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TRANSCRIPT
1
Methodology for Comparing Two Carrier Phase Tracking Techniques
Dina Reda Salem, Cillian O’Driscoll and Gérard Lachapelle
Position, Location And Navigation (PLAN) Group
Department of Geomatics Engineering
Schulich School of Engineering, University of Calgary
Email: dina.salem(at)ucalgary(dot)ca website: http://plan.geomatics.ucalgary.ca/
Received on 10 June 2010
Accepted on 22 April 2011
Abstract
The carrier phase tracking loop is the primary focus of the current work. In particular, two carrier phase
tracking techniques are compared, the standard phase tracking loop, i.e. the Phase Lock Loop (PLL) and the
Extended Kalman filter (EKF) tracking loop. In order to compare these two different techniques and taking
into consideration the different models adopted in each, it is important to bring them to one common ground.
In order to accomplish this, the equivalent PLL for a given EKF has to be determined in terms of steady-state
response to both thermal noise and signal dynamics. A novel method for experimentally calculating the
equivalent bandwidth of the EKF is presented and used to evaluate the performance of the equivalent PLL.
Results are shown for both the L1 and L5 signals. Even though the two loops are designed to track equivalent
dynamics and to have equivalent carrier phase standard deviations, the EKF outperforms the equivalent PLL
both in terms of the transient response and sensitivity.
Keywords
Phase lock loop; Kalman filter tracking; L1 signal; L5 signal
1. Introduction
GPS signal tracking has been a widely studied issue in the literature. It is critical to ensure accurate tracking
before attempting to use the navigation measurements. Several methods were previously introduced and
commonly used for tracking. The most common method, which is referred to in the paper as standard
tracking, was originally used for various communication systems. It models the incoming system as having
deterministic dynamics. Examples of these loops are Phase Lock Loop (PLL), frequency lock loop (FLL) and
delay lock loop (DLL). Another method, which is also widely used, is based on the Kalman filter theory.
"The final publication is available at www.springerlink.com".
2
Rather than assuming deterministic dynamics, the Kalman filter assumes that the signal dynamics follow a
linear stochastic model.
Even though both approaches have been widely used, few attempts were made to compare the performance
of both. Several works in the literature have attempted to find the steady-state model of the Kalman filter
(O’Driscoll and Lachapelle 2009 and Yi et al. 2009). Their idea is to obtain the steady-state Kalman gain and
derive the equivalent loop bandwidth. By getting the noise equivalent bandwidth, the equivalent PLL should
give the same phase jitter and steady-state responses of the Kalman filter.
However, there is no work in the literature, to the authors’ knowledge, that attempted to obtain the equivalent
of an EKF. This is most probably due to the nonlinearity introduced by the nature of the models used in the
EKF. In order to find the equivalent of the EKF, an experimental methodology is developed and presented in
the paper. The proposed methodology takes into account the different signal model assumptions and
parameters dealt with. The equivalent model is derived based on the 1-sigma carrier phase error estimates
obtained from each of the two models. This methodology is shown only for the separate tracking of a single
GPS signal to compare the two tracking techniques in general, which is the main contribution and focus of
the paper. Results are presented for both the L1 signal, to represent the conventional signal, and the L5
signal, as an example of the modern signals. The comparison could be expanded similarly for other signals.
Moreover, other cases, where collaboration between signals is considered, are discussed in the conclusions
section.
The paper begins with a discussion of the signals and systems model in section 2. Then the two carrier phase
tracking loops under consideration, namely the standard tracking loop and Kalman filter tracking, are
discussed in sections 3 and 4. Section 5 discusses the proposed comparison methodology with an applied
example showing also the experiment and simulations setup. Since the real data collected from SVN 49 does
not comply with the specifications of the IS-GPS-2006 as discussed in Erker et al. 2009, and to ensure a fair
comparison, simulated data had to be used for the two signals. The use of simulated data was essential to
eliminate any errors that might be introduced from the signal itself or from any inadequate parameters.
Section 6 shows the simulation results. Finally, the conclusions are given in section 7.
2. Signal and System Models
The signals under consideration are the L1 C/A signal and the L5 signal. The signal models are given as
follows (Mongrédien et al. 2006).
For the L1 signal:
( ) ( ) ( ) ( )1 1 1 1 1 1. . cos 2L L L L L LS t P D t C t f tπ ϕ= + (1)
3
For the L5 signal:
( )( ) ( ) ( )
( ) ( ) ( )5 10 5 5
5 5
20 5 5
( ). . .cos 2.
. .sin 2
L data L L
L L
pilot L L
D t C t NH t f tS t P
C t NH t f t
π ϕ
π ϕ
+=
+ + (2)
where PL1, PL5 are the total received power for the L1 and L5 signal, respectively, DL1, DL5 are the navigation
data bits for the L1 and L5 signal, respectively, CL1 is the L1 C/A code, Cdata, Cpilot are the Pseudo Random
Noise (PRN) codes for the data and pilot channels respectively of the L5 signal, NH10, NH20 are the 10 and 20
bit Neuman-Hoffman (NH) codes applied to the data and pilot channels respectively of the L5 signal, fL1, fL5
are the L1 and L5 carrier frequencies, respectively and 1Lϕ , 5Lϕ are the L1 and L5 carrier phases,
respectively.
The incoming signal passes through the receiver. The receiver starts by wiping off the carrier frequency by
multiplying the incoming signal with the locally generated carrier phase, the in-phase and 90o shifted. The
result is then multiplied by the locally generated code phase. The output is passed to integrate and dump
blocks, creating the correlator outputs, as shown in Fig. 1 (Van Dierendonck 1996). After the code,
secondary code and carrier wipe-off of the two signals, the resulting correlator outputs are given in (3) and
(4). Only the pilot channel of the L5 signal is used in the paper to simplify calculations without loss of
generality.
For the L1 signal
( )( )
1 1 1 1 1 1
1 1 1 1 1 1
. . . .cos( )
. . . .sin( )
L L L L L L
L L L L L L
IP A N D R
QP A N D R
δτ δϕ
δτ δϕ
=
= (3)
For the L5 pilot channel
( )( )
5 5 5 5 5
5 5 5 5 5
. . .cos( )
. . .sin( )
L L L L L
L L L L L
IP A N R
QP A N R
δτ δϕ
δτ δϕ
=
= (4)
and
( ) ( )1 551
1 5
1 5
sin sin,
2 2
L LLL
L L
L L
f T f TPPA A
f T f T
πδ πδ
πδ πδ= = (5)
where AL1, AL5 are the effective amplitude of the L1 and the L5 pilot signals, NL1, NL5 are the number of
accumulated samples for both the L1 and the L5 signals, ( ) ( )L1 L5R ,Rδτ δτ are the correlation of the
filtered incoming code with the local generated code for the L1 and the L5 signals, respectively, T is the
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coherent integration time, L1 L5,δτ δτ are the L1 and the L5 local code phase error in units of chips,
L1 L5f , fδ δ are the L1 and the L5 local carrier frequency error in units of rad/s and L1 L5,δϕ δϕ are the L1
and the L5 average local carrier phase error over the integration interval in units of rad.
The average phase error is expanded as (Psiaki and Jung 2002):
2
0 0 02 6
T Tfδϕ δϕ δ α= + + (6)
where 0α is the phase acceleration in units of rad/s2. The subscript zero indicates the value at the start of the
integration. The early and late correlators for the two signals have the same parameters, each with early-late
spacing set to one chip. The three correlator outputs, prompt, early and late, of both signals are then fed to the
tracking errors estimation block. The output of this block is the update for the numerically controlled
oscillators (NCO).
Fig. 1 Signal tracking loop
Carrier
NCO
Carrier
Disc. &
Loop
QP
Sin
-90
IP
Code
NCO
Integrate
& Dump
Integrate
& Dump
Integrate
& Dump
Integrate
& Dump
P E
L Code Disc.
& Loop
filter
Integrate
& Dump
Integrate
& Dump
IE
IL
QL
QE
Quadrature arm
In-phase arm
5
3. Standard Carrier Phase Tracking Loops
The standard tracking technique assumes that the signal dynamics tracked by the loop are of a deterministic
nature. The order of the tracking loop arises from the assumption of the type of dynamics from which the
signal is undergoing: constant phase, constant frequency or constant acceleration. This paper focuses only on
the carrier phase tracking loops.
Assuming that acquisition of the GPS signal has been achieved, the receiver switches to signal tracking.
These correlator outputs pass through the appropriate discriminator function and loop filter to generate the
NCO inputs. Finally, these NCO inputs are applied to the NCO to update the locally generated carrier
frequency. The carrier phase tracking loop is shown in Fig. 2.
Fig. 2 Standard carrier phase tracking loop
This same loop structure is used for tracking either the L1 or L5 signals, however for the L5 signal an extra
step is required after the code wipe-off to wipe-off the secondary NH codes. The loop parameters used are
shown in Table 1. The associated errors are illustrated as follows.
Table 1: Standard tracking loop parameters
Parameters L1 Signal PLL L5 Signal PLL
Order
Bandwidth
Integration Time
3
10 Hz
20 ms
3
10 Hz
20 ms
3.1. Phase tracking loop errors
The performance of the PLL is measured by the variance of the total phase jitter2
PLLσ . It is defined as
(Egziabher et al. 2003)
2 2 2
PLL t δφσ = σ + σ (7)
where PLL
σ is the 1-sigma phase jitter from all sources (in degrees), t
σ is the 1-sigma thermal noise (in
degrees) and δφσ is the 1-sigma input tracking error (in degrees).
GPS
Signal
Carrier NCO
Integrate
& Dump Carrier
Discriminator Loop filter
6
Often the thermal noise t
σ is treated as the only source of carrier tracking error. The input tracking
error δφσ , on the other hand, can be classified into correlated sources and the dynamic stress error as shown
below (Egziabher et al. 2003)
2
3
e,cδφ δφ
θσ = σ + (8)
2 2 2
,c v Aδφσ = σ + θ (9)
where ,cδφσ denotes the correlated sources of phase error,
eθ is the dynamic stress error,
vσ is the 1-sigma
vibration-induced oscillator phase noise and A
θ is the Allan deviation-induced oscillator jitter (in degrees).
The correlated sources include the vibration-induced oscillator noise and Allan deviation. These errors
constitute the total phase jitter, which can be written as (Ward et al. 2006)
( )2 2 2 3
ePLL t v A
θσ = σ + σ + θ + (10)
A rule of thumb is to maintain the total phase jitter below 15° for a data channel and below 30° for a pilot
channel to ensure reliable tracking (Ward et al. 2006). The following sections provide a brief review on each
of these error sources.
Thermal noise
The thermal noise jitter for an arctangent PLL is computed as (Ward et al. 2006)
( )0 0
180 11 degrees
2
nt
B
C N .T .C N
σ = +
π (11)
where Bn is the loop filter noise bandwidth and C/N0 is carrier-to-noise ratio. The carrier thermal noise error
is independent of the carrier frequency when it is expressed in degrees.
Vibration-induced oscillator phase noise
In some cases, the oscillator is installed in environments where it is subjected to mechanical vibrations. The
equation for the vibration-induced oscillator jitter is given in (Ward et al. 2006) as
7
( ) ( ) ( )degrees 2
360 max
min
2
2
∫=f
f
m
m
m
mv
L
v dff
fPfS
f
πσ (12)
where fL is the L-band input frequency in Hz, Sv(fm) is the oscillator vibration sensitivity of ∆f/fL per g as a
function of fm, fm is the random vibration modulation frequency in Hz, P(fm) is the power curve of random
vibration in g2/Hz as a function of fm and g is the gravitational acceleration ≈ 9.8 m/s
2.
Allan deviation oscillator phase noise
Receiver oscillator instabilities can cause errors in the phase as well as the code measurements, resulting in a
range of dynamics that should be tracked by the tracking loops. Thus, an adequate model for the oscillator
used in the receiver should be developed to account for these instabilities. The Allan deviation oscillator
phase jitter is usually used as a measure of the oscillator noise. It consists of three distinct segments, the
white frequency noise, flicker noise and integrated frequency noise.
For a third order PLL, the Allan deviation phase jitter is given by (Irsigler and Eissfeller 2002)
( )2
2 2 02 1
3 2
1802 degrees
3 63 3A carrier
L LL
hh hf − −
π πθ = π + + π ω ωω
(13)
where fcarrier is the carrier frequency, L
ω is the loop filter natural radian frequency, h0 is white frequency
noise, h-1 is flicker frequency noise and h-2 is random walk frequency noise.
Dynamic stress error
The dynamic stress characterizes the transient response of the PLL to a non-continuous input signal, e.g., a
step, acceleration or jerk input phase. It can be obtained from the following steady-state error, shown for a
third order loop filter sensitive to jerk stress (Ward et al. 2006):
3 3
30 4828
e
n
d R / dt.
Bθ = (14)
where d3R/dt
3 is the maximum line of sight jerk dynamics (
o/s
3). As shown in (14), the dynamic stress error
depends on the noise bandwidth. Increasing the bandwidth decreases the total dynamic stress error. The third-
order loop filter error model is chosen to accommodate the typical car dynamics. The reader is referred to
Ward et al. 2006 for further details on the dynamic stress error models.
8
4. Extended Kalman Filter-based Tracking Loops
An alternative architecture for signal tracking arises from the assumption that the signal dynamics follow a
linear stochastic model (O’Driscoll and Lachapelle 2009). The general Kalman filter (KF) structure used
consists of two models, namely the system dynamic model, which describes the dynamics of a continuous
time system and the measurement model, which includes the set of observations available to estimate the
states. The general Kalman filter structure has been used extensively in the literature (Petovello and
Lachapelle 2006 and Psiaki and Jung 2002).
Further to note, the KF used is an iterated EKF due to the nonlinear nature of the measurement models used,
which calls for linearization of the measurement model in each iteration. The details of these two steps can be
found in Salem et al. 2009. Fig. 3 shows the steps for tracking each of the L1 and the L5 signals. For the L1
signal, the incoming signal undergoes a carrier wipe-off followed by a code wipe-off. The output of this stage
is then applied to the Kalman filter to extract the tracking errors and use these to update the NCOs. The L5
signal undergoes the exact same steps, namely a carrier wipe-off followed by a code wipe-off. An extra step
is required to wipe-off the NH codes. Since the pilot L5 signal only is used, wipe-off of the NH20 codes is
required. Then the steps proceed similar to those for the L1 signal.
Fig. 3 Extended Kalman filter tracking
QPL5
IPL5
Kal
man
Fil
ter
Tra
ckin
g
Up
dat
e N
CO
s
Car
rier
NC
O
I&D
SL5
Co
de
NC
O
NH
20
cod
e
I&D
Co
de
NC
O
Up
dat
e N
CO
s
Car
rier
NC
O
I&D
QPL1
SL1
I&D
Kal
man
Fil
ter
Tra
ckin
g
IPL1
9
4.1. Dynamic model
The states to be estimated in single signal tracking are the amplitude of the signal, the code phase error, the
carrier phase error, the frequency error and the carrier acceleration error. The amplitude is modelled as a
random walk and its process noise is expected to absorb the signal level variations (Psiaki and Jung 2002).
The code phase error is estimated from the carrier frequency error, with two process noise components,
1wτ and
011β wφ . The random walk component
1wτ accounts for any ionospheric error divergence and
multipath. The carrier frequency and phase process noises account for the oscillator jitter effects. The carrier
acceleration process noise accounts for the remaining signal dynamics.
The system dynamic model can be written as
( ) ( ) ( ) ( ) ( )x t F t x t G t w t= +� (15)
where x is the set of states of the dynamic system, F(t) is the coefficient matrix describing the dynamics of
the system, G(t) is shaping matrix for the white noise input and w is the random forcing function, zero-mean
additive white Gaussian noise.
The states to be estimated and the corresponding state space equations can be written as
( )
( )1 1 1 01 01 01
5 5 5 05 05 05
T
L
T
L
x A f
x A f
=
=
δτ δφ δ α
δτ δφ δ α (16)
1
0
0
0
0 0
0 0
0 0
0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 1 0 0
0 0 0 0 1 0 0 0 1 0
0 0 0 0 0 0 0 0 0 1
s
s s s
s ss
s ss
As s
s ss s
f
wA A
wβ βd wdt
f f w
w
τ
φ
α
δτ δτ
δφ δφ
δ δ
α α
= +
(17)
where β converts units of radians into units of chips for the subscripted signal s. The remaining terms were
described in (3) to (5).
4.2. Measurement model
The measurement model includes the set of observations z available to estimate the states x. The state vector
x is known to relate to the observation vector z as
10
k k k kz H x v= + (18)
where zk is the measurement vector, Hk is linearized design matrix and vk is the measurement noise vector.
For each signal, the observations are formed from the six correlator outputs available, namely the in-phase
and quadra-phase prompt, early and late (IP, QP, IE, QE, IL, QL) correlators. These correlators are then used
to estimate the state parameters shown in (16).
The z vector can thus be written as
( )
( )1 1 1 1 1 1 1
5 5 5 5 5 5 5
T
L L L L L L L
T
L L L L L L L
z IP IE IL QP QE QL
z IP IE IL QP QE QL
=
= (19)
The details of the measurement model can be found in Salem (2010).
5. Standard Tracking Loops versus Kalman Filter
Tracking Loop
The main objective of this paper is to compare the two tracking methods, namely the Kalman filter and the
standard tracking loop. Focusing on the carrier phase tracking, two approaches might be considered, the first
is to find the equivalent PLL that gives the same performance as the steady-state EKF, and the second is to
find an equivalent EKF that gives the same performance as the PLL. The second approach requires a prior
knowledge of the EKF bandwidth. For the EKF it is hard to find the bandwidth due to the nonlinearity
introduced by the models used. Thus, the adopted approach is to find the equivalent PLL that gives the same
performance as the steady-state EKF. The equivalent model has to take into account the different signal
model assumptions and parameters dealt with.
The equivalent model is thus derived based on the 1-sigma carrier phase error estimates obtained from each
of the two models. Note that the 1-sigma carrier phase error for the standard tracking loop is the one
expressed in (7), which includes both the thermal error and the steady-state error. For the EKF tracking loop,
the 1-sigma carrier phase error estimate is obtained from the appropriate element of the estimated state
covariance matrix.
For the tracking, the comparison is directly performed for each of the two signals between the 1-sigma carrier
phase errors. The steps of calculating the EKF equivalent bandwidth are illustrated with an applied example
of a strong signal and a static receiver. First the proposed comparison methodology is explained, followed by
the experiment setup and the data analysis criteria, and finally an applied example is shown for illustration.
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5.1. Equivalent bandwidth calculation steps
As was discussed in the introduction, it is required to find an equivalent PLL for the Kalman filter. For the
standard tracking loops, the 1-sigma carrier phase error includes the four error sources discussed in section
3.1. The vibration-induced oscillator phase noise is ignored in the simulations; however, it can easily be
incorporated in the proposed technique. The remaining three sources are:
• thermal noise, which characterizes the PLL and is a function of the bandwidth and C/N0.
• oscillator phase noise, which can be set using (13).
• dynamic stress error, which can be set using (14).
For the EKF, the 1-sigma carrier phase error includes:
• thermal noise, entering the filter through the measurements as shown in (18).
• oscillator noise, which is set using the carrier phase and frequency noise spectral densities shown in
Table 2.
• dynamics experienced by the signal, which can be set using the LOS acceleration spectral density shown
in Table 2.
Thus, when comparing to the EKF, the equivalent PLL has to be able to:
1. track the same dynamics.
2. have the same oscillator noise.
3. have the same thermal noise, configured by the bandwidth.
The first two requirements, the oscillator noise and equivalent dynamics, can be set in the parameters of both
loops using (13), (14) and Table 2. The third requirement, the equivalent bandwidth, has to be calculated.
Two methods could be suggested, namely an analytical method, which is complicated and will call for
several approximations for the extended Kalman filter used and an experimental method, which is chosen
and proposed in this paper.
Table 2: Kalman filter parameters
Extended Kalman Filter
L1 Signal L5 Signal
Process noise Spectral density Process noise Spectral density
Amplitude AL1 dB/s/√Hz 1 AL5 dB/s/√Hz 1
Code phase δτL1 m/s/√Hz 0. 1 δτL5 m/s/√Hz 0. 1
carrier phase δφL1 cycles/s/√Hz (2πfs).√(h0/2) δφL5 cycles/s/√Hz (2πfs).√(h0/2)
carrier
frequency δfL1 Hz/s/√Hz (2πfs).√( (2π
2h-2) δfL5 Hz/s/√Hz (2πfs).√( (2π
2h-2)
LOS
acceleration αL1 m/s
3/√Hz 5 αL5 m/s
3/√Hz 5
12
Summing up, if the dynamics and oscillator noise are configured in the two loops, the remaining effort
should be focused on finding the equivalent bandwidth. Fig. 4 shows the steps required to calculate the EKF
equivalent bandwidth. First, set the process noise parameters in the EKF, i.e. oscillator and LOS acceleration
spectral density then run the EKF and calculate the average of the estimated carrier phase standard deviation
of both the L1 and L5 signals. These should be the target standard deviations of the PLL. Next, using (10),
set the dynamic stress in the PLL to be the same value as the LOS acceleration spectral density and choose
the bandwidth that yields the same standard deviation as the target standard deviation calculated in the first
step. To test the procedure, the EKF and its equivalent PLL are used to process the same data. The standard
deviation from the EKF is extracted from the estimated state covariance matrix, while the PLL standard
deviation is calculated using (10). In order to explain these steps, a simplified running example is shown
below for a strong signal and a static user.
Fig. 4 EKF equivalent bandwidth calculation
5.2. Experiment setup
In order to test and compare the two methods, a controllable environment is required to enable the simulation
of different environments. Although an L5 signal is now available, since a demo payload has been launched
on SVN 49 in March 2009, the signal does not comply with the specifications of the IS-GPS-2006 as
discussed in Erker et al. 2009.
In order to obtain correct performance measures using the original L5 signal specifications, a GSS7700
Spirent GPS signal simulator (Spirent 2006) has been used. It has also the advantage of being able to
simulate different environments for the L1 and the L5 signals. The RF output of the GSS7700 simulator is
then passed to a National Instruments front-end (NI PXI-5661 2006) which logs raw IF samples for each of
Set acceleration PSD in EKF Calculate Mean of 1-sigma
Carrier phase error (Mσ,kf)
Set Dynamic stress in PLL Set target standard
deviation = Mσ,kf
Calculate
equivalent
bandwidth
Calculate actual 1-sigma
error using equivalent
bandwidth
Compare results of EKF and
PLL
13
the two signals, which are subsequently processed by the software receiver. Fig. 5 describes the experiment
setup.
Fig. 5 Experiment Setup
Data analysis criteria
In order to evaluate the signal carrier phase tracking performance, the phase lock indicator is used. It is
implemented to determine carrier phase lock and it inherently contains the code lock information. The phase
lock can be detected using the normalized estimate of the cosine of twice the carrier phase. The phase lock
indicator (PLI) can be written as (Van Dierendonck 1996)
( )2PLI cos≈ δφ (20)
The values of the lock indicator will range from -1, where the locally generated signal is completely out of
phase with the incoming signal and 1 that indicates perfect match.
5.3. EKF versus Standard Tracking Loops
For illustrating the methodology used, the equivalent bandwidth of the EKF is calculated first for a strong
signal. A strong signal for a static user is simulated, with an L1 C/N0 of 42 dB-Hz.
First, set the acceleration PSD in the EKF to 0.01 g/s/√Hz and calculate the average value of the estimated
phase standard deviation of both the L1 and L5 signals. Note that the estimated phase standard deviation is
extracted from the appropriate element of the estimated state covariance matrix.
Second, set the oscillator parameters to be the same as the EKF and the dynamic stress of the PLL to the
same value of the acceleration PSD, 0.01 g/s. It is found by experiment that setting the dynamic stress in the
PLL and the acceleration PSD in the EKF results in the same response to a certain level of dynamics. The
PLL tracking error is plotted as a function of the loop bandwidth using (10). The next step is to use the
calculated mean of the L1 and L5 1-sigma carrier phase error from step 1 (i.e. those of the EKF) as the target
standard deviations for the PLL as shown in Fig. 6. Using these targets, the equivalent bandwidth for each
EKF can be found, as marked in the figure.
GSS7700 Simulator NI Front-end Software Receiver
RF Samples IF Samples
14
Fig. 6 PLL tracking errors for L1 and L5 signals: strong signal
Finally, using the equivalent bandwidth in the standard PLL, the actual 1-sigma phase error is calculated.
Fig. 7 shows the results from both the EKF and standard PLL, where it is observed that the two tracking
loops give the same standard deviations for each signal separately. For the EKF, the estimated standard
deviations are obtained from the appropriate element of the state covariance matrix from the Kalman filter,
whereas the PLL standard deviations are the theoretical standard deviations given in (7) with the C/N0
estimated and averaged over one second period using the software receiver. The smoothing of the C/N0
estimate is what leads to smoother estimates of the phase error standard deviation for the standard tracking
case, when compared to the EKF. To that end, the equivalent PLL for each Kalman filter has been calculated.
Fig. 7 Carrier phase error: strong signal
15
The tracking performance of the EKF and the standard PLL can now be compared in a fairer manner as
shown in the following figures. Fig. 8 shows the L1 PLI and Doppler frequencies calculated using each
method. It shows an important advantage of the EKF, namely the faster transient response noticed in faster
adaptation to the change of bandwidth when switching from the FLL, compared to the PLL. The faster
response is significant and is more apparent in the scenario where motion is simulated as shown later.
Moreover, once the two loops settle, they both have similar PLI. Similar results for the L5 pilot channel are
shown in Fig. 9.
Fig. 8 L1 PLI: strong signal
Fig. 9 L5 PLI: strong signal
6. Results
The method used to calculate the equivalent bandwidth has been verified using a strong signal for a static
user. However, the difference between the Kalman filter tracking loops and the standard PLL tracking loops
is of interest in real scenarios, e.g., dynamic vehicle, weak signals, etc. This section shows a comparison
between the EKF and the standard tracking loops in two scenarios, the first is a car experiencing high
16
accelerations and abrupt turns and the second is for a static user that suffers from a continuous decrease in its
received power levels.
6.1. Motion
The scenario used here simulates a dynamic vehicle, moving with varying speed. The scenario uses the
following parameters:
• Dynamic vehicle, with a velocity ranging between 25 km/hr and 100 km/hr, moving in the rectangular
path shown in Fig. 10 and accelerating within a distance of 100 m.
• Moderate signal power levels (The received L1 C/N0 is 36 dB-Hz).
Fig. 10 Dynamic vehicle model
Following the same steps, Fig. 11 shows the equivalent bandwidth, followed by the 1-sigma carrier phase
errors in Fig. 12.
Fig. 11 PLL tracking errors for L1 and L5 signals: moving vehicle test
20 m
vmin = 25 km/hr
vmax = 100 km/hr
acc. distance = 100 m Start Point
700 m
30
0 m
17
Fig. 12 Estimated Phase Standard Deviations for Standard and Kalman Filter Tracking: moving
vehicle test
Fig. 13 L1 PLI and Doppler frequency: moving vehicle test
In the static user case, a strong agreement has been observed between the Kalman filter tracking loop and the
equivalent PLL; however, when motion is introduced, the equivalent PLL shows deviation from the Kalman
filter. The Kalman filter adapts its gains according to changes in operating point and C/N0, which points out
its second advantage, whereas the PLL assumes a deterministic signal with pre-known dynamics and uses a
fixed bandwidth which results in a slower transient response. The L1 PLI illustrates these results as shown in
Fig. 13. The standard PLL shows lower PLI in instances where the vehicle changes its speed and that is
18
reflected in the Doppler frequency shown in the same figure. The L5 PLI shows the same results as the L1 as
shown in Fig. 14.
Fig. 14 L5 PLI and Doppler frequency: moving vehicle test
6.2. Sensitivity analysis
This scenario simulates a signal for a static user that suffers from a continuous decrease in its power levels by
a rate of 0.5 dB per second. Fig. 15 shows the C/N0 calculated from the software receiver.
Fig. 15 Carrier-to-Noise ratio-sensitivity analysis
Fig. 16 shows the equivalent bandwidth calculation. It is interesting to note that these calculations are based
on the mean of the 1-sigma carrier phase error of the Kalman filter, which is continuously increasing due to
the decrease of the C/N0 as shown in Fig. 17. That is why the PLL shows more error at the start of the
19
scenario and less at the end; it is on the average the same as that of the Kalman filter. This figure also
emphasises the second advantage of the EKF: it adapts its bandwidth to changes in scenario (most
particularly changes in C/N0) whereas the standard tracking is a fixed bandwidth approach.
Fig. 16 PLL tracking errors for L1 and L5 signals: sensitivity analysis
Fig. 17 Phase error: sensitivity analysis
Fig. 18 shows the 1-sigma carrier phase error for the both signals using the equivalent PLL and Kalman filter
tracking. For the L1 signal at high to moderate values of C/N0, the two methods yield almost the same error.
However, they start to deviate at lower C/N0. The PLL crossed the 15o rule of thumb threshold at a C/N0
value of 19 dB-Hz. The Kalman filter does not cross the threshold even at low C/N0 values.
Similarly, for the L5 signal, at high to moderate values of C/N0, the two methods also yield almost the same
error. The PLL starts to deviate from the Kalman filter at lower C/N0 values and crossed the 30o rule of
thumb threshold at a C/N0 value of 12 dB-Hz. The Kalman filter, similar to the L1 signal, does not cross the
threshold even at low C/N0 values. Note that these improvements in tracking errors translate directly to
20
improvements in the position error. However, testing and analysis for such improvements are left for a future
work.
Fig. 18 Estimate phase error standard deviation versus carrier-to-noise ratio
7. Conclusions
A methodology to compare two carrier phase tracking approaches has been developed and presented. It is
based on experimentally finding the equivalent steady-state bandwidth of the EKF and plugging it into the
standard PLL. Having the two tracking approaches come to an equivalent level, a fair comparison can be
made. The methodology was applied for separate tracking of the L1 and L5 signals. The Kalman filter shows
an improvement over the standard PLL tracking loops in two main aspects:
1) The transient response of the tracking loop, which is noticed in faster adaptation to the change of
bandwidth when the tracking loop was switching from the FLL to the Kalman filter-based tracking as
compared to the slower adaptation when switching to PLL.
2) The improved sensitivity of the tracking loop, which was noticed when standard tracking loops crossed the
15o and 30
o rule of thumb tracking thresholds, contrary to the Kalman filter which did not cross these
thresholds even at low C/N0 values (down to 10 dB-Hz).
The results shown in the paper are for the separate tracking loops, i.e. no collaboration between the two
signals was assumed. These results can be further expanded to compare the aided standard tracking loops
(presented in Qaisar 2009 and Salem 2010) and the combined Kalman filter-based tracking loops (proposed
in Salem et al. 2009). The reader is referred to Salem (2010) for comparison results.
21
Acknowledgments
The financial support of General Motors of Canada, the Natural Science and Engineering Research Council
of Canada, Alberta Advanced Education and Technology and the Western Economic Diversification Canada
is acknowledged.
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