RADIO PROPAGATIONMEASUREMENT ANDCHANNEL MODELLING

RADIO PROPAGATIONMEASUREMENT ANDCHANNEL MODELLING

Sana SalousDurham University, UK

A John Wiley & Sons, Ltd., Publication

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Library of Congress Cataloging-in-Publication Data

Salous, Sana.Radio propagation measurement and channel modelling / Sana Salous.

pages cmIncludes bibliographical references and index.ISBN 978-0-470-75184-8 (cloth)

1. Shortwave radio–Transmitters and transmission–Measurement. 2. Radio wavepropagation–Measurement. 3. Wireless communication systems. I. Title.

TK6553.S214 2013621.3841′1–dc23

2012037692

A catalogue record for this book is available from the British Library.ISBN: 9780470751848

Set in 9/11 Times by Laserwords Private Limited, Chennai, India

To the memory of my parents, Mariam and Hasan and to my latebeloved nephew Mounther Salous.

Contents

Foreword xiii

Preface xv

List of Symbols xvii

Acronyms and Abbreviations xix

1 Radio Wave Fundamentals 11.1 Maxwell’s Equations 11.2 Free Space Propagation 31.3 Uniform Plane Wave Propagation 31.4 Propagation of Electromagnetic Waves in Isotropic and Homogeneous Media 51.5 Wave Polarization 81.6 Propagation Mechanisms 11

1.6.1 Reflection by an Isotropic Material 121.6.2 Reflection/Refraction by an Anisotropic Material 181.6.3 Diffuse Reflection/Scattering 191.6.4 Diffraction 20

1.7 Propagation in the Earth’s Atmosphere 211.7.1 Properties of the Earth’s Atmosphere 211.7.2 Radio Waves in the Ionosphere 25

1.8 Frequency Dispersion of Radio Waves 291.8.1 Phase Velocity versus Group Velocity 301.8.2 Group Path versus Phase Path 311.8.3 Phase Path Stability: Doppler Shift/Dispersion 32

References 33

2 Radio Wave Transmission 352.1 Free Space Transmission 35

2.1.1 Path Loss 352.1.2 Relating Power to the Electric Field 37

2.2 Transmission Loss of Radio Waves in the Earth’s Atmosphere 382.2.1 Attenuation due to Gases in the Lower Atmosphere and Rain: Troposphere 382.2.2 Attenuation of Radio Waves in an Ionized Medium: Ionosphere 41

2.3 Attenuation Due to Propagation into Buildings 43

viii Contents

2.4 Transmission Loss due to Penetration into Vehicles 462.5 Diffraction Loss 49

2.5.1 Fundamentals of Diffraction Loss: Huygen’s Principle 492.5.2 Diffraction Loss Due to a Single Knife Edge: Fresnel Integral Approach 50

2.6 Diffraction Loss Models 542.6.1 Single Knife Edge Diffraction Loss 542.6.2 Multiple Edge Diffraction Loss 55

2.7 Path Loss Due to Scattering 572.8 Multipath Propagation: Two-Ray Model 57

2.8.1 Two-Ray Model in a Nondispersive Medium 582.8.2 Two-Ray Model due to LOS and Ground Reflected Wave: Plane

Earth Model 592.8.3 Two-Ray Propagation via the Ionosphere 63

2.9 General Multipath Propagation 662.9.1 Time Dispersion due to Multipath Propagation 662.9.2 Effects of Multipath Propagation in Frequency, Time and Space 69

2.10 Shadow Fading: Medium Scale 772.11 Measurement-Based Large-Scale Path Loss Models 78References 82

3 Radio Channel Models 853.1 System Model for Ideal Channel: Linear Time-Invariant (LTI) Model 853.2 Narrowband Single Input–Single Output Channels 87

3.2.1 Single-Path Model 873.2.2 Multipath Scattering Model 88

3.3 Wideband Single Input–Single Output Channels 933.3.1 Single-Path Time-Invariant Frequency Dispersive Channel Model 933.3.2 Single-Path Time-Variant Frequency Dispersive Channel 983.3.3 Multipath Model in a Nonfrequency Dispersive Time-Invariant Channel 993.3.4 Multipath Propagation in a Nonfrequency Dispersive Time-Variant Channel 1043.3.5 Multipath Propagation in a Frequency Dispersive Time-Variant Channel 106

3.4 System Functions in a Linear Randomly Time-Variant Channel 1063.5 Simplified Channel Functions 108

3.5.1 The Wide-Sense Stationary (WSS) Channel 1083.5.2 The Uncorrelated Scattering Channel (US) 1093.5.3 The Wide-Sense Stationary Uncorrelated Scattering Channel (WSSUS) 109

3.6 Coherence Functions 1103.7 Power Delay Profile and Doppler Spectrum 1113.8 Parameters of the Power Delay Profile and Doppler Spectrum 111

3.8.1 First and Second Order Moments 1113.8.2 Delay Window and Delay Interval 1143.8.3 Angular Dispersion 115

3.9 The Two-Ray Model Revisited in a Stochastic Channel 1153.10 Multiple Input–Multiple Output Channels 115

3.10.1 Desirable Channel Properties for Narrowband MIMO Systems 1163.10.2 MIMO Capacity for Spatial Multiplexing 118

3.11 Capacity Limitations for MIMO Systems 1203.12 Effect of Correlation Using Stochastic Models 120

3.12.1 Capacity Expressions Based on Stochastic Correlation Models 121

Contents ix

3.12.2 Capacity Expressions Based on Uniform and ExponentialCorrelation Models 122

3.12.3 The Kronecker Stochastic Model 1233.13 Correlation Effects with Physical Channel Models 123

3.13.1 Distributed Scattering Model 1243.13.2 Single-Ring Model 1253.13.3 Double-Ring Model 1263.13.4 COST 259 Models 1273.13.5 Multidimensional Parametric Channel Model 1273.13.6 Effect of Antenna Separation, Antenna Coupling and Angular

Spread on Channel Capacity 1283.13.7 Effect of Mutual Coupling 130

3.14 Effect of Number of Scatterers on Channel Capacity 1343.14.1 Free Space Propagation 1353.14.2 Limited Number of Multipath Components 136

3.15 Keyholes 1373.16 Rician Channels 1413.17 Wideband MIMO Channels 143

3.17.1 Wideband Channel Model 145References 145

4 Radio Channel Sounders 1494.1 Echoes of Sound and Radio 1494.2 Definitions and Objectives of Radio Sounders and Radar 151

4.2.1 Modes of Operation 1514.2.2 Basic Parameters 152

4.3 Waveforms 1524.4 Single-Tone CW Waveforms 153

4.4.1 Analysis of a Single-Tone System 1534.5 Single-Tone Measurements 158

4.5.1 Measurement Configurations 1584.5.2 Triggering of Data Acquisition 1604.5.3 Strategy of CW Measurements 162

4.6 Spaced Tone Waveform 1644.7 Pulse Waveform 166

4.7.1 Properties of the Pulse Waveform 1674.7.2 Factors Affecting the Resolution of Pulse Waveforms 1714.7.3 Typical Configuration of a Pulse Sounder 1714.7.4 Practical Considerations for Pulse Sounding 171

4.8 Pulse Compression Waveforms 1744.8.1 Ideal Correlation Properties of Pulse Compression

Sounding Waveforms 1754.8.2 Pulse Compression Detectors 1774.8.3 Comment on Pulse Compression Detectors 180

4.9 Coded Pulse Signals 1824.9.1 Barker Codes (1953) 1824.9.2 PRBS Codes 1844.9.3 PRBS Related Codes: Gold Codes 1924.9.4 Kasami Code 1944.9.5 Loosely Synchronous Codes 196

x Contents

4.10 Serial Correlation Detection of Coded Transmission 1964.10.1 Sliding Correlator 1964.10.2 Stepped Cross Correlator 198

4.11 Comment Regarding Coded Transmission 1984.12 Frequency Modulated Continuous Wave (FMCW) Signal 199

4.12.1 Matched Filter Detector 1994.12.2 Heterodyne Detector of FMCW Signals 2034.12.3 Practical Consideration of Detection Methods of FMCW Signals 207

4.13 Range Doppler Ambiguity of Chirp Signals: Advanced Waveforms 2074.13.1 Three-Cell Structure 2084.13.2 Multiple WRF Structure 2104.13.3 Target Movement 2114.13.4 Doppler Shift Estimation 211

4.14 Architectures of Chirp Sounders 2134.15 Monostatic Operation of FMCW Sounder/Radar 217

4.15.1 Reduction of Effective Mean Received Power 2184.15.2 Spreading of the Spectrum and Interference 2194.15.3 Blind Ranges and Range Ambiguity 2204.15.4 Selection Criteria for Switching Sequences 2214.15.5 Considerations for Edge Weighting 2244.15.6 Length of the Window 2244.15.7 Window Functions 2244.15.8 Interpolation and Quantization 225

4.16 Single and Multiple Antenna Sounder Architectures 2254.16.1 Single Input Single Output (SISO) Sounders 2264.16.2 MISO, SIMO and MIMO Measurements with SISO Sounders 2274.16.3 Semi-Sequential MIMO Sounders 2284.16.4 Parallel MIMO Sounders 228

4.17 Ultra-wideband (UWB) Channel Sounders 2324.18 Sounder Design 233

4.18.1 Sounder for Indoor Radio Channels in the UHF Band 2394.18.2 Sounder for UHF Frequency Division Duplex Links for

Outdoor Radio Channels 2394.18.3 Sounder for Multiple Frequency Links for Outdoor Radio Channels 239

4.19 Performance Tests of a Channel Sounder and Calibration 2394.19.1 Ambiguity Function 2414.19.2 Linearity Test 2424.19.3 Frequency Response 2434.19.4 Calibration of Automatic Gain Control 2434.19.5 Isolation between Multiple Channels 2454.19.6 Sensitivity and Dynamic Range 2464.19.7 Effect of Interference on the Dynamic Range 2494.19.8 Stability of Frequency Sources 2514.19.9 Temperature Variations 251

4.20 Overall Data Acquisition and Calibration 251References 251

5 Data Analysis 2555.1 Data Validation 2555.2 Spectral Analysis via the Discrete Fourier Transform 256

Contents xi

5.3 DFT Analysis of the FMCW Channel Sounder Using a Heterodyne Detector 2595.3.1 Snapshot Impulse Response Analysis 2605.3.2 Frequency Response Analysis 2635.3.3 Estimation of the Delay Doppler Function 266

5.4 Spectral Analysis of Network Analyzer Data via the IDFT 2685.5 DFT Analysis of CW Measurements for Estimation of the Doppler Spectrum 2685.6 Estimation of the Channel Frequency Response via the Hilbert Transform 2695.7 Parametric Modelling 269

5.7.1 ARMA Modelling 2715.7.2 AR Modelling 2715.7.3 Practical Implementation of Parametric Modelling 2715.7.4 Parametric Modelling for Interference Reduction 2725.7.5 Parametric Modelling for Enhancement of Multipath Resolution 274

5.8 Estimation of Power Delay Profile 2765.8.1 Noise Threshold 2775.8.2 Stationarity Test 280

5.9 Small-Scale Characterization 2865.9.1 Time Domain Parameters 2875.9.2 Estimation of the Coherent Bandwidth 2885.9.3 Statistical Modelling of the Time Variations of the Channel Response 291

5.10 Medium/Large-Scale Characterization 2925.10.1 CDF Representation 2925.10.2 Estimation of Path Loss 2935.10.3 Relating RMS Delay Spread to Path Loss and Distance 2965.10.4 Frequency Dependence of Channel Parameters 299

5.11 Multiple Antenna Array Processing for Estimation of Direction of Arrival 3015.11.1 Theoretical Considerations for the Estimation of Direction of Arrival 3035.11.2 Spectral-Based Array Processing Techniques 3085.11.3 Parametric Methods 3125.11.4 Joint Parametric Techniques 316

5.12 Practical Considerations of DOA Estimation 3195.12.1 Choice of Antenna Array 3205.12.2 Array Calibration 3225.12.3 Estimation of Direction of Arrival 3265.12.4 Estimation of Direction of the Arrival/Direction of Departure 331

5.13 Estimation of MIMO Capacity 333References 333

6 Radio Link Performance Prediction 3376.1 Radio Link Simulators 3376.2 Narrowband Stochastic Radio Channel Simulator 338

6.2.1 Quadrature Amplitude Modulation Simulator 3396.2.2 Filtered Noise Method 3396.2.3 Sum of Sinusoids Method (Jakes Method) 3416.2.4 Frequency Domain Method 3436.2.5 Reverberation Chambers (or Mode-Stirred Chambers) 344

6.3 Wideband Stochastic Channel Simulator 3466.3.1 Time Domain Channel Simulators 3466.3.2 Frequency Domain Simulators 348

6.4 Frequency Domain Implementation Using Fast Convolution 349

xii Contents

6.5 Channel Block Realization from Measured Data 3516.6 Theoretical Prediction of System Performance in Additive White Gaussian Noise 353

6.6.1 Matched Filter and Correlation Detector 3546.6.2 Bit Error Rate of the Matched Filter Detector in AWGN 3566.6.3 Bit Error Rate with Noncoherent Detectors 3576.6.4 Comparison of BER of Coherent and Noncoherent Detectors 3586.6.5 Higher Order Modulation 358

6.7 Prediction of System Performance in Fading Channels 3616.7.1 Narrowband Signals 3616.7.2 Wideband Signals 363

6.8 Bit Error Rate Prediction for Wireless Standards 3646.8.1 IEEE 802.16-d Standard 3656.8.2 IEEE 802.11-a Standard 3716.8.3 Third Generation WCDMA Standard 372

6.9 Enhancement of Performance Using Diversity Gain 3766.9.1 Diversity Combining Methods 3776.9.2 Diversity Gain Prediction of Rayleigh Fading Channels

from Measurements in a Reverberation Chamber 382References 383

Appendix 1 385A.1 Probability Distribution Functions 385A.2 The Gaussian (Normal) Distribution 385A.3 The Rayleigh Distribution 387A.4 The Rician Distribution 388A.5 The Nakagami m-Distribution 389A.6 The Weibull Distribution 390A.7 The Log-Normal Distribution 390A.8 The Suzuki Distribution 391A.9 The Chi-Square Distribution 391References 391

Appendix 2 393

Index 395

Foreword

A full understanding of radio wave propagation is fundamental to the efficient operation of manysystems, including cellular communications, radio detection and ranging (RADAR) and globalpositioning system (GPS) navigation to name a few. It is essential in these, and other systems, tobe able to measure or ‘sound out’ the channel and collect the channel impulse or frequency responsecharacteristic. This may then be used in the transmitter and/or receiver to ensure that data or othertraffic is transferred in the most effective manner with minimal distortion, interference and signalloss. With relative motion between the transmitter or receiver, for example from a moving vehicle,these responses will vary with time and the channel characteristic will require to be continuouslyupdated.

Radio propagation has been a well-studied topic in laboratories worldwide, over many decades,most probably starting in earnest with the advent of wireless communication systems in the 1950s.

This new volume, from a recognized UK expert, provides an excellent summary of the state ofthe art in channel models, sounders, propagation and data analysis with application examples tocurrent wireless standards and will be an essential addition to the library collection of many oftoday’s practitioners in wireless communications.

Peter GrantEmeritus Regius Professor of Engineering,

The University of EdinburghAugust 2012

Preface

Radio propagation measurements and channel modelling continue to be of fundamental importanceto radio system design. As new technology enables dynamic spectrum access and higher data rates,radio propagation effects such as shadowing, the presence of multipath and frequency dispersionare the limiting factors in the design of wireless communication systems. While there are severalbooks covering the topic of radio propagation in various frequency bands, there appears to be nobooks on radio propagation measurements, which this book addresses at length. To provide thereader with a comprehensive and self-contained book, some background material is provided inthe first two chapters, which cover the fundamentals of radio transmission including propagation inionized media. The aim here is to bring two different communities together, namely those workingon communication via the ionosphere in the high frequency (HF) band with those working at ultra-high frequency (UHF) through examples that illustrate that although the medium of transmissionis different the principles are similar. Thus the two-ray model commonly used in mobile radiopropagation studies is shown to be applicable to the two magneto-ionic waves that propagate viathe ionosphere. The distortion effects on wideband signals as they travel through a frequencydispersive medium is studied for both narrow pulses and for frequency modulated continuouswave signals to illustrate the principles of transmission. Some basic path loss models are brieflydescribed at the end of Chapter 2 including a discussion on shadow fading and location variability.Chapter 3 addresses various stochastic channel models and relates them to system models startingfrom single input–single output to multiple input–multiple output models. Chapter 4 explains atlength the principles of design of a radio channel sounder and relates them to radar principles. Thedifferent waveforms and architectures are contrasted and calibration techniques and performancemeasures are detailed to aid the practising engineer in the design and realization of appropriate radiomeasurement systems. Chapter 5 addresses the important topic of data analysis starting from themost basic discrete Fourier transform to more advanced parametric estimation methods. Multipleantenna processing techniques to extract angle of arrival information including suitable antennaarrays and array calibration as well as multiple antenna channel capacity are detailed. Chapter 6discusses the prediction of link performance of digital communication systems starting from thebasic principles of the matched filter and correlation detector. This is followed by a descriptionof various channel simulators and application of extracted channel parameters to the simulation oflink performance of two wireless standards, namely the wireless metropolitan area network and theWi-Fi standard. Finally, diversity combining methods are briefly outlined.

Throughout the book examples from propagation measurements in the HF band and higherfrequency bands have been either specifically reprocessed for presentation or used as appearedin publications. The higher frequency band measurements have been generated by my researchstudents using custom designed radio channel sounders. The wideband HF measurements relate tomy earlier work at Birmingham University and here a special gratitude is due to Professor RamsayShearman who inspired my interest in radio science and set the direction of my professional career.

xvi Preface

The move to the UHF band occurred while working with Professor David Parsons at LiverpoolUniversity. Working in these two frequency bands enabled me to have a broader outlook on radiopropagation. Hence, when multiple antenna technology was being mainly investigated in the UHFband, its application to the HF band seemed a natural extension.

In addition I would like to acknowledge the kind assistance and encouragement of ProfessorLouis Bertel of Rennes University 1 and Dr Sean Swords of Trinity College Dublin who providedme with their laboratory facilities. Finally, I would like to thank my colleague and friend, Dr RobertBultitude from the Communications Research Centre, Ottawa, for his contribution to the originaloutline of the book.

Sana Salous

List of Symbols

αn Azimuth angleαR Unit vector along Rβ Wave numberαβ Unit vector along the direction of wave propagationdel Differential vector operator∇ grad , gradient of a scalar∇. div , divergence of a vector∇× Curl of a vector. Dot productδ(t) Impulse function, dirac delta functionh(t) Impulse responseh(t ,τ ) Time-variant impulse responseH (ω) Frequency responseH (ω,t) Time-variant frequency responseH (ω,ν) Frequency Doppler functionρ Charge density∈ Permittivityμ PermeabilityC ConductivityD Electric fluxB Magnetic fluxJ Current densityf Frequency, Hzfc Carrier frequencyfH Gyro-frequencyfi Instantaneous frequencyfD Doppler frequencyϕn Phase shift with respect to an arbitrary referenceI In-phaseJo Zero order Bessel function of the first kindω Angular frequency, radians/secβ Wave numberλ Wavelength

xviii List of Symbols

n Impedance* Complex conjugateS Poynting vector� Ohmδs Skin depthn Phase refractive indexn′ Group refractive indexN Refractivityp Pressureν Diffraction coefficientm MeterNo Single-sided noise power spectrum densityS (τ ,ν) Delay Doppler functionτ Time delay variableτ g Group time delayτmax Maximum time delayQ Quadraturevec( ) Vector operator stacking all elements of a matrix column-wise into a single vector,⊗ Kronecker productE {} Expected valueT Transpose

Acronyms and Abbreviations

A AmpereAIC Akaike information criterionAR AutoregressiveARMA Autoregressive moving averageAUV Autonomous underwater vehicleAWGN Additive white Gaussian noiseBER Bit error rateBLAST Bell labs layered space timeBPF Bandpass filterC Coulombsc Speed of lightCDF Cumulative distribution functionCLR Correlated low rankCUBA Circular uniform beam arrayCW Continuous waveformdB DecibeldBm Decibel relative to 1 mWdBW Decibel relative to 1 WDFT Discrete Fourier transformDOA Direction of arrivalDOD Direction of departureE EnergyE Electric fieldEDOF Effective degrees of freedomEHF Extra high frequencyEIRP Effective (or equivalent) isotropic radiated powerELF Extremely low frequencyEM ElectromagneticEM Expectation maximizationESPRIT Estimation of signal parameters via rotational invarianceF FaradFB Forward backwardFFT Fast Fourier transform

xx Acronyms and Abbreviations

FMCW Frequency modulated continuous waveformFT Fourier transformGPS Global positioning systemH HenryH Magnetic fieldHF High frequencyHz HertzI Identity matrixIDFT Inverse discrete Fourier transformIEEE Institute of Electronic and Electrical EngineersIFT Inverse Fourier transformIID Independent and identically distributedIIR Infinite impulse responseITU International Telecommunications UnionKS Kolmogorov–SmirnovLF Low frequencyLO Local oscillatorLTI Linear time invariantLUF Lowest usable frequencyMA Moving averageMBES Multibeam echo sounderMDL Minimum description lengthMEA Multielement arrayMF Medium frequencyMIMO Multiple input–multiple outputMISO Multiple input–single outputMUF Maximum usable frequencyMUSIC Multiple signal classificationPDF Probability density functionPDP Power delay profilePLL Phase locked loopPPI Plan position indicatorPRBS Pseudo random binary sequencesPRF Pulse repetition frequencyRADAR Radio detection and rangingRMS Root mean squareSAGE Space alternating generalized expectation maximization algorithmSAW Surface acoustic devicesSHF Super high frequencySIMO Single input–multiple outputSISO Single input–single outputSOS Sum of sinusoidsSSB Single side bandTTL Transistor transistor logicUHF Ultra high frequencyUHR Uncorrelated high rankULR Uncorrelated low rankUMTS Universal mobile telecommunication systemUS Uncorrelated scatteringUWB Ultra wideband

Acronyms and Abbreviations xxi

VHF Very high frequencyVLF Very low frequencyWCDMA Wideband code division multiple accessWiMAX Worldwide interoperability for microwave accessWSS Wide-sense stationaryWSSUS Wide-sense stationary uncorrelated scattering

1Radio Wave Fundamentals

Radio wave propagation is governed by the theory of electromagnetism laid down by the Scottishphysicist and mathematician James Clerk Maxwell (13 June 1831 to 5 November 1879) whodemonstrated that electricity, magnetism and light are all manifestations of the same phenomenon.Electromagnetic wave propagation depends on the properties of the transmission medium in whichthey travel. Classifications of transmission media include linear versus nonlinear, bounded versusunbounded, homogeneous versus nonhomogeneous and isotropic versus nonisotropic. Linearityimplies that the principle of superposition can be applied at a particular point, whereas a mediumcan be considered bounded if it is finite in extent or unbounded otherwise. Homogeneity refers tothe uniformity of the physical properties of the medium at different points and an isotropic mediumhas the same physical properties in different directions.

In this chapter we start by a revision of the fundamentals of Maxwell’s wave equations andpolarization. This is followed by a discussion of the different propagation phenomena includingreflection, refraction, scattering, diffraction, ducting and frequency dispersion. These are discussedin relation to different transmission media such as propagation in free space, the troposphere andthe ionosphere. For a more detailed treatment of the subject, the reader is referred to [1, 2].

1.1 Maxwell’s Equations

Originally Maxwell’s equations referred to a set of eight equations published by Maxwell in 1865.In 1884 Oliver Heaviside, concurrently with other work by Willard Gibbs and Heinrich Hertz,modified four of these equations, which were grouped together and are nowadays referred to asMaxwell’s equations. Individually, these four equations are known as Gauss’s law, Gauss’s law formagnetism, Faraday’s law of induction and Ampere’s law with Maxwell’s correction.

Fundamental to Maxwell’s four field equations is the differential vector operator ∇ (pronounceddel ) and the bold denotes a vector given by:

∇ = ∂

dxax + ∂

dyay + ∂

dzaz (1.1)

For a scalar V and a vector function A with components along the xyz axes:

A = Axax + Ayay + Azaz (1.2)

Radio Propagation Measurement and Channel Modelling, First Edition. Sana Salous.© 2013 John Wiley & Sons, Ltd. Published 2013 by John Wiley & Sons, Ltd.

2 Radio Propagation Measurement and Channel Modelling

there are three possible operations related to the ∇ operator, defined as follows:

1. The gradient of a scalar V is a vector given by:

∇ V = grad V = ∂V

dxax + ∂V

dyay + ∂V

dzaz (1.3)

2. The divergence of A is a scalar given by:

∇ .A = div A = ∂Ax

∂x+ ∂Ay

∂y+ ∂Az

∂z(1.4)

3. The curl of A is a vector given by:

∇ ×A = curl A =(

∂Az

∂y− ∂Ay

∂z

)ax +

(∂Ax

∂z− ∂Az

∂x

)ay +

(∂Ay

∂x− ∂Ax

∂y

)az (1.5)

or

∇ ×A =

∣∣∣∣∣∣∣∣∂

∂x

∂

∂y

∂

∂z

Ax Ay Az

ax ay az

∣∣∣∣∣∣∣∣(1.6)

Related to these operators are the following identities:

divcurl A = ∇ .(∇ ×A) = 0 (1.7)

curlgrad V = ∇ ×(∇ V ) = 0 (1.8)

divgrad = ∇ .(∇ V ) =2∇V (1.9)

where2∇ = ∂2

∂x2+ ∂2

∂y2+ ∂2

∂z2

∇ ×∇ ×A = ∇(∇ .A) −2∇A (1.10)

Using the ∇ operator, Maxwell’s four equations relate the electric field E volts per m (V/m) andmagnetic field H amperes per metre (A/m), as given in Equations (1.11 to 1.14):

Gauss’s electric law: ∇ .E = ρ

∈ (1.11)

Gauss’s law for magnetism: ∇ .H = 0 (1.12)

Maxwell–Faraday equation (induction law) : ∇ ×E = −μ∂H∂t

(1.13)

Ampere’s circuital law (with Maxwell’s correction) : ∇ ×H = σE+ ∈ ∂E∂t

(1.14)

where ρ is the charge density in coulombs per cubic metre (C/m3), ∈ is the permittivity in faradsper metre (F/m), μ is the permeability in henrys per metre (H/m) and σ is the conductivity ofthe medium in mho per metre or siemens per metre (S/m), which is assumed to be homogenous,

Radio Wave Fundamentals 3

isotropic and source-free. The permittivity and permeability of the medium are usually expressedrelative to vacuum as εr and μr and are given by:

∈ = εr εo, εo = 8.85 × 10−12 F/m (1.15a)

μ = μrμo, μ0 = 4π × 10−7 H/m (1.15b)

Note that, in general, the permittivity in Equation (1.15a) is complex but in many representationsthe imaginary part is not included. Maxwell’s equations can also be represented in terms of theelectric flux D in C/m2, magnetic flux B in tesla (V s/m2) and current density J in amperes persquare metre (A/m2) given by:

B = μH (1.16a)

D = εE (1.16b)

J = σE (1.16c)

If the medium is anisotropic then the medium properties become tensors. For example, the rela-tionship in Equation (1.16b) becomes:⎡

⎣Dx

Dy

Dz

⎤⎦ =

⎡⎣∈11 ∈12 ∈13

∈21 ∈22 ∈23∈31 ∈32 ∈33

⎤⎦

⎡⎣Ex

Ey

Ez

⎤⎦ (1.17)

1.2 Free Space Propagation

In transmission media where the electric and current charges in Equations (1.11 to 1.14) are zerothe solution of Maxwell’s equations gives the following relationships:

Gauss’s electric law: ∇ ·E = 0 (1.18)

Gauss’s law for magnetism: ∇ ·H = 0 (1.19)

Maxwell–Faraday equation (induction law) : ∇ ×E = −μ∂H∂t

(1.20)

Ampere’s circuital law (with Maxwell’s correction) : ∇ ×H =∈ ∂E∂t

(1.21)

Taking the curl of Equations (1.20) and (1.21), and using the identity in Equation (1.10), we obtainthe free space wave equations:

∇2E = μoεo

∂2E∂t2

(1.22a)

∇2H = μoεo

∂2H∂t2

(1.22b)

1.3 Uniform Plane Wave Propagation

In uniform plane wave propagation the electric and magnetic field lines are perpendicular to eachother and to the direction of propagation, as illustrated in Figure 1.1. This condition is satisfiedif the electric and current charges are zero and the electric and magnetic fields are of a singledimension.

4 Radio Propagation Measurement and Channel Modelling

X

Y

Z

E

H

S

Figure 1.1 Electric field and magnetic field and direction of propagation of a plane wave.

For example, for a uniform plane wave propagating in the x direction, the electric and magneticfield lines can be along the y axis and the z axis respectively and the curl of Equation (1.20) gives:

2∇E = μ ∈ ∂2E∂t2

= ∂2E∂x2

or∂2E y

∂x2= μ ∈ ∂2E y

∂t2(1.23)

A solution that satisfies Equation (1.23) is of the form:

Ey = Eocos(ωt − βx) (1.24)

where Eo is the amplitude of the wave, ω = 2πf is the angular frequency in radians per second(rad/s) and β = 2π /λ is the wave number where f is the frequency in Hz and λ is the wavelengthin m. Similarly, the magnetic field can be given by:

Hz = Ho cos(ωt − βx) (1.25)

where Ho is the amplitude of the magnetic field.The ratio of the electric field to the magnetic field is an impedance η given by:

η = Ey

Hz

=√

μ

∈ (1.26)

For a vacuum, Equation (1.26) becomes:

η = Ey

Hz

=√

μo

εo

= 377 � (1.27)

which is called the intrinsic impedance of free space.If we select a point along the wave satisfying the condition:

ωt − βx = constant (1.28)

then the phase velocity of the wave, vp is given by:

dx

dt= ω

β= f λ = 1√

μ ∈ = vp m/s (1.29)

In a vacuum the phase velocity is equal to the speed of light c = 3 × 108 m/s.The direction of propagation of a plane wave is determined from the Poynting vector S given by:

S = E × H W/m2 (1.30a)

Radio Wave Fundamentals 5

where the mean rate of flow of energy is the real part of S , that is:

Re (S ) = 1

2Re(E × H ∗), where the star (*) indicates complex conjugate (1.30b)

The direction of propagation is therefore obtained by turning E into H and proceeding as witha right-handed screw, as shown in Figure 1.1, where the dot inside the circle indicates the tip ofan arrow; that is the Poynting vector S is perpendicular to the plane of the electric and magneticfields and gives the magnitude and direction of the energy flow rate as given in Equations (1.30a)and (1.30b).

1.4 Propagation of Electromagnetic Waves in Isotropic and HomogeneousMedia

For propagation in matter we would need to solve Maxwell’s Equations (1.11 to 1.14), which caninclude the conduction current and the charge. We start by studying plane wave propagation indifferent media, which are both isotropic and homogeneous, since these two properties apply tomost gases, liquids and solids provided that the electric field is not too high. The combination ofthese two properties implies that the relative permittivity, relative permeability and conductivity areconstant and that the permittivity is a scalar. This results in both D and E having the same direction.

Here we will consider two media of propagation: dielectric material and conductors.For time-harmonic fields at x = 0 the electric and magnetic fields can be written as:

E (t) = E o cos(ωt + ϑE) (1.31a)

H (t) = H o cos(ωt + ϑH ) (1.31b)

Rewriting Equations (1.31a) and (1.31b) as a complex exponential and taking the derivative gives:

∂E (t)

∂t= jωE (1.32a)

∂H (t)

∂t= jωH (1.32b)

Equations (1.13) and (1.14) can now be expressed as:

∇ ×E = −jμωH (1.33)

∇ ×H = (σ + j ∈ ω)E = jω

(σ

jω+ ∈

)E =

(∈ −j

σ

ω

) ∂E∂t

(1.34)

Taking the curl of Equation (1.33) and using Equation (1.34) gives:

∇ ×E × E = ∇(∇ .E ) −2∇E = −jμω(∇ ×H ) = −jμω(σ + j ∈ ω)E (1.35)

If the charge is zero, then the divergence of E = 0 and Equation (1.35) reduces to:

2∇E = jμω(σ + j ∈ ω)E = γ 2E (1.36)

whereγ 2 = jμω(σ + j ∈ ω) = (α + jβ)2

6 Radio Propagation Measurement and Channel Modelling

Solving for α and β, we obtain:

α = ω

√√√√μ ∈2

(√1 + σ 2

(∈ ω)2 − 1

)(1.37a)

β = ω

√√√√με

2

(√1 + σ 2

(εω)2 + 1

)(1.37b)

A possible solution for Equation (1.36) gives the following form for the electric field:

Ey(x, t) = Re(Eoe−γ x+jωt ) (1.38)

Similarly, a solution for H is:

H z(x, t) = Re(H oe−γ x+jωt ) (1.39)

Using Equation (1.39) in Equation (1.33) and taking the curl of E in Equation (1.38) gives:

∇ ×E = −jμωH oe−γ x+jωt = −jμωH = −γ E (1.40)

The ratio of the electric field to the magnetic field is again the intrinsic impedance η given by:

EH

= jμω

γ= η =

√jμω

σ + j ∈ ω(1.41)

Using the definition of η in Equation (1.36) the electric field in Equation (1.38) can be expressed as:

E y(x, t) = e−αx Re(Eoe−jβx+jωt ) (1.42)

Equation (1.42) is the general equation for transverse propagation, which can be simplified forvarious special cases. In the equation α represents the attenuation factor, with units per metre (m−1)and indicates an exponential decay in the field strength with distance.

• Case 1: Free space propagationFree space propagation can be considered a special case of Equation (1.42) where σ = 0, andhence, from Equations (1.37a) and (1.37b), α = 0 and β = ω

√μoεo = ω

c= 2π

λwhich gives an

intrinsic impedance of 377 � as in Equation (1.27).• Case 2: Perfect dielectric

In this case both σ = 0, α = 0 and β = ω√

μoμrε oεr . This gives a phase velocity of the wavein the medium as:

vp = ω

β= f λ (1.43)

and intrinsic impedance:

η =√

μ

∈ (1.44)

• Case 3: Good dielectricFor this case, σ/εω � 1.