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

4

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.

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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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