i

i

Summary

Global Navigation Satellite Systems (GNSS) provide user position and velocity by processing

signals transmitted by the satellites of one or more constellations. The quality of GNSS

signals can potentially be degraded by interfering signals. Hence, many receivers are

equipped with interference mitigation units to reduce the impact of interfering signals.

However, interference mitigation techniques can degrade the performance of GNSS receivers.

The impact of a particular interference mitigation technique, i.e. the notch filter, is analyzed in

this project. Notch filters are linear devices that place a narrow notch in correspondence of a

specific frequency in the bands recovered by the receiver front-end. For this reason, they are

particularly indicated for the removal of Continuous Wave (CW) interference.

Notch filters can introduce a bias in the code delay measurements. This bias results in errors

in the final position solution. Thus, it is necessary to analyze this bias in order to predict and

possibly correct these errors.

The first objective of this project was to provide a theoretical analysis of the bias on the code

delay measurements due to notch filters and to validate the obtained results by simulations

and real data experiments. The following results are obtained. The bias due to Infinite Impulse

Response (IIR) notch filters depends on the Doppler frequency of the received signal, and

thus cannot be completely removed by the navigation solution. The bias due to Finite Impulse

Response (FIR) notch filters with linear phase is frequency independent and is removed by

the navigation solution.

The second objective was to compare the position errors due to the IIR notch filters and the

FIR notch filters with linear phase in the presence of noise and front-end filtering using real

data experiments. Real data were collected in open sky conditions and processed using the

University of Calgary Software Receiver, GSNRxTM. The following results were found. IIR

and FIR notch filters introduce mean position errors equal to few tens of centimeters. The

position estimate with linear phase FIR notch filters seems to be more accurate than the one

ii

with IIR notch filters. In view of this result, it is recommended to use complex FIR notch

filters with linear phase.

iii

Résumé Les systèmes de navigation par satellites (Global Navigation Satellite Systems - GNSS)

permettent d’estimer la position et la vitesse d’un utilisateur en traitant les signaux transmis

par les satellites d’une ou plusieurs constellations. La qualité des signaux GNSS peut être

potentiellement dégradée par des signaux interférents. Ainsi, beaucoup de récepteurs sont

équipés d’unités de mitigation d’interférences permettant de réduire l’impact des signaux

interférents. Toutefois, les techniques de mitigation d’interférences peuvent dégrader les

performances des récepteurs GNSS. Ce projet a pour but d’analyser l’impact d’une technique

de mitigation d’interférence particulière, c'est-à-dire le filtre notch. Les filtres notch sont des

systèmes linéaires plaçant une encoche étroite à une fréquence spécifique de la bande traitée

par le filtre frontal du récepteur. Pour cette raison, ils sont particulièrement indiqués dans la

suppression d’interférences caractérisées dans le domaine temporelle par un signal continu.

Les filtres notch peuvent introduire un biais dans les mesures des retards de code. Ce biais

engendre des erreurs dans l’estimation finale de la position. Il est donc nécessaire d’analyser

ce biais afin de prédire et de corriger ces erreurs.

Le premier objectif de ce projet était d’apporter une analyse théorique du biais sur les mesures

des retards de code dû aux filtres notch et de valider les résultats obtenus par simulations et

par des expériences basées sur des données réelles. Les résultats obtenus sont les suivants. Le

biais dû aux filtres notch à Réponse Impulsionnelle Infinie (RII) dépend de la fréquence

Doppler du signal reçu, et ne peut donc pas être totalement supprimé par la solution de

navigation. Le biais dû aux filtres notch à Réponse Impulsionnelle Finie (RIF) et à phase

linéaire est indépendant de la fréquence Doppler et est supprimé par la solution de navigation.

Le second objectif était de comparer les erreurs de position dues aux filtres notch à RII et aux

filtres notch à RIF et à phase linéaire en présence de bruit et de filtrage frontal et en utilisant

des expériences basées sur des données réelles. Les données réelles ont été collectées à ciel

ouvert et traitées par le récepteur GNSS interne au Position, Location and Navigation (PLAN)

Group, GSNRxTM. Les résultats obtenus sont les suivants. Les filtres notch à RII et à RIF

iv

introduisent des erreurs de position de moyenne égale à quelques dizaines de centimètres.

L’estimation finale de la position en présence des filtres notch à RIF et à phase linéaire

semble plus précise que celle en présence des filtres notch à RII. De part ces résultats, il est

recommandé d’utiliser les filtres à notch à RIF et à phase linéaire.

v

Acknowledgements I would like to thank:

- Prof. Gérard Lachapelle who has welcomed me in the PLAN Group and has provided

me the financial resources I needed to work in the laboratory,

- Dr. Daniele Borio for the constant support and the technical advices during the

internship,

- Nicola and Martin for the technical discussions,

- all the members of the PLAN group for the friendly atmosphere in the laboratory, and

particularly Leila, Melania, Anshu, Daniele, Peng, Da Wang, Martin, Cyril, Nicola,

Pierre, Florian, Salvatore, Antonio, Vahid…

vi

List of figures

Figure 2-1: GNSS receiver architecture ................................................................................... 10

Figure 2-2: GPS and GALILEO frequency allocation............................................................. 11

Figure 2-3: Modulated signal structure for a DSSS signal....................................................... 13

Figure 2-4: Down-converter operations ................................................................................... 15

Figure 2-5: Normalized auto-correlation function of a GPS C/A signal.................................. 18

Figure 2-6 : Normalized auto-correlation function of a GPS C/A signal close to the main peak

.................................................................................................................................................. 18

Figure 2-7: Normalized cross-correlation function of two GPS C/A signals........................... 20

Figure 2-8: PSD of the GPS C/A signal ................................................................................... 21

Figure 2-9 : Normalized power spectral density of a CW signal and GPS L1 C/A signal ...... 23

Figure 2-10: General structure of a DLL.................................................................................. 25

Figure 2-11: PSD of the GPS-L1 C/A signal and of the noise component before de-spreading

.................................................................................................................................................. 28

Figure 2-12: PSD of the GPS-L1 C/A signal and of the noise component after de-spreading 28

Figure 2-13: Representation of early, prompt and late correlator outputs on a ideal auto-

correlation function .................................................................................................................. 31

Figure 3-1: Scheme of a GNSS receiver including interference detection and mitigation units

.................................................................................................................................................. 38

Figure 3-2: Amplitude response of a notch filter ..................................................................... 39

Figure 3-3: amplitude response of a real IIR notch filter ......................................................... 41

Figure 3-4: Phase of a real IIR notch filter............................................................................... 42

Figure 3-5: Amplitude response of a complex IIR notch filter ................................................ 43

Figure 3-6: Phase of a complex IIR notch filter....................................................................... 44

Figure 3-7: Example of notch filter impulse response symmetric with respect to for N = 6

.................................................................................................................................................. 46

vii

Figure 3-8: Example of coefficients corresponding to the notch filter

impulse response coefficients shown in Figure 3-7 ................................................................. 47

Figure 3-9: Amplitude response of the low pass prototype for a 0.5dB cut-off frequency

corresponding to , and for N=500 ............................................................................... 51

Figure 3-10: Amplitude of the transfer function of a real FIR notch filter .............................. 52

Figure 3-11: Phase of the transfer function of a real FIR notch filter ...................................... 52

Figure 3-12: Amplitude of the transfer function of the complex low-pass prototype for a

0.5dB cut-off frequency corresponding to , and for N=500........................................ 53

Figure 3-13: Amplitude response of a complex FIR notch filter designed with the low-pass

prototype method...................................................................................................................... 55

Figure 3-14: Phase of a complex FIR notch filter designed with the low-pass prototype

method...................................................................................................................................... 55

Figure 3-15: Amplitude of the transfer function of a complex FIR notch filter designed using

the method based on the IIR notch filter .................................................................................. 57

Figure 3-16: Phase of a complex FIR notch filter designed using the method based on the IIR

notch filter ................................................................................................................................ 58

Figure 3-17: Correlation function between the incoming GPS L1 signal (PRN 29) and its local

replica ....................................................................................................................................... 60

Figure 3-18: The correlation performed between the incoming signal and the local PRN code

replica in the presence of notch filter ....................................................................................... 61

Figure 3-19: Representation of the normalized correlation functions in the absence and

presence of IIR notch filter....................................................................................................... 67

Figure 4-1: Normalized correlation functions in the absence and presence of IIR notch filter75

Figure 4-2: Approximation of the correlation function in the absence of notch filter by a third

order Taylor expansion............................................................................................................. 76

Figure 4-3: Approximation of the correlation function in the presence of the complex IIR

notch filter by a third order Taylor expansion.......................................................................... 76

Figure 4-4: Approximation of the correlation function in the absence of notch filter by a 9-th

order polynomial ...................................................................................................................... 78

viii

Figure 4-5: Approximation of the correlation function in the presence of the complex IIR

notch filter by a 9-th order polynomial .................................................................................... 78

Figure 4-6: Correlation functions in the absence and presence of the real notch filter with

linear phase designed with the windowed Fourier series approach (real data) ........................ 83

Figure 4-7: Correlation functions in the absence and presence of the real notch filter with

linear phase designed with the the series expansion approach (real data) ............................... 83

Figure 4-8: Correlation functions in the absence and presence of the real notch filter with

linear phase designed with the windowed Fourier series approach (simulation)..................... 86

Figure 4-9: Correlation functions in the absence and presence of the real notch filter with

linear phase designed with the series expansion approach (simulation) .................................. 86

Figure 4-10: Correlation functions in the absence and presence of the complex notch filter

with linear phase designed with the series expansion approach (simulation).......................... 87

Figure 4-11: Block diagram of the experimental setup............................................................ 88

Figure 4-12: Amplitude response of the complex FIR and IIR notch filters (coefficients

quantized over 11 bits) ............................................................................................................. 90

Figure 4-13: for PRN 11 in the absence and presence of the IIR notch filter ............... 91

Figure 4-14: for PRN 11 in the absence and presence of the FIR notch filter .............. 92

Figure 4-15: Pseudo-range error due to the presence of the IIR notch filter an of the FIR notch

filter for PRN 11 and 32 ........................................................................................................... 94

Figure 4-16: Position error due to the IIR notch filter and the FIR notch filter (east direction)

.................................................................................................................................................. 97

Figure 4-17: Position error due to the IIR notch filter and the FIR notch filter (north direction)

.................................................................................................................................................. 97

Figure 4-18: Position error due to the IIR notch filter and the FIR notch filter (up direction) 98

Figure 4-19: Clock bias error due to the IIR notch filter and the FIR notch filter ................... 98

ix

List of tables

Table 2-1: Standard DLL discriminators.................................................................................. 32

Table 2-2: Analytical tracking jitter expression for standard DLL discriminators .................. 35

Table 4-1: Bias on the code delay estimate due to the amplitude distortion of the complex IIR

notch filter ................................................................................................................................ 76

Table 4-2: Bias on the code delay estimate due to the amplitude distortion of IIR notch filters

.................................................................................................................................................. 79

Table 4-3: Additional delay on the correlation peak due to the real and complex FIR notch

filters with linear phase designed with the series expansion approach .................................... 84

Table 4-4: Mean for different PRNs in the presence of the IIR and the FIR notch filter

for different PRNs .................................................................................................................... 92

Table 4-5: Mean pseudo-range errors due to the FIR notch filter and the IIR notch filter for

different PRNs.......................................................................................................................... 95

x

Acronyms ACF Auto-Correlation Function

C/A code Coarse/Acquisition code

CCF Cross-Correlation Function

CCIT Calgary Centre for Innovative Technologies

CDMA Code Division Multiple Access

Carrier-to-Noise density power ratio

CW Continuous Wave

CWI Continuous Wave Interference

DFT Digital Fourier Transform

DSSS Direct Sequence Spread Spectrum

DLL Delay Locked Loop

FFT Fast Fourier Transform

FIR Finite Impulse Response

FM Frequency Modulation

FT Fourier Transform

GIS Geospatial Information Systems

GNSS Global Navigation Satellite Systems

GPS Global Positioning System

IF Intermediate Frequency

IFT Inverse Fourier Transform

IIR Infinite Impulse Response

xi

LMS Least Mean Square

LNA Low Noise Amplifier

NCO Numerically Controlled Oscillator

P code Precision code

PED Personal Electronic Devices

PLAN Position, Location, Navigation

PLL Phase Locked Loop

PRN Pseudo Random Noise

PSD Power Spectral Density

RF Radio Frequency

UHF Ultra High Frequency

VHF Very High Frequency

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Table of contents

Preface .......................................................................................................................................... 1

1. Introduction........................................................................................................................... 4

1.1. Interference Sources and Interference Mitigation Techniques...................................... 4

1.2. Objectives and Organization of the Thesis.................................................................... 7

2. Signals and system................................................................................................................ 8

2.1. GNSS signals ................................................................................................................. 9

2.1.1. GNSS signals structure........................................................................................... 9

2.1.2. Temporal and spectral characteristics of GNSS signals....................................... 16

2.2. Temporal and spectral characteristics of interfering signals ....................................... 22

2.3. In phase/quadrature sampling ...................................................................................... 24

2.4. Code tracking loop summary....................................................................................... 24

3. Impact of notch filters on a GNSS receiver ........................................................................ 36

3.1. Functional description and implementation of notch filters ........................................ 36

3.1.1. IIR notch filters .................................................................................................... 40

3.1.2. Linear phase FIR notch filters .............................................................................. 44

3.2. Theoretical analysis of the impact of notch filters on the correlation function ........... 59

3.2.1. IIR notch filters .................................................................................................... 66

3.2.2. Linear phase FIR notch filters .............................................................................. 69

3.3. Theoretical analysis of the impact of notch filters on tracking jitter........................... 71

4. Simulation and real data analysis........................................................................................ 73

4.1. Theoretical results validation....................................................................................... 73

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4.1.1. Impact of IIR notch filter on the correlation function .......................................... 73

4.1.2. Impact of linear phase FIR notch filter on the correlation function..................... 80

4.2. Real data analysis ........................................................................................................ 87

4.2.1. Experimental setup ............................................................................................... 88

4.2.2. Experimental results and analysis ........................................................................ 90

5. Conclusions....................................................................................................................... 101

Appendix .................................................................................................................................. 104

A.1. Symmetric amplitude distortion due to real FIR notch filters with linear phase.......... 104

A.2. Symmetric amplitude distortion due to complex FIR notch filters with linear phase .. 108

References ................................................................................................................................ 110

1

Preface

This section aims at describing the organization of the University of Calgary (Alberta,

Canada), the Schulich School of Engineering, the Geomatics department and the Position,

Location and Navigation (PLAN) Group.

University of Calgary [11]

The University of Calgary was created in 1966. The University is a member of the 13 most

research intensive universities in Canada. It hosts 17 faculties, more than 60 departments and

more than 30 research institutes and centers. The University has graduated 135.000 students

over its 44-year history and 29.000 students are currently enrolled in the undergraduate,

graduate and professional degree programs. The University has 2761 academic staff engaged

in research and teaching and more than 3.000 staff. It is one of the 4 largest employers in

Calgary.

Schulich School of Engineering [11]

The Schulich School of Engineering is one of the 17 faculties of the University of Calgary. It

offers 9 academics programs: Chemical, Civil, Computer, Electrical, Geomatics,

Manufacturing, Mechanical, Oil and Gas and Software Engineering. The 9 departments are

accredited by the Canadian Council of Professional Engineers.

Geomatics Department [8]

The Geomatics Department belongs to the Schulich School of Engineering. It deals with the

acquisition, modeling, analysis and management of spatial data and includes applications,

such as positioning by satellites. The Geomatics Department is broken down into 4 areas:

2

- Geospatial Information Systems (GIS) and land tenure. Land tenure is the manner

in which people live on the land, how they relate to their environment. GIS are software

to manage land and natural resources,

- Earth observation is an interdisciplinary research filed aiming at solving science

and engineering questions, such as climate change, natural hazards and evolution of the

Earth’s oceans and land surface,

- Digital imaging is the manipulation and the interpretation of the digital images

from a wide variety of the sensors onboard terrestrial, airborne and space-borne

platforms. These procedures are used for a variety of applications, such as change

detection,

- Positioning, navigation and wireless location deals with the ability to locate a user

on the earth and navigate over its surface. It covers applications, such as airborne

positioning and navigation. 5 areas are studied:

- GNSS,

- Inertial navigation systems,

- Multi-sensors systems,

- Wireless location,

- Atmosphere remote sensing.

PLAN Group [3]

The PLAN Group is a research centre belonging to the Geomatics Department. It is leaded by

Professors G. Lachapelle and E. Cannon. The PLAN Group has 7 professors, 6 research

engineers and about 40 Master students, PhD and visiting students. It is dedicated to the

research, development and improvement of positioning, navigation and wireless location

technologies. It covers most of the issues related to:

- Sensor augmentation. It includes integration of GNSS with inertial and other sensors,

- The applications, such as indoor, pedestrian and vehicular navigation,

- GNSS signal processing, such as signal acquisition and tracking algorithm

development.

3

This project on the impact of interference mitigation techniques on a GNSS receiver with a

specific focus on signal tracking belongs to the last research area listed above. This thesis has

been supervised by Dr. D. Borio and Professor G. Lachapelle.

4

1. Introduction

The goal of GNSS is to provide user position and velocity everywhere at any time. GNSS

include one or more satellite constellations. Each satellite simultaneously broadcasts a Radio

Frequency (RF) signal. The signals transmitted by the satellites are recovered by a receiver

that tries to estimate the travel time of the received signals. The receiver uses the information

broadcast by each satellite, i.e, its position and velocity, and the estimated travel time to

determine the position and velocity of the user.

The ability of GNSS to provide navigation information depends on the quality of the received

signals. The signals received by a user on the surface of the Earth are characterized by low

power levels [1]. As a consequence, different factors, such as RF interference, can easily

degrade the quality of these low power signals and, thus, the quality of the position and

velocity estimates provided by the receiver. One particular type of interference source is

treated in this project, and is presented in the next section.

1.1. Interference Sources and Interference Mitigation

Techniques

Different interference sources can degrade GNSS signal reception. One of the most common

interference types are Continuous Wave (CW) signals that can be approximated as pure

sinusoids. Indeed, almost every electronics device and communication system generates CW

signals, which can have harmonics in the GNSS bands. As an example, TV ground stations

can emit CWs in the L1 frequency band [2] which is a frequency band used by the Global

Positioning System (GPS), the American GNSS. This is the reason why CWs constitute an

extremely common type of interference. The analysis of this type of signals, with specific

focus on the impact of mitigation techniques on GNSS receivers, is the main subject of this

project. The spectral and temporal characteristics of a CW can be summarized as follows.

CWs can be approximated as pure sinusoids in the time domain [2], whereas in the frequency

5

domain, their spectrum is concentrated around a specific frequency. Hence, the power of CWs

is concentrated on a narrow band. Since the CW band is narrow with respect to the band of

GNSS signals, CWs belong to the class of narrow band interference for GNSS signals.

In the literature, CWs have been widely treated. More precisely, different techniques aiming

at limiting the degradation caused by CWs have been designed and implemented.

Notch filtering is a common CW mitigation technique. The goal of a notch filter is to

attenuate the specific frequency around which the CW spectrum is concentrated [4].

Consequently, a notch filter is essentially a narrow stop band filter around this specific

frequency.

The efficiency of mitigation techniques have been studied and discussed in the literature.

Particularly, it has been shown that the limitation of these techniques is the following: even if

interference mitigation techniques remove the interference signal, they cause some distortions

on GNSS signals. More specifically biases can be introduced in the measurements and in the

final position solution [5].

Interference mitigation techniques can be linear devices, such as notch filter, non-linear

memory-less techniques or non-linear algorithms with memory. In all cases, a distortion can

be introduced in the received GNSS signal. This distortion can lead to biases in the final

measurements produced by a GNSS receiver. Three different types of measurements are

usually provided by a GNSS receiver [6]:

- pseudo-ranges are a measure of the travel time of the GNSS signals between

satellites and receiver and are obtained using the property of the pseudo random noise

code used to modulate the transmitted GNSS signal,

- carrier-phase measurements are an ambiguous measure of the signal travel time and

are obtained by exploiting the properties of the carrier waves used to broadcast the

GNSS signal,

- Doppler frequency measurements are the projection of the satellite-user relative

velocity along the satellite-receiver line of sight.

Several types of distortions can be introduced by interference mitigation techniques and have

an impact on the final measurements:

6

- amplitude distortion: the cross-correlation function between incoming signal and its

local replica can be distorted, leading to errors in the travel time estimates. It is noted that a

GNSS receiver estimates the travel time of a signal by correlating it with a local replica

opportunely delayed. The delay that maximizes the cross-correlation function is used to

generate the travel time estimate. Distortions in the cross-correlation function can introduce

biases in the travel time estimates and, consequently, in the pseudo-range measurements.

- additional delay in the received GNSS signal due to filtering and different

processing time can bias the travel time estimate and lead to biases in the pseudo-range

measurements.

- phase distortions can introduce some biases in the carrier phase measurements and

in the Doppler frequency measurements.

If the received GNSS signal is distorted, the information carried by this signal is degraded.

Hence, in the presence of interference mitigation techniques, the ability of GNSS to provide

navigation information can be limited because of these distortions. The impact of CW

interference mitigation techniques on GNSS signals has to be investigated in order to predict

the GNSS signal distortions and prevent errors due to these distortions.

A GNSS receiver consists of different parts [7]. The antenna at first recovers RF GNSS

signals. The RF front-end amplifies, down-converts and digitizes the received GNSS signals.

Then, the signals are processed in the signal processing block which aims at extracting the

information included in the different received GNSS signals. This information is used by the

navigation processing block which provides an estimate of the user position and velocity. In

this project, the impact of the interference mitigation techniques on GNSS signals will be

treated with a special focus on the code tracking loop. The code tracking loop, the Delay

Locked Loop (DLL), belongs to the signal processing block of a GNSS receiver. The DLL

has to provide a precise estimate of the code delay, which is directly related to the

propagation time of the signal between satellite and receiver. This project focuses on the

study of the impact of CW mitigation techniques on code tracking loops. More specifically,

the following algorithms are considered: FIR linear phase notch filters are analysed and

implemented using the design approach suggested by [4], [9] and [28]. IIR notch filters are

also studied with specific focus on the structure proposed by [2].

7

This subject has already been partly treated in the literature. More specifically, in [5], the

impact of notch filtering on code delay estimation in code tracking loops has been considered.

A formula for the code delay variance in the presence of notch filtering is proposed and

validated. Moreover, it has been shown that notch filtering causes a bias in the code delay

measurements. Although it was shown that this bias depends on the notch filter

characteristics, no analytical expression of the code delay bias caused by notch filtering was

proposed and the analysis was essentially empirical.

1.2. Objectives and Organization of the Thesis

As it has been underlined in the last paragraph, there is a lack of analysis concerning the

impact of notch filtering on GNSS measurements, and especially concerning the expression

and quantification of:

- the variance of the code delay measurements given by the code tracking loop in the

presence of notch filtering,

- the bias introduced by the mitigation techniques on the code delay measurements.

The variance analysis is completed with respect to results already presented in [5]. The

expression of the variance is important to quantify the additional noise introduces by

interference mitigation techniques. An analysis of the bias introduced on the code delay

measurements needs to be developed in order to identify the causes leading to these errors.

It is noted that an increase of the measurements variance can be compensated by filtering the

obtained observations whereas biases are more difficult to eliminate. For this reason, the

developed analysis is useful in predicting and possibly correcting systematic errors and for the

design of bias-free mitigation techniques. In this respect, the analysis on the notch filter is

used to determine the conditions leading to bias free measurements when a linear device is

used to excise an interfering signal.

The rest of this thesis is organized as follows:

8

- In Chapter 2, an overview of the GPS signal structure and of its time-domain and

frequency-domain characteristics is provided. Then, a model for narrowband and CW

interference is provided. A brief summary of the delay tracking loop is also presented.

- Chapter 3 deals with notch filters. More precisely, a functional description of

different notch filters is given. The analytical expressions for the variance of the

tracking loop outputs and for the measurement bias in the presence of notch filters is

provided.

- In Chapter 4, simulations are provided in order to validate the analytical

expressions derived in Chapter 3. Real data experiments are presented to quantify the

impact of notch filters on GNSS receiver performance.

- Finally, some conclusions are provided in Chapter 5 with a summary of the main

results achieved during the project and some possible future directions.

2. Signals and system

9

Since this project aims at investigating the impact of CW interference mitigation techniques

on a GNSS receiver, it is necessary to study:

- the main temporal and spectral characteristics of CW signals in order to choose a

technique suitable for CW signals mitigation,

- the structure and temporal and spectral characteristics of the GNSS signals in order

to investigate the impact of the mitigation technique on the GNSS signals.

These aspects are treated in the first and second sections of this chapter. Since this project

focuses on the impact of interference mitigation techniques on code tracking loops, the second

section of the chapter provides a functional description of code tracking loops.

2.1. GNSS signals

2.1.1. GNSS signals structure

In this thesis only pre-correlation mitigation techniques working on the raw input samples are

considered, thus, the expression of GNSS signals at the receiver input is required to

investigate the impact of mitigation techniques. Hence, it is necessary to provide the structure

and analytical expression of the GNSS signal at the processing block input. Figure 2-1 shows

the architecture of a GNSS receiver [7]. It is shown that the recovered signal is at first pre-

amplified, down-converted and sampled by the RF front-end before entering in the signal

processing block. As a consequence, the expression of the signal recovered by the receiver

antenna is firstly provided. Next, the effects of the RF front-end block on the recovered signal

are investigated and the expression of the signal after the RF front-end block is provided.

10

Figure 2-1: GNSS receiver architecture

In order to provide the user position and velocity, a receiver needs to recover signals from the

satellites in view.

The centre frequencies allocated for GNSS transmission have been chose in the Ultra High

Frequency (UHF, 300 MHz to 3 GHz) band. The advantage of the UHF is the reduced

interference with other RF systems. Figure 2-2 shows the frequency band allocation for two

GNSS, the American system GPS and the future European system GALILEO system [26].

11

Figure 2-2: GPS and GALILEO frequency allocation

Only the GPS L1 (1575.42 MHz) and L2 (1227.6 MHz) carrier frequencies are utilized, and

only L1 carrier frequency can be fully exploited for civil applications. This justifies why the

impact of interference mitigation techniques on the GPS L1 signals is specifically investigated

in this project. The modulation technique of GPS L1 signals is presented in the next

paragraph.

GPS L1 signals are Direct Sequence Spread Spectrum (DSSS) signals. In DSSS signals case,

two waveforms are combined by multiplication and the resulting signal is multiplied by the

carrier. These signals are:

- the navigation message. The GPS L1 navigation message is 1500 bits long and

contains clock corrections, orbit parameters and health of each satellite [20]. The

navigation message is characterized by a data rate equals to 50 bits per second. Hence,

each bit has a duration Tb = 20 ms.

- the Pseudo Random Noise (PRN) signal. The PRN signal is a periodic signal and

each sequence is called PRN code. PRN codes are generated from tapped feedback shift

registers. More details about registers are provided in [20]. Two types of PRN codes

exist on L1: the Coarse/Acquisition (C/A) code and the Precision (P) code. Since 1994,

P code encryption is enabled, so this project takes only into account the C/A code. The

length of the C/A code is 1023 chips and the rate of the PRN signal, called chip rate and

denoted Rc, is equal to 1.023 MHz [20]. A chip is the basic duration over which the C/A

code assumes a constant value. The duration of each chip is Tc = 1/Rc, where Tc is

12

called chip period. As a consequence, the total duration of the C/A code is 1 ms. The

C/A code is periodically repeated. A unique PRN code is assigned to each satellite, so

its signal can be identified by the unique C/A code.

The waveforms corresponding to the navigation message and the PRN code broadcast by the

satellite and denoted respectively and are:

(2.1)

where:

○ , is the navigation message binary sequence,

○ is the duration of a navigation bit,

○ corresponds to the materialization of the navigation message sequence, and is given

by [10]:

(2.2)

○ , is the PRN C/A binary sequence,

○ is the chip period,

○ corresponds to the materialization of the PRN C/A sequence, and is given by [10]:

(2.3)

13

Figure 2-3 illustrates the structure of the modulated signal for a DSSS signal.

Figure 2-3: Modulated signal structure for a DSSS signal

Many factors explain why DSSS signals are used for satellite navigation application. DSSS

waveforms are adopted for satellite navigation mainly because [10]:

- The good correlation properties of PRN signals allows precise range estimation,

- in a GNSS constellation, each satellite has its own PRN code and transmits its signal

on the same frequency band: DSSS signals exploit Code Division Multiple Access

(CDMA) to transmit multiple signals simultaneously and on the same frequency,

- the Power Spectral Density (PSD) of a DSSS signal occupies a large band and with

density levels often below the noise floor with reduced interference with other

communication systems.

The base-band bipolar signal described above is multiplied by a carrier and up-converted to

the RF centre frequency. Hence, each satellite transmits a signal which is a multiplication

between the RF carrier and the data waveform. Between transmitter and receiver, the signal

passes through a transmission channel. As a consequence, the receiver antenna recovers a

signal which is different from the transmitted signal. Indeed, the received signal is mainly

affected by:

- a delay corresponding to the propagation time,

- a phase offset,

14

- a carrier frequency offset due to the relative motion of the satellite with respect to

the receiver (Doppler effect),

- additive noise.

Moreover, some RF signals interfere with the GNSS signals and the receiver antenna also

recovers these interfering components. Hence, the analytical expression of the signal

recovered at the receiver antenna is [5]:

(2.4)

Where:

○ is the received signal power from the i-th satellite,

○ is the navigation message emitted by the i-th satellite and given by equation (2.1),

○ is the delay introduced by the transmission channel on the i-th satellite signal,

○ is the PRN signal emitted by the i-th satellite and given by equation (2.1),

○ is the carrier frequency, for GPS-L1 signals,

○ is the Doppler frequency of the i-th satellite signal,

○ is the phase offset of the i-th satellite signal,

○ is the noise component,

○ is the interfering signal described in Section 2.2,

○ is the number of satellites in view.

As said before, the recovered signal is then pre-amplified, down-converted and sampled by

the RF front-end before entering in the signal processing block.

The pre-amplifier is the first active component after the antenna. The purpose of the pre-

amplifier is to amplify the signal at the output of the antenna for further processing. Usually,

15

the amplifier is referred to as a Low Noise Amplifier (LNA) the aim of which is amplifying

the signal while limiting the additional noise introduced by the device.

The GNSS signal needs to be down-converted to an Intermediate Frequency (IF) before

sampling and further processing. The down-conversion is required to lower signal processing

complexity [21] and is performed by mixing the incoming signal with a local carrier. Low-

pass filtering is then required to keep only the frequency band of interest and remove local

oscillator harmonics. Figure 2-4 shows the operations performed during down-conversion

[7].

Figure 2-4: Down-converter operations

The incoming signal is given by equation (2.4). The down-converted signal is then given by:

(2.5)

where:

○ is the IF.

Finally, the signal is digitized by a sampler that transforms the continuous time signal

into a discrete time sequence. The digitizer also quantizes the continuous

valued signal into a sequence with values from a discrete alphabet. Quantization leads to

some losses in the signal quality.

16

Finally, the expression of the signal at the signal processing block input at is thus

given by:

(2.6)

where:

○ is the sampling period.

In equation (2.6) the effect of quantization is neglected since it is assumed that the front-end

is using a high number of bits for the digital representation of the input analog signal.

In the following, the notation is used to denote a digital sequence sampled at

the frequency .

2.1.2. Temporal and spectral characteristics of GNSS signals

In the time domain, a PRN signal is characterized by its auto-correlation and cross-correlation

functions. The expression of the autocorrelation function of a GPS PRN signal, , is:

(2.7)

where:

○ * denotes complex conjugation.

It is demonstrated in [22] that the autocorrelation function given by equation (2.7) can also be

written as:

(2.8)

17

where:

○ is the chip period,

○ is the triangular function:

(2.9)

○ is the auto-correlation function of the PRN sequence defined as

, where is the length of the PRN code.

The normalized auto-correlation function of a GPS C/A PRN code, that is a Gold code,

assumes only four values [5]:

(2.10)

where:

- is the length of the tapped feedback shift register used to generate the PRN code

( )

-

Figure 2-5 shows the normalized auto-correlation function of a GPS C/A signal. ACF stands

for Auto Correlation Function. In this figure, it is clearly shown that the auto-correlation

function has four values. Since and for a GPS C/A code, these values are

equal to . It is also illustrated that the auto-correlation is

chips periodic.

18

Figure 2-5: Normalized auto-correlation function of a GPS C/A signal

Close to its main peak, and according to equation (2.8), the normalized autocorrelation

function can be modelled as a triangular function. Figure 2-6 shows the zoom performed on

the auto-correlation function of a GPS C/A signal (Figure 2-5) close to the main peak.

Figure 2-6 : Normalized auto-correlation function of a GPS C/A signal close to the main peak

19

The expression of the cross-correlation function between two GPS C/A signals and

( ) is:

(2.11)

In the same way as for the auto-correlation function, the cross-correlation function can be

written as [22]:

(2.12)

where:

○ is the cross-correlation function of two PRN sequences defined as

, where is the length of the PRN code.

The normalized cross-correlation function of a periodic GPS C/A PRN codes assumes

only three values [5]:

(2.13)

Figure 2-7 shows the normalized cross-correlation function of two different GPS C/A signals.

CCF stands for Cross Correlation Function. . In this figure, it is clearly shown that the cross-

correlation function has three values. Since and for a GPS C/A code, these

values are equal to . It can also be concluded that the cross-

correlation function does not present any significant peak, so any C/A code is almost

uncorrelated with any other PRN code.

20

Figure 2-7: Normalized cross-correlation function of two GPS C/A signals

This project focuses mainly on the impact of interference mitigation techniques on GPS L1

signals. As a consequence, the spectral characteristics of GPS L1 signals are presented herein.

In the frequency domain, the PRN signal is characterized by its PSD. The expression of the

PSD for the PRN signal with an autocorrelation given by equation (2.11) is:

(2.14)

where:

○ FT denotes the Fourier Transform.

If it is assumed that the C/A code is random biphase sequence and if the effects of finite-

length C/A codes are neglected ( is thus considered close to zero), the analytical

expression of the normalized autocorrelation function close to the main peak is given by:

21

(2.15)

Computing the FT of equation (2.15) leads the PSD of the C/A signal:

(2.16)

Figure 2-8 shows the shape of the PSD defined in equation (2.16). The power is mainly

concentrated around the central frequency. The width of the main lobe is equal

to .

Figure 2-8: PSD of the GPS C/A signal

Since the C/A signal is multiplied with the RF carrier (equation (2.4)), the C/A signal PSD is

then translated around .

22

2.2. Temporal and spectral characteristics of

interfering signals

The aim of this section is to provide a brief description of CW signals. Firstly, CW

interference sources are presented. Secondly, the main temporal and spectral characteristics of

CW signals are provided.

Examples of CW interference sources are:

- electronic devices and communication systems that generate CW signals for the

transmission of communication signals [2]. Because of the imperfections of the

electronic components, harmonics can be generated in the GNSS frequency bands. In

[1], the impact of Frequency Modulation (FM), Very High Frequency (VHF) and UHF

emitters on a GNSS receiver is analyzed. It is concluded that almost every RF emitters

generate harmonics that can enter the L1 band,

- Personal Electronic Devices (PED), such as cell phones or laptops connected to a

wireless network, also transmit CW signals which can potentially interfere with GPS L1

signals [2].

As a conclusion, every CW signals generated by communication systems or personal

electronic devices can potentially interfere with GPS L1 signals.

Interfering signals are frequently classified in the literature relative to their temporal and

spectral characteristics [2]. The class of CW signals is modeled in the time domain as a pure

sinusoid given by [2]:

(2.17)

where:

○ is the CW signal at ,

○ is the amplitude of the sinusoid,

○ is the frequency of the sinusoid,

23

○ is the phase of the sinusoid.

As a consequence, CW signals are ideally modelled in the frequency domain as two spectral

lines located at . Figure 2-9 is an example of a CW signal PSD superimposed to the

GPS L1 signal PSD.

Figure 2-9 : Normalized power spectral density of a CW signal and GPS L1 C/A signal

In Figure 2-9, it is shown that the CW signal is narrow with respect to the GNSS signal band,

so CW interference belongs to the class of narrow band interference.

The presence of CW interference generated by FM, VHF, UHF emitters or PED means that

two spectral lines have to be mitigated. In order to attenuate these two spectral lines, different

CW interference mitigation techniques have been proposed in the literature. However, CW

interference mitigation techniques have an impact on the GNSS receiver: they result in a

degradation of the code delay tracking accuracy. Code delay tracking is performed by the

code tracking loop, also called DLL. A functional description of the code tracking loop and

the evaluation of its performance is presented in Section 2.4.

24

2.3. In phase/quadrature sampling

In the previous sections, real signals have been considered. More specifically, it has been

assumed that the receiver front-end uses a real sampling technique [27] to generate a digital

sequence at the input of the processing block. It is noted that a different down-

conversion/sampling technique can be used. This technique is called in-phase/quadrature

sampling [27] and produces a base-band complex signal. In this case, equation (2.6) becomes:

(2.18)

In a similar way, equation (2.17) becomes:

(2.19)

In the following, both signal models, real and complex, will be used.

2.4. Code tracking loop summary

To determine its position, a GNSS receiver needs to estimate precisely the parameters of the

signals transmitted by the different GNSS satellites. Indeed, the signal parameters are then

converted to distances. Next, position is estimated by triangulation based on the estimated

distances [7]. The parameters of the signals are:

- the propagation time,

- the Doppler frequency,

- the carrier phase.

The estimation of these parameters is performed in two steps:

- The signal acquisition block involves detection of the signals from satellites in view

and rough estimation of the code delay and Doppler frequency of each satellite,

25

- The signal tracking block refines the estimation of the code delay and Doppler

frequency. It also estimates the carrier phase. By maintaining continuously updated

estimates of the signals parameters, the signal tracking block provides a dynamic

estimation of these parameters.

This project aims at investigating the impact of interference mitigation techniques on a GNSS

receiver with a special focus on the signal tracking block. As a consequence, an overview of

the signal tracking block is provided herein.

In order to refine the estimation of the code delay and Doppler frequency, and to estimate the

carrier phase, two separate locked loops are used:

- the DLL aims at precisely estimating the code delay and its changes over time,

- the Phase Locked Loop (PLL) aims at precisely estimating the Doppler frequency

and carrier phase and their changes over time.

A DLL is a feedback system that is able to track the delay of a PRN signal. DLL is able to

synchronize its own local PRN replica with the incoming PRN signal, so that code delay

estimate can be derived from estimate of the local PRN code. Figure 2-10 depicts the general

structure of the DLL.

Figure 2-10: General structure of a DLL

26

DLL is an iterative process: at iteration , the loop updates the code delay estimate by adding

the code delay estimated by the loop at iteration and the estimated code delay error

provided by the loop during iteration (the code delay estimate at iteration is given by the

acquisition stage). To estimate the code delay error at iteration , several steps are needed

and are presented in the following.

Multiplication by the local carrier

In the first step performed by the DLL, the incoming signal (given by equation (2.6)),

which has been pre-amplified, down-converted, quantized and sampled by the RF front-end

block, is multiplied by the local carrier estimated by the PLL. The analytical expression if the

local carrier is:

(2.20)

where:

○ is the Doppler frequency of the j-th satellite signal,

○ is the carrier frequency of the j-th satellite signal estimated by the PLL,

○ is the IF,

○ is the sampling period.

The goal of this multiplication is to eliminate the Doppler frequency and carrier phase present

in the incoming signal. Hence, the multiplication by the local carrier converts the incoming

signal to baseband.

De-spreading

After, the baseband signal is correlated by three copies of a local code replica, the early

replica , the prompt replica and the late replica , each with a different delay.

27

Indeed, the early and late components are spaced by a time interval denoted and called

correlator spacing [7]. The analytical expressions of these copies are given by:

(2.21)

where:

○ is the delay introduced by the transmission channel on the j-th satellite signal

estimated by the DLL during iteration .

The correlation is performed by:

- multiplying the baseband signal by the three replicas. The multiplication by the

local code leads to reduce the bandwidth of the signal and remove the PRN code: the

signal is converted to a narrow-band signal concentrated around the 0Hz frequency

while the noise bandwidth is unchanged. It leads also to isolate the contribution of the

signal broadcast by satellite from the signals broadcast by the other satellites.

- applying Integrate and Dump filters. Integrate and Dump filters are integrators. In

the digital domain, this integration is performed by adding together samples of

signals , and obtained after the multiplication with the three local codes.

Let be the integration time and . The impact of Integrate and Dump

filters on the incoming signal is then to reduce the noise bandwidth. Since the noise

power is proportional to the noise bandwidth, the noise power is also reduced because

of Integrate and Dump filters.

Figure 2-11 and Figure 2-12 illustrate the signal and noise power spectral density

respectively before and after multiplication by the local carrier and de-spreading.

28

Figure 2-11: PSD of the GPS-L1 C/A signal and of the noise component before de-spreading

Figure 2-12: PSD of the GPS-L1 C/A signal and of the noise component after de-spreading

In Figure 2-11 and Figure 2-12, it is shown that, before de-spreading, the signal is well

below the noise level. After de-spreading, the signal is above the noise level. The reduction of

the noise bandwidth leads to decrease the noise power. Indeed, the noise power before de-

spreading denoted is equal to the integral of the noise PSD calculated on the

interval :

(2.22)

where:

29

○ is the noise PSD level [W/Hz],

○ is the sampling frequency.

The noise power after de-spreading denoted is equal to the integral of the noise

PSD calculated on the interval :

(2.23)

Since , equations (2.22) and (2.23) lead to:

(2.24)

Equation (2.24) shows that the noise power is reduced by a factor due to de-spreading.

As a conclusion, the main functions of the de-spreading process are:

- to isolate the contribution of the signal from satellite ,

- to make the signal strong enough for being utilizable (otherwise the signal would be

completely hidden by noise).

According Figure 2-10, the analytical expressions of the three correlator outputs for the

iteration (for the kth interval of seconds of the incoming signal) are the following:

(2.25)

If it is assumed that:

30

- the data navigation message has a constant value during the N instants needed for

the integration,

- the code and carrier phase errors are constant during the the N instants needed for

the integration,

- the double frequency terms are negligible due to the low pass filtering performed

by the Integrate and Dump filters,

and by using the property of almost uncorrelation (see Section 2.1.2) of the different PRN

codes, equations (2.25) can be approximated as [24]:

(2.26)

Where:

○ is the noise component,

○ is the interference component,

○ is the residual carrier phase error,

○ is the cross correlation function between the incoming PRN code of the satellite

and its local replica,

○ is the code delay estimate error (for the satellite ),

○ the subscripts , and are used to denote early, late and prompt components.

Figure 2-13 proposes an example of the values of the early, prompt and late correlator

outputs. In this case, the effects of front-end filtering and interference have been neglected.

31

Figure 2-13: Representation of early, prompt and late correlator outputs on a ideal auto-correlation function

DLL discriminator

The goal of the discriminator is to extract the code delay estimate error from early, prompt

and late correlator outputs given by equations (2.26). In order to obtain a control signal that is

proportional to the residual delay error on the estimated code delay, different discriminators

are proposed in [7].The analytical expression of the error signals are provided in [22]. The

error signals have been derived assuming that:

- early, prompt and late correlator outputs belong to the main peak of the cross-

correlation function which is modeled as a triangular function given by equation (2.15),

- and (or, equivalently, the code delay estimate error is

smaller than half of the correlator spacing ).

32

As a consequence, the analytical expression of the error signals presented in Table 2-1 are

only valid for . is the residual carrier phase error.

Table 2-1: Standard DLL discriminators

Name Algorithm Error signal

Early minus late

Early minus late power

Dot product

The error signals provided by Early minus late and Early minus late power discriminators are

linear functions of for , and the error signals provided by the dot

product discriminator can be linearized for small code delay estimate errors.

The early minus late discriminator provides the simplest implementation and is a linear

device. As it is explained in the last paragraph of this section, this is an advantage since non-

linear devices lead to additional noise in the code delay estimate error. The problem with

early minus late discriminator is that the carrier phase error is still present in the error

signal expression. In order to remove the carrier phase error in the error signal

expression, two solutions are proposed:

- using an accurate carrier phase estimation technique (PLL) to decrease the carrier

phase error,

- applying non-linear functions to the correlator outputs in order to obtain a phase

independent discriminator output. Early minus late power and dot product

discriminators both provide phase independent discriminator outputs. Indeed, the phase

dependence is removed by:

- the complex prompt correlator for the Early minus late power discriminator,

33

- the absolute value operator for the dot product discriminator.

However, as explained in the last paragraph of this section, non-linear discriminators

disadvantage is to increase the variance of the code delay estimate.

Loop filter

The discriminator output is noisy (because of the noise present in the input signal) and varies

with time. The objective of the loop filter is to reduce the noise present at the discriminator

output in order to produce an accurate estimate of the code delay error. To do that, the loop

filter has to:

- average the noise effect,

- respond effectively to signal dynamics [7].

There are many design approaches to digital filters. The most commonly used consists in two

steps. Firstly, the loop filter is designed in the analog domain. The loop filter design in the

analog domain is based on the choice of two main parameters:

- the order of the filter which determines the ability of the loop filter to respond to

different types of signal dynamics,

- the noise bandwidth (determined by the quantity of noise transferred from the input

signal to the final estimate) which determines the ability of the loop filter to reduce the

noise present at the discriminator output.

Secondly, this filter is transformed into the digital domain [10] using transformation methods

which are detailed in [7] and [10].

Numerically Controlled Oscillator (NCO)

The NCO is used to generate the local signal replicas and is made of two parts [7]:

34

- an accumulator: the output of the loop filter is accumulated to generate a new

estimate of the code delay. This new code delay estimate is used to generate a new time

scale for the generation of the local codes.

- a delay to amplitude converter: it is used to generate the local replicas of the code

signal to be correlated with the incoming signal.

DLL performance

The performance of the DLL can be evaluated in terms of tracking jitter. The tracking jitter is

a measure of the variance of the code delay error estimated by the loop. Under the following

assumptions [25]:

- the noise component recovered by the receiver antenna is white,

- the GNSS receiver does not contain any filtering stages placed before the

processing block,

the tracking jitter can be expressed as [7]:

(2.27)

where:

○ is the speed of light,

○ is the Carrier-to-Noise density power ratio of the signal at the input of the loop,

○ is the squaring loss [7]. It is a function of the correlator spacing , the chip

period and the signal . Its analytical expression depends on the discriminator used,

○ is the loop bandwidth. The loop bandwidth is defined as the quantity of noise

transferred from the input signal to the final delay estimate, and is expressed as [7]:

(2.28)

35

where:

○ is the variance of the noise at the input of the loop,

○ is the variance of the noise at the output of the loop.

Equation (2.27) can be interpreted as follows [7]. Non-linear discriminators (such as early

minus late power and dot product discriminators) introduce additional noise components

which causes an increase in the variance of the code delay estimated error. In fact, the

additional noise generated by the non-linearity degrades the tracking jitter by the squaring

loss . Table 2-2 provides the expressions of tracking jitters for the three

standard DLL discriminators presented in Table 2-1. The degradation of the tracking jitter

due to non-linear discriminators is a function of the input and increases when the input

decreases.

Table 2-2: Analytical tracking jitter expression for standard DLL discriminators

Name Algorithm Tracking jitter expression

Early minus late

Early minus late power

Dot product

In the following chapters, the impact of a CW interference technique on the DLL tracking

jitter is analyzed. In addition to this, the bias introduces on the final delay estimated by the

loop is also considered.

36

3. Impact of notch filters on a

GNSS receiver

Several CW interference mitigation techniques have been proposed and investigated in the

literature. [15] presents the advantages and disadvantages of CW interference mitigation

techniques. The most efficient techniques have a complex architecture and have high cost.

Computationally effective techniques are simpler to implement but are not efficient against

in-band interference (the GPS L1 has centre frequency equal to 1575.42 MHz and a total

bandwidth of 20 MHz). In this project, notch filters have been chosen as CW interference

mitigation technique since they effectively attenuate CW signals with a limited impact on

GNSS signals [2]. This chapter aims at investigating the theoretical impact of notch filters on

a GNSS receiver. The first section provides a functional description and the design approach

of notch filters. Next, the theoretical impact of IIR notch filters and FIR notch filters on a

GNSS receiver is studied in Sections 3.2. and 3.3.

3.1. Functional description and implementation of

notch filters

The goal of notch filters is to attenuate a certain frequency of the input signal spectrum. As

explained in Chapter 2, CW signals can be modeled in the time domain as:

37

- a real sinusoid with a frequency (equation (2.17)) if a real sampling

technique is used. As a consequence, in the frequency domain, CW signals are ideally

modelled as two spectral lines located at .

- a complex exponential with a frequency (equation(2.19)) if in

phase/quadrature sampling is used. As a consequence, in the frequency domain, CW

signals are ideally modelled as a single spectral line located at .

In a GNSS receiver, notch filters are implemented such that the frequency to attenuate

corresponds to the CW signal frequency or . In the following:

- are called notch frequencies,

- a notch filter attenuating two frequencies is called real notch filter since it

deals with real sinusoids,

- a notch filter attenuating one frequency is called complex notch filter, since

its impulse response will have complex coefficients.

In a GNSS receiver, the notch filter is placed after the front end signal and before the signal

processing block [2]. The receiver able to cope with interfering signals is generally equipped,

in addition to the interference mitigation unit, with a detection unit which detects the

interference. In some cases, detection, interference parameter estimation and mitigation are

jointly performed.

Figure 3-1 depicts the scheme of a GNSS receiver including interference detection and

mitigation units [2].

38

Figure 3-1 Scheme of a GNSS receiver including interference detection and mitigation units

In the literature, several algorithms aiming at estimating the frequency of a sinusoid wave are

proposed, and their performances are analyzed [13, 14]. For example, a Least Mean Square

(LMS) algorithm which converges on the central frequency of the interference by minimizing

the power of the resulting filtered signal is proposed in [15]. In this project, it is assumed that

the interference has been detected and its frequency has been estimated by the detection

unit.

Since the goal of a notch filter is to attenuate the frequencies , its ideal transfer function

can be expressed as [4]:

(3.1)

where:

○ for a real notch filter

(3.2)

39

is the amplitude of the notch filter transfer function,

○ for a complex notch filter

(3.3)

○ is the phase of the notch filter transfer function.

In practice, notch filters are band-stop filters with two (one) very narrow stop-bands at

( ). In real filter, the amplitude of the transfer function deviates for the ideal definition

provided above. In [12], the maximum deviation in the pass-band region of the transfer

function is called the pass-band ripple and is denoted by . The notch width is defined as the

width of the stop band when the notch filter magnitude is equal to . Figure 3-2 shows the

magnitude function of a notch filter as a function of the digital frequency normalized by the

sampling frequency of the system. The concepts of pass-band ripple and notch width are also

shown.

Figure 3-2: Amplitude response of a notch filter

40

In this project, two types of notch filters are considered:

- IIR notch filters,

- FIR notch filters.

3.1.1. IIR notch filters

Firstly, IIR notch filters are presented. It is noted that in the literature, several types of IIR

filters have been considered [29]. In this thesis, only a specific type of filter used for GNSS

applications is detailed.

Real IIR notch filter design approach

According to Chapter 2, two poles notch filters aim at removing interference signals

characterized by a real sinusoid given by:

(3.4)

As a consequence, two complex conjugate zeros are required in the notch filter transfer

function to remove the two complex exponentials in equation(3.4). In this case, the transfer

function of a real IIR notch filter is thus given by [2]:

(3.5)

where:

○ and are the zeros of the notch filter corresponding to the interference frequencies,

○ is called the pole contraction factor. For stability reason, ,

○ denotes the real part operator.

41

Figure 3-3 shows the amplitude response of this filter for different values of . In this case,

the notch frequency is chosen such as its normalized value is equal to:

(3.6)

where:

○ is the sampling frequency.

Figure 3-3: amplitude response of a real IIR notch filter

By comparing the amplitude responses obtained for two different values of pole contraction

factor, it is concluded that allows the regulation of the notch width. This factor has to be

[15]:

- small enough to obtain acceptable CW interference attenuation (large notch width)

- large enough to obtain acceptable GPS signal degradation caused by the notch

filter.

Figure 3-4 shows the phase of this type of IIR notch filter. The phase of a IIR notch filter is

non-linear.

42

Figure 3-4: Phase of a real IIR notch filter

Complex IIR notch filter design approach

A complex notch filter aims at removing interference signals of the form:

(3.7)

As a consequence, only a single zero is required, in the notch filter transfer function, to

remove the interference described in equation(3.7). The transfer function of a complex IIR

notch filters is thus given by [2]:

(3.8)

where:

○ is the notch filter zero corresponding to the interference frequency,

○ is the pole contraction factor defined in the previous section.

43

It is noted that the real notch filter described in the previous section can be obtained by

cascading two complex notch filters of the form (3.8) where the zero of the second notch filter

is equal to , the complex conjugate of .

Figure 3-5 shows the amplitude response of this type of filter for different values of . As

for the real notch filter, the notch frequency is chosen such that its normalized value is equal

to:

(3.9)

Figure 3-5: Amplitude response of a complex IIR notch filter

Concerning the pole contraction factor, the same conclusions as for real IIR notch filters can

be made. Figure 3-6 shows the phase response of the complex IIR notch filter. Its phase is

non-linear.

44

Figure 3-6: Phase of a complex IIR notch filter

3.1.2. Linear phase FIR notch filters

Secondly, FIR notch filters are presented. The transfer function of any N-th order FIR notch

filter has the following form [5]:

(3.10)

where:

○ are the coefficients defining the impulse response of the FIR notch

filter.

The impact of notch filters on a GNSS receiver causes a bias in the code delay estimation.

This bias has a predictable value in the case of linear phase notch filter. Consequently, FIR

notch filters are considered in this project with a special focus on linear phase FIR notch

filters.

45

The condition to have linear phase FIR notch filter is that the impulse response coefficients of

the notch filter are symmetric with respect to . Here, the proof is given for N even, but the

demonstration is similar for N odd.

The transfer function presented in equation (3.10) is equivalent to [16]:

(3.11)

By defining:

(3.12)

It is possible to restate equation (3.11) as:

(3.13)

At first, the filter characterized by the transfer function is analyzed. It is assumed that

the impulse response coefficients of the notch filter are symmetric with respect to , so the

impulse response coefficients are such that:

(3.14)

Equations (3.12) and (3.14) lead to:

(3.15)

Hence, the coefficients, , are symmetric with respect to zero

and has a zero phase [9]. Figure 3-7 shows the coefficients for

a 6th order FIR notch filter with symmetric impulse response coefficients with respect to .

46

Figure 3-7: Example of notch filter impulse response symmetric with respect to for N = 6

Figure 3-8 illustrates the coefficients obtained from the coefficients

.

47

Figure 3-8: Example of coefficients corresponding to the notch filter

impulse response coefficients shown Figure 3-7

From the term of equation (3.13), it is possible to obtain the phase of the filter transfer

function. More specifically, clearly shows that the filter phase is

linear and equal to . In addition to this, it is possible to evaluate the group delay

introduced by the filter. The group delay caused by a filter with transfer function

is defined as [17]:

(3.16)

In this case, it is equal to:

48

(3.17)

Equation (3.17) shows that the group delay introduced by the filter is a constant.

As a conclusion, the transfer function of any N-th order FIR notch filter with

symmetric impulse response coefficients with respect to can be considered as the product

of:

- a zero phase FIR filter ( ),

- a constant magnitude filter ( ) with a constant group delay equal to .

Real FIR linear phase notch filter design approach

A technique frequently used to implement a real FIR notch filter with a linear phase

is based on equation(3.13). A zero phase FIR notch filter with the same notch

frequencies as is at first designed. The transfer function of the zero phase filter

is then multiplied by the transfer function of the linear phase filter to achieve the

FIR linear phase notch filter . It is noted that a zero phase notch filter is not physical

implementable since it would violate causality restriction.

The zero phase notch filter, , can be designed using the difference between a zero phase

low-pass prototype filter (with a cut-off frequency corresponding to the notch

frequency) and its amplitude complementary [9]. Hence, the transfer function of the filter

is given by:

(3.18)

where:

49

○ is the transfer function of the zero phase low-pass prototype filter.

Using equations (3.13) and (3.18), it is possible to express the transfer function of the FIR

notch filter as:

(3.19)

From [9], the impulse response of the FIR linear phase notch filter can be given as:

(3.20)

where :

○ is the impulse response of the low-pass prototype filter.

Hence, real FIR notch filters with linear phase can be designed using a zero phase low-pass

prototype filter. Different techniques are available to design this prototype filter [4]:

- the frequency sampling design approach consists in specifying the amplitude

response of the prototype filter at a finite number of frequency samples. Next, the

coefficients of the prototype filter are computed from these samples,

- the optimal FIR linear phase filter design approach aims at achieving an equiripple

behavior of the magnitude response in the pass-band and in the stop-band,

- the windowed Fourier series design approach is detailed herein.

In [4], a comparison between the performances of these techniques is provided. Windowed

Fourier series design approach provides acceptable design results in terms of notch width and

pass-band ripple. It is also the simpler technique to implement. As a consequence, the

windowed Fourier series design approach has been chosen to implement the zero phase low-

pass prototype.

To design the low pass prototype filter with the windowed Fourier series design

approach, the ideal transfer function of this zero phase filter is needed:

50

(3.21)

Where:

○

○ is the cut-off frequency.

The impulse response corresponding to the transfer function (3.21) is infinite and is given by:

(3.22)

The transfer function of the non-ideal realizable filter is given by [16]:

(3.23)

where:

○ is the truncated version of the coefficients .

○ is a weighting window, which has to be symmetric with respect to in order to

preserve the linear phase property of .

For example, the rectangular window, characterized by the following function, is frequently

used:

(3.24)

Rectangular window has been used to design a 500th order FIR low pass prototype. The

amplitude function of the obtained low-pass filter is shown in Figure 3-9.

51

Figure 3-9: Amplitude response of the low pass prototype for a 0.5dB cut-off frequency

corresponding to , and for N=500

The magnitude and phase of the transfer function of the notch filter obtained from the

prototype shown in Figure 3-9 are shown in Figure 3-10 and Figure 3-11, respectively.

Since the coefficients of the low-pass prototype are symmetric with respect to zero, the

low-pass prototype is a zero phase filter and the FIR notch filter has a linear

phase and a group delay equal to .

52

Figure 3-10: Amplitude of the transfer function of a real FIR notch filter

Figure 3-11: Phase of the transfer function of a real FIR notch filter

Complex FIR linear phase notch filter design approach

Complex FIR notch filters with linear phase can be designed as follows. Firstly, a zero phase

low-pass prototype with a cut-off frequency corresponding to the notch frequency

to be attenuated is designed using one of the techniques described in the previous

53

section. Next, the Hilbert transform is applied to the low-pass prototype . The transfer

function of the low-pass prototype after Hilbert transform is given by:

(3.25)

where:

○ is the Heaviside function defined as:

(3.26)

Hence, the negative frequency part of the spectrum of the low-pass prototype is set to

zero due to the Hilbert transform. Figure 3-12 shows the magnitude of the transfer function of

the filter, , corresponding to the Hilbert transform of a 500th order low-pass

prototype with a cut-off frequency corresponding to .

Figure 3-12: Amplitude of the transfer function of the complex low-pass prototype for a

0.5dB cut-off frequency corresponding to , and for N=500

54

It is noted that the actual transition at is not vertical. Indeed, in practice, the

Heaviside function can only be approximated by a function with a non-vertical

transition.

Next, the following transformation is applied to filter in order to obtain the complex

linear phase FIR notch filter :

(3.27)

By computing the inverse Fourier transform of equation (3.27), the impulse response of the

FIR linear phase notch filter is given by:

(3.28)

where:

○ is the analytic impulse response of the low-pass prototype .

A complex FIR notch filter has been designed using equation (3.27) with the low pass

prototype presented in . Figure 3-12. The magnitude and phase of its transfer function are

shown in Figure 3-13 and Figure 3-14, respectively.

55

Figure 3-13: Amplitude response of a complex FIR notch filter designed with the low-pass prototype method

Figure 3-14: Phase of a complex FIR notch filter designed with the low-pass prototype method

56

It is observed in Figure 3-13 that the notch at is removed from the spectrum of

filter . The phase of the complex FIR notch filter is linear. It is also noted that,

because of the impossibility to perfectly implement the Hilbert transform in practice, the

amplitude response of has a notch at . For this reason, another method

is considered to design complex linear phase FIR notch filters [28]. In the following, this

design approach is denoted “series expansion design approach”. This method consists in:

- implementing the one pole IIR notch filter defined with equation (3.8) and with a

notch frequency

- deriving the one pole linear phase FIR notch filter from the one pole IIR notch filter

by keeping a finite number of IIR notch filter impulse response coefficients.

Equation (3.8) is equivalent to:

(3.29)

By keeping only N impulse response coefficients, equation (3.29) becomes:

(3.30)

where:

57

○ is the transfer function of a Nth order complex FIR notch filter.

It is noted that the transfer function defined by equation (3.30) does not have a linear phase,

since the coefficients of the filter impulse response are not symmetric. A linear phase FIR

filter of order 2N can be obtained by simply applying a back-forward filtering. In this way,

the final impulse response becomes:

(3.31)

The group delay introduced by applying the back-forward is equal to . The group

delay is thus equal to the order of the FIR notch filter divided by . From equation (3.17),

the theoretical expression of the group delay does not change for real and complex FIR filters.

The magnitude and phase of the transfer function of this type of filter are shown in Figure

3-15 and Figure 3-16, respectively. The contraction factor of the adopted IIR notch filter is

equal to 0.9 and the FIR notch filter is a 200-th order FIR notch filter.

Figure 3-15: Amplitude of the transfer function of a complex FIR notch filter designed using the method based on the IIR notch filter

58

Figure 3-16: Phase of a complex FIR notch filter designed using the method based on the IIR notch filter

It is observed that in Figure 3-15 the notch at is removed from the spectrum of filter

designed with the low-pass prototype method. The phase of the complex FIR

notch filter is linear. For these reasons, the complex FIR linear phase notch filter approach

design based on the IIR notch filter is used in the following.

It is noted that this method can be used for the design of real notch FIR filters as well.

3.1.3. Comparison between IIR notch filters and FIR notch

filters

The advantages of IIR notch filters over FIR notch filters are discussed in [18]. IIR notch

filters can provide frequency responses closer to an ideal notch filter than FIR notch filters

requiring the same computational load. As a consequence, IIR notch filters are employed for a

more efficient suppression of narrowband interferences with a lower computational

59

complexity compared to FIR notch filters. However, the phase of the transfer function of a

IIR filter is difficult to whereas it is always possible to design a FIR filter with linear phase.

IIR and FIR notch filters have a double impact on the code delay tracking loop. Firstly, they

modify the shape and the location of the correlation function between the incoming code and

its local replica on the time axis. Secondly, they increase the tracking jitter of the loop. Next

section presents an analytical analysis of the impact of notch filters on the correlation

function. In Section 3.3., the expression of the tracking jitter in the presence of notch filter is

provided.

3.2. Theoretical analysis of the impact of notch filters

on the correlation function

Code delay determination is based on the correlation function between the incoming signal

and a local code generated by the DLL. In the absence of filtering, the correlation function

presents a peak at the instant corresponding to the true code delay. Detecting the correlation

peak leads to an estimate of the code delay. Figure 3-17 is an example of correlation function

between a real GPS L1 signal and a local replica of the PRN code. In this example, the code

delay modulo the code duration (1 ms) is roughly equal to 0.82 ms.

60

Figure 3-17: Correlation function between the incoming GPS L1 signal (PRN 29) and its local replica

In the presence of notch filter, the correlation function is distorted and translated on the time

axis. The goal of this section is to derive the analytical expression of the correlation function

in presence of a CW interference mitigation unit placed before the delay and phase tracking

loops.

After the down-conversion and the digitalization of the incoming signal performed by the

front-end filter, the analytical expression of the GNSS signal is (equation (2.6)):

(3.32)

Let:

○ be the signal part of the incoming signal given by equation:

(3.33)

○ be the notch filter impulse response,

61

○ be the signal part of the input signal filtered by the notch filter,

○ be the filtered incoming signal multiplied by the local carrier.

It is assumed that the interfering component is totally filtered out by the notch filter.

To derive the expression of the correlation function in the presence of notch filter, the

expression of the prompt correlator output has to be computed. Mathematically, the

expression of the prompt component is derived by using Figure 3-18.

Figure 3-18: The correlation performed between the incoming signal and the local PRN code replica in the presence of notch filter

In Figure 3-18:

- is the local carrier generated by the PLL and given by equation (2.20),

- is the local code (prompt component) generated by the DLL and given by equation

(2.21).

At first, the signal passes through the notch filter placed at the phase and delay tracking loops

input. The analytical expression of is:

62

(3.34)

The cosine term of the incoming signal can be written as the sum of two exponentials, so

equations (3.33) and (3.34) lead to:

(3.35)

The signal is then multiplied by the local carrier. The expression of is thus given by:

(3.36)

For the following, it is assumed that:

- the carrier phase and Doppler frequency errors are negligible, or, equivalently, the

PLL tracks perfectly the carrier phase and Doppler frequency of the satellite,

- double frequency terms are negligible due to the low pass filtering performed by

the Integrate and Dump filters (summation in equation (3.36)).

Thus, becomes:

(3.37)

The multiplication by the local code leads to:

63

(3.38)

and the prompt correlator assumes the following form:

(3.39)

According to section 2.1.2, any C/A code is almost uncorrelated with any other PRN code. As

a consequence, equation (3.39) can be simplified as:

(3.40)

It is assumed that the integration time is less than 20 ms (duration of a navigation bit) and that

the data navigation term is constant during the integration time. Hence, the data navigation

term is neglected and equation (3.40) becomes:

(3.41)

The analytical expression of the correlation function between the incoming and local PRN

code in the continuous time domain is given by equation (2.7). In the discrete time domain,

equation (2.7) becomes:

(3.42)

Hence, equation (3.41) can be written as:

64

(3.43)

where:

○ is the code delay estimate error (for the satellite ).

According to the mathematical definition of the convolution product:

(3.44)

Hence, equation (3.44) in equation (3.43) leads to:

(3.45)

Any auto-correlation function of a real signal is a real and even function [19]. Since is

the auto-correlation function of local code, is even. Equation (3.45) is thus equivalent

to:

(3.46)

where:

○ is the noise component,

○ .

65

It is concluded that the correlator output in the presence of notch filter can be viewed as the

ideal correlation function filtered by a filter characterized by the impulse

response .

In the frequency domain, the signal part of equation (3.46), denoted , can be

written as:

(3.47)

where:

○ FT denotes the Fourier Transform.

Substituting equation (3.1) into equation (3.47) leads to:

(3.48)

The Inverse Fourier transform of equation (3.48) leads to:

(3.49)

where:

○ IFT denotes the Inverse Fourier Transform.

According to equation (3.49), two types of distortions are introduced by notch filters on the

correlation function:

- amplitude distortion (due to ). It can potentially make the

correlation function asymmetric with respect to the correlation peak and introduce a

bias in the code delay estimate,

- translation of the correlation peak on the time axis and phase distortion (due to

66

). This can lead to an additional delay on the code

delay estimate possibly depending on the estimated Doppler frequency.

The impact of IIR notch filters and of linear phase FIR notch filters on the correlation

function are analyzed in the next paragraphs.

3.2.1. IIR notch filters

As explained in the last paragraph, there is a double impact of notch filters on the correlation

function.

Firstly, they cause a translation of the correlation function on the time axis. Hence, the

correlation peak is also translated. This translation depends on the phase characteristics of the

utilized notch filter. If IIR notch filters are utilized, it is not possible to easily characterize the

correlation peak translation on the time axis since the phase of IIR notch filters is not linear. It

is assumed that the additional delay on the correlation peak due to IIR notch filtering is .

Hence, the peak of the correlation function in the presence of IIR notch filter is located at:

(3.50) where:

○ is the code delay in the absence of IIR notch filtering (true code delay of satellite ).

Secondly, the amplitude of the correlation function is distorted by the presence of notch filter.

In case of IIR notch filter, this distortion is, in general, asymmetric. In other words, the

correlation function in the presence of IIR notch filter is not symmetric with respect to the

correlation peak. That leads to introduce a bias in the code delay estimate. To illustrate that,

consider Figure 3-19. This figure shows:

- the normalized correlation function in the absence of IIR notch filter denoted

(blue curve),

- the normalized correlation function in the presence of IIR notch filter denoted

(red curve).

67

The early, prompt and late correlator outputs in the absence of IIR notch filter are denoted

whereas they are denoted when a IIR notch filter is present.

Figure 3-19: Representation of the normalized correlation functions in the absence and presence of IIR notch filter

It is assumed that the DLL is locked. Indeed, the early and late components have the same

value. As a consequence, according to the algorithm of the standard discriminators provided

in Section 2.3., the discriminator output is equal to zero.

In the absence of notch filtering, the correlation function is symmetric. For a given correlator

spacing, the final code delay estimate matches the correlation peak.

In the presence of notch filtering, the correlation function is asymmetric. For the same

correlator spacing, the final code estimate delay does not match the correlation peak. Indeed,

68

the final code delay estimate corresponds to , where is the bias introduced by

the amplitude distortion.

The asymmetry of the correlation function caused by the presence of IIR notch filtering

introduces a bias in the code delay estimate.

The correlation function in the presence of IIR notch filter is not symmetric anymore. Due to

its complexity, it is thus useful to approximate the distorted correlation function by a

polynomial. This function can be used to predict the asymmetry. A third order Taylor

expansion around the correlation peak located at leads to approximate the absolute value

of the correlation function in presence of notch filter by the following expression:

(3.51)

Using the convention:

(3.52)

Equation (3.51) becomes:

69

(3.53)

The code delay estimate is solution of the following equation:

(3.54)

Equations (3.53) and (3.54) can be used to derive the analytical expression of the code delay

estimate:

(3.55)

Hence, the bias due to amplitude distortions is given by:

(3.56)

As a conclusion, the bias on the code delay estimate caused by the asymmetry of the

correlation function in the presence of IIR notch filter can be approximated as a function of:

- the coefficients of the polynomial function which approximates the absolute value

of correlation function in the presence of IIR notch filter,

- the correlator spacing.

However, since the phase of IIR notch filter is not linear (see Figure 3-4), the additional delay

caused by the phase distortion can not be easily predicted. This additional delay can be

determined in the presence of FIR linear phase notch filters, as discussed below.

3.2.2. Linear phase FIR notch filters

According to section 3.1.2, any Nth order linear phase FIR notch filter has a constant group

delay equal to . This is equivalent to express the phase function of any linear phase FIR

filter as:

70

(3.57)

Substituting equation (3.57) into equation (3.48) leads to:

(3.58)

In the time domain, equation (3.58) becomes:

(3.59)

where:

○ is the Dirac delta defined as:

(3.60)

From equation (3.59), it clearly emerges that linear phase filters introduce an additional delay

equal to . This delay does not depend on the estimated signal Doppler frequency. Thus,

this bias will affect in the same way all the received signals and will be removed by the

navigation solution. More specifically, it will be absorbed by the clock bias.

Moreover, it can be demonstrated that the amplitude distortion caused by a linear phase FIR

notch filter on the correlation function does not affect the symmetry of the correlation

function. In other words, if the correlation function in the absence of notch filter is a

symmetric function, the correlation function in the presence of FIR notch filter with linear

phase remains a symmetric function. This demonstration is provided in Section A.1. for real

FIR notch filters with linear phase and in Section A.2. for complex FIR notch filters with

linear phase.

71

In this section, the bias introduced by a notch filter has been discussed. In the next section, the

expression of the tracking jitter in the presence of notch filter is provided.

3.3. Theoretical analysis of the impact of notch filters

on tracking jitter

The analytical expression of the tracking jitter without filtering stages and with white noise is

given by equation (2.27). The same formula can be used to approximate the tracking jitter in

the presence of a wideband receiver front-end filter [25]. In the literature, the tracking jitter

analytical expression in the presence of limited receiver pre-correlation bandwidth and non-

white noise is proposed [25]. This formula is quite difficult to evaluate. An approximated

expression of this formula is proposed in [5]. This approximation is achieved as follows. The

tracking jitter expression without filtering stages depends on the at the input of the

DLL. In the presence of filtering stages, there is a degradation of the at the input of the

tracking block [5]. Indeed, the in the presence of filtering stages is:

(3.61)

where:

○ is the at the input of the loop in the absence of filtering stages,

○ is the filtering loss. This loss can be expressed as follows [5]:

72

(3.62)

where:

○ is the sampling frequency,

○ is the PSD of the GNSS signal given by equation(2.16),

○ is the equivalent transfer function of the input filters. In this project, two input filters

have to be taken into account:

- the front-end filter. Front-end filter is the low-pass filter performed by the RF front-

end block (see section 2.1.1). Let be its transfer function.

- the notch filter.

Hence, the equivalent transfer function of the input filters is:

(3.63)

The tracking jitter in the presence of filtering stages is approximated by substituting into

equation (2.27) the effect of the filtering loss studied in the last paragraph (equation (3.62)).

Finally, the approximated tracking jitter in the presence of filtering stages is:

(3.64)

Hence, there is a double impact of notch filters on DLL. Firstly, they introduce a bias on the

code delay estimate. Secondly, they modify the analytical expression and the measurements

of the tracking jitter. Chapter 4 aims at validating the analytical results on the impact of notch

filters on DLL derived in this Chapter.

73

4. Simulation and real data

analysis

In Chapter 3, a theoretical analysis of the impact of IIR and FIR notch filters on the

correlation function is provided. The first section of this Chapter aims at validating the results

derived in Chapter 3. Next, experimental results about the effect of IIR and FIR notch filters

in terms of , pseudo-range error and position error are shown and analyzed .

4.1. Theoretical results validation

4.1.1. Impact of IIR notch filter on the correlation function

As explained in Section 3.2.1., the amplitude of the correlation function is distorted by the

presence of a notch filter. In the case of IIR notch filter, this distortion is, in general,

asymmetric. That leads to a bias in the code delay estimate. A theoretical approximation of

this bias is given by equation(3.56). This section aims at comparing the theoretical and

experimental values of the bias on code delay estimate caused by the amplitude distortion of

IIR notch filters (denoted ).

The approach to quantify the experimental value of the bias is as follows. At first, the ideal

correlation function in the presence of a front-end filter is simulated. Substituting the front-

end filter impulse response, , into equation (3.46) leads to:

(4.1)

74

where:

○ is the normalized correlation function in the presence of the front-end filter.

For the numerical simulations:

- the sampling frequency is set to ,

- the IF is set to ,

- the Doppler frequency of the simulated satellite signal is set to .

This choice is conventional and does not introduce any loss of generality.

The front-end filter is a 8-th order Butterworth filter with a 3dB cut-off frequency equal to

. The experimental bias due to the asymmetric amplitude distortion of

the front-end filter is graphically evaluated. The early-late components technique is used to

estimate the estimated code delay. This technique consists in solving equation (3.54) with a

correlator spacing fixed and equal to in this simulation. In the following,

denotes the bias due to the amplitude distortion of the front-end filter.

Secondly, the ideal correlation function in the presence of the front-end filter and an IIR notch

filter is simulated. In this case, the normalized correlation function in the presence of front-

end and notch filter, , is:

(4.2)

The notch filter is a complex IIR notch filter. The notch frequency is set to ,

and the pole contraction factor is equal to . Hence, the notch frequency belongs to

the main lobe of the spectrum of the incoming signal after the front-end block. The

experimental bias due to the asymmetric amplitude distortion of both front-end and notch

filters is evaluated. The same technique as for the determination of the bias due to the front-

end filter alone is used. In the following, denotes the bias due to the amplitude

distortion of the front-end filter and notch filter.

Thirdly, the experimental value of is achieved by comparing and :

75

(4.3)

The normalized correlation functions in the presence of the front-end filter only (blue curve)

and front-end and notch filter (red curve) are shown in Figure 4-1.

Figure 4-1: Normalized correlation functions in the absence and presence of IIR notch filter

Third order Taylor expansions of the correlation functions and close to their

correlation peaks are performed in order to achieve the theoretical value of the bias. The

Taylor approximations of the correlation functions in the absence and presence of the notch

filter are shown in Figure 4-2 and in Figure 4-3, respectively. In these figures, the

experimental biases due to the amplitude distortion of the front-end filter only and of both

front-end and notch filters have been evaluated using the early-late components technique

(dashed lines).

76

Figure 4-2: Approximation of the correlation function in the absence of notch filter by a third order Taylor expansion

Figure 4-3: Approximation of the correlation function in the presence of the complex IIR notch filter by a third order Taylor expansion

It is observed that inFigure 4-2 and in Figure 4-3 that the Taylor approximation does not

match the absolute value of the correlation function in the absence and presence of notch

filter. Table 4-1 shows the experimental and theoretical values of the bias due to the

amplitude distortion of the notch filter, .

Table 4-1: Bias on the code delay estimate due to the amplitude distortion of the complex IIR

77

notch filter

Experimental value 156.4

Theoretical value 61.3

As a conclusion, the theoretical approximation used for determining the bias due to the

amplitude distortion of the complex IIR notch filter does not match the experimental value.

This is due to the fact that the third order polynomial approximation of the correlation

functions is not accurate enough. Hence, the theoretical value of the bias based on the

coefficients of the Taylor expansion (denoted in equation(3.53)) does not match

the experimental bias achieved by the early-late components technique. The Taylor series

approach has been tested for higher order expansion. However, this polynomial

approximation was slowly converging to the true correlation function. For this reason, a

different approach has been adopted.

Another technique consists in determining the coefficients of the polynomial approximation

the correlation functions using the LMS algorithm. In this case, the coefficients of the

approximating polynomial are found by minimizing the Mean Square Error on the whole

support of the function to be interpolated. It has been found that a 9-th order polynomial is

required for accurately approximate filtered correlation functions. These correlation functions

and their polynomial approximations are shown in Figure 4-4 and in Figure 4-5.

78

Figure 4-4: Approximation of the correlation function in the absence of notch filter by a 9-th order polynomial

Figure 4-5: Approximation of the correlation function in the presence of the complex IIR notch filter by a 9-th order polynomial

It is observed in Figure 4-4 and in Figure 4-5 that the 9-th order polynomial approximation

matches the absolute value of the correlation function in the absence and presence of notch

filter. Hence, this approximation can be used to compute the bias due to the amplitude

distortion introduced by IIR notch filters. It is noted that the order of the polynomial

approximation is quite high. This shows that analytical methods for the determination of the

asymmetry bias hardly provide satisfactory results unless high order approximations are used.

For this reason, the direct computation of the filtered correlation and the use of the numerical

79

techniques for finding the correlation peak can be directly used as an alternative of the

polynomial approximation described below.

The adopted polynomial approximation method consists in the following steps. At first, the

coefficients of the 9-th order polynomial approximation of the correlation function,

, are computed using the LMS algorithm. Thus, the correlation function closed to

the correlation peak can be expressed as:

(4.4)

Next, a numerical technique is used to determine the estimated code delay, which is solution

of equations (3.54) and (4.4). The estimated delay is then subtracted to the delay

corresponding to the correlation peak in order to derive the value of bias due to the amplitude

distortion. This technique is used to determine precisely the bias due to the amplitude

distortion of:

- the complex IIR notch filter presented above,

- a real IIR notch filter with the same parameters of the complex IIR notch filter

(notch frequency equal to , and pole contraction factor is equal to

).

The biases are determined for different values of Doppler

frequency: in Table 4-2.

Table 4-2: Bias on the code delay estimate due to the amplitude distortion of IIR notch filters

Doppler frequency due to the amplitude

distortion of the complex IIR notch filter

due to the amplitude distortion of the real IIR notch

filter

80

-5 kHz 156.56 168.53

-2.5 kHz 156.49 168.42

0 Hz 156.43 168.31

2.5 kHz 156.38 168.20

5 kHz 156.33 168.09

From Table 4-2 it is possible to observe that:

- the biases due to the amplitude distortion of the real IIR notch filter are larger than

the biases due to the complex IIR notch filter. Indeed, real notch filters aim at

attenuating two notch frequencies ( ), whereas only one frequency is attenuated by

complex notch filters ( ), so the signal is more distorted by real notch filters than

by complex notch filters.

- as predicted in equation (3.49), the bias due to the amplitude distortion of IIR notch

filters depends on the estimated signal Doppler frequency. Thus, this bias will not affect

in the same way all the received signals and will not be completely removed by the

navigation solution. The difference between the determined biases is however of the

order of a few tenths of nanoseconds. Thus, the impact of this bias will be quite small.

This fact is further investigated in the following.

4.1.2. Impact of linear phase FIR notch filter on the correlation function

From Chapter 3, there is a double impact of FIR notch filters with linear phase on the

correlation function. At first, they introduce a constant additional delay on the correlation

peak. Next, they generate a symmetric amplitude distortion on the correlation function. These

properties are validated in this section.

Additional delay on the correlation peak

Theory proves that the peak of the correlation function in the presence of FIR notch filter with

linear phase is affected by a constant delay. This result is validated as follows. The correlator

81

outputs of a real signal in the absence and presence of a FIR notch filter with linear phase are

compared and the experimental additional delay on the correlation peak due to the notch filter

is measured. The correlation functions in the presence and absence of notch filter have been

obtained using “time domain FFT method” [28]. This technique is briefly described herein.

From Figure 3.16, the correlator output in the presence of notch filter can be expressed as:

(4.5)

where:

○ is the input signal after Doppler removal,

○ is the locally generated code delayed by .

Equation (4.5) is equivalent to:

(4.6)

Using the convention:

(4.7)

Equations (4.6) and (4.7) lead to:

(4.8)

Equation (4.8) can be viewed as the convolution between the signals and :

(4.9) The calculation of this convolution product can be achieved in the frequency domain using

the observation that convolution in the time domain corresponds to multiplication in the

frequency domain. In the frequency domain, equation (4.9) becomes:

(4.10)

where:

82

(4.11)

Hence, the correlator output is obtained by multiplying the Fourier transform of the

signal , product of the incoming signal filtered by the notch and the local carrier, and the

Fourier transform of the local PRN code signal :

(4.12)

This can be approximately achieved by using the FFT algorithm. This algorithm leads to a

circular correlation. Details relative to this technique can be found in [31].

In the absence of notch filter, the expression of the correlator output is similar to

equation(4.12).

The incoming signal is a GPS L1 signal with a sampling frequency is equal to .

The intermediate frequency is 0.42 MHz. The Doppler frequencies of the satellites in view

have been estimated by the PLL. Three signals with different Doppler frequencies are

considered: PRN2, PRN 29, PRN 30.

Two FIR notch filter with linear phase are considered:

- a 200-th order real filter designed by using the windowed Fourier series design

approach.

- a 200-th order real filter designed by using the series expansion approach.

The windowed Fourier series design approach and the series expansion approach are

described in Section 3.2.2.

The notch frequency is equal to 0.42 MHz for both filters.

The correlator output in the absence and presence of the real FIR notch filter designed with

the windowed Fourier series design approach for PRN 29 is shown in Figure 4-6. The same

curves have been plotted for the FIR notch filter designed with the series expansion approach

in Figure 4-7.

83

Figure 4-6: Correlation functions in the absence and presence of the real notch filter with linear phase designed with the windowed Fourier series approach (real data)

Figure 4-7: Correlation functions in the absence and presence of the real notch filter with

84

linear phase designed with the the series expansion approach (real data)

It is observed in Figure 4-6 and in Figure 4-7 that:

- the particular shape of the correlation function in the presence of the real FIR notch

filter designed with the windowed Fourier series design approach makes the detection

of the correlation peak impossible. Indeed, the correlation function has a notch at its

maximum and it is split in two lobes. The risk is that the receiver locks on one of those

two secondary peaks located 0.6 µs before and after the correlation peak. As a

conclusion, the windowed Fourier design approach does not provide satisfactory results

for navigation purpose. For this reason, it is recommended to use FIR notch filter based

on the series expansion design approach,

- the additional delay on the correlation peak due to the real FIR notch filters is equal

to 20.0 µs. From Chapter 3, the theoretical value of the additional delay is equal to the

group delay of the filter, which is , where is the order of the

real notch filters. The theoretical prediction of the additional delay on the correlation

peak due to FIR notch filters with linear phase is thus validated.

The experimental delays on the correlation peak have been measured for two other satellite

signals (PRN 2 and PRN 30). Two FIR notch filters with linear phase designed with the series

expansion approach are considered:

- a 200-th order real filter,

- a 200-th order complex filter.

The results are shown in Table 4-3.

Table 4-3: Additional delay on the correlation peak due to the real and complex FIR notch filters with linear phase designed with the series expansion approach

Additional delay on the correlation peak [µs]

PRN

Doppler frequency

estimate (by the PLL) [Hz]

Real FIR notch filter with linear phase designed with

the series expansion approach

Complex FIR notch filter with linear phase

designed with the series expansion approach

85

2 -2600 20.0 20.0

29 1754 20.0 20.0

30 -1750 20.0 20.0

From Table 4-3, and as predicted in Chapter 3, the additional delay due to the FIR notch

filters with linear phase does not depend on the estimated signal Doppler frequency. Thus,

this bias will affect in the same way all the received signals and will be removed by the

navigation solution.

Bias due to the amplitude distortion

Theory proves in Chapter 3 that the amplitude distortion caused by a linear phase FIR notch

filter on the correlation function does not affect the symmetry of the correlation function. To

validate this fact, the ideal correlation function is simulated using equation (2.15). The same

real FIR notch filters (designed with the windowed Fourier series approach and with the series

expansion approach) and complex FIR notch filter (designed with the series expansion

approach) as those used in the previous paragraph have been implemented. The sampling

frequency is also unchanged. The Doppler frequency is zero. The correlation function in the

presence of the FIR notch filters is simulated using equation (3.46). The correlation functions

in the absence and presence of the real FIR notch filters designed with the windowed Fourier

series approach and the series expansion approach are shown in Figure 4-8 and Figure 4-9,

respectively. The same curves have been plotted for the complex FIR notch filter in Figure

4-10.

86

Figure 4-8: Correlation functions in the absence and presence of the real notch filter with linear phase designed with the windowed Fourier series approach (simulation)

Figure 4-9: Correlation functions in the absence and presence of the real notch filter with linear phase designed with the series expansion approach (simulation)

87

Figure 4-10: Correlation functions in the absence and presence of the complex notch filter with linear phase designed with the series expansion approach (simulation)

As predicted in Chapter 3, it is observed in Figure 4-8,Figure 4-9 and Figure 4-10 that the

absolute value of the correlation function in the presence of real and complex FIR notch

filters with linear phase is symmetric with respect to the correlation peak. Moreover, the

shape of the correlation function in the presence of the real notch filter designed with the

windowed Fourier series approach obtained by simulation is similar to the shape of the

correlation function obtained by real data experiment (see Figure 4-6). This confirms the fact

that the amplitude distortion introduced by FIR notch filters designed with the windowed

Fourier series approach is not compatible with a precise code delay estimate.

4.2. Real data analysis In the first section of this chapter, some results on the bias introduced by a notch filter on the

code delay estimate have been validated. The goal of this section is to evaluate the impact of

notch filters on the position solution provided by processing real data with the University of

Calgary Software Receiver (GSNRxTM) [23]. In the first part of this section, a brief

presentation of the experimental setup used for the data collection is provided. In the second

part of this section, the results obtained by processing the data with GSNRxTM are shown and

analyzed.

88

4.2.1. Experimental setup

Firstly, the experimental setup is presented. The experiment consists in:

‐ collecting the data,

‐ down-converting, sampling and quantifying the collected signal with a signal

analyzer, ‐ computing the coefficients of the transfer function of the notch filter to implement,

quantifying these coefficients and entering the computed coefficients into GSNRxTM,

‐ processing the quantified data with GSNRxTM.

Figure 4-11 shows the experimental setup described above.

Figure 4-11: Block diagram of the experimental setup

The data have been collected on the 14th of May 2010 using a fixed antenna positioned on the

roof of the Calgary Center for Innovative Technologies (CCIT). The data are live data from

the GPS L1 C/A signal. The antenna is connected to a signal analyzer (National Instrument

NI-PXI-5660) composed of three front-end filters which aim at:

89

‐ down-converting the collected signal to a selectable IF and with a selectable

bandwidth (see Section 2.1.1. for down-converter operations). The IF is equal to 0.42

MHz in this experiment, and the front-end bandwidth is equal to 5.0 MHz.

‐ digitalizing the collected signal with a selectable sampling frequency (real or in

phase/quadrature sampling).The sampling frequency is equal to 5.0 MHz in this

experiment. The collected data are sampled with the in phase/quadrature sampling

technique.

‐ quantizing the collected signal with a selectable number of quantization bits. The

number of quantization bits is equal to 16 in this experiment.

Next, the collected data are analyzed using GSNRxTM developed by the Position and

Navigation (PLAN) Group of the University of Calgary. GSNRXTM is able to provide user

position solution in the presence of real or complex notch filters. The coefficients of the

transfer function of the notch filter have to be quantized over bits and entered as inputs into

GSNRxTM. The quantization consists in approximating each coefficient by an integer between

and . It is noted that GSNRxTM adopts a fixed-point arithmetic. In order not to

exceed the size of the data type used by GSNRxTM (C++ integers on 32 bits), the number of

quantization bits used to quantize the coefficients of the transfer function of the notch filter

coefficients is limited to 11. Because of the particular shape of the correlation function in the

presence of real FIR notch filters with linear phase (see previous Section), this type of filter

cannot be used as CW interference mitigation technique in a GNSS receiver. More

specifically, the results obtained using this type of filter were too poor to obtain any valid

navigation solution. This confirm the fact that the design technique described in Chapter 3

based on the windowed Fourier series design approach does not provide satisfactory results

foe navigation purposes. Hence, the impact of complex notch filters on the position solution is

considered in this experiment. Two complex notch filters have been chosen:

‐ a complex IIR notch filter. In [5], the impact of the notch frequency on the position

solution has already been investigated. Hence, the notch frequency is fixed in this

project and equal to . The pole contraction factor equal to ,

‐ a complex 200-th order FIR notch filter with the same notch frequency

. This FIR notch filter is designed with the series expansion approach

and is derived from a IIR notch filter with a pole contraction factor equal to 0.95 (see

Section 3.1.2. for the series expansion design approach).

90

It is noted that the contraction factor of the IIR and FIR notch filters have been chosen to

provide IIR notch filter and FIR notch filter similar amplitude responses.

Figure 4-12 shows the amplitude response of these filters after quantization of their transfer

function coefficients over bits.

Figure 4-12: Amplitude response of the complex FIR and IIR notch filters (coefficients quantized over 11 bits)

It is observed in Figure 4-12 that the amplitude response of IIR and FIR notch filters are

similar. Indeed, the IIR and FIR filters have been chosen to have the same notch width (0.18

MHz). In this way, the impact of the IIR and the FIR on the position solution can be

compared. It is also noted that the attenuation of the notch frequency due to the IIR notch

filter is smaller than the attenuation due to the FIR notch filter (-50.0 dB attenuation for the

IIR notch filter and -70 dB for the FIR notch filter).

Finally, the position solution in the presence of these filters is provided by GSNRxTM and the

results are shown in the next section.

4.2.2. Experimental results and analysis

The impact of FIR and IIR notch filters on the:

‐ ,

‐ pseudo-range error,

91

‐ position error,

is reported in this section.

At first, the effect of the FIR and IIR filters in terms of is investigated. Figure 4-13 and

Figure 4-14 show the for satellite PRN 11 in the absence and presence of the IIR notch

filter and of the FIR notch filter, respectively.

Figure 4-13: for PRN 11 in the absence and presence of the IIR notch filter

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Figure 4-14: for PRN 11 in the absence and presence of the FIR notch filter

It is observed in Figure 4-13 and Figure 4-14 that there is a degradation of the due to

the presence of IIR or FIR notch filters: on average, the decreases by 1.2 dB-Hz due to

the presence of the IIR notch filter and by 0.2 dB-Hz due to the presence of the FIR notch

filter. The attenuation of the in the presence of notch filters can be interpreted as

follows. From Section 2.1.2., almost all the power of the GNSS signal is concentrated in the

main lobe of the PSD of the GNSS signal. The notch frequency belongs to the main lobe of

the spectrum of the GNSS signal. As a consequence, useful signal components are excised by

the notch filter from the spectrum of the GNSS signal. The power of the useful signal is thus

reduced by the presence of notch filters. This causes degradation in terms of due to the

notch filters.

Table 4-4 shows the mean in the presence of the IIR notch filter and the FIR notch

filter for different PRNs.

Table 4-4: Mean for different PRNs in the presence of the IIR and the FIR notch filter for different PRNs

IIR notch filter FIR notch filter

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PRN Mean [dB-Hz] Mean [dB-Hz]

11 47.4 48.4

20 41.4 42.4

30 40.7 41.9

32 48.5 49.5

It is observed from Table 4-4 that the IIR notch filter degrades the more than the FIR

notch filter. Indeed, in the presence of the IIR notch filter, the mean is smaller than the

mean in the presence of the FIR notch filter (about 1.0 dB-Hz less).

The use of back-forward filtering seems to lead to a sharper notch and thus a lower loss in the

effective . Although the loss difference is not significant, further theoretical

investigation are required to confirm this result. More specifically, the theoretical formula

reported in equations (3.61) and (3.62) should be used to support this result. This analysis is

left for future work.

Next, the effect of the FIR and IIR filters in terms of pseudo-range error is investigated.

Figure 4-15 shows the pseudo-range error for two PRNs (PRN 32 and 11) due to the presence

of the IIR and FIR notch filter.

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Figure 4-15: Pseudo-range error due to the presence of the IIR notch filter an of the FIR notch filter for PRN 11 and 32

It is observed that in Figure 4-15 there is a jump in the pseudo-range error after 18 s from the

beginning of the experiment. This can be interpreted as follows. Before , the pseudo-

ranges of the j-th satellite signal in the absence and presence of a notch filter are, respectively:

(4.13)

where:

○ is the geometric range between the j-th satellite and the receiver at the instant ,

○ is the clock bias at the instant ,

○ is the bias on the pseudo-range due to the notch filter at the instant .

Hence, from equation(4.13), the pseudo-range error due to the notch filter before is

given by:

(4.14) can be expressed as:

(4.15) where:

○ is the component of the bias due to the notch filter which depends on the PRN,

○ is the component of the bias due to the notch filter which is common to all PRNs. This

bias is absorbed by the clock bias.

Substituting equation (4.13) in equation(4.15) leads to:

(4.16)

95

where:

○ is the equivalent clock bias in the presence of notch filter.

At , a clock steering is performed: the clock bias is removed from the pseudo-range

measurements. Hence, after , the pseudo-ranges of the j-th satellite signal in the

absence and presence of a notch filter are, respectively:

(4.17)

Hence, from equation (4.17), the pseudo-range error due to the notch filter after is

given by:

(4.18)

From equations (4.14), (4.15) and (4.18), the jump in the pseudo-range error shown in Figure

4-15 corresponds to the component of the bias due to the notch filter which does not depend

on the single PRN, . Table 4-5 shows the mean pseudo-range errors due to the presence of

the IIR notch filter and of the FIR notch filter before and after the clock steering and for

different PRNs.

Table 4-5: Mean pseudo-range errors due to the FIR notch filter and the IIR notch filter for different PRNs

Before the clock steering

Mean pseudo-range error [m]

After the clock steering

Mean pseudo-range error [cm]

PRN IIR notch filter FIR notch filter IIR notch filter FIR notch filter

11 5.45 1.62 11.83 11.01

20 5.98 1.63 -9.70 23.63

30 4.96 1.75 -4.08 9.78

32 5.30 1.78 1.29 12.72

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It is observed in Table 4-5 that the jump in the pseudo-range error, , due to:

‐ the IIR notch filter is in average equal to 5.4 m (1.8.10-2 chip),

‐ the FIR notch filter is in average equal to 1.6 m (5.3.10-3 chip).

It is theoretically proved in Section 3.2.2. and validated in Section 4.1.2. that FIR notch filters

with linear phase cause a constant bias on the code delay estimate equal to for

the FIR notch filter used in this experiment. This bias on the code delay estimate corresponds

to a bias on the pseudo-range measurements equal to 6.0 km. However, the common bias on

the pseudo-range measurements provided by GSNRxTM is equal to 1.6 m in the presence of

the FIR notch filter. This difference is due to the way the receiver clock is initialized in

GSNRxTM. At first, the receiver clock is initialized to 0 and progressively incremented by

number of samples the receiver is processing. When the receiver has extracted the GPS time

for the first time, the receiver time is set to:

(4.19)

A time of 69 ms is added to the GPS time, denoted in equation (4.19), for

numerical reasons. More specifically, this initialization allows a faster convergence of the

algorithm used for solving for the navigation solution. Thus, in the presence of the FIR notch

filter, both receiver time and transmit time are affected by a constant delay equal to .

Hence, the pseudo-range measurements in the presence of the FIR notch filter does not take

account on this bias. That is the reason why the jump in the pseudo-range error is not equal to

6.0 km. The jump of 1.6 m is probably caused by a residual common bias due to different

errors.

Finally, the effect of the FIR and IIR filters in terms of position error is investigated. Figure

4-16 and Figure 4-17 show the position errors due to the IIR notch filter and the FIR notch

filter in the east and north directions, respectively.

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Figure 4-16: Position error due to the IIR notch filter and the FIR notch filter (east direction)

Figure 4-17: Position error due to the IIR notch filter and the FIR notch filter (north direction) It is observed in Figure 4-16 and Figure 4-17 that the mean position errors in the east and

north directions is smaller in the presence of the FIR notch filter than in the presence of IIR

notch filter. Moreover, the variance of the position error due to the FIR notch filter is about

four times smaller than the variance of the error due to the IIR notch filter for both east and

north directions. As a conclusion, the position estimate in the east and north directions is more

accurate in the presence of the FIR notch filter than in the presence of the IIR notch filter. It is

98

also noted that the mean position error due to the presence of a notch filter is a few

centimeters.

Figure 4-18 shows the position errors due to the IIR notch filter and the FIR notch filter in the

up direction. Figure 4-19 shows the clock bias error due to the IIR and FIR notch filters.

Figure 4-18: Position error due to the IIR notch filter and the FIR notch filter (up direction)

Figure 4-19: Clock bias error due to the IIR notch filter and the FIR notch filter

99

It is observed in Figure 4-18 that the variance of the position error due to the FIR notch filter

is about ten times smaller than the variance of the error due to the IIR notch filter for the up

direction. Hence, the position estimate in the up direction is, as for the east and north

directions, more accurate in the presence of the FIR notch filter than in the presence of the IIR

notch filter. Moreover, the variances of the position error due to the IIR and FIR notch filters

in the up direction are larger than the variances of the position errors in the east and north

direction. As a consequence, the notch filters degrade the accuracy of the position estimate in

the three directions (east, north and up), but particularly in the up direction. This is expected

because of poorer geometry with respect to the up direction. As for the east and north

directions, the mean position error due to the presence of a notch filter is a few centimeters for

the up direction.

It is observed that the shape of the position error in the up direction and the clock bias error

are similar. [30] explains that there is a strong correlation between the up position error and

the clock bias error. Indeed, moving the antenna along the vertical axis is very similar to a

clock bias in terms of impact on the pseudo-range measurements. The clock bias adds or

subtracts the same amount from each satellite, and moving the antenna will change each

pseudo-range of about the same amount. Hence, the up position error and the clock bias are

related. Thus, the errors introduces by the notch filter seems to be distributed between these

two components.

As a conclusion, the main results provided by simulation and real data analysis are

summarized as follows.

The bias due to the amplitude distortion of IIR notch filters can be hardly predicted using a

theoretical approach based on low order polynomial approximations of the correlation

function. A 9-th polynomial approximation of the correlation function based on the LMS

algorithm is required for obtaining accurate results. It is validated that this bias depends on the

satellite PRN. The bias is roughly equal to a few of tenths of nanoseconds and is larger in the

presence of real IIR notch filters (two poles) than in the presence of complex IIR notch filters

(one pole).

100

Concerning the impact of FIR notch filters with linear phase on the correlation function, it is

validated that:

‐ the additional delay on the correlation peak is equal to the group delay of the

adopted notch filter,

‐ the amplitude distortion is symmetric.

The real data processing obtained using GSNRxTM shows that:

‐ IIR and FIR notch filters degrade the , and the degradation due to IIR notch

filters is larger than the degradation due to FIR notch filters (1dB-Hz less) when similar

notch width are used,

‐ IIR and FIR notch filters introduce a bias on the pseudo-range measurements. The

bias shared by all the received signals is removed by clock steering. After clock

steering, the bias depending on the PRN remains and is roughly equal to a few of tenths

of centimeters,

‐ IIR and FIR notch filters introduce a position error in the three directions east, north

and up. The position solution in the presence of the FIR notch filter is more accurate

than the position solution in the presence of the IIR notch filter. The mean position error

is roughly equal to a few of centimeters in the three directions.

101

5. Conclusions

This project aimed at investigating the impact of notch filters on a GNSS receiver with a

special focus on their impact on code delay measurements. In the following, the obtained

results are summarized and future works are proposed.

At first, a theoretical analysis of the bias in the code delay measurements due to notch filters

has been provided. Code delay estimate is based on the detection of the cross-correlation peak

between the incoming GNSS signal and the local code replica. Notch filters have a double

impact on the correlation function. Firstly, they cause an amplitude distortion on the

correlation function. Secondly, they translate the correlation peak on the time axis. Amplitude

distortion and translation of the correlation peak on the time axis can potentially introduce

biases in the code delay measurements. The analytical expressions of these biases have been

investigated for two types of notch filters:

‐ Real (two poles) and complex (one pole) IIR notch filters cause an asymmetric

amplitude distortion on the correlation function. This leads to a bias on the code delay

estimate. A theoretical prediction of this bias based on low order polynomial

approximation of the correlation function has been proposed, but simulations show that

this prediction is not accurate enough. A 9-th polynomial approximation of the

correlation function based on the LMS algorithm is required for obtaining accurate

results. It has been proven that the bias due to asymmetric amplitude distortion depends

on the signal Doppler frequency and thus it is not completely removed by the navigation

solution. The introduced bias can be expressed as the sum of two terms. The first one is

common to all input PRN signals whereas the second component is frequency

dependent. The first bias is removed by the navigation solution whereas the remaining

component is quite small (few of tenths of nanoseconds). Simulations show that, for a

given satellite PRN, this bias is larger in the presence of a real IIR notch filter than in

102

the presence of a complex IIR notch filter. The analytical expression of the bias due to

the translation of the correlation peak caused by IIR notch filters is left as future work.

‐ Real and complex FIR notch filters with linear phase cause a symmetric amplitude

distortion on the correlation function. Thus, there is no bias in the code delay estimate

due to the amplitude distortion. Two different design approaches have been considered.

The first one is based on the windowed Fourier series approach. The filter is at first

designed in the frequency domain and its coefficients, in time domain, are determined

using an inverse DFT. This type of filter introduces a significant distortion in the

correlation function and thus cannot be used as CW interference mitigation technique in

a GNSS receiver. The second type of filters is obtained by expanding in series the

transfer function of the IIR notch filter considered above. This series is truncated and

back-forward processing is adopted to make the impulse response of the resulting filter

symmetric. This second class of filters provided satisfactory results. For this type of

filters, the analytical expression of the bias due to the translation of the correlation peak

on the time axis has been derived. Since this bias is common to all PRNs, it will affect

in the same way all the received signals and will be removed by the navigation solution.

The University of Calgary Software Receiver (GSNRxTM) has been used to quantify the

impact of a complex IIR notch filters and a complex FIR notch filter with linear phase on the

, on the pseudo-range measurements and on the position solution. The collected signal

was a GPS L1 signal. The results can be summarized as follows. The degradation due

to notch filters is not significant, and the degradation due to the IIR notch filter is

larger than the one due to the FIR notch filter. The pseudo-range measurement error due to

IIR and FIR notch filters is of the order of a few centimeters whereas the mean position

estimate error is roughly equal to a few tens of centimeters. The position estimate seems to be

more accurate in the presence of the FIR notch filter. However, further investigations are

required.

Finally, the advantages of complex FIR notch filter with linear phase over IIR notch filter are

the following:

103

‐ the bias in the code delay measurements due to FIR notch filter is common to all

satellite signals,

‐ the bias in the final position due to FIR notch filter is thus smaller than the bias due

to IIR notch filter.

From this analysis, it emerges that, even if complex IIR notch filters are simpler to

implement, it is recommended to use complex FIR notch. Further data analyses in the

presence of a real CW interference are required to generalize these conclusions. These

analyses are recommended for future work.

104

Appendix

A.1. Symmetric amplitude distortion due to real FIR

notch filters with linear phase

To prove that the amplitude distortion due to real FIR notch filters with linear phase is

symmetric, it is assumed that the correlation function in absence of notch filter, , is

symmetric with respect to 0. As explained in Section 3.2.2., the correlation function peak is

affected by an additional delay in the presence of linear phase FIR notch filter. Hence,

the correlation peak in the presence of linear phase FIR notch filter is located at . The goal

of the proof is thus to demonstrate that the absolute value of the correlation function in the

presence of real FIR notch filter with linear phase is symmetric with respect to the correlation

peak located at . That is equivalent to demonstrate:

(A.1)

The proof is proposed for N even, but a similar demonstration can be provided for N odd.

In equation (3.46), is defined as:

(A.2)

105

Equation (A.2) evaluated at leads to:

(A.3)

The impulse response coefficients of any N-th order real FIR notch filter are such as [16]:

(A.4)

Equations (A.3) and (A.4) lead to:

(A.5)

From equation(A.5), the squared absolute value of the complex term is:

(A.6)

That is:

(A.7)

Using the following trigonometric relationships,

106

(A.8)

equation (A.7) is equivalent to:

(A.9)

Using the convention ,the second term of the equation (A.9), denoted

as , becomes:

(A.10)

Since is a symmetric function with respect to zero:

(A.11)

In Section 3.1, it is demonstrated that the impulse response coefficients of any linear phase

FIR notch filter are symmetrical with respect to , so:

(A.12)

Hence, substituting equations (A.11) and (A.12) into equation (A.10) leads to:

(A.13)

The first term of equation (A.9) is:

107

(A.14)

that, using the convention, , can be written as:

(A.15)

Since , equation (A.15) further simplifies as:

(A.16)

Finally, substituting equations (A.13) and (A.16) into equation (A.9) gives:

(A.17)

Similarly, equation (A.9) evaluated at assumes the following form:

(A.18)

108

Finally, by comparing equations (A.17) and (A.18), the following result is proven:

(A.19)

As a conclusion, real FIR notch filters with linear phase do not affect the symmetry of the

absolute value of the correlation function.

A.2. Symmetric amplitude distortion due to complex

FIR notch filters with linear phase

As explained in Section A.1. for real notch filters, proving the symmetry of the amplitude

distortion due to complex FIR notch filters with linear phase is equivalent to demonstrate that:

(A.20)

The proof is proposed for N even, but a similar demonstration can be provided for N odd.

As in Section A.1., in the presence of a complex notch filter is defined as:

(A.21)

The impulse response coefficients of any N-th order complex FIR notch filter are such as:

(A.22)

Equations (A.21) and (A.22) lead to:

(A.23)

By using a similar computation method as in Section A.1., equation (A.17) in case of complex

109

FIR notch filter with linear phase becomes:

(A.24)

Equation (A.18) in case of complex FIR notch filter with linear phase becomes:

(A.25)

Finally, by comparing equations (A.24) and(A.25), the following result is proven:

(A.26)

As a conclusion, complex FIR notch filters with linear phase do not affect the symmetry of

the absolute value of the correlation function.

110

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[3] PLAN Group Website http://plan.geomatics.ucalgary.ca/index.php

[4] T. Yu, S. K. Mitra, H. Babic, “Design of linear phase FIR notch filters”, Sadhana, Vol15. Part3, November 1990

[5] G. Giodanengo, “Impact of Notch Filtering on Tracking Loops for GNSS Applications”, 2009

[6] H. Van der Marel, “Lecture notes on Introduction to Satellite Navigation systems (GPS, Glonass and Galileo)”, Delft University of Technology, September 2007

[7] D. Borio, C. O’Driscoll, “Engo 638 lecture notes on GNSS Receiver Design”, PLAN Group UoC, Summer 2009

[8] Geomatics Department website http://www.geomatics.ucalgary.ca/

[9] P. Jarske, Y. Neuvo, S. K. Mitra, “A simple approach to the design of linear phase FIR digital filters with variables characteristics”, Signal processing 14, 1988

[10] P. W. Ward, J. W. Betz, Eds.,“Understanding GPS, Principles and Applications, second edition”, Artech House Publishers, 2006

[11] University of Calgary website http://ucalgary.ca/about/

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