y2view(crossindex)), 'g*')
hold off
titletext2 = sprintf('Peak signal at RANGE(%g)km and CROSSRANGE(%g)m', … tx_range, cross4max);
titletext = strvcat(titletext1, titletext2); title(titletext)
phase_movie(id_loop) = getframe(gcf);
% update waitbar waitbar( id_loop / nrg, wbh)
end close(wbh)
% ask user if they wish to save the phase movie questext = 'Do you wish to save the Phase propagation movie'; questitle = 'Spatial Phase Propagation Movie'; savemovie = questdlg(questext, questitle, 'Yes', 'No', 'No');
switch savemovie % allow gui specification for saving movie case 'Yes'
% want name like "Baldon_e2w_07_08_09_PHASEMOVIE.mat" d_in_name = find(quadcoefname == 'd'); d_index = max(d_in_name);
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namepart = quadcoefname(1 : d_index - 2); name2use = cat(2, namepart, 'MOVIE.mat');
% user gui selection ui_title = 'Select directory to save PHASE movie'; [moviefilename, moviepath] = uiputfile(name2use, ui_title);
% check if user has changed filename and allow change changetest = isequal(name2use, moviefilename); if changetest == 0
name2use = strcat(moviefilename, '.mat'); end
% save PHASE_MOVIE and DISPLAY so current range can be seen moviepathname = strcat(moviepath, name2use);
save(moviepathname, 'phase_movie') end
%******************************************************************************************* %* Estimate range (and corresponding cross-range) to the Tx source based on %* propagated field %******************************************************************************************
% find the magnitude (and maximum value) of the propagated field displaymag = abs(display); normaliser = max(max(displaymag));
% find range and crossrange to the Tx site [rangeindex, xrangeindex] = find(displaymag == normaliser); range_estimate = (rangeindex - 1) .* rginc;
% crossrange will be specified wrt the START of measurement xrange_fraction = xrangeindex / index2showtally; metric_width = abs(rhlim - lhlim); xrange_estimate = xrange_fraction * metric_width;
% account for start of measurement xrange_estimate = xrange_estimate + lhlim;
% remove extension from filename dotindex = find(quadcoefname == '.'); name2show = quadcoefname(1 : dotindex - 1);
% generate text indicating where Tx is located (with specified crossrange limits) rangetext = sprintf('Range estimate to source with % s is %g (km)', ...
name2show, range_estimate); disp(rangetext);
xrangetextp1 = 'Cross range value for the range estimate '; xrangetextp2 = '(with respect to beginning of measurement) is '; xrangetext2show = cat(2, xrangetextp1, xrangetextp2); xrangetext = sprintf('%s %g (m)', xrangetext2show, xrange_estimate); disp(xrangetext);
xrangelimits = sprintf('LH limit is %g (m) and RH limit is %g (m)',lhlim, rhlim); disp(xrangelimits);
% ask user if they wish to view the waterfall plot
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questext = 'Do you wish to view the waterfall plot'; questitle = 'Waterfall plot'; viewplot = questdlg(questext, questitle, 'Yes', 'No', 'No'); switch viewplot case 'Yes'
range = (1 : 1 + nrg) .* rginc - rginc; figure waterfall(z(xrangeindex2show), range, displaymag ./ normaliser )
% select view angles view_azi = -65; view_elv = 54;
view(view_azi, view_elv) axis tight idpelsxticklabels % function to include crossrange of Tx in xticks idpelsyticklabels % function to include range of Tx in yticks
end
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B.3 NEWSINTR function s=newsintr(x) %*********************************************************************** % computes discrete sine transform of complex vector x % SIN transform maintains quadratic variation of spatial-phase % during transform and inverse transform % % Reference: % J. R. Kuttler, "Computing Discrete Sine and Cosine Transforms with a % Discrete Fourier Transform", J. A. Krill, Ed. Laurel, Maryland, USA: % The John Hopkins University Applied Physics Laboratory, 1994. % % requires x to be odd length (else crash ... in calculation) %*********************************************************************** n=length(x)+1; n2=n/2; fsglob=ones(1,n2-1)./(8*sin((pi/n)*[1:n2-1])); y=x(3:2:n-1) - x(1:2:n-3); y=fft([2*x(1),x(2:2:n-2)+y,-2*x(n-1),fliplr(y-x(2:2:n-2))]); a=(i/4)*(y(2:n2)-y(n:-1:n2+2)); b=(y(2:n2)+y(n:-1:n2+2)).*fsglob; s=[a+b,.25*y(n2+1),fliplr(b-a)];
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B.4 IDPELS XTICKLABELS %******************************************************************************** % IDPELSXLABEL apply the specified width values to the waterfall plot, % where there user can change the number of X-AXES % % the cross-range corresponding to the peak field value is highlighted by % one of the axes & want peak cross-range to lie between the limits % %********************************************************************************
%********************************************************************************* %* get the default x-ticks %********************************************************************************* % % use the default number of xaxes % xtick_default = get(gca, 'xtick'); % xaxis_tally = length(xtick_default);
%*************************************************** %* to have limits on edge use below code %*************************************************** % find the default MIM and MAX value in X-axes xlimits_default = get(gca, 'Xlim'); lhlim_default = ceil(xlimits_default(1)); rhlim_default = floor(xlimits_default(2));
% apply a specified number of X-axes (aka 'xtick') % adjust value so peak axis doesn overlap others xaxistitle = 'Specification of cross-range axes'; xaxisinput = 'Enter the number of desired X axes'; xaxisdef = {'5'}; xaxistr = inputdlg(xaxisinput, xaxistitle, 1, xaxisdef); xaxis_tally = str2double(xaxistr);
xtick_spec = linspace(lhlim_default, rhlim_default, xaxis_tally); xtick_spec = round(xtick_spec);
% round to nearest integer values (more precise) lhlabel = floor(lhlim); rhlabel = ceil(rhlim);
% need to check if first measurement is included in the plot label_span = lhlabel : rhlabel; zero_test = find(label_span == 0); empty_zero_check = isempty(zero_test);
% find the corresponding metre valued labels to the xticks init_xrangelabels = linspace(lhlabel, rhlabel, xaxis_tally); init_xrangelabels = round(init_xrangelabels);
% apply the estimated XRANGE value to the labels and arrange in ascending order if empty_zero_check == 0
xrangelabels = [init_xrangelabels round(xrange_estimate) 0]; else
xrangelabels = [init_xrangelabels round(xrange_estimate)]; end
xrangelabels = sort(xrangelabels);
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% now adjust the XTICKS
% find how many initial crossrange indexs match the new entry ie (XRANGE_ESTIMATE) % 1) apply the LHLIM to convert to metre units % 2) find the width fraction corresponding to XRANGE estimate
l2rmove = abs(lhlim - xrange_estimate); % have METRIC_WIDTH peakxrangefraction = l2rmove / metric_width; xrange_default = abs(lhlim_default - rhlim_default); peakxrange_default = peakxrangefraction * xrange_default + lhlim_default;
% include the zero axis (if present) in the XTICKS if empty_zero_check == 0
move2zero = abs(lhlim); zeroxrangefraction = move2zero / metric_width; zeroxrange = zeroxrangefraction * xrange_default + lhlim_default; xtick_spec = [xtick_spec peakxrange_default zeroxrange];
else xtick_spec = [xtick_spec peakxrange_default];
end
% arrange in ascending order and apply xtick_spec = sort(xtick_spec); set(gca, 'xtick', [xtick_spec], 'linewidth', 2)
set(gca,'xticklabel', {xrangelabels}) % DO NOT CHANGE COLOUR, only want the xrange_estimate to be different colour % set(gca, 'Xcolor', 'm') init_xticklen = get(gca, 'ticklength'); set(gca, 'ticklength', [init_xticklen(1) 0.05])
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B.5 IDPELS Y TICKLABEL %********************************************************************************* % IDPELSYTICKLABEL highlights the range to the Tx site by including value in YTICKS % % the default YTICKS (i.e. y-axes) remain in the waterfall plot because they are at each KM % (but steps of 0.1km) % % this one WONT account for other stepping distances
% only adjust the YTICKS if range_estimate is NOT one of the default YTICKLABELS % %*********************************************************************************
% get inital YTICK settings init_ytick = get(gca, 'ytick'); % are numerical values
% find number of YTICKS yaxis_tally = length(init_ytick);
% find the default MIM and MAX value in Y-axes yticklimits = get(gca, 'Ylim'); min_ytick = floor(yticklimits(1)); max_ytick = ceil(yticklimits(2));
% default ytick setting is shown below (DONT DELETE) % ytick_init = linspace(0, rgmax, yaxis_tally );
%**** WITH ABOVE TICK SETTING, want labels to match yticklabels = linspace(0, rgmax, yaxis_tally);
% now add the RANGE_ESTIMATE to the labels % ONLY if its value is not already being used
% check if estimate is already specified by default estimate_check = []; estimate_check = find(range_estimate == yticklabels);
% if no match than isempty will be `1` if isempty(estimate_check)
yticklabels = [yticklabels range_estimate]; % arrange in ascending order yticklabels = sort(yticklabels);
% apply the RANGE_ESTIMATE to the yticks yticks = [init_ytick range_estimate]; yticks = sort(yticks);
% apply the ytick and yticklabels set(gca, 'ytick', [yticks]) set(gca, 'yticklabel', {yticklabels})
end
% have pre-specified other settings in IDPELSXTICKLABEL
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B.6 Load GPS field data file
%***************************************************************************** %* GPSREAD will read Rojone GPS field data file %* and is incorporated with other field trial programs programs %* %*****************************************************************************
%* Load GPS file %-----------------------------------------------------------------------------------------------------------% NMEA GGA format :-% FIELD 1 - GMT time, 2 - latitude, 3 - lat hemisphere, 4 - longitude % 5 - long hemisphere, 6 - Rx mode[0 - N/A, 1 non-diff, 2 - diff] % 7 - No. of sats, 8 - HDOP, 9 - Altitude, 10 - altitiude units % 11 - Geoidal seperation, 12 - seperation unit, 13 - Age of diff % corrections, 14 - diff station, 15 - NMEA checksum % example :-% $GPGGA,183805,3722.3622,N,2159.8274,W,2,03,02.8,16.6,M,20.2,M,5,80*XX %------------------------------------------------------------------------------------------------------------
fclose('all'); gpstitle1 = 'Select GPS data file with NMEA GGA data'; gpstitle2 = 'for correct input signal positioning'; gpstitle = strcat(gpstitle1, gpstitle2); [gpsname, gpspath] = uigetfile('*.txt', gpstitle); gps_filename = strcat(gpspath, gpsname); disp(sprintf('uigetfile `%s`', gps_filename)); gps_fin = fopen(gps_filename);
% find number of rows in file nmea = textread(gps_filename,'%s'); frewind(gps_fin); gps_filesize = size(nmea); gps_rows = gps_filesize(1);
%**************************************************************** % store TIME, LATITUDE & LONGITUDE %**************************************************************** gga_data = []; for gga_loop = 1 : gps_rows gps_buffer = fgetl(gps_fin);
[nmea_format] = strread(gps_buffer,'$%5c,%*[^\n]'); if nmea_format == 'GPGGA'
[time, lat, long] = strread(gps_buffer, $GPGGA, %f, %f, %*c, %f, %*[^\n]'); % assign to a numerical array gga_input = [floor(time), lat, long]; gga_data = [gga_data ; gga_input];
end end
fclose('all'); gga_len = length(gga_data(:,1));
%**************************************************************** % find time in units of seconds time_sec = 1 : gga_len; time_sec = time_sec - time_sec(1);
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%**************************************************************** % calculate lat and long while van is moving in units of seconds gga_lat = gga_data(:,2); gga_lon = gga_data(:,3);
lat_min = []; lat_deg = [];
lon_min = []; lon_deg = [];
% need to convert to string format to assign minute and degress % done sequentially, hence `for` loop for i = 1 : gga_len
%********************* LATTITUDE ************************ current_lat = gga_lat(i); lat_str = num2str(current_lat);
% need to account for different number of digit before and % after decimal sign deci = find(lat_str == '.');
% define field position of min and deg min_char = []; deg_char = [];
% must account for when dimension of minutes or degree change str_len = length(lat_str); decimal_points = str_len - deci;
% last digital at any decimal point could be zero switch decimal_points case 0
min_char = [deci - 2, deci - 1]; case 1
min_char = [deci - 2, deci - 1, deci, deci + 1]; case 2
min_char = [deci - 2, deci - 1, deci, deci + 1, deci + 2]; case 3
min_char = [deci - 2, deci - 1, deci, deci + 1, deci + 2, deci + 3]; case 4
min_char = [deci - 2, deci - 1, deci, deci + 1, deci + 2, deci + 3, deci + 4]; end
% only 2 digits for minutes up to 60, then degree change % check how many characters to left of 2nd character on lhs of % decimal point deg_check = deci - 1 - 2; switch deg_check case 1
deg_char = [deci - 3]; case 2
deg_char = [deci - 4, deci - 3]; end
lat_min = [lat_min ; str2num(lat_str(min_char))]; lat_deg = [lat_deg ; str2num(lat_str(deg_char))];
%********************** LONGITUDE ************************
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current_lon = gga_lon(i); lon_str = num2str(current_lon);
% need to account for different number of digit before and % after decimal sign deci = find(lon_str == '.');
% be aware there may be no fractional value(i.e no '.') if isempty(deci) == 1
deci = 0; end
% define field position of min and deg min_char = []; deg_char = [];
% must account for when dimension of minutes or degree change % eg whether deg are > or < 100 deg str_len = length(lon_str); decimal_points = str_len - deci;
% last digital at any decimal point could be zero switch decimal_points case 0
min_char = [deci - 2, deci - 1]; case 1
min_char = [deci - 2, deci - 1, deci, deci + 1]; case 2
min_char = [deci - 2, deci - 1, deci, deci + 1, deci + 2]; case 3
min_char = [deci - 2, deci - 1, deci, deci + 1, deci + 2, deci + 3]; case 4
min_char = [deci - 2, deci - 1, deci, deci + 1, deci + 2, deci + 3, deci + 4]; end
% only 2 digits for minutes up to 60, then degree change % check how many characters to left of 2nd character on lhs of % decimal point, degree can go over 100 (3 numbers) deg_check = deci - 1 - 2; switch deg_check case 1
deg_char = [deci - 3]; case 2
deg_char = [deci - 4, deci - 3]; case 3
deg_char = [deci - 5, deci - 4, deci - 3]; end
lon_min = [lon_min ; str2num(lon_str(min_char))]; lon_deg = [lon_deg ; str2num(lon_str(deg_char))];
end
latitude = lat_deg + lat_min./60; longitude = lon_deg + lon_min./60;
first_lat = latitude(1); first_lon = longitude(1);
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B.7 Frequency Shift Code
%******************************************************************************** %* FREQUENCY_SHIFT provides a spectral analysis of a single data set %* %* It first provides the FREQUENCY SHIFT over the 60second period %* using a large time-bin size, than a spectral plot for each (fine) time bin %* %* Troy Spencer – December 2005 %******************************************************************************** close all clear all
fclose('all'); [wavname, wavpath] = uigetfile('*.wav', 'Select WAV data file for frequency analysis'); wavfile = strcat(wavpath, wavname); disp(sprintf('uigetfile `%s`, please wait ...', wavfile));
% load the wav data wavdata = wavread(wavfile);
% find length of wav data file wavlen = length(wavdata);
%******************************************************************** %* FREQUENCY SHIFT DIAGARM %******************************************************************** % Define parameters fs = 44100; coarse_bin_size = 2.^15; no_of_coarse_bins = fix(wavlen/coarse_bin_size); bins_per_wavsample = coarse_bin_size./fs;
if no_of_coarse_bins >= 1
% initalise vector for signal FREQUENCY and AMPLITUDE freq_center = zeros(1, no_of_coarse_bins); amp_sig = zeros(1, no_of_coarse_bins);
for block_loop = 1 : no_of_coarse_bins
% specify WAV indexs for each bin bin_start = 1 + (block_loop - 1) * coarse_bin_size; bin_stop = block_loop * coarse_bin_size; bin_wavindexs = bin_start : bin_stop;
%---- below removes the spectral mirror -----fft4bin = fft((wavdata(bin_wavindexs, 1)+...
i.*wavdata(bin_wavindexs, 2)).*hanning(coarse_bin_size));
absfft4bin = abs(fft4bin); [fftmax, fftmax_index] = max(absfft4bin);
% Frequency interpolation y0 = abs(absfft4bin(fftmax_index - 1));
y1 = fftmax; y2 = abs(absfft4bin(fftmax_index + 1));
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interp_freq_index = fftmax_index + (y0-y2)/(2*(y0+y2-2*y1));
% Add new value to FREQ and AMP vector freq_center(1, block_loop) = (coarse_bin_size - interp_freq_index + 1) .*…
fs ./ coarse_bin_size; amp_sig(1, block_loop) = fftmax;
end end
alltimesamples = no_of_coarse_bins * bins_per_wavsample; timesamples = linspace( 1, alltimesamples, no_of_coarse_bins ); figure plot(timesamples, freq_center, 'r', 'linewidth', 2) region = wavname(4); setnumber = wavname(1:2); title(‘Frequency Shift’) xlabel('Time (sec)') ylabel('Frequency (Hz)') grid on
%******************************************************************** %* Spectral plot (per time bin) %******************************************************************** %* Display only the real spectrum, i.e. positive spectrum bin_size = 2.^15;
% Frequency range via Nyquist freq_bw = linspace( -fs./2, fs./2, bin_size);
% Number of time-blocks for calculating frequency freqblocs = fix(wavlen / bin_size);
% Step centre frequency to the middle of each time-bin for freqloop = 1 : freqblocs
% step in spectrum to centre of next time-bin if freqloop == 1
freqcentre = bin_size / 2; else
freqcentre = freqcentre + bin_size; end
% define indexs of time-bin about centre frequency spec_bloc = (freqcentre - (bin_size / 2) + 1 : freqcentre + (bin_size / 2));
% Take real spectrum (i.e. mirror image removed) pos_spec = fft(wavdata(spec_bloc,2)+ i.*wavdata(spec_bloc,1)); spec_bw = linspace( -fs./2, fs./2, bin_size);
figure plot(spec_bw, fftshift(abs(pos_spec)), 'b', 'linewidth', 2)
title('Real Spectrum') xlabel('Frequency (Hz)')
ylabel('Magnitude') grid on
axis([200 400 0 30000]) pause(0.1)
end
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B.8 Law of Cosine %****************************************************************************** %* The Law of Cosine is used with observed GPS data to provide %* the correct CROSSRANGE and RANGE value to the Transmitter %* The Cosine values are based only the measurement data %* %* A GUI is only returned with the cross-range and range values %* to the Tx, with the Rx being based at the start of signal %* measurement %* %* EQ = a^2 = b^2 + c^2 - 2bc cosA %* a = RANGE(1) %* b = RANGE(end) %* c = DISTANCE %*******************************************************************************
close all clear all
% get directory with code, to allow proper exit code_dir = cd;
%*************************************************************************************** %* user to specify how many data sets to analysis and extension factor %*************************************************************************************** inprompt{1} = 'How many consectutive GPS data sets will be evaluated'; inprompt{2} = 'Enter measurement path extention factor'; inlines = 1; inpdef ={'3', '2'}; inptitle = 'Crossrange based on COSINE Rule (GPS Data Analysis)'; inputstr = inputdlg(inprompt, inptitle, inlines, inpdef); no_of_files = str2double(inputstr(1)); ext_factor = str2double(inputstr(2));
% define vector to store all data obs_lat = []; obs_long = []; setnum = {}; alldatafiles = {};
% analysis all the gps data to determine a linear path
for fileloop = 1 : no_of_files gpsread; % use sub-program to load gps data file
if fileloop == 1 % analysis GPS file name to find direction of movement twosinfname = find(gpsname == '2');
two2use = max(twosinfname); going_to = gpsname(two2use + 1);
% select first 5 characters from gps filename to allow code % to know where transmission source is positioned
firstpart = gpsname(1:5); underscoreindex = find(firstpart == '_'); setnumber = firstpart(1 : underscoreindex - 1); region = firstpart(underscoreindex + 1);
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% St Kilda region if (region == 'M') | (region == 'P')
% St Kilda Tx site tx_lat = 34 + 43./60 + 26.2./3600; tx_lon = 138 + 32./60 + 15.6./3600;
end
% Mt Lofty ranges (Truro) at base if region == 'B'
% Baldon Rd, Truro Tx site tx_lat = 34 + 25./60 + 2.85./3600; tx_lon = 139 + 14./60 + 10./3600;
end end
end % fileloop
% change directory back to original setting cd(code_dir)
%******************************************************************************** % find distance (metric) travelled while OBSERVING signal % (from the start of measuring) %******************************************************************************** % WGS-84 semi-major and semi-minor a_earth = 6378.137; b_earth = 6356.752; N_earth = (a_earth.^2)./ ...
sqrt((a_earth .* cos(pi * first_obs_lat ./ 180)) .^2 + ... (b_earth .* sin(pi * first_obs_lat ./ 180)) .^ 2);
obs_ns_dist = abs(first_obs_lat - obs_lat) .* N_earth .* pi ./ 180; obs_ew_dist = abs(first_obs_long - obs_long) .* N_earth .* pi .* ...
cos(first_obs_lat .* pi ./ 180) ./ 180;
obs_dist = 1000 .* sqrt(obs_ns_dist .^ 2 + obs_ew_dist .^ 2); time_sec = 1 : length(obs_lat);
%******************************************************************************** % find RANGE between Tx and each OBSERVED receiver position %******************************************************************************** a_earth = 6378.137; b_earth = 6356.752; N_earth = (a_earth.^2)./ ...
sqrt((a_earth .* cos(pi * tx_lat ./ 180)) .^2 + ... (b_earth .* sin(pi * tx_lat ./ 180)) .^ 2);
obs_ns_range = abs(tx_lat - obs_lat) .* N_earth .* pi ./ 180; obs_ew_range = abs(tx_lon - obs_long) .* N_earth .* pi .* ...
cos(tx_lat .* pi ./ 180) ./ 180;
obs_range = 1000.*sqrt(obs_ns_range.^2 + obs_ew_range.^2);
%******************************************************************************* %- Determine linear polynomial for the EXTENDED measurement path %******************************************************************************* % find linear coefficients for straight path % NB LATITUDE is a function of LONGTIUTE, i.e lat = f(long) linpath = polyfit(obs_long, obs_lat, 1);
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% find min and max OBSERVED longitude min_obs_long = min(obs_long); max_obs_long = max(obs_long);
% extend longitude value (on both sides) and find corresponding % latitude values max_obs_dist = ceil(max(obs_dist));
delta_long = max_obs_long - min_obs_long; min_ext_long = min_obs_long - (delta_long * ext_factor); max_ext_long = max_obs_long + (delta_long * ext_factor); % long_step = 0.0001; % approx 10m long_step = 0.00001; % approx 1m ext_long = min_ext_long : long_step : max_ext_long; ext_lat = polyval(linpath, ext_long);
%----------------- EXTENDED MEASUREMENT PATH --------------------%**************************************************************** % find distance (metric) of EXTENDED linear measurement path * %**************************************************************** a_earth = 6378.137; b_earth = 6356.752; N_earth = (a_earth.^2)./ ...
sqrt((a_earth .* cos(pi * ext_lat(1) ./ 180)) .^2 + ... (b_earth .* sin(pi * ext_lat(1) ./ 180)) .^ 2);
ext_ns_dist = abs(ext_lat(1) - ext_lat) .* N_earth .* pi ./ 180; ext_ew_dist = abs(ext_long(1) - ext_long) .* N_earth .* pi .* ...
cos(ext_lat(1) .* pi ./ 180) ./ 180;
ext_dist = 1000 .* sqrt(ext_ns_dist .^ 2 + ext_ew_dist .^ 2);
%******************************************************************************** % find RANGE between Tx and each EXTENDED Rx position %******************************************************************************** a_earth = 6378.137; b_earth = 6356.752; N_earth = (a_earth.^2)./ ...
sqrt((a_earth .* cos(pi * tx_lat ./ 180)) .^2 + ... (b_earth .* sin(pi * tx_lat ./ 180)) .^ 2);
ext_ns_range = abs(tx_lat - ext_lat) .* N_earth .* pi ./ 180; ext_ew_range = abs(tx_lon - ext_long) .* N_earth .* pi .* ...
cos(tx_lat .* pi ./ 180) ./ 180;
ext_range = 1000.*sqrt(ext_ns_range.^2 + ext_ew_range.^2);
%--------------------- Apply COSINE RULE ------------------------%**************************************************************** %* Law of Cos is based only on the ACTUAL measurement path. %* It does not need the EXTENDED measurement path %**************************************************************** a = obs_range(1); % first range b = obs_range(end); % last range c = obs_dist(end); % 2 use as only based on actual obs
ang_rads = acos( (b .^ 2 + c .^ 2 - a .^ 2) ./ (2 .* b .* c) ); dist_ew = b .* cos(ang_rads) - c;
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dist_ns = b .* sin(ang_rads);
% apply reciprical sign to COSINE cross-range dist_ew = -dist_ew;
% Account for number of data files opened to display in MSGBOX switch no_of_files case 1
msg1 = cat(2, 'Data Set -> ', gpsname4title); case 2
switch region case 'B'
msg1 = sprintf('Baldon Data Sets - %s-%s', … cell2mat(setnum(1)), cell2mat(setnum(2))); case 'M'
msg1 = sprintf('McEvoy Data Sets - %s-%s', ... cell2mat(setnum(1)), cell2mat(setnum(2)));
case 'P' msg1 = sprintf('Pt Gawler Data Sets - %s-%s', ...
cell2mat(setnum(1)), cell2mat(setnum(2))); end
case 3 switch region case 'B'
msg1 = sprintf('Baldon Data Sets - %s-%s-%s', ... cell2mat(setnum(1)), cell2mat(setnum(2)), cell2mat(setnum(3)));
case 'M' msg1 = sprintf('McEvoy Data Sets - %s-%s-%s', ...
cell2mat(setnum(1)), cell2mat(setnum(2)), cell2mat(setnum(3))); case 'P'
msg1 = sprintf('Pt Gawler Data Sets - %s-%s-%s', ... cell2mat(setnum(1)), cell2mat(setnum(2)), cell2mat(setnum(3)));
end case 4
switch region case 'P'
msg1 = sprintf('Pt Gawler Data Sets - %s-%s-%s-%s', .... cell2mat(setnum(1)), cell2mat(setnum(2)), cell2mat(setnum(3)), ...
cell2mat(setnum(4))); end
end
%*********************************************************************** %* Show COSINE results in msgbox %*********************************************************************** msg2 = 'Dimension of RX is relation to TX at start of signal measurement'; msg3 = ' '; msg4 = sprintf('In an E-W orientation, Rx is %g (m) from Tx', ...
dist_ew); msg5 = sprintf('In a N-S orientation, Rx is %g (m) from Tx', ...
dist_ns); msg = strvcat(msg1, msg2, msg3, msg4, msg5);
% limitation on amount of characters in title msgtitle = 'Law of Cosine - Geodetic Spatial Relationship'; icondata = 1 : 64; icondata = (icondata'*icondata)/64; msgbox(msg, msgtitle, 'custom', icondata, hot(64)); % uiwait(msgbox(msg, msgtitle, 'custom', icondata, hot(64)));
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%********************************************************************* % find the index corresponding to the COS RULE crossrange on the % EXTENDED DISTANCE (ext_dist) array % need to account for 4 difference scenarios, i.e, % 1) direction travelled and 2) +ve or -ve crossrange specification %**********************************************************************
% find index value of EXTENDED position that matches % the FIRST Rx position on actual measurement path
% Can NOT exactly match positions, so must use minimum residue % i.e. find index closest to zero, based on some difference % ----- chosen longitude difference -------long_dif = ext_long - first_obs_long;
% find index with value closest to zero neg_long_dif = find(long_dif < 0); pos_long_dif = find(long_dif > 0); % have decided to take index of first positive value (only approximate) obs_start_index = pos_long_dif(1);
% find metric distance to each geodetic position on extended % measurement path (ext-long, ext-lat) a_earth = 6378.137; % WGS-84 semi-major and semi-minor b_earth = 6356.752; N_earth = (a_earth.^2)./ ...
sqrt((a_earth .* cos(pi * first_obs_lat ./ 180)) .^2 + ... (b_earth .* sin(pi * first_obs_lat ./ 180)) .^ 2);
dist_store = []; % store distance to each geodetic position
switch going_to case 'e'
if dist_ew < 0 disp(sprintf('Heading (%s) and -ve crossrange (%g)', going_to, dist_ew)); % because given CROSSRANGE is -ve (& moving E), have NO need % to evaluate EXTENTED PATH past FIRST obs point for obsloop = 1 : obs_start_index - 1
current_ns_dist = abs(first_obs_lat - ext_lat(obs_start_index - … obsloop)) .* N_earth .* pi ./ 180;
current_ew_dist = abs(first_obs_long - ext_long(obs_start_index… - obsloop)) .* N_earth .* pi .* ... cos(first_obs_lat .* pi ./ 180) ./ 180;
current_dist = 1000.*sqrt(current_ns_dist.^2 + current_ew_dist.^2); dist_store = [dist_store ; current_dist];
end
% subtract COS-RULE crossrange distance from each calculated % crossrange distance (from first observation point) and % find its INDEX value so RANGE can be found less_indexs = find(abs(dist_ew) > dist_store); deltaxrangeindex = max(less_indexs) + 1; dist_ewindex = obs_start_index - deltaxrangeindex;
end
if dist_ew > 0 disp(sprintf('Heading (%s) and +ve crossrange (%g)', going_to, dist_ew)); % because given CROSSRANGE is +ve (& moving east), evaluate % all EXTENDED PATH from first obs point to end
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for obsloop = obs_start_index : length(ext_long) - 1 current_ns_dist = abs(first_obs_lat - ext_lat(obsloop)) .* …
N_earth .* pi ./ 180; current_ew_dist = abs(first_obs_long - ext_long(obsloop)) .* …
N_earth .* pi .* ... cos(first_obs_lat .* pi ./ 180) ./ 180; current_dist = 1000.*sqrt(current_ns_dist.^2 + current_ew_dist.^2); dist_store = [dist_store ; current_dist];
end
% subtract COS-RULE crossrange distance from each calculated % crossrange distance (from first observation point) and % find its index value so RANGE can be found less_indexs = find(abs(dist_ew) > dist_store); deltaxrangeindex = max(less_indexs) + 1; dist_ewindex = obs_start_index + deltaxrangeindex;
end
case 'w' if dist_ew < 0
disp(sprintf('Heading (%s) and -ve crossrange (%g)', going_to, dist_ew) % because given CROSSRANGE is -ve (& moving west), evaluate
% all EXTENDED PATH from first obs point to end of EXTENDED PATH for obsloop = obs_start_index : length(ext_long) - 1
current_ns_dist = abs(first_obs_lat - ext_lat(obsloop)) .* … N_earth .* pi ./ 180;
current_ew_dist = abs(first_obs_long - ext_long(obsloop)) .* … N_earth .* pi .* ...
cos(first_obs_lat .* pi ./ 180) ./ 180; current_dist = 1000.*sqrt(current_ns_dist.^2 + current_ew_dist.^2); dist_store = [dist_store ; current_dist];
end
% subtract COS-RULE crossrange distance from each calculated % crossrange distance (from first observation point) and % find its index value so RANGE can be found less_indexs = find(abs(dist_ew) > dist_store); deltaxrangeindex = max(less_indexs) + 1; dist_ewindex = obs_start_index + deltaxrangeindex;
end
if dist_ew > 0 disp(sprintf('Heading (%s) and +ve crossrange (%g)', going_to, dist_ew)); % because crossrange is +ve (& moving W), evaluate EXTENTION up to % first obs point for obsloop = 1 : obs_start_index - 1
current_ns_dist = abs(first_obs_lat - ext_lat(obs_start_index -… obsloop)) .* N_earth .* pi ./ 180;
current_ew_dist = abs(first_obs_long - ext_long(obs_start_index… - obsloop)) .* N_earth .* pi .* ... cos(first_obs_lat .* pi ./ 180) ./ 180;
current_dist = 1000.*sqrt(current_ns_dist.^2 + current_ew_dist.^2); dist_store = [dist_store ; current_dist];
end
% subtract COS-RULE crossrange distance from each calculated % crossrange distance (from first observation point) and % find its index value so RANGE can be found less_indexs = find(abs(dist_ew) > dist_store);
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deltaxrangeindex = max(less_indexs) + 1; dist_ewindex = obs_start_index - deltaxrangeindex;
end end
%********************************************************************** %* apply the crossrange found by COSINE RULE to plot %********************************************************************** % show extened measurement path first (since its longer) % (so it wont overlap observed path) figure plot(ext_long, ext_lat, 'r', 'linewidth', 2)
% show observed and esimated measurement positions hold on plot(obs_long, obs_lat, 'linewidth', 2)
% use 2 lines below to show start / stop of measurements plot(obs_long(1), obs_lat(1), 'o') plot(obs_long(end), obs_lat(end), '*')
legend('Extended Measurement Path', 'Signal Measurement Path') grid on
% now show Tx site hold on plot(tx_lon, tx_lat, 'm*') axis ij
% highlight the COS-RULE crossrange value hold on plot(ext_long(dist_ewindex), ext_lat(dist_ewindex), 'm*')
%************************************************************************** %* draw line between COS-RULE crossrange site, and tx site %************************************************************************* xrange2Txlon = [tx_lon ext_long(dist_ewindex)]; xrange2Txlat = [tx_lat ext_lat(dist_ewindex)]; line(xrange2Txlon, xrange2Txlat, 'color', 'm')
% line from tx to first measuremnt position first2Txlon = [tx_lon first_obs_long]; first2Txlat = [tx_lat first_obs_lat]; line(first2Txlon, first2Txlat, 'color', 'g')
xlabel(['Longitude deg (East \rightarrow)']) ylabel(['Latitude deg (North \rightarrow)'])
%*************************************************** %* title of the plot %*************************************************** % add data set numbers used switch no_of_files case 1
setnos = setnum{1}; case 2
setnos = cat(2, setnum{1}, '-', setnum{2}); case 3
setnos = cat(2, setnum{1}, '-', setnum{2}, '-', setnum{3}); case 4
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setnos = cat(2, setnum{1}, '-', setnum{2}, '-', setnum{3}, '-', setnum{4}); end
% Apply region in title and join data set numbers switch region case 'B'
if no_of_files == 1 ftxtp1 = 'Mt Lofty Base Data Set (';
else ftxtp1 = 'Mt Lofty Base Data Sets (';
end filetxt = cat(2, ftxtp1, setnos ,')');
case 'M' if no_of_files == 1
ftxtp1 = 'McEvoy Road Data Set ('; else
ftxtp1 = 'McEvoy Road Data Sets ('; end filetxt = cat(2, ftxtp1, setnos ,')');
case 'P' if no_of_files == 1
ftxtp1 = 'Pt Gawler Rd Data Set ('; else
ftxtp1 = 'Pt Gawler Rd Data Sets ('; end filetxt = cat(2, ftxtp1, setnos ,')');
end % switch
titlepart1 = 'Geodetic Position of Tx and Rx Measurement path '; titletext = cat(2, titlepart1, ' - ', filetxt); title(titletext)
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290
Appendix C Matlab Code - Simulation
C.1 PEM
%********************************************************************************* %* PEM performs forward signal propagation based on the Parabolic %* Wave Equation Model. Multiple sources at various range and heights %* can be specified by the user. This is performed by the control %* file CF_TERRAIN. After the control file has initialised the PEM %* environment, a vertically polarised signal is propagated with the %* Fourier spilt-step method (via FFT). When the propagation range %* corresponds to another user specified source, a new signal profile %* is added in vector fashion via the PROFILE2ADD code. %* %* NB. CF_TERRAIN allows the user to specify a Wedged Terrain Profile %* for the signal to propagate over. A Non line-of-sight scenario %* can be generated with a wedge terrain profile being specified %*********************************************************************************
close all clear all
%********************************************************************** %* generate terrain profile and relative position of source with %* CF_WEDGE control file %**********************************************************************
sim_file = uigetfile('cf_*.m','Select Sim File'); dot_m = '.m'; sim_len = findstr(sim_file, dot_m); eval(sim_file(1 : sim_len - 1));
if (arb_terrain == 1) TP = TP - min(TP) + 1; Xmax = (length(TP) * dtm_step) - dx;
end
c = 3e8; % speed of light a = 6375000000; % radius of earth (m)
%********************************************************************** %* computed required PEM variables * %********************************************************************** lambda = c / f; % wavelength k = 2 * pi / lambda; % vacuum wave number p_max = k * sin(theta_max * pi / 180); % vertical field profile Tz = ((2 * pi) / (2 * p_max));
% Increased domain height by 10% to allow signal to propagate entire range z_adjust = (Xmax + Xmax / 10) * tan(theta * pi / 180); z_max = desired_z + z_adjust; Zmax = 2 * z_max;
%********************************************************************** %* Define the input field profile for PEM propagation * %**********************************************************************
% need length of `u` to be an index of 2 for FFT
291
close_power = log(Zmax/Tz)/log(2); close_power = ceil(close_power); N_minus_1 = close_power; N = N_minus_1 + 1;
u = zeros(2^N_minus_1, Xmax / dx); u_len = length(u(:,1));
% field is defined from z = 1*Tz as z=0 is modeled by the Sin FFT % insertion of a 0 z = Tz : Tz : ((2^N_minus_1)*Tz); z = reshape(z, u_len, 1);
% antenn beam pattern beam_width = ceil(ant_beam/Tz); han_beam = hanning(beam_width);
% determine number of grids points up to height of all antennas % antennas can NOT be closer than their beam width temp_hgt = src_at_same_range{1,1}; ant_hgt = temp_hgt(:,2); ant_range = temp_hgt(1,1);
% need to check if a field profile is being added, so initalise % CHECK_CHANGE is used in PROFILE2ADD check_change = 0;
% check if any source is on left hand SIDE boundary of domain & create % input field, else PROFILE2ADD will generate field profiles if ant_range == 0
% begin an index progresion through `range_change_index` check_change = check_change + 1;
% find no of PE grid points to each antenna in inital field for startloop = 1 : length(ant_hgt) S(startloop) = ceil(ant_hgt(startloop)/Tz); end
% assign value of `1` to sources, cause 0 x .34 = 0 for asignloop = 1 : length(ant_hgt) u(S(asignloop) + 1 : S(asignloop) + beam_width , 1) = ...
u(S(asignloop) + 1 : S(asignloop) + beam_width , 1) + 1; end
% apply signal strength to each antenna for sigloop = 1 : length(ant_hgt) u(S(sigloop) + 1 : S(sigloop) + beam_width , 1) = ... u(S(sigloop) + 1 : S(sigloop) + beam_width , 1).*han_beam; end
end
%********************************************************************** %* Determine Hanning Window of propagation domain * %********************************************************************** hn = Hanning(u_len); hn(1 : u_len/2) = ones(size(hn(1 : u_len/2))); hn = reshape(hn, 1, u_len);
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%********************************************************************** %* Define the Propagator * %**********************************************************************
dp = 1 / ((2^N)*Tz); p = 0 : dp : (((2^N_minus_1) - 1) * dp); q = p.*p; propagator = exp(-i*k*dx*(1-(1-((4*pi*pi*q)/(k*k))).^(1/2))); mirror_prop = fliplr(propagator(2 : ((2^N_minus_1)))); combined_prop = [propagator, 0, mirror_prop];
%********************************************************************** % rearrange indexing for sin transform to be performed via FFT * %**********************************************************************
odd_fft_len = (2^N_minus_1) - 1; fft_size = 2^N; range_step = dx/dtm_step;
%********************************************************************** %* Propagation Loop - Boundary Shifting * %**********************************************************************
% create hangle to waitbar for signal propagation h = waitbar(0,'Please wait...');
x_index = 1; for x = 0 : dx : (Xmax - 2 * dtm_step);
% a check for another source must be made prior to boudary shift check_range = []; check_range = find(x_index == src_pos(:,1)); empty_flag = isempty(check_range);
% if EMPTY_FLAG = 0, there was a match if empty_flag == 0
% increment check through 'RANGE_CHANGE_INDEX` % (used in profile2add) check_change = check_change + 1;
% add new vertical signal profile with PROFILE2ADD profile2add;
% the added profile will now be propagated u(:, x_index) = u(:, x_index) + toadd; disp(sprintf('added signal profile at range %d', x*dx'));
end
prev_field = u(:, x_index); ps = size(prev_field); prev_field = reshape(prev_field, 1, ps(1));
% Apply boundary Shift if (arb_terrain == 1)
% linear interpolation is used to `dx` step is a fraction of `dtm_step' step_1 = (x_index) * range_step; if (ceil(step_1) == step_1);
t_1 = TP(step_1 + 1); else
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step_1c = ceil(step_1) + 1; step_1f = floor(step_1) + 1; m = (TP(step_1c) - TP(step_1f)) / ... ((step_1c - step_1f) * dtm_step); t_1 = TP(step_1f) + m * (step_1 - floor(step_1));
end
step_2 = (x_index + 1) * range_step;
if (ceil(step_2) == step_2); t_2 = TP(step_2 + 1);
else step_2c = ceil(step_2) + 1; step_2f = floor(step_2) + 1; m = (TP(step_2c) - TP(step_2f)) / ...
((step_2c - step_2f) * dtm_step); t_2 = TP(step_2f) + m * (step_2 - floor(step_2));
end
% +ve dh means upward slope % -ve dh means downward slope dh = t_2 - t_1;
% convert change in terrain height to z-domain units delta_bins = round(dh/Tz); bin_count(x_index) = delta_bins;
% if delta_bins == 0, no padding if (delta_bins > 0)
prev_field = prev_field((abs(delta_bins) + 1) : ... length(prev_field));
prev_field(length(prev_field) + abs(delta_bins))=0; end
if (delta_bins < 0) clear temp; temp(abs(delta_bins)) = 0; prev_field = [temp, prev_field(1 : ... (length(prev_field) - length(temp)))];
end end
% apply Hanning window prev_field = prev_field .* hn;
% SIN FFT odd_part = fliplr(-prev_field(1 : odd_fft_len)); combined = [0, prev_field, odd_part]; U_x = fft(combined, fft_size); U_x = U_x .* (1/(2*j));
% multiply propagator for forward propagation U_x = U_x .* combined_prop;
% INVERSE SIN FFT u_x = fft(U_x,fft_size); u_x = u_x / (2*j);
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% NORMALISE the solution u_x = u_x(2 : (2^(N_minus_1) + 1)); u_x = u_x .* (2/(2^N_minus_1)); u(:, x_index + 1)= u_x(:); x_index = x_index + 1;
% display propagation progression waitbar(x / Xmax, h)
end % end for close(h)
%********************************************************************** %* Graphical display * %**********************************************************************
% Modify u matrix so that the arbitary terrain heights appear correct if (arb_terrain == 1)
x_index = 1; num_zeros = round(TP(x_index)/Tz); for x = 0 : dx :(Xmax - 2*dtm_step);
if (x_index == 1) delta_bins = 0;
else delta_bins = bin_count(x_index - 1);
end
num_zeros=num_zeros+delta_bins;
if ~(num_zeros == 0) clear temp; temp(abs(num_zeros))=0; u_field = u((1 : (length(u(:, x_index)) - ...
length(temp))), x_index); us = size(u_field);
% Reshape to a vector u_field = reshape(u_field, 1, us(1)); temp_field = [temp, u_field]; u(:, x_index) = temp_field(:);
end
x_index = x_index + 1; end
end
hold off;
NumZelem = round(desired_z / Tz); X = 0 : dx : Xmax - 2*dtm_step; Z = 0 : Tz :(NumZelem)*Tz; figure pcolor(X, Z, 20*log(abs(u(1 : length(Z), 1 : length(X))))); shading interp; colorbar;
if (arb_terrain==1) hold on;
terrain_x = 0 : dtm_step : (Xmax - 2 * dtm_step); plot(terrain_x, TP(1 : length(terrain_x)),'w');
end
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xlabel('Range x (metres)') ylabel('Height z (metres)') title(['PEM - frequency = ',num2str(f/1e9),' (Hz), theta = ',...
num2str(theta),' °']);
%********************************************************************** %* save the workspace * %********************************************************************** save_workspace = questdlg('Do you want to save the worksspace?'); switch save_workspace case 'Yes'
fn_date = date; fn_type = '.mat'; fname = strcat('various_ranged_src_', fn_date, fn_type); ui_title = 'Select directory to save workspace'; [havefn, pathname] = uiputfile(fname, ui_title); filename = strcat(pathname, fname);
disp(sprintf('Saving `%s`',filename)); save(filename)
case 'No' disp('Not saving workspace');
end
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C.2 Signal Profile-to-Add
%********************************************************************************* %* PROFILE2ADD generates a new signal profile corresponding to a new %* source. This source is added to the PEM signal already being %* propagation in a vector fashion %* %*********************************************************************************
% initialise vertical signal profile that will vectored together toadd = zeros(2^N_minus_1, 1);
% find height of interference sources temp_hgt = src_at_same_range{check_change, 1}; ant_hgt = temp_hgt(:,2);
% find number of grid elements to height of all antennas % N.B. antennas can NOT be closer than their beam width for sloop = 1 : length(ant_hgt) S(sloop) = ceil(ant_hgt(sloop)/Tz);
end
% assign value of `1` to all sources for aloop = 1 : length(ant_hgt) toadd(S(aloop) + 1 : S(aloop) + beam_width , 1) = ...
toadd(S(aloop) + 1 : S(aloop) + beam_width , 1) + 1; end
% apply signal strength (hanning window) to each antenna for sigloop = 1 : length(ant_hgt) toadd(S(sigloop) + 1 : S(sigloop) + beam_width , 1) = ...
toadd(S(sigloop) + 1 : S(sigloop) + beam_width , 1).*han_beam; end
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C.3 Control Test File
%********************************************************************************* %* CONTROL TEST FILE - WEDGE terrain profile %* This file generates required variables for PEM propagation %* Propagation is performed in PEM %* N.B. NLOS environments can be generated with this test file %* %* PARAMETERS that are user specified are shown below with their UNITS %* and VARIABLE NAME in this code %* %* 1) Antenna tilt (deg) - `theta` %* (aka grazing angle - ISBN 1891121014, p88, Stimson) %* 2) Maximum propagation angle (deg)- `theta_max` %* (used to determined vertical spatial sampling, i.e. `Tz` %* and accounts for signal interaction with terrain) %* 3) Frequency (GHz) - `f` %* 4) Uniform Antenna Beam Width (m) - `ant_beam` %* 5) Distance between terrain elements (m) - `dtm_step` %* 6) Propagation step (m) - `dx` %* 7) Domain height (m) - `desired_z` %* 8) Range of propagation domain (m) - 'range' %* 9) Flag to use arbitrary terrain - `arb_terrain` %* %* Operating procedure of this test file is indicated below:-%* 1) User specified of above parameter (defaults are provided) %* 2) User specification of terrain wedge %* 3) User to specify location of sources (via mouse) on diagram %* %*********************************************************************************
%********************************************** %* PEM parameter specification * %********************************************** heading = 'User specification of PEM parameters'; pem_prompt{1} = 'antenna tilt (deg) (a.k.a. grazing angle) - '; pem_prompt{2} = 'maximum propagation angle (deg) - '; pem_prompt{3} = 'signal frequency (GHz) - '; pem_prompt{4} = 'width of antenna sources (m) - '; pem_prompt{5} = 'distance between terrain elements (m) - '; pem_prompt{6} = 'PEM propagation step (m) - '; pem_prompt{7} = 'height of propagation domain (m) - '; pem_prompt{8} = 'range of propagation domain (m) - '; pem_prompt{9} = 'flag for wedge terrain (1 = wedge, 0 = flat) - ';
pem_default = {'0', '70', '1.399', '1', '1', '1', '100', ... '600', '1'};
pem_answer = inputdlg(pem_prompt, heading, 1, pem_default);
% convert cell array values to numeric values theta = str2double(pem_answer(1)); theta_max = str2double(pem_answer(2)); f = str2double(pem_answer(3))*1e9; ant_beam = str2double(pem_answer(4)); dtm_step = str2double(pem_answer(5)); dx = str2double(pem_answer(6)); desired_z = str2double(pem_answer(7)); range = str2double(pem_answer(8));
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arb_terrain = str2double(pem_answer(9));
%********************************************************************** %* Terrain Profile (TP) specification * %**********************************************************************
% **** define flat Terrain Profile (TP) range_grids = range / dtm_step; TP = ones(1, range_grids);
% ***** User Specification of Wedge (if requested) ***** if arb_terrain == 1
heading = 'User Specification of Wedge Parameters'; part1 = 'range to the beginning of the wedge ascent (m) - '; part2 = 'apex height of wedge (m) - '; part3 = 'range dimension of wedge (m) = '; wedge_prompt = {part1, part2, part3};
% default range to wedge will be approx 1/2 range of domain start_str = num2str(round(range / 2));
% default height of wedge will be 1/5 of domain height height_str = num2str(round(desired_z / 5));
% default range dimension of wedge will be 1/10 of domains range deltar_str = num2str(round(range / 10));
wedge_default = {start_str, height_str, deltar_str}; wedge_ans = inputdlg(wedge_prompt, heading, 1, wedge_default);
% convert cell array values to numeric values wedge_start = str2double(wedge_ans(1)); wedge_height = str2double(wedge_ans(2)); wedge_range = str2double(wedge_ans(3));
% calculate some values for determination of wedge dtm_tally = round(wedge_range / dtm_step); delta_range = round(wedge_range / 2); wedge_grad = wedge_height / delta_range;
% GRADIENT_COUNTER is used to descend height of wedge gradient_counter = 1;
for wedge_loop = 1 : dtm_tally range_check = wedge_loop * dtm_step; if range_check <= delta_range
% postive gradient TP(wedge_start + wedge_loop) = wedge_grad * wedge_loop;
else % negative gradient less_height = gradient_counter * wedge_grad; TP(wedge_start + wedge_loop) = wedge_height - less_height; gradient_counter = gradient_counter + 1;
end end
end
%********************************************************************** %* User specification of number of sources * %**********************************************************************
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heading = 'Source Specification'; source_prompt = 'Enter the number of source required for simulation'; source_default = {'3'}; source_tally_cell = inputdlg(source_prompt, heading, 1, source_default);
% convert the cell array to a numerical value fieldname = {'tally'}; % create structure will fieldname source_struct = cell2struct(source_tally_cell, fieldname); source_tally = str2num(source_struct.tally);
% allow gui for specification of source locations temp_ter = 0 : (length(TP) * dtm_step) - dx; plot(temp_ter, TP) axis([0 range 0 desired_z]) title('Specify source locations via mouse (PEM propagation)'); grid on hold on
source_record = []; for src_loop = 1 : source_tally
spec_loc = []; while isempty(spec_loc)
% must use DRAWNOW to avoid `Segmentation Violation` % www.mathworks.com/support/solutions/data/25049.shtml drawnow spec_loc = round(ginput(1));
end
% check user hasnt specified negative range value if spec_loc(1) < 1
spec_loc(1) = 1; end
% highlight the specified locations of sources plot(spec_loc(1), spec_loc(2), 'r*')
% store the source locations source_record = [source_record ; spec_loc];
end hold off
%********************************************************************** %* calculations based on source locations * %********************************************************************** % make a matrix for location of multiple sources % format of `src_pos` -> [longitude height latitude] src_pos = [];
% ginput specification src_pos(:, :) = source_record;
% arrange SRC_POS in ascending order based on range src_pos = sortrows(src_pos);
% user has specified 2 or more sources if source_tally > 1
% ***** Determine number of different ranges to sources ***** % there must be at least one source, hence at least one range range_tally = 1;
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% record index value of SRC_POS when there has been a range variation range_change_index = 1;
% start range comparison, intially based on first range range2compare = src_pos(1, 1);
% do a loop and find how many different ranges there are for rloop = 2 : length(src_pos)
% only interested if range is different if src_pos(rloop, 1) ~= range2compare
% adjust 1) tally of different ranges % 2) record of index where range changes % 3) record new range to compared with range_tally = range_tally + 1; range_change_index = [range_change_index rloop]; range2compare = src_pos(rloop, 1);
end end
%***** STORE SOURCES AT SAME RANGE IN SAME CELL INDEX ***** % define dimension of cell array to store source locations src_at_same_range = cell(range_tally, 1);
% consider if all sources are at same range if range_tally == 1
src_at_same_range{1, 1} = src_pos(:,:); end
% consider if sources are at different ranges if range_tally > 1
% the TLOOP (through-loop) requires data in ascending orders % it loops through `src_at_same_range` for tloop = 1 : range_tally - 1
src_at_same_range{tloop, 1} = ... src_pos(range_change_index(tloop) : ...
range_change_index(tloop + 1) - 1, :); end
% must specify last input outside of loop src_pos_len = length(src_pos);
src_at_same_range{range_tally, 1} = ... src_pos(range_change_index(range_tally) : src_pos_len, :);
end else
%******* only one source has been specified by user ******** range_tally = 1; src_at_same_range = cell(range_tally, 1); src_at_same_range{1, 1} = src_pos(:,:);
end
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C.4 Inverse Diffraction Localisation %********************************************************************************* %* IDPELS_SIM performs IDPELS localisation on field data recorded %* under simulation by PEM. An array or continuous configuration %* can be applied to the input field profile. Both the array %* and continuous profiles are graphically specified by the user %* %* The Fourier Split/Step method is used with boundary shifting. %* IDPELS divides the propagator for inverse propagation, as %* opposed to multiplication for forward propagation in PEM %* Terrain is modelled as a perfect conductor with a vacuum %* atmosphere %* %* N.B this code will load the input field profile saved by PEM %* program. Control of the input has been made with PEM %* %*********************************************************************************
% clear workspace close all clear all
%********************************************************************** %* load the data developed by forward propagation (i.e. PEM) %********************************************************************** heading = 'Select saved workspace for Inverse Propagation analysis '; [filename, filepath] = uigetfile('*.mat', heading); file_path_name = strcat(filepath, filename); disp(sprintf('loading `%s`', file_path_name)); disp(' Please wait ....'); load(file_path_name)
%********************************************************************** %* ask if user wants an array or sar configuration of input field %********************************************************************** heading = 'Continuous or Array field configuration'; query1 = 'Do you want the input field profile to be configured '; query2 = 'as an Array ?'; query = cat(2, query1, query2); array_config = questdlg(query, heading);
% set flag based on array setting arraystatus = strcmp('Yes', array_config);
%********************************************************************** %* user specification of various idpels parameters %********************************************************************** if arraystatus == 1
heading = 'Specification of IDPELS parameters'; part1 = 'Enter range extension for inverse propagation'; part2 = 'Enter the number of array elements '; part3 = 'Specify uniform width (m) of array elements';
idpels_prompt = {part1, part2, part3}; idpels_def = {'100', '2', '1'}; idpels_ans = inputdlg(idpels_prompt, heading, 1, idpels_def);
extra_length = str2double(idpels_ans(1)); no_ip_ant = str2double(idpels_ans(2));
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elements = round(str2double(idpels_ans(3)) / Tz); else
heading = 'Specification of IDPELS parameters'; idpels_prompt = {'Enter range extension for inverse propagation'}; idpels_def = {'100'}; idpels_ans = inputdlg(idpels_prompt, heading, 1, idpels_def); extra_length = str2double(idpels_ans(1)); no_ip_ant = 1;
end
%********************************************************************** %* terrain profile must be reversed and apply range extension %**********************************************************************
ip_terrain = fliplr(TP);
% apply range extension extra_ones = ones(1, extra_length) .* 2; ip_terrain = cat(2, ip_terrain, extra_ones); ip_Xmax = (length(ip_terrain) * dtm_step) - dx;
% define dimensions of IDPELS field back_u = zeros(2^N_minus_1, ip_Xmax / dx);
%********************************************************************************** %* user specification concerning location of IDPELS array elements %**********************************************************************************
% display reversed terrain profile and allow user specification temp_ter = 0 : (length(ip_terrain) * dtm_step) - dx; plot(temp_ter, ip_terrain) axis([0 ip_Xmax 0 desired_z])
if arraystatus == 1 titletext1 = 'Specify locations of array elements ';
else titletext1 = 'Specify range of desired input signal profile ';
end titletext2 = '(IDPELS Propagation direction -->)'; titletext = cat(2, titletext1, titletext2); title(titletext)
grid on hold on
% format of IP_ANT_RECORD is [range height] ip_ant_record = []; for src_loop = 1 : no_ip_ant
spec_loc = []; while isempty(spec_loc)
% must use DRAWNOW to avoid `Segmentation Violation` % www.mathworks.com/support/solutions/data/25049.shtml drawnow spec_loc = round(ginput(1));
end
% check user hasnt specified negative range if spec_loc(1) < 1
spec_loc(1) = 1; end
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if arraystatus == 1 % highlight the specified locations of sources plot(spec_loc(1), spec_loc(2), 'r*')
else plot(spec_loc(1), 1 : desired_z, 'mx') % use RANGE_GRIDS name as it allows simpler integration of
% SAR code (arrays) range_grids = spec_loc(1);
end
% store the source locations ip_ant_record = [ip_ant_record ; spec_loc];
end hold off
%************************************************************************************** % arrange IP_ANT_RECORD into ascending order based on RANGE % this allows code to determine if any antennas are at the % same range %************************************************************************************** if arraystatus == 1
ip_ant_record = sortrows(ip_ant_record);
% assign height and range to IDPELS antennas as vectors ip_ant_height = ip_ant_record(:, 2); ip_ant_range = ip_ant_record(:, 1);
% % ensure heights are in ascending order % ip_ant_height = sort(ip_ant_height);
% determine no. of grid to each antenna height ip_ant_grids = ceil(ip_ant_height / Tz);
%********************************************************************** %* Define signal profile at each
%********************************************************************** % have the entire forward propagation field in `u` % so select appropriate signal profiles and store in array % that will be indexed at correct range during propagation
% check if there are any IDPELS antenna at same range and record % in an array with the format -> [range number_of_antennas] ants_at_same_range = [];
% initalise with first antenna at nearest to inverse propagation % origin
tally = 1; ants_at_same_range = [ip_ant_range(1) tally];
tally_size = size(ants_at_same_range); tally_rows = tally_size(1);
if no_ip_ant > 1 for i = 2 : no_ip_ant
% if there is another antenna at same range, increase tally if ip_ant_range(i) == ip_ant_range(i - 1)
tally = tally + 1; % have tally of antennas in second column ants_at_same_range(tally_rows, 2) = tally;
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end
% if there AREN`T other antennas at same range, add new row if ip_ant_range(i) ~= ip_ant_range(i -1)
% reset the count of antennas to `1` tally = 1; % increase the number of rows in ANT_SAME_RANGE tally_rows = tally_rows + 1; ants_at_same_range = ...
[ants_at_same_range ; ip_ant_range(i) tally]; end
end end
%***** make vector of range and antenna tally for easier coding ***** % find number of range grids to each idpels antenna range_grids = ants_at_same_range(:,1) ./ dtm_step
% make tally vector tally_vec = ants_at_same_range(:,2);
% configue signal profile at each range with IDPELS antenna % maximum of 5 antennas at same range
% determine how many different ranges to IDPELS antennas same_range_size = size(ants_at_same_range); different_ranges = same_range_size(1);
% define array that stores the input field profile stored_inputs = [];
for rangechangeloop = 1 : different_ranges
% initally obtain entire field profile and specified range current_range = range_grids(rangechangeloop); field2adjust = u(:, Xmax - current_range);
% adjust field profile according to number of antennas tally_value = tally_vec(rangechangeloop);
get_rows = [];
switch tally_value case 1
field2adjust(1 : ip_ant_grids(1)) = 0; field2adjust(ip_ant_grids(2) + elements : end) = 0;
case 2 % find antenna heights and arrange in ascending order get_rows = find(current_range == ip_ant_record); current_heights = sort(ip_ant_record(get_rows, 2));
% apply array configuration to signal profile field2adjust(1 : ip_ant_grids(1)) = 0; field2adjust(ip_ant_grids(1) + elements : ip_ant_grids(2)) = 0; field2adjust(ip_ant_grids(2) + elements : end) = 0;
case 3 % find antenna heights and arrange in ascending order get_rows = find(current_range == ip_ant_record); current_heights = sort(ip_ant_record(get_rows, 2));
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% apply array configuration to signal profile field2adjust(1 : ip_ant_grids(1)) = 0; field2adjust(ip_ant_grids(1) + elements : ip_ant_grids(2)) = 0; field2adjust(ip_ant_grids(2) + elements : ip_ant_grids(3)) = 0; field2adjust(ip_ant_grids(3) + elements : end) = 0;
case 4 % find antenna heights and arrange in ascending order get_rows = find(current_range == ip_ant_record); current_heights = sort(ip_ant_record(get_rows, 2));
% apply array configuration to signal profile field2adjust(1 : ip_ant_grids(1)) = 0; field2adjust(ip_ant_grids(1) + elements : ip_ant_grids(2)) = 0; field2adjust(ip_ant_grids(2) + elements : ip_ant_grids(3)) = 0; field2adjust(ip_ant_grids(3) + elements : ip_ant_grids(4)) = 0; field2adjust(ip_ant_grids(4) + elements : end) = 0;
case 5 % find antenna heights and arrange in ascending order get_rows = find(current_range == ip_ant_record); current_heights = sort(ip_ant_record(get_rows, 2));
% apply array configuration to signal profile field2adjust(1 : ip_ant_grids(1)) = 0; field2adjust(ip_ant_grids(1) + elements : ip_ant_grids(2)) = 0; field2adjust(ip_ant_grids(2) + elements : ip_ant_grids(3)) = 0; field2adjust(ip_ant_grids(3) + elements : ip_ant_grids(4)) = 0; field2adjust(ip_ant_grids(4) + elements : ip_ant_grids(5)) = 0; field2adjust(ip_ant_grids(5) + elements : end) = 0;
end
% store the array configured profile in new column stored_inputs = [stored_inputs field2adjust];
end else
% SAR option will use entire vertical field profile % STORED_INPUTS also used with array code, but used for % simpler integration of SAR code stored_inputs = []; stored_inputs = u(:, Xmax - range_grids);
end
%********************************************************************** %* Define the INPUT signal profile %********************************************************************** if arraystatus == 1
% if there is no antenna located on left-hand side boundary, input % field will be zeros check_boundary = find(1 == range_grids);
if isempty(check_boundary) back_u(:, 1) = zeros(2^N_minus_1, 1);
else back_u(:, 1) = stored_inputs(:, 1);
end else
if range_grids == 0 back_u(:, 1) = stored_inputs;
else back_u(:, 1) = zeros(2^N_minus_1, 1);
end
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end
%********************************************************************** %* inverse diffraction propagation loop %********************************************************************** % display waitbar for propagation process h = waitbar(0,'Inverse diffraction propagation progress ...');
% cant index matrix with '0', so index input with `1` x_index = 1;
% initialise indexing value of 'STORED_INPUTS' input_index = 0;
% define propagation limit prop_limit = ip_Xmax - 2 * dtm_step;
for x = 0 : dx : prop_limit %****************************************************** %* check if another source is being added %****************************************************** check_range = []; check_range = find(x_index == range_grids); empty_flag = isempty(check_range);
% if EMPTY_FLAG = 0, there was a match if empty_flag == 0
% increment check through 'STORED_INPUTS' input_index = input_index + 1;
% add new vertical signal profile toadd = stored_inputs(:, input_index);
% the added profile will now be propagated back_u(:, x_index) = back_u(:, x_index) + toadd; disp(sprintf('added signal profile at range %d', x_index*dx'));
end
prev_field = back_u(:,x_index); ps = size(prev_field); prev_field = reshape(prev_field, 1, ps(1));
%*********************************************** %* apply boundary shift to account for terrain * %***********************************************
%****** input signal field ***** % specify distance (via index) to inital field step_1 = (x_index)*range_step; %N.B >> range_step = dx/dtm_step ... pem_beam
% check if stepping range is an integer if (ceil(step_1) == step_1)
% if an integer, then dont interpolate t_1 = ip_terrain(step_1 + 1);
else % if not an integer, then do interpolate step_1c = ceil(step_1) + 1; step_1f = floor(step_1) + 1; % find gradient between terrain elements at limits
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m = (ip_terrain(step_1c) - ip_terrain(step_1f)) / ... ((step_1c - step_1f) * dtm_step);
% find terrain height for inital field by applying gradient t_1 = ip_terrain(step_1f) + m * (step_1 - floor(step_1));
end
%***** propagated signal field ***** % code is same as above, except distance corresponds to where % next field will be calculated step_2 = (x_index+1)*range_step; if (ceil(step_2) == step_2)
t_2 = ip_terrain(step_2+1); else
step_2c = ceil(step_2) + 1; step_2f = floor(step_2) + 1; m = (ip_terrain(step_2c) - ip_terrain(step_2f)) / ...
((step_2c - step_2f) * dtm_step); t_2 = ip_terrain(step_2f)+m*(step_2-floor(step_2));
end
% find the change in terrain height (metres) % +ve dh implies an upward slope % -ve dh implies a downward slope dh = t_2 - t_1;
% find number of field elements in the height change delta_bins = round(dh/Tz); bin_count(x_index) = delta_bins;
% Boundary Shifting is employed in this code, so padding of % zeros is required if `delta_bins` does not equal zero % N.B padding is applied to the PREVIOUS FIELD if (delta_bins > 0)
% removing lower elements, and pad with zeros at top prev_field = prev_field((abs(delta_bins) + 1) : ...
length(prev_field)); prev_field(length(prev_field) + abs(delta_bins)) = 0;
end
if (delta_bins < 0) % remove higher elements, and pad with zeros at floor
clear temp; temp(abs(delta_bins)) = 0;
prev_field=[temp, prev_field(1 : ... (length(prev_field) - length(temp)))];
end
% hanning window is applied before each propagation step prev_field = prev_field.*hn;
%********************************************** %* apply sin transform %********************************************** % N.B >> odd_fft_len = (2^N_minus_1)-1 ... pem_beam % N.B >> fft_size = 2^N ... pem_beam odd_part = fliplr(-prev_field(1:odd_fft_len)); % Flip combined = [0, prev_field, odd_part]; % make odd Back_U_x = fft(combined, fft_size); % transform Back_U_x = Back_U_x.*(1/(2*j)); % / 2j
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%**************************** %* apply inverse propagator * %**************************** % Back_U_x is the angular spectrum from the previous step % So DIVIDE the Propagator for inverse diffraction zero_index = find(combined_prop == 0); combined_prop(zero_index) = 1.0e-5 + 1.0e-5i; Back_U_x = Back_U_x./combined_prop;
%******************************* %* apply inverse sin transform * %******************************* % Inverse Sin Transform is same as Sin transform, but requires % multiplication of the factor - N/2 (or 2^N-1 is this code)
back_u_x = fft(Back_U_x, fft_size); % is odd so FFT ~ SIN FFT back_u_x = back_u_x / (2*j); % / 2j to get SIN FFT
% NORMALISE the solution % ignore the domain extension back_u_x = back_u_x(2 : (2^(N_minus_1) + 1)); % start at 1 not 0 back_u_x = back_u_x.*(2 / (2^N_minus_1)); % multiple the factor
% save the time domain information back_u(:, x_index + 1) = back_u_x(:); x_index = x_index + 1;
% update waitbar waitbar(x / prop_limit, h)
end close(h)
% ============== GRAPHICAL DISPLAY =================== % Modify u matrix so that the arbritary terrain heights are % included so that plot appears correct. x_index = 1; num_zeros = round(ip_terrain(x_index) / Tz); for x = 0 : dx : (ip_Xmax - 2 * dtm_step);
% there can be NO padding with the input field if x_index == 1
delta_bins = 0; else
delta_bins = bin_count(x_index-1); end
num_zeros = num_zeros + delta_bins;
if ~(num_zeros == 0) clear temp; temp(abs(num_zeros)) = 0;
back_u_field = back_u((1:(length(back_u(:,x_index)) ... - length(temp))), x_index);
back_us = size(back_u_field); back_u_field = reshape(back_u_field, 1, back_us(1)); temp_field = [temp, back_u_field]; back_u(:, x_index) = temp_field(:);
end
x_index = x_index + 1;
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end
NumZelem = round(desired_z / Tz); X = 0 : dx : ip_Xmax - 2 * dtm_step; Z = 0 : Tz : (NumZelem) * Tz; figure pcolor(X, Z, 20 * log(abs(back_u(1 : length(Z), 1 : length(X))))); shading interp colorbar grid on hold on
% add terrain profile terrain_x = 0 : dtm_step : (ip_Xmax-2*dtm_step); plot(terrain_x, ip_terrain(1 : length(terrain_x)),'w');
xlabel('Range x(metres)') ylabel('Height z (metres)') title(['IDPLES - frequency = ',num2str(f/1e9), ' (Hz), theta = ',num2str(theta),' °']);
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Appendix D - Huygens Principle Model Code %****************************************************************************************************** % Huygens Principle Model (HPM)- Simple free space program with diffraction over edge % Time or Freq Domain option %******************************************************************************************************
close all clear all tic eps = 1e-10; %eps = 1e-15;
% Initialise variables - distances all in m
rginc = 2000;
%nzpt = 8192; %nzpt = 4096; nzpt = 2048; %nzpt = 1024; htmax = 200; htinc = 4*htmax/nzpt;
rfreq = 1e9; cvel = 3e8; wavl = cvel/rfreq;
% Set up variables
z = -2*htmax:htinc:2*htmax; nhpt = length(z);
% Calculate propagator
kwavn = 2*pi/wavl; zp = -4*htmax:htinc:4*htmax; pdif = zeros(1, 2*nzpt+1); delta = sqrt(rginc.^2 + zp.^2); %pdif = exp(-i.*kwavn.*delta)./delta; pdif = exp(-i.*kwavn.*delta)./sqrt(delta);
figure plot(zp, abs(pdif),'b') title('Propagator') xlabel('Height(m)') ylabel('Amplitude') grid on figure plot(zp, unwrap(angle(pdif)),'b') title('Propagator') xlabel('Height(m)') ylabel('Unwrapped Phase(rad)') grid on
ppdif = fftshift(fft(pdif));
figure plot(abs(ppdif),'b')
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title('Propagator - frequency domain') ylabel('Amplitude') grid on figure plot(unwrap(angle(ppdif)),'b') title('Propagator - frequency domain') ylabel('Unwrapped Phase(rad)') grid on
% Window for field
windfrac = 0.5 - htmax /(nhpt*htinc); window = ones(1,nhpt) + i*zeros(1,nhpt); window(1) = 0.0 + i*0.0; for jw = 0:nhpt*windfrac
arg = 0.5*pi*(-1+2*jw/(nhpt*windfrac)); win = (0.5 + 0.5*sin(arg)); window(2+jw) = win + i*0.0; window(nhpt-jw) = win + i*0.0;
end figure clf plot(z, abs(window)) title('FFT window - absolute value') xlabel('Height (m)') figure clf plot(z, unwrap(angle(window)) ) title('FFT window - unwrapped phase(rad)') xlabel('Height (m)')
% Initial field
thetabw = 1; thetael = 0;
y = zeros(1, nhpt) + i*zeros(1, nhpt); yadd = zeros(1, nhpt) + i*zeros(1, nhpt); yadd2 = zeros(1, nhpt) + i*zeros(1, nhpt); psamp = pi / (2*htmax); p = (-(nhpt/2)*psamp:psamp:(nhpt/2-1)*psamp) + i*zeros(1,nhpt);
% Set up njam sources njam = 4
% Source positions radhts = [ 30 -50 80 -130 ];
for ijam = 1:njam
yjam = ones(1, nhpt) + i*zeros(1, nhpt); % Sin(x)/x antenna pattern
pk = asin(p./kwavn) - thetael.*pi./180; afac = 1.39157 ./ sin(0.5*thetabw*pi/180.0); arg = afac.*sin(pk);
% Avoid divide by zero chks = find(abs(arg) <= eps);
% y(1, chks) = 1 + i.*eps; chks = find(abs(arg) > eps); ns = length(chks);
% y(1,chks) = sin(arg(1,chks))./arg(1,chks) + i.*ones(1,ns).*eps;
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antstr = 'SINC'; radht = radhts(ijam); yjam = yjam.*exp(-i*p*radht);
% y = y + yjam; if ijam == 1
% y = yjam; end if ijam == 1 | ijam == 2
y = y + yjam; end if ijam == 3
yadd = yjam; yadd = fftshift(ifft(yadd));
end if ijam == 4
yadd2 = yjam; yadd2 = fftshift(ifft(yadd2));
end
end % y = y./njam; % y = y./2;
y = y.*1.5;
% Omni antenna y = fftshift(ifft(y));
% Antenna tilt in deg tilt = 20; pe = kwavn.*sin(-tilt.*pi./180); %y = y.*exp(i.*pe.*z);
figure clf plot(z, abs(y) ) title('Initial field - absolute value') xlabel('Height (m)')
y = y.*window; % y = abs(y);
y1 = ones(1, nhpt) + i*zeros(1, nhpt);
% Propagate field
% Number of steps nstep = 3; % Select integration method - 1 = Time Domain, 2 = Frequency Domain isel = 2; % Obliquity factor obqufac = abs( rginc./sqrt(rginc.^2 + z.^2) );
% For frequency domain method nd = 4*(nzpt+1); yd = zeros(1,nd) + i*zeros(1,nd); pd = zeros(1,nd) + i*zeros(1,nd); pd(1,1+nzpt:length(pdif)+nzpt) = pdif; pd = fft(pd);
for istep = 1:nstep
% Obliquity factor
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y = y.*obqufac;
% Frequency domain method
if isel == 2
yd = zeros(1,nd); yd(1,1+fix(3*nzpt/2):fix(3*nzpt/2)+length(y)) = y; yd1 = fft(yd).*pd; yd1 = fftshift( ifft(yd1) ).*htinc; y1 = yd1( 1+fix(3*nzpt/2):1+fix(5*nzpt/2) );
end
% Time domain method if isel == 1
for iht = 1:nhpt
y1(iht) = sum( pdif(nzpt-iht+2:2*nzpt-iht+2).*y ).*htinc;
end
end
if istep ~= nstep % Allows us to plot 2 steps below
y = y1.*window; % y = abs(y); % Diffraction over a knife edge % y(1:fix(nhpt/2)) = 0 + i*0; % Add extra signals if istep == 1
y = y + yadd/2; end if istep == 2
y = y + yadd2/6; end
end
end
chk = find(abs(y) < eps); y(chk) = eps; chk = find(abs(y1) < eps); y1(chk) = eps;
figure plot(z, 20.*log10(abs(y)),'b', z, 20.*log10(abs(y1)),'r') title('Field') xlabel('Height(m)') ylabel('Amplitude') grid on figure plot(z, unwrap(angle(y)),'b', z, unwrap(angle(y1)),'r')
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title('Field') xlabel('Height(m)') ylabel('Unwrapped Phase(rad)') grid on
% Diffraction Inversion
y = y1.*window;
mdet = 20;
plus = 10;
dind = round(nhpt/4):round(3*nhpt/4); sind = length(dind);
display = zeros( 1+mdet*2+plus, sind);
nstep = nstep*mdet; rginc = rginc./mdet;
display(1,:) = y(dind);
for istep = 1:nstep + plus
% Use Frequency domain method delta = sqrt( (istep.*rginc).^2 + zp.^2 ); pdif = exp(-i.*kwavn.*delta)./sqrt(delta); pd = zeros(1,nd) + i*zeros(1,nd); pd(1,1+nzpt:length(pdif)+nzpt) = pdif; pd = fft(pd);
yd = zeros(1,nd); yd(1,1+fix(3*nzpt/2):fix(3*nzpt/2)+length(y)) = y;
% Because this is INVERSE DIFFRACTION we divide here intead of multiply yd1 = fft(yd)./pd; yd1 = fftshift( ifft(yd1) ).*htinc; y1 = yd1( 1+fix(3*nzpt/2):1+fix(5*nzpt/2) );
if istep ~= nstep % Allows us to plot 2 steps below % y = y1.*window;
y2 = y1; end
display(1+istep,:) = y1(dind);
end
chk = find(abs(y2) < eps); y2(chk) = eps; chk = find(abs(y1) < eps); y1(chk) = eps; chk = find(abs(display) < eps); display(chk) = eps;
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figure %plot(z, 20.*log10(abs(y2)),'b', z, 20.*log10(abs(y1)),'r') plot(z, abs(y2),'b', z, abs(y1),'r') title('Field') xlabel('Height(m)') ylabel('Amplitude') grid on figure plot(z, unwrap(angle(y)),'b', z, unwrap(angle(y1)),'r') title('Field') xlabel('Height(m)') ylabel('Unwrapped Phase(rad)') grid on
figure waterfall(abs(display)) title('INVERSE PROPAGATION') axis off view(-65, 54) toc
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