Multipath Mitigation Techniques Suitable For Low Cost GNSS ...€¦ · Multipath Mitigation Techniques Suitable For Low Cost GNSS Receivers Tiago Roque Peres ... Waveforms (CCRWs)

Multipath Mitigation Techniques

Suitable For Low Cost GNSS Receivers

Tiago Roque Peres

A Thesis submitted in fulfilment of the requirements for theDegree of Master of Science in

Aerospace Engineering

Jury

President: Prof. João Manuel Lage de Miranda Lemos

Advisor: Prof. Fernando Duarte Nunes

Vogal: Prof. José Eduardo Charters Ribeiro da Cunha Sanguino

September 2008

Acknowledgements

I would like to thank Deimos Engenharia for giving me the opportunity of training in dynamic and

enterprising company, while developing this exciting subject. I, also, thank everyone who work there,

for their politeness and for the excellent work environment. My special thanks to Eng. João Silva for

his time, availability, technical feedback, suggestions and good moods.

No less important, I am very grateful to my advisor at Instituto Superior Técnico, Prof. Fernando

Nunes, for his time, availability, and priceless corrections and suggestions for this work.

To all my teachers, without whom I would not be the person that I am today, I sincerely thank them.

I am, also, very thankful to all my friends and colleagues at Salesianos de Manique, Mem Martins and

Instituto Superior Técnico. I write here a special word to Catarina and Paulo, not only for their good

friendship but also for helping me achieving my goals several times.

Finally, but not least, I am deeply thankful to my parents and sister, who I owe the most. It is very

difficult, if not impossible, to describe in words my acknowledges for their love, support and

comprehension. To them I dedicate this work.

I

Abstract

Today, Global Navigation Satellite Systems (GNSS) have become available to civilian population.

Because of that, the market of low cost GNSS receivers is very active.

There are several limitations to the use of GNSS receivers: signal delays in the ionosphere, receiver's

clock bias, multipath, etc. Nevertheless, the effect of multipath is very significant, specially in urban

environments, where there are many and large building surfaces that can reflect GNSS signals. So, it

is required low cost receivers that are capable to mitigate the multipath effect in difficult situations, like

urban navigation. Thereby, it is necessary to find multipath mitigation techniques, suitable for use in

low cost GNSS receivers.

In the search for techniques capable of meeting those requirements, several techniques were found:

the Narrow Correlator, the High Resolution Correlator (HRC), several Code Correlation Reference

Waveforms (CCRWs) and a Teager Kaiser operator based technique. The different techniques were

initially analysed using the multipath error envelopes and the steady-state noise. For more

representative results, it was developed a GNSS receiver simulator, using Simulink and GRANADA

FCM Blockset.

From the tested techniques, it is one of the CCRWs that is considered the more suitable for

implementation in low cost receiver.

KEYWORDS: GNSS, Multipath Mitigation, Narrow Correlator, High Resolution Correlator, Code

Correlation Reference Waveforms, Teager-Kaiser.

II

Resumo

Hoje em dia, os Sistemas Globais de Navegação por Satélite (GNSSs) tornaram-se acessíveis à

população em geral. Actualmente, o mercado dos receptores de baixo custo é bastante activo.

Existem várias limitações ao uso dos receptores de GNSS: atrasos dos sinais na ionoesfera, erros de

relógio do receptor, multipercurso, etc. No entanto, o multipercurso destaca-se, sobretudo para

ambientes urbanos, onde existem inúmeras e grandes superficies reflectoras como edifícios. É, então,

exigido que receptores de baixo custo sejam capazes de cumprir requisitos exigentes como a

navegação urbana. Torna-se então necessário encontrar técnicas eficientes a mitigar o efeito do multi-

percurso mas que cuja implementação seja possível em receptor de baixo custo.

Iniciou-se então uma pesquisa por técnicas capazes de cumprir os requisitos e encontraram-se

algumas técnicas: Narrow Correlator, High Resolution Correlator (HRC), algumas Code Correlation

Reference Waveforms (CCRWs) e ainda um técnica baseada no operador Teager Kaiser. As técnicas

foram inicialmente analisadas tendo em conta as envolventes de erro na presença de multipercurso e

o ruído em regime estacionário.

Para resultados mais fidedignos, foi implementado um simulador de receptor GNSS, utilizando o

software Simulink e GRANADA FCM blockset. O receptor implementado fez uso de ajudas em cadeia

fechada.

Concluiu-se que das técnicas utilizadas a que apresenta um maior potencial para um receptor de

baixo custo é uma das CCRWs.

Palavras chave: GNSS, Mitigação de multipercuso, Narrow Correlator, High Resolution Correlator,

Code Correlation Reference Waveform, Teager-Kaiser.

III

Table of Contents

Chapter 1 - Introduction.........................................................................................................................1

1.1 Motivation.......................................................................................................................................1

1.2 Research Objectives......................................................................................................................1

1.3 Thesis outline.................................................................................................................................2

Chapter 2 - Basic Concepts...................................................................................................................3

2.1 GNSS Core Constellations............................................................................................................3

2.2 Principles of GNSS........................................................................................................................4

2.3 Signals...........................................................................................................................................5

2.3.1 Modulations...........................................................................................................................6

2.3.1.1 Binary Phase Shift Keying.............................................................................................6

2.3.1.2 Binary Offset Carrier......................................................................................................7

2.3.1.3 Multiplexed Binary Offset Carrier..................................................................................9

2.3.2 Considered Signals..............................................................................................................12

2.3.2.1 GPS L1 signal..............................................................................................................12

2.3.2.2 GPS L1C......................................................................................................................12

2.3.2.3 Galileo E1 Open Service Signal..................................................................................13

Chapter 3 - Receiver.............................................................................................................................14

3.1 Receiver Architecture Overview..................................................................................................14

3.2 Baseband Signal Processing.......................................................................................................15

3.2.1 Aided Tracking Loops..........................................................................................................19

3.2.1.1 Aided PLL....................................................................................................................19

3.2.1.2 Aided DLL....................................................................................................................19

Chapter 4 - Multipath Effect.................................................................................................................20

4.1 Tracking error due to Multipath....................................................................................................21

4.2 Multipath error envelope..............................................................................................................23

Chapter 5 - Multipath Mitigation..........................................................................................................25

5.1 Narrow Correlator........................................................................................................................26

5.2 High Resolution Correlator..........................................................................................................30

5.3 Code Correlation Reference Waveform.......................................................................................34

5.3.1 Rectangular CCRW.............................................................................................................37

5.3.2 W1 CCRW...........................................................................................................................41

5.3.3 W2 CCRW...........................................................................................................................44

5.3.4 W3 CCRW...........................................................................................................................47

5.3.5 W4 CCRW...........................................................................................................................50

III

5.4 Teager-Kaiser Operator...............................................................................................................52

Chapter 6 - Simulation Setup...............................................................................................................57

6.1 GRANADA FCM Blockset............................................................................................................57

6.1.1 Using FCM with CBOC signals and Strobe Correlator........................................................58

6.1.2 FCM with correlated correlators noise.................................................................................59

6.2 Multipath channel model..............................................................................................................60

6.2.1 Direct Path...........................................................................................................................61

6.2.2 Near Echoes........................................................................................................................61

6.2.3 Far Echoes..........................................................................................................................62

6.2.4 Multipath model with FCM...................................................................................................62

6.3 Simulation Plan............................................................................................................................63

Chapter 7 - Results Discussion..........................................................................................................68

7.1 Comparison between different techniques..................................................................................68

7.2 Aiding signal.................................................................................................................................71

7.3 Simulation Results.......................................................................................................................72

Chapter 8 - Conclusion and Final Remarks.......................................................................................77

8.1 Summary......................................................................................................................................77

8.2 Conclusion...................................................................................................................................77

8.3 Future work..................................................................................................................................78

References.............................................................................................................................................79

Appendix A - Cross-Correlation Function..........................................................................................82

Appendix B - Noise Analysis...............................................................................................................83

Appendix C - Tables of Results...........................................................................................................85

IV

Index of Tables

Table 2.1: Main technical characteristics of GPS L1, GPS L1C and Galileo E1 OS signals [7].............13

Table 6.1: Multipath mitigation techniques selected for simulation.........................................................64

Table 6.2: Multipath Model Parameters..................................................................................................67

Table 7.1: Tested conditions....................................................................................................................71

Table 7.2: Computational complexity......................................................................................................76

Table C.1: BPSK signals and bandwidth BWTc = 5..................................................................................85

Table C.2: BOC(1,1) signals and bandwidth BWTc = 5............................................................................85

Table C.3: CBOC(6,1,1/11) signals and bandwidth BWTc = 5..................................................................86

Table C.4: BPSK signals and bandwidth BWTc = 12................................................................................86

Table C.5: BOC(1,1) signals and bandwidth BWTc = 12. ........................................................................86

Table C.6: CBOC(6,1,1/11) signals and bandwidth BWTc = 12................................................................87

V

Illustration Index

Figure 2.1: Baseline GPS Satellite Constellation.....................................................................................3

Figure 2.2: The principle of satellite navigation........................................................................................4

Figure 2.3: Indirect distance determination...............................................................................................5

Figure 2.4: Power Spectral Density of BPSK, BOC(1,1) and MBOC(6,1,1/11) signals............................9

Figure 2.5: Example of TMBOC (6,1,p) Sub-carrier...............................................................................10

Figure 2.6: Possible CBOC(6,1,1/11) sub-carriers..................................................................................11

Figure 2.7: Auto-correlation functions for unfiltered BPSK, BOC(1,1) and CBOC(6,1,1/11) signals......11

Figure 2.8: GPS and Galileo frequency plan..........................................................................................12

Figure 3.1: A conceptual view of a GNSS receiver.................................................................................15

Figure 3.2: Typical diagram of the receiver code and carrier loops........................................................15

Figure 3.3: Code discriminator outputs for BOC(1,1) with unlimited pre-correlation bandwidth............18

Figure 3.4: Carrier discriminator output for the arc tangent discriminator..............................................19

Figure 4.1: Possible multipath situation..................................................................................................20

Figure 4.2: Normalized in-phase prompt correlator for BOC signals, with α = 0.5 and τ = 0.4Tc.........22

Figure 4.3: E-L power code discriminator outputs for BOC(n,n).............................................................22

Figure 4.4: Multipath error envelopes for the Narrow Correlator and BOC signals...............................24

Figure 5.1: Code discriminator outputs and multipath error envelopes for the NELP discriminator......27

Figure 5.2: Multipath error envelopes for the NELP discriminator and normalized bandwidth BWTc = 5

and BWTc = 12 .........................................................................................................................................28

Figure 5.3: Normalized code error variances for the NELP discriminator..............................................29

Figure 5.4: Multipath error envelopes as function of E-L spacing and attenuation coefficient, for the

NELP discriminator..................................................................................................................................30

Figure 5.5: HRC code discriminator response for BPSK signals for several λ values...........................31

Figure 5.6: Code discriminator outputs and multipath error envelope for HRC, with unlimited pre-

correlation bandwidth and λ = 2..............................................................................................................32

Figure 5.7: Multipath error envelopes for the HRC, with normalized bandwidth BWTc = 5 and BWTc=12 .

................................................................................................................................................................32

Figure 5.8: Normalized code error variances for the HRC.....................................................................33

Figure 5.9: Multipath error envelopes as function of E-L spacing and attenuation coefficient, for the

HRC discriminator with λ=2....................................................................................................................34

Figure 5.10: Block diagram of the receiver code and phase loops using the strobe correlator.............35

Figure 5.11: CCRW waveforms for a BPSK signal.................................................................................37

Figure 5.12: W1 CCRW pulse.................................................................................................................37

Figure 5.13: Code discriminator outputs and multipath error envelopes for the RECT CCRW ...........38

Figure 5.14: Code discriminator outputs and multipath error envelopes for the New RECT CCRW.....39

Figure 5.15: Normalized code error variances versus pre-correlation bandwidth RECT CCRW...........40

VI

Figure 5.16: Multipath error envelopes as function of CCRW pulse width and attenuation coefficient,

for RECT CCRW with BPSK signals and for New RECT CCRW with BOC and CBOC signals............40


Figure 5.18: Code discriminator outputs and multipath error envelopes for W1 CCRW........................42

Figure 5.19: Multipath error envelope for several signals for the W1 CCRW and pre correlation

bandwidth BWTc = 5 and BWTc = 12 .........................................................................................................42

Figure 5.20: Code discriminator outputs and multipath error envelopes for the New W1 CCRW..........43


for W1 CCRW with BPSK signals and for New W1 CCRW with BOC and CBOC signals....................43

Figure 5.22: Normalized code error variances for the W1 CCRW ........................................................44

Figure 5.23: Code discriminators outputs and multipath error envelopes for W2 CCRW......................45

Figure 5.24: Multipath error envelopes for the W2 CCRW and pre correlation bandwidth BWTc = 5 and

BWTc = 12.................................................................................................................................................46

Figure 5.25: Normalized code error variances for the W2 CCRW..........................................................46


for W2 CCRW..........................................................................................................................................47


Figure 5.28: Code discriminator output for several signals and multipath error envelopes for several

signals for the W3 CCRW.......................................................................................................................48


BWTc = 12.................................................................................................................................................48

Figure 5.30: Normalized code error variances versus pre-correlation bandwidth and CCRW pulse

width, for W3 CCRW and several signal modulations: BPSK (top), BOC(1,1) (bottom left) and

CBOC(6,1,1/11) (bottom right)................................................................................................................49


for W3 CCRW..........................................................................................................................................49


Figure 5.33: Code discriminator output for several signals (left) and multipath error envelopes for

several signals (right) for the W4 CCRW................................................................................................50


BWTc = 12.................................................................................................................................................51

Figure 5.35: Multipath error envelopes as function of CCRW pulse duration and attenuation coefficient,

for W4 CCRW..........................................................................................................................................51

Figure 5.36: Normalized code error variances for the W4 CCRW..........................................................52

Figure 5.37: TK operator output for BPSK, BOC(1,1) and CBOC(6,1,1/11) signals..............................53

Figure 5.38: TK operator output for BOC(1,1) for a LOS signal and one reflect ray with different delays

relative the LOS signal............................................................................................................................54

Figure 5.39: Code discriminator response (left) and multipath error envelopes (right) for several signals

and unlimited pre-correlation bandwidth.................................................................................................56

VII

Figure 5.40: Multipath error envelopes for several signals BWTc = 5 and BWTc =12 ...............................56

Figure 6.1: Typical signal processing chain for the tracking of GNSS signals [35]................................58

Figure 6.2: Possible implementation of a multi-channel GNSS receiver's tracking loops [35]...............58

Figure 6.3: Simulink block diagram of the implement receiver simulator...............................................63

Figure 6.4: Receiver trajectory................................................................................................................64

Figure 6.5: Receiver dynamics...............................................................................................................65

Figure 6.6: Echoes amplitudes versus echoes delays, for the different scenarios. ..............................66

Figure 7.1: Multipath error envelopes for BPSK signals, for the different techniques and bandwidths. 68

Figure 7.2: Multipath error envelopes for BOC(1,1) signals, for the different techniques and

bandwidths..............................................................................................................................................69

Figure 7.3: Multipath error envelopes for CBOC(6,1,1/11) signals, for the different techniques and

bandwidths..............................................................................................................................................69

Figure 7.4: Normalized code error variances for BPSK signals,for the different techniques and

bandwidths..............................................................................................................................................70

Figure 7.5: Normalized code error variances for BOC(1,1) signals, for the different techniques and

bandwidths..............................................................................................................................................70

Figure 7.6: Normalized code error variances for CBOC(6,1,1/11) signals for the different techniques

and bandwidths.......................................................................................................................................70

Figure 7.7: Code tracking error for aided PLL versus not aided PLL without multipath (left), and with

multipath..................................................................................................................................................71

Figure 7.8: Code tracking error mean and variance (average of the eight scenarios) for normalized

bandwidth BWTc = 5, different techniques and modulations....................................................................73

Figure 7.9: Code tracking error mean and variance (average of the eight scenarios) for normalized

bandwidth BWTc = 12, different techniques and modulations..................................................................74

Figure 7.10: Code tracking error mean and variance (average of the four worst scenarios) for

normalized bandwidth BWTc = 5, different techniques and modulations. ...............................................74

Figure 7.11: Code tracking error mean and variance (average of the four worst scenarios) for

normalized bandwidth BWTc = 12, different techniques and modulations...............................................75

Figure A.1: Signals a(t) and b(t)..............................................................................................................82

Figure A.2: Cross-correlation function....................................................................................................82

VIII

List of Acronyms

ACF Auto-Correlation Function

BOC Binary Offset Carrier

BPSK Binary Phase-Shift Keying

C/A Coarse / Acquisition

CBOC Complex BOC

CCF Cross-Correlation Function

CCRW Cross Correlation Reference Waveforms

CDMA Code Division Multiple Access

DLL Delay Lock Loop

DSSS Direct Sequence Spread Spectrum

E Early

FCM Factored Correlator Model

FPGA Field-Programmable Gate Array

FT Fourier Transform

GNSS Global Navigation Satellite System

GPS Global Positioning System

HRC High Resolution Correlator

IF Intermediate Frequency

IFT Inverse Fourier Transform

IMU Inertial Measurement Units

L Late

LOS Line-of-Sight

MBOC Multiplex BOC

NCO Numerically-Controlled Oscillator

NELP Narrow Early-minus-Late Power

OS Open Service

IX

P Prompt

PDF Probability Density Function

PLL Phase Lock Loop

PRN Pseudo-Random Noise

PSD Power Spectral Density

PVT Position, Velocity and Time

RF Radio Frequency

SNS Satellite Navigation System

TK Teager-Kaiser

TMBOC Time MBOC

VE Very Early

VL Very Late

VVE Very Very Early

VVL Very Very Late

WSSUS Wide-Sense Stationary with Uncorrelated Scatterers

X

List of Symbols

∗ convolution

)(tΠ rectangular pulse

)(tΛ triangular pulse

c speed of light

ρ pseudorange

b bias

)(tA signal amplitude

)(tD navigation message

)(tC spreading code

cT Chip duration

chipN Spreading code length in chips

codeT code period

XR auto-correlation function of signal X

XYR cross-correlation function between signals X and Y

)(tX spread sequence

)(~

tX filtered spread sequence

WB pre-correlation bandwidth

0ω Carrier central frequency

0θ Initial carrier Phase

dω offset frequency due to the Doppler effect

eω frequency error

nN , noise

2σ variance

∆ early-late spacing

T integration period

ε code tracking error

( )εd code discriminator

)(tW CCRW

0NC carrier-to-noise density

α attenuation coefficient

XI

τ Reflected ray delay relative to the LOS

φ carrier phase excess

DLLB equivalent code loop bandwidth

PLLB equivalent carrier loop bandwidth

XII

Chapter 1

Introduction

1.1 Motivation

Global Navigation Satellite System (GNSS) is defined as satellite navigation systems (SNS) capable of

providing position, speed and time (PVT) with global coverage. The original motivations behind GNSS

were military applications (e.g. precise location of forces on the field, weaponry guidance, etc.).

However, today, GNSS cover a wider range of applications, like transportation systems, agriculture

and fisheries, several sciences or leisure applications.

Currently, the location of persons and vehicles has become more and more important and a large

number of GNSS receivers were made available for that purpose. However, GNSS application in

urban scenarios is limited by low satellites' visibility, signal interference and multipath. In fact, today,

multipath is one of the dominant error sources in GNSS applications [1]. Thereby, it is necessary to

search for suitable techniques capable of mitigating the multipath effect.

1.2 Research Objectives

This thesis investigates the multipath mitigation techniques for urban scenarios, which are suitable for

low cost GNSS receivers. Specifically, the main objectives are

● To study the GNSS principles, some navigation signals available for civilian use, typical GNSS

receiver architecture and the effect of multipath;

● To investigate different multipath mitigation techniques: special attention will be given to the

correlation based techniques;

● To evaluate several multipath mitigation techniques: this analysis will be done comparing

simulation results obtaining for several multipath scenarios;

● Identify the multipath mitigation techniques suitable for low cost GNSS receivers.

1

1.3 Thesis outline

This thesis will be composed of eight chapters.

Chapter 2 will explain the principles behind GNSS and present the navigation signals considered in

the thesis.

In Chapter 3, an overview of a GNSS receiver architecture will be presented. In this chapter the

receiver's baseband processing will be described and the concepts of code delay and carrier phase

loops will be introduced. Chapter 4 discusses multipath and its effects on the tracking loops.

Chapter 5 provides a description of the multipath mitigation techniques considered in this thesis. The

concepts of Narrow Correlator, High Resolution Correlator, different Code Correlation Reference

Waveforms (CCRWs) and the Teager-Kaiser operator are explained in this chapter.

Chapter 6 will describe the simulation set up: the used software, the workarounds required, the

correlated noise generator, the multipath scenarios considered and the simulation plan. In Chapter 7,

the simulation results will be presented and analysed.

Finally, in Chapter 8 will summarize the work described in this thesis, the major conclusions will be

drawn and future work guidelines and proposals will be also presented.

2

Chapter 2

Basic Concepts

2.1 GNSS Core Constellations

The United States' Global Positioning System (GPS) is the most well-known GNSS system,

operational since 1978, and was the first GNSS system available worldwide for civil use. Currently it is

still the most utilized GNSS by civilians. GPS is composed of three segments: space segment, control

segment and user segment. The space segment consists of 24 to 32 medium Earth orbit satellites in

six different orbital planes (Figure 2.1), the control segment comprises the worldwide network of

monitor stations and the user segment consists of the GPS receivers and the user community [2].

Currently GPS is being modernized in order to meet the requirements of the most critical civil

applications, as civil aviation operations [3].

Beside GPS, the Russian GLONASS (Global'naya Navigatsionnaya Sputnikovaya Sistema) is the

other active GNSS. Since the collapse of the Soviet Union, GLONASS has fallen into disrepair and,

currently, it has only partial availability, but it is planned to restore it to full global availability by 2010

with the collaboration of India.

Figure 2.1: Baseline GPS Satellite Constellation [4].

3

In the future, new GNSS are scheduled to become operational. Europe’s own GNSS system, Galileo ,

is currently being developed and it is expected to be compatible with the modernized GPS [5] and

working from 2013 [6]. Future receivers should be able to combine both modernized GPS and Galileo

signals to increase overall system performance. The China's Compass is the other satellite navigation

system planned to have a global coverage.

2.2 Principles of GNSS

The primary goal of a GNSS receiver is the determination of Position, Velocity and Time (PVT). The

basic principle behind the determination of position and velocity is trilateration (as illustrated in Figure

2.2). The receiver needs to solve the so called Navigation Equation for, at least four satellites:

( ) ( ) ( ) bzzyyxx iiii −−+−+−=222ρ (2.1)

where iρ is the pseudorange between the receiver and the ith satellite, [ ]zyx is the receiver

position, [ ]iii zyx is the ith satellite position and b depends on the user receiver clock bias.

Figure 2.2: The principle of satellite navigation [4].

The receiver measurements are called pseudorange because they include the clock offset. The

pseudorange, ρ , between the receiver and a satellite is directly proportional to the signal propagation

time:

btc −∆= .ρ (2.2)

where c is the speed of light and xx rt ttt −=∆ is the propagation time defined by the difference

between the time of transmission and the time of reception (as illustrated in Figure 2.3). The range

measured by the receiver is affected by several error sources: receiver's clock errors, ionosphere

delays, multipath,etc. [4].

4

Figure 2.3: Indirect distance determination.

2.3 Signals

The simplest navigation signal component is named channel [7]. A navigation signal component is one

of the spreading sequences modulated onto one common carrier. Each navigation signal component

has its own spreading code and can carry its own data modulation.

A single channel signal, )(ts , can be given as

( ) ( )0cos)()()()( θω += ttxtCtDtAts , (2.3)

where:

● )(tA is the signal amplitude;

● )(tD is the navigation message;

● ( )tC is the spreading code;

● )(tx is the sub-carrier;

● ( )0cos θω +t is the carrier, a Radio Frequency (RF) sinusoidal with a known frequency ω and

initial carrier phase 0θ ;

There are two type of channels: data and pilot. In a data channel, the navigation message, )(tD ,

contains information, for instance like precise orbits of the satellites (ephemeris) and the signal's time

of transmission. In a pilot channel, the navigation data carries no data, 1)( =tD .

The spreading code of a signals is a unique Pseudo Random Noise (PRN) assigned to one only

satellite. It is named spreading code because of the wider bandwidth occupied by the signal after

modulation by the high-rate PRN waveform [8]. The minimum interval of time between transitions in

the spreading code is commonly referred to as the chip duration, cT , and the portion of the spreading

code over one chip duration is named chip. PRN sequences approximate the properties of truly

random sequences:

∑=

+

≠

==

chipN

k

nkk

chipn

nCC

N 1 00

0,11 (2.4)

5

where chipN is the period in chips ( cT ) and )( ck kTCC = , and

∑=

+ ∀=′chipN

k

nkk

chip

nCCN 1

,01

(2.5)

where kC and kC′ are different PRN sequences.

The product between the spreading code and the sub-carrier, ( ) )(txtC , is known as spread spectrum

or spread sequence. For that reason, the signal given by (2.3) is named Direct Sequence Spread

Spectrum (DSSS) [8].

The use of DSSS signals in satellite navigation has three main reasons [8]:

1. The spreading code introduces frequent phase transitions in the signal. These transitions help

the receiver to make precise range measurements.

2. The use of different PRN sequences allows multiple signals, from multiple satellites, to be

transmitted simultaneously and with the same RF carrier. Then, these signals can be

distinguished, based on their different codes. In other words, it allows Code Division Multiple

Access (CDMA).

3. Offers significant rejection of narrow band interference.

2.3.1 Modulations

There are several modulations employed by GNSSs. In, this thesis only the modulations described in

sub sub sections 2.3.1.1 to 2.3.1.3 will be considered.

2.3.1.1 Binary Phase Shift Keying

A Binary Phase Shift Keying (BPSK) signal can be defined as

∑−

=

−Π=

1

0

)(chipN

n c

cnBPSK

T

nTtCts , (2.6)

where chipN is number of chips of the spreading code period, ( )cn nTCC = is the code sign for the nth

chip and

≤

=

Π

otherwise,0

2,1 Tt

T

t

denotes the rectangular pulse.

The signal's Fourier Transform (FT) is given by

∑−

=

−=1

0

2)()(

chip

c

N

n

nfTi

ccnBPSK eTfncsiTCfSπ . (2.7)

6

with ttt ππ )sin()(sinc = .

The auto-correlation function (ACF) of the signal )(ts is given by [9]:

dttstsssRs )()()()()( εττε ∫+∞

∞−

−=−∗= , (2.8)

where ∗ denotes convolution. And, in the frequency domain, the FT of the ACF will be [9]

)()()( fSfSfGS −= (2.9)

Thereby, from (2.7) and (2.9), the ACF of a BPSK signal is given by:

∑ ∑−

=

−

=

−−=

1

0

1

0

)(222 )()(chip chip

c

N

n

N

k

knfTi

ckncBPSK eTfncsiCCTfGπ (2.10)

Let ( )∑=

−=′chip

chip

N

n

Nlnnl CCC0

,mod , with knl −= and where ( )chipNln ,mod − is the circular chip operator. Now

(2.10) can be rearranged as

∑−

+−=

−′=

1

1

222 )()(chip

chip

c

N

Nl

lfTi

clcBPSK eTfncsiCTfGπ

(2.11)

If 0>l then 0≈′lC and for 0=l then chipNC =′

0 . Thereby, applying the Inverse FT (IFT) to (2.11), the

ACF of a BPSK signal will be:

Λ=

c

cchipBPSKT

TNRε

ε )( (2.12)

where

denotes the triangular pulse.

2.3.1.2 Binary Offset Carrier

The (sine-shaped) BOC(m,n) modulation consists of multiplying the spreading code waveform, )(tC ,

by the square-wave subcarrier, ( )[ ]tfsigntB sπ2sin)( = to obtain )()()( tCtBtX BOC = . The chip rate and the

subcarrier frequency are given, respectively, by gcc nfTf == 1 and gs mff = , where m and n are two

positive integers and MHz023.1=gf [10].

A Binary Offset Carrier (BOC(pn,n)) signal can be defined BOC(pn,n) signal, where 2p is a positive

integer, can be written as:

7

<−

=

Λ

otherwise,0

1 Tt,Tt

T

t

∑−

=

−=1

0

)()(chipN

n

BOCnBOC ntxCts (2.13)

where chipN is the number of chips per code period, nC is the spreading code sign for the nth chip and

∑=

+

−Π−=

p

l c

cl

BOCpT

lTttx

2

1

1

2)1()( .

From (2.9), the FT of the signal )(txBOC ACF will be

)()()( fXfXfG BOCBOCBOC −= (2.14)

where )( fX BOC is the FT of the signal )(txBOC :

∑=

−=chip

c

N

n

fnTi

BOCnBOC efXCfX1

2)()(

π (2.15)

and )( fX BOC is the FT of the signal )(txBOC

( ) plfiT

c

R

l

l

cBOCcefTpincspTfXπ−

=

+∑ −= 2)1(2)(2

1

1 (2.16)

Thereby the ACF in frequency domain will be

( ) ( )

( ) ( ) pmfiTp

l

p

m

mplfTil

ccchip

pmfiTp

l

p

m

c

m

c

plfiT

c

l

cchip

chip

N

Nk

lfi

l

N

n

N

k

knfi

knBOC

cc

cc

chip

chip

chip chip

eefpTincspTN

efTpincspTefTpincspTN

fXfXN

efXfXC

efXfXCCfXfXfG

ππ

ππ

π

π

∑ ∑

∑ ∑

∑

∑∑

= =

+−+

= =

+−+

−

−=

−

= =

+−

−−=

−−−=

−=

−=

−=−=

2

1

2

1

1122

2

1

2

1

11

1

1

2'

1 1

)(2

)1()1(22

2)1(22)1(2

)()(

)()(

)()()()()(

(2.17)

There so, after apply the IFT, the ACF in time domain will be:

( )∑−

+−=

−−−

Λ= →

−12

12

)1(22

2)()(1

p

pm

pmiTm

chip

cchipBOC

TF

BOCcemp

pTpTNRfG

επεε (2.18)

A BPSK signal can be seen as (2.13) with p = 0.5, so (2.12) is a particular case of (2.18) with p = 0.5.

It can be shown that for BOC(pn,n) with p integer the ACF is given by [10]:

( )( ) ( )

<

+−−−++−−

=

+

otherwise,0

,12421

)1(21

c

c

k

BOC

TT

kppkkpkp

R

εε

ε (2.19)

8

where ( )cTpceilk ε2= , with the ceiling function, ( )xceily = , denoting the smallest integer such that

xy ≥ . And the Power Spectral Density is [11]

( ) even.2

,

2cos

sin2

sin

2

nf

f

f

ff

f

f

f

f

ffGc

s

s

cs

cBOC =

=π

π

ππ

(2.20)

2.3.1.3 Multiplexed Binary Offset Carrier

Multiplexed BOC (MBOC) signal results from adding or multiplexing BOC signals. For instance,

MBOC(6,1,1/11) is the result of a wideband BOC(6,1) signal multiplexed with a narrow-band BOC(1,1)

signal, with 1/11 of the signal power allocated on the BOC(6,1) component [12]. The MBOC

normalized Power Spectral Density (PSD) is given by

)(11

10)(

11

1)( )1,1()1,6()11/1,1,6( fGfGfG BOCBOCMBOC += (2.21)

where )(),( fG npnBOC is the unit-power spectrum density of a sine-phased BOC modulation.

In Figure 2.4, the MBOC(6,1,1/11) normalized PSD is compared to the BPSK and BOC(1,1)

normalized PSDs. Compared to the BOC(1,1) PSD, in the MBOC(6,1,1/11) PSD is evident the

increased power at the frequency shifted about 6MHz from the central frequency, due to the presence

of the BOC(6,1) component. However for bandwidths narrower than about 5 MHz the BOC(1,1) and

MBOC(6,1,1/11) PSDs are very similar. Because of that, it is expected that the MBOC(6,1,1/11)

behaves similarly to the BOC(1,1) for bandwidths narrower than about 5 MHz.

Figure 2.4: Power Spectral Density of BPSK, BOC(1,1) and MBOC(6,1,1/11) signals

There are different ways to generate a MBOC signal. Two possible solution are Time MBOC (TMBOC)

and Complex BOC (CBOC).

9

-20 -15 -10 -5 0 5 10 15 20-40

-35

-30

-25

-20

-15

-10

-5

0

pow

er s

pect

ral d

ensi

ty, d

B/M

hz

frequency, Mhz

BPSKBOC(1,1)CBOC(6,1,1/11)

The TMBOC(6,1,4/33) is characterized by using a binary sub-carrier component that results from the

time-multiplexing of the BOC(1,1) and BOC(6,1) sub-carriers according to a deterministic pattern [13].

The TMBOC(6,1,4/33) sub-carrier is defined as [12]

∈

∈=

2)1,6(

1)1,1(

)33/4,1,6( )(Stifx

Stifxtx

BOC

BOC

TMBOC (2.22)

where )1,1(BOCx and )1,1(BOCx are respectively BOC(1,1) and BOC(6,1) sub-carriers and the 1S and 2S

are the union of the segments of time when the BOC(1,1) and the BOC(6,1) sub-carriers are used,

respectively. 1S and 2S are chosen so that 4/33 of the signal total duration will be assigned to the

BOC(6,1) sub-carrier and the remaining time to the BOC(1,1) sub-carrier. Figure 2.5 shows a possible

portion of a TMBOC sub-carrier.

The ACF of a TMBOC(6,1,4/33) signal can be shown to be

)(33

4)(

33

29)( )1,6()1,1()11/1,1,6( εεε BOCBOCCBOC RRR += (2.23)

0 1 2 3 4 5 6-1

0

1

time, Tc

Figure 2.5: Example of TMBOC (6,1,p) Sub-carrier

A CBOC(6,1,1/11) signal uses four-level sub-carrier formed by the weighted sum of BOC(1,1) and

BOC(6,1) signals and is given by:

±= )(

11

1)(

11

10).()( )1,6()1,1()11/1,1,6( txtxtCts BOCBOCCBOC (2.24)

where )()1,1( txBOC and )()1,6( txBOC are the sub-carrier of BOC(1,1) and BOC(6,1), respectively. The two

possible sub-carriers of (2.24) are plotted in Figure 2.6.

10

0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5

time, Tc

0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5

time, Tc

Figure 2.6: Possible CBOC(6,1,1/11) sub-carriers.

The ACF of a CBOC(6,1,1/11) signal can be shown to be

[ ])()()(11

1)(

11

10)( )1,1()1,6()1,6()1,1()1,6()1,1()11/1,1,6( εεεεε BOCBOCBOCBOCBOCBOCCBOC RRRRR +±+= (2.25)

Let [ ]aTtta Π=)( and [ ]bTttb Π=)( stand for two unity rectangular pulses of duration chipa TpT 12= and

chipb TpT 22= . The cross-correlation between two BOC signals can be written as:

−−−

−−= ∑ ∑∑ ∑

−−

=

−

==

−

=

1 21 1

21

0

12

0 20

12

0 1

),(),(2

)1(2

)1()(chipchip N

n

p

l chip

l

n

N

n

p

l chip

l

nnnpBOCnnpBOCTp

ltbC

Tp

ltaCR εε (2.26)

Applying (A.1):

( )( )( )

∑−+

−+−=

+−=

1

1 21

),(),(

21

21

21

pp

ppl chip

ABnnpBOCnnpBOCTpp

lRR εε (2.27)

Figure 2.7 shows the auto-correlation function for BPSK, BOC(1,1) and CBOC(6,1,1/11). It is possible

to see that the CBOC(6,1,1/11) curve is not too different from the BOC(1,1) curve, confirming the

similarities between the power spectrum of the two signals. In Figure 2.4, however, the sharper peak

of the CBOC signal will have benefits in the code tracking accuracy.

-1.5 -1 -0.5 0 0.5 1 1.5-0.5

0

0.5

1

ε/Tc

RX( ε

)

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 2.7: Auto-correlation functions for unfiltered BPSK, BOC(1,1) and CBOC(6,1,1/11) signals.

11

2.3.2 Considered Signals

There are several GNSS signals currently available or planned for the future. GNSS signals are

divided by bands, as Figure 2.8 shows for the case of GPS and Galileo [7]. However, as this thesis

scope only covers low cost civilian GNSS receivers, the signal considered must be freely worldwide

available for civil use. Because of that, three signals will be considered: GPS L1, Galileo E1 Open

Service (OS) and GPS L1 Civil (L1C).

Figure 2.8: GPS and Galileo frequency plan [7].

2.3.2.1 GPS L1 signal

The L1 signal can be modelled as [4]:

{ } ( )001/1 cos)()(2)()(2)( θω ++= ttDtyPtDtxPts YACL (2.28)

where ACP / and 1YP are the signal powers for signals carrying C/A-code and P(Y)-codes and x and y

are the Coarse / Acquisition (C/A) and P(Y)-code sequences assigned to a unique satellite; D

denotes the navigation data bit stream; 10 2 Lfπω = are the carrier frequencies corresponding to L1

signal; and 0θ is the carrier phase offset.

The channel carrying P(Y)-code is encrypted and restricted for military use; so from now on, the focus

in legacy GPS signals will be on the L1 signal carrying C/A-code and which uses a BPSK modulation.

2.3.2.2 GPS L1C

The GPS L1C baseband signal has two channels and can be modelled as [12], [13]:

{ } ( )00)33/4,1,6(1)1,1(111 cos)()()()()(2

1)( θω ++= −−− ttxtetxtdtets TMBOCQCLBOCICLICLCL (2.29)

where )(1 te ICL − and )(1 te QCL − are the data and pilot spreading sequences respectively, )(1 td ICL − is the

GPS L1C navigation message, and )()1,1( txBOC and )()33/4,1,6( txTMBOC are the BOC(1,1) and

TMBOC(6,1,4/33) sub-carriers respectively.

12

The selected GPS implementation of MBOC places 75% of the total power on the pilot channel and

25% on the data channel. Thereby, the PSD of the signal )(1 ts CL will be equal to the PSD of a

MBOC(6,1,1/11) signal [5], [13].

2.3.2.3 Galileo E1 Open Service Signal

Composite BOC (CBOC) was the selected MBOC implementation of Galileo E1 Open Service (OS)

Signal [5], [7]. Both pilot and data components are modulated onto the same carrier component, with a

power split of 50%. The normalized base-band Galileo E1 OS composite signal model is given by [7]:

[ ] [ ] ( )001111 cos)()()()()()(2

1)( θω +

−−+= −−− ttQytPxetQytPxtdtets CEBEBECE (2.30)

where )(1 te BE − and )(1 te CE − are the data and pilot spreading sequences respectively, )(1 td BE − is the

Galileo E1-B navigation message, and )(tx and )(ty are the sine-BOC(1,1) and sine-BOC(6,1) sub-

carriers respectively. The parameters P and Q are chosen as 11/10 and 11/1 respectively, so

the power associated with BOC(6,1) sub-carrier components equals 1/11 of the total power.

From (2.30) it can be noted that the data and pilot are in anti-phase, with respect to the BOC(6,1)

component.

Table 2.1 summarizes the main technical characteristics of the signals considered in this thesis.

Signal GPS L1 GPS L1C Galileo E1

SignalComponent

C/A I Q B C

Channel type Data Data Pilot Data Pilot

Componentpower

- 25% 75% 50% 50%

RF carrierfrequency

1575.42 MHz

PRN codelength

1024 10230 4092

Chip rate 1.023 Mbps

SpreadingModulation

BPSK BOC(1,1) TMBOC(6,1,4/33) CBOC(6,1,1/11)

Table 2.1: Main technical characteristics of GPS L1 [4], GPS L1C [13] and Galileo E1 OS signals [7].

13

Chapter 3

Receiver

Chapters 2 gave a first preview of GNSS systems and the signals considered in this thesis. This

chapter will now address to the receiver operation.

The basic functions of a GNSS receiver are [4]:

● to capture RF signals transmitted by the satellites spread out in the sky;

● to separate the signals from satellites in view;

● to perform measurements of signal transit time and Doppler shift;

● to decode the navigation message in order to determine the satellite position, velocity, and

clock parameters;

● to estimate the user position, velocity, and time.

3.1 Receiver Architecture Overview

A generic GNSS receiver is given in Figure 3.1, with four blocks:

● Antenna

The signals are gathered by the antenna. Several antenna types can be used, however, usually, the

antenna is right-hand circularly polarized, to match the incoming signals and reject reflected signals,

and the pattern is hemispherical, to allow tracking of satellites from zenith almost down to the horizon

for all azimuths [14].

● Front End

The Front End is responsible for RF and all the intermediate frequency (IF) signal processing: the

signal received by the antenna is filtered, amplified and down converted to baseband or near

baseband. The final conversion to baseband involves converting the (IF) signal to the in-phase (I) and

quadrature (Q) components of the signal envelope [14].

● Tracking Loops

Two loops per each receiver's channel fit in this block: the Delay Lock Loop (DLL) and the Phase Lock

Loop (PLL). The DLL is responsible to track the code delay, while the PLL is responsible to track the

14

carrier phase. The code delay and carrier phase tracking loops will be analysed with more detail in the

next section. When both the code delay and the carrier phase loops are locked, the navigation

message can be extracted from the received signal [14].

● Navigation Processing

This block is responsible to provide the PVT solution. Normally it relies on the pseudorange, rate

measurements and navigation message to solve the navigation equation [14], defined in (2.1).

Figure 3.1: A conceptual view of a GNSS receiver

3.2 Baseband Signal Processing

For better analysing the code delay and carrier tracking loops, it is useful to describe a GNSS receiver

with the equivalent baseband receiver as sketched in Figure 3.2 [2].

It is assumed that the received signal is given by )()()( tntstr += , where )(tn is Gaussian noise and

)(ts is the navigation signal given by

15

Figure 3.2: Typical diagram of the receiver code and carrier loops.

Phase

Rotation

I&D

I&D

I&D

I&D

Code

Discriminator

Signal

Generator

Local Phase

Generator

Carrier

Discriminator

d(ε)

IE

IP

QE

QP

cos (t)φ̂ φ̂ sin (t)

z (t)i

z (t)q

y (t)q

y (t)i

3-bit Shift Register

I&D

I&DQL

LI

L P E

r(t)

sin(ω t) 0

cos(ω t) 0

[ ],)(cos)()(~

)( 00 θωω ++= ttDtXAts d (3.1)

where A depends on the power )(ts , )(~

tX is the filtered spread sequence, )(tD is the navigation

data, 0ω is the nominal carrier frequency, dω is an offset frequency due to the Doppler effect and

oscillators misalignments and 0θ is the initial phase.

The filtered spread sequence can be defined as )()()(~

thtXtX ∗= , where )(th is the impulse response

of the receiver's input filter. Considering a raised-cosine filter with bandwidth WB and roll-off factor

10 ≤≤ ξ , the impulse response is [15]:

−=

222161

)2cos()2sinc()(

tB

tBtBth

W

WW

ξ

ξπ (3.2)

The in-phase and quadrature components of the received signal are given by

+

=

)(

)(

)(sin

)(cos)()(

~

)(

)(

tn

tn

t

ttDtXA

ty

ty

q

i

q

i

ϕ

ϕ, (3.3)

with 0)( θωϕ += tt d and where )(tni and )(tnq are the in-phase and quadrature noise components. The

signals )(tzi and )(tzq results from rotating [ ]Tqi tyty )()( by the phase estimate )(ˆ tϕ provided by the

Phase-Lock Loop (PLL):

.)(

)(

)(ˆcos)(ˆsin

)(ˆsin)(ˆcos

)(

)(

=

ty

ty

tt

tt

tz

tz

q

i

q

i

ϕϕ

ϕϕ (3.4)

Assuming that the phase error in the integration interval [ ]T,0 may be written as

eee tttt θωϕϕϕ +=−≡ )(ˆ)()( , where ee fπω 2= is the frequency error and eθ is the initial phase error,

leads to

′

′+

=

)(

)(

)(sin

)(cos)()(

~

)(

)(

tn

tn

t

ttDtXA

tz

tz

q

i

e

e

q

i

ϕ

ϕ, (3.5)

where )(tD is the navigation data in the integration, and )(tni′ and )(tnq

′ are the noise components

after phase rotation.

The components )(tzi and )(tzq are then multiplied by Early (E) and Late (L) version of the locally

generated )(tX , respectively ( )2∆+tX and ( )21 ∆−X , where ∆ is the E-L spacing (with cT≤∆ ). By

integrating in the interval ],0[ T , leads to

( ) ( ) EieeeXX

T

iE NTfTfADRdttXtzT

I ,~

0

)cos()(sinc2

2)(1

++

∆−=∆+−= ∫ θπεεε (3.6)

16

and similarly

( ) ( )

( )

( )

( ) ( )

( ) LqeeeXXL

PqeeeXXP

EqeeeXXL

LieeeXXL

PieeeXXP

NTfTfADRQ

NTfTfADRQ

NTfTfADRQ

NTfTfADRI

NTfTfADRI

,~

,~

,~

,~

,~

)sin()(sinc2

)sin()(sinc

)sin()(sinc2

)cos()(sinc2

)cos()(sinc

++

∆+=

++=

++

∆+=

++

∆+=

++=

θπεε

θπεε

θπεε

θπεε

θπεε

(3.7)

where the cross-correlation function between )(tX and )(~

tX can be defined as

( ) ( ) ( ) )()(0

~ εεεεε hRdttXXR X

T

XX∗=−= ∫ , ∆ is the early-late spacing, and

( )

∫

∫

∫

−

∆−

=

−

=

−

∆+

=

T

q

i

Lq

Li

T

q

i

Pq

Pi

T

q

i

Eq

Ei

dttXn

n

TN

N

dttXn

n

TN

N

dttXn

n

TN

N

0,

,

0,

,

0,

,

2

1

1

2

1

ε

ε

ε

(3.8)

are zero-mean, Gaussian, random variables with equal variance 2σ and cross-correlation

{ } { } )(2

,,,, ∆== XLqEqLiEi RNNENNE σ and { } { } )2/(2

,,,, ∆== XPqLqPiEi RNNENNE σ .

The output of the Code Discriminator block depends on the selected discriminator. Three common

discriminators are:

● E-minus-L Coherent

( ) ( ) ( )εεε LE IId −= (3.9)

● Non-coherent Early-minus-Late Power

( ) ( ) ( )[ ] ( ) ( )[ ]εεεεε 2222

LELE QQIId −+−= (3.10)

● Non-coherent dot-product

( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( )εεεεεεε PLEPLE QQQIIId −+−= (3.11)

The three code discriminator functions are plotted in Figure 3.3. These three functions, as all the

discriminator functions, present a zero crossing for a tracking error 0=ε and a linear or near linear

behaviour in the neighbour region, where the code discriminator output will be proportional to the code

tracking error. Thereby the output of the code discriminator, after filtered with a low pass filter, is used

17

as a estimation of the code tracking error to feed back the signal generated (see Figure 3.2). In this

way, the receiver can keep the locally generated signal synchronized with the received signal.

It is important that the zero crossing point be 0=ε , because, for the receiver's point of view, the code

of the locally generated signal is synchronized with the code of the received signal, when the code

discriminator output is ( ) 0=εd . If the zero crossing point is shifted, then the receiver stable lock point

would not be 0=ε and a code tracking error will occur.

-1 -0.5 0 0.5 1

-0.05

-0.025

0

0.025

0.05

code tracking error, ε/TC

disc

rimin

ator

out

put,

d ( ε

)

E-L CoherentDot ProductE-L Power

Figure 3.3: Code discriminator outputs for BOC(1,1) with unlimited pre-correlation bandwidth.

The output of the Carrier Discriminator depends on the carrier discriminator function. A common carrier

discriminator function is the arc tangent discriminator, that is insensitive to the phase transition of the

navigation data bit transitions [8]:

=

)(

)(arctan)(

eP

ePePLL

Q

Id

ϕ

ϕϕ (3.12)

The discriminator for the carrier loop is showed in Figure 3.4, and, similar to the code discriminator, it

is used to give a phase tracking error estimate to feed back the Local Phase Generator. Thereby the

carrier discriminator functions must have a stable lock point for 0=eϕ .

18

-1 -0.5 0 0.5 1-0.5

0

0.5

dis

crim

ina

tor

ou

tpu

t, d

PLL

( φ

/ π

)

phase tracking error, φ/π

Figure 3.4: Carrier discriminator output for the arc tangent discriminator.

3.2.1 Aided Tracking Loops

The receiver illustrated in Figure 3.2 uses unaided tracking loops. However, today low-cost micro-

electromechanics allows the integration of miniature inertial measurement units (IMUs) and GNSS in

low cost receivers. This enables the use of IMU measurements for helping the receiver tracking loops.

[16].

3.2.1.1 Aided PLL

In a conventional receiver, the local phase generator output is only controlled by the phase

discriminator output. However, when IMU's measurements are available it is possible to estimate the

receiver Doppler signal, based on satellite ephemeris and position estimates calculated with both

GNSS and INS measurements, and use it as an aiding signal [16]:

daid ωω ˆ= . (3.13)

Therefore the output frequency will be given by

aidPPL ωωω ˆ+= (3.14)

The aiding signal allows the local phase generator to generate a more accurate frequency and thereby

it will be easier for the PLL to track the carrier phase, giving also a more accurate phase.

3.2.1.2 Aided DLL

Now, with the more accurate phase measurements determined by the PLL, an aided DLL can use the

phase measurements to aid the code delay tracking [16]. In this way, the aided DLL increases

robustness to dynamics, noise, interference and multipath.

19

Chapter 4

Multipath Effect

In Chapter 3 was assumed that there was an unobstructed path from to the receiver to the satellite

and that the received signal was a single undisturbed signal, the so called Line-of-Sight signal (LOS).

However in a realistic environment multipath may occur.

Multipath is defined as the propagation of a wave from one point to another by more than one path. By

that means, the received signal will be the result of the sum of a first path plus one or several reflected

echoes. The first path is usually a direct, unobstructed path from the satellite to the antenna (LOS),

and the reflected rays are the result of reflections from a nearby objects or the ground. These reflected

paths will be delayed relative to the LOS and, usually, will be weaker than the LOS, due to the energy

loss from the reflection. However, when the LOS is partially obstructed or when the reflecting surface

has a large area (e.g. a building), the LOS may be weaker than some echoes. Figure 4.1 shows a

possible multipath situation, where three rays reach the receiver: LOS, a ray reflected on a building

and another ray reflected on the ground.

Figure 4.1: Possible multipath situation.

20

LOS

As it will be seen in the following section, the reflections will confound the receiver by distorting the

correlation peak and hence the code discriminator will not estimate correctly the code delay. The

effects of multipath in the receiver tracking loops will depend on:

● the amplitudes of the reflected signals relative to the LOS;

● the delays of the reflected signals relative to the LOS;

● the phases of the reflected signals relative to the LOS;

● the rate of change of the relative phases.

4.1 Tracking error due to Multipath

The simplest multipath scenario occurs with the simultaneous reception of the LOS signal and a

reflected ray delayed, attenuated and with a excess of phase. In this situation the receiver's input may

be written as

( ) ( )[ ] ( ) ( )[ ] )(coscos)( 0000 tnttAXttAXtr dd ++++−+++= φθωωταθωω (4.1)

where 10 << α is the attenuation coefficient and stands for the relative amplitude of the reflected ray,

and τ and φ are respectively the delay and the extra phase of the reflected signal relative the LOS.

Following the proceeding described in Section 3.2, now the correlator outputs will be given by:

( )

( ) ( ) ( ){ }

( )

( )

( ) ( ) ( ){ }

( ) LqeeXXeeXXeL

PqeeXXeeXXeP

EqeeXXeeXXeE

LieeXXeeXXeL

PieeXXeeXXeP

EieeXXeeXXeE

NTfRTfRTfADQ

NTfRTfRTfADQ

NTfRTfRTfADQ

NTfRTfRTfADI

NTfRTfRTfADI

NTfRTfRTfADI

,~~

,~~

,~~

,~~

,~~

,~~

)sin(2

)sin(2

)(sinc

)sin()sin()(sinc

)sin(2

)sin(2

)(sinc

)cos(2

)cos(2

)(sinc

)cos()cos()(sinc

)cos(2

)sin(2

)(sinc

+

++

−

∆+++

∆+=

+++−++=

+

++

−

∆−++

∆−=

+

++

−

∆+++

∆+=

+++−++=

+

++

−

∆−++

∆−=

φθπτεαθπεε






(4.2)

According to (4.2), in the presence of multipath, the correlator outputs can be viewed as a

superposition of shifted and distorted versions of (3.6) and (3.7). Figure 4.2 shows the effect of a

single reflected ray on the output of the in-phase prompt correlator, )(εPI . In the Figure, it is

considered that the reflected signal has a attenuation coefficient of 5.0=α and is in-phase, 0=φ , and

delayed cT4.0=τ regarding the LOS.

As the output of the correlators is distorted in the presence of multipath, the zero-crossing of the

discriminator function will be shifted from the correct position (see Figure 4.3). This leads to an error

21

in the code delay measurement between the received signal code and the local generated code.

-1.5 -1 -0.5 0 0.5 1 1.5 2-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

ε/Tc

I P(ε

)

direct signal

reflected signal

received signal

Figure 4.2: Normalized in-phase prompt correlator for BOC signals, with α = 0.5 and τ = 0.4Tc.

In Figure 4.3 is plotted the discriminator function for BOC(1,1) signals with unlimited bandwidth and for

three situations: E-L spacing cT2.0=∆ and no reflected path; E-L spacing cT2.0=∆ a in-phase echo

with attenuation coefficient 5.0=α and delay cT4.0=τ ; and the same as the second situation, but now

with an E-L spacing cT05.0=∆ . It is evident that the discriminator's function is shifted when 0≠α and

that a narrower E-L spacing decreases the error due to multipath.

Figure 4.3: E-L power code discriminator outputs for BOC(n,n).

22

-1 -0.5 0 0.5 1

-0.1

-0.05

0

0.05

0.1

tracking error, ε/Tc

disc

rimin

ator

out

put,

d ( ε

)

φ=0 τ/Tc=0.4

∆/Tc=0.2 α=0

∆/Tc=0.2 α=0.5

∆/Tc=0.05 α=0.5

A common approach to describe the effects of multipath consists of determining the multipath error

envelopes, as it will be explained in the next section.

4.2 Multipath error envelope

The multipath error envelopes are plots that illustrate the extreme values of the code tracking error. In

a multipath error envelope, it is assumed that the receiver input signal is given by (4.1) and the

tracking error is function of the reflected path delay τ and the attenuation coefficient α relative to the

LOS.

It is considered that the output of the correlators is given by (4.2), then the discriminator's output can

be written as a function of (4.2), ( )LPELPE QQQIIId ,,,,, . As the output of the correlators depends of the

tracking error, ε , the reflected path delay relative to the LOS, τ , the attenuation coefficient, α and the

extra phase relative the LOS, φ ; the discriminator's function can be defined as a function of those

variables, ( )φατε ,,,d .

The tracking error can be determined by solving ( ) 0,,, =φατεd for ε . The worst tracking errors will

occur when the reflected path is in-phase (constructive interference), 0=φ , or out-of-phase

(destructive interference), πφ = . The multipath error envelope consists of two curves given by

( ) 00,,, =ταεd and ( ) 0,,, =πταεd , that define the tracking error limits for a constant attenuation

coefficient, α , and as a function of the reflected path delay τ relative to the LOS.

Depending of the discriminator's function and the signal's modulation, there may exist more than one

zero-crossing in the discriminator's function. When this happens, it must be selected the zero-crossing

correspondent to the stable lock point near 0=ε .

Figure 4.4 displays the multipath error envelopes for BOC(1,1) obtained for the E-L power

discriminator, with two E-L spacing values and infinite bandwidth. It can be seen that the use of

narrower correlator architecture decreases the code tracking error in the presence of multipath. Since

the error given in Figure 4.4 is in units of chip ( cT ), it is important to state that signals with higher chip

rates will offer better multipath performance (less code tracking error) than signals with lower chip

rates. Other important conclusion is that for reflected rays delayed by more than about one chip, the

tracking error due to multipath will drop to zero. The reason for this event is the the fact that the auto-

correlation function be null for misalignments equal or greater than one chip.

23

0 0.2 0.4 0.6 0.8 1 1.2-0.03

-0.02

-0.01

0

0.01

0.02

0.03

multipath delay, τ/TC

track

ing

erro

r, ε /

T C

α=0.5 ∆=0.1Tc

α=0.5 ∆=0.05Tc

α=0.2 ∆=0.05Tc

Figure 4.4: Multipath error envelopes for the Narrow Correlator and BOC signals.

Despite the simplifications assumed in the multipath error envelopes, it is a very suited tool to compare

the multipath performance of different signals and different tracking techniques. Therefore, in the next

chapter, where it will be discuss several methods to mitigate the effects of multipath, the multipath

error envelope will be used to assess the performance of each technique.

24

Chapter 5

Multipath Mitigation

As it was seen in Chapter 4, the multipath error is a limiting factor of the GNSS positioning accuracy.

Several techniques have been studied in the past to mitigate the multipath tracking error. These

different techniques can be classified into three main categories:

1) Pre-processing techniques.

These techniques are applied before the satellite signals enters the receiver's processing

chain. In this category it is possible to find:

● Methods that make assumptions about the geometry of the multipath. We find in this

category the Choke-Ring antenna, which works quite well for under the horizon

multipath (which covers most of the survey situations), but falls short for above the

ground configurations (building reflections for example) [17];

● Methods that assume repeatability of the multipath from one day to another, and

characterize the multipath environment by azimuth and elevation [18]. These have the

disadvantage of requiring a significant time to calibrate the environment, and are

limited to the specific location where the calibration was made. This limits the use of

methods to static cases such as reference stations. The other drawback is that, unless

another calibration is made, these methods will not take into account any change in

the multipath environment with the time.

2) Receiver signal processing techniques.

Signal processing techniques occur within the code and frequency tracking loops.

● Techniques that make no assumption about the multipath model, like the Narrow

Correlator [19] and Code Correlation Reference Waveforms [20], [10];

● Techniques that assume a multipath model and try to identify the parameters of this

model. Some examples are: DLL with interference cancellation and/or interference

mitigation, Kalman Filtering, Multipath Estimating DLL, Pulse Subtraction, Least

Squares, Subspace-based algorithms and Quadratic Optimization Methods [21].

25

3) Post-processing techniques.

Post-processing techniques are applied after the pseudo-range measurements have been

produced. In this category, it can be found the methods that analyse the consistency of the

different code measurements, and eliminate the satellites with a too large bias (Receiver

Autonomous Integrity Monitoring). Although this might work well if only one or two satellite

signals present multipath and if the multipath level is quite significant (20 meters level), it has

severe limitations otherwise [18].

However, not all the mitigation techniques are suitable for implementation in low cost GNSS receivers.

Low cost receivers can not afford to use high performance processor and expensive hardware. These

receivers have limited computational capabilities. Thereby, the techniques suitable for such type of

receiver must minimize the computing load. In this category, it can be found: the Narrow Correlator,

the CCRWs and the Teager-Kaiser operator. These techniques will be analysed in the following

sections.

5.1 Narrow Correlator

The Narrow Correlator is known for quite a few years [19]. The structure of the Narrow Correlator's

receiver is sketched in Figure 3.2 and the correlator outputs are given by (4.2). Historically, the first

generation of GPS receivers used large E-L spacings (e.g. cT1=∆ ). The main concept behind the

Narrow Correlator is narrowing the E-L correlator spacing. This has the advantage of reducing the

tracking errors in the presence of both noise and multipath [19]. Although, the Narrow Correlator was

developed for BPSK signals, the BOC and MBOC signals can also be used, as shown in [12] and [5].

Several discriminator functions can be used with the Narrow Correlator, as, for instance, coherent, E-L

power and dot-product discriminators. However, these different discriminators have a similar

performance in the presence of multipath, and then, in this section, it only will be considered the

Narrow E-L Power (NELP) discriminator, given by (3.10).

The noise reduction is achieved with narrower E-L spacings because the noise components of the E

and L correlator outputs are correlated and the discriminator output tend to cancel. The multipath

effects are reduced because the code discriminator is less distorted by the delayed multipath signal. In

Figures 4.3 and 4.4, the reduction of tracking errors in the presence of multipath with narrower E-L

spacings is well visible.

Figure 5.1 (left) shows the NELP code discriminator response for unlimited bandwidth and E-L spacing

cT05.0=∆ . For the BPSK signal, the code discriminator response has only one stable lock point at

0=ε . However for BOC(1,1) and CBOC(6,1,1/11) it has three stable lock points: at 0=ε and two

false-lock points at 2cT±=ε . The false-lock points are a disadvantage because they introduce

ambiguity in the code discriminator, which means that the receiver may lock at the wrong point and

estimate wrongly the code tracking delay.

26

On the right side of Figure 5.1 (right) are plotted the multipath error envelopes for NELP code

discriminator, attenuation coefficient 5.0=α , E-L spacing cT05.0=∆ and unlimited bandwidth. There is

a performance improvement in changing from a BPSK signal to a BOC(1,1) signal and from a

BOC(1,1) signal to a CBOC(6,1,1/11) signal. The reason for this is that CBOC(6,1,1/11) signals have

sharper peaks in their ACF than BOC(1,1) signals, and the last ones also have sharper peaks in their

ACF than BPSK signals(as it was seen in Figure 2.7). The sharper peaks in the ACF allows the code

discriminator to be less distorted by delayed reflected rays.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

tracking error, ε/TC

disc

rimin

ator

out

put,

d (

ε

)

∆=0.05Tc

BW

Tc=∞

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.015

-0.01

-0.005

0

0.005

0.01

0.015


trac

king

err

or,

ε / T

C

∆=0.05Tc

BW

Tc=∞

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.1: Code discriminator outputs (left) and multipath error envelopes (right) for the NELPdiscriminator, with unlimited pre-correlation bandwidth.

The impact of the input filter bandwidth on the multipath performance should not be neglected. The

input filter bandwidth affects the correlator outputs, the code discriminator output and, thereby, the

multipath error envelope. Figure 5.2 shows the multipath error envelopes for the NELP code

discriminator, E-L spacing cT05.0=∆ , attenuation coefficient 5.0=α and two different bandwidths:

5=cW TB and 12=cW TB . In general, the larger the input bandwidth, the better will be the multipath

performance. This behaviour was expected, because narrower input bandwidth implies smoother

correlation peaks and, therefore, code discriminator response more sensitive to reflected rays.

27

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04


trac

king

err

or,

ε / T

C

∆=0.05Tc

BW

Tc=5

α=0.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.015

-0.01

-0.005

0

0.005

0.01

0.015


trac

king

err

or,

ε/T C

∆=0.05Tc

BW

Tc=12

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.2: Multipath error envelopes for the NELP discriminator and normalized bandwidth BWTc = 5

(left) and BWTc = 12 (right).

For pre-correlation bandwidths 5≤cW TB , the behaviour of CBOC(6,1,1/11) will be similar to BOC(1,1).

This is explained because the power density of the two types of signals is very similar when the

bandwidth is limited to that set of values (see Figure 2.4). Or in other words, the high frequency

BOC(6,1) is neglected for input bandwidths 5≤cW TB . Thus, to take full advantage of the wideband

CBOC(6,1,1/11) it is necessary to use a input bandwidth larger than 7≥cW TB so that the wide

frequency BOC(6,1) component would not be neglected.

The multipath error envelopes plotted in Figures 5.1 and 5.2 are important, because they can give a

good idea how the NELP discriminator behaves in the presence of multipath. Nevertheless, the

contribution of the thermal noise in the correlator outputs should not be neglected. The discriminator

performance is deeply affected with the amount of channel noise.

Assuming weak noise conditions and linearising (3.10) around 0=ε (stead-state), from (B.3) and

(B.4), the steady-state normalized code error variance, ( )2/ cTε , is given by:

( ) ( )[ ]( )

2

~~

~

2

0

2

)2()2(

)2(12

∆′∆

∆∆−=

cXXXX

XXXPLLc

TRR

R

NC

RBTε . (5.1)

Figure 5.3 shows the evolution of the normalized code error variance, ( )2/ cTε , versus the pre-

correlation bandwidth, WB , and the E-L spacing, ∆ , for the fixed carrier-to-noise density ratio

dB/Hz400 =NC . It can be seen that narrower E-L spacings reduce the code discriminator noise, but

requires wider pre-correlation bandwidths. This is the primary disadvantage of the Narrow Correlator,

because wider bandwidths require higher sample rates, higher digital signal processing rates and

more expensive receivers.

28

So, for each bandwidth value there will be an optimal E-L spacing, that minimizes the steady-state

normalized code error variance. The selected E-L spacing does not need to be one that minimizes the

code error variance, but it must assure low variances while providing better multipath performance.

Thereby, it is preferable to use a narrower E-L spacing that allow better multipath mitigation (see

Figure 5.4), but higher steady-state code error variance, than the optimal value. In this way, cT1.0=∆

and cT05.0=∆ are suitable E-L spacings, for the normalize pre-correlation bandwidths 5=cW TB and

12=cWTB , respectively. These two pre-correlation bandwidths will be the ones picked up in

Subsection 6.3 and used in the simulation, whose results will be presented in Chapter 7.

29

multipath delay, τ/Tc

code

trac

king

erro

r, ε

/ Tc

BWTc=∞

-α∆/2

0

α∆/2

0 0.5+∆/4 1+∆/2

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.4: Multipath error envelopes as function of E-L spacing and attenuation coefficient, for theNELP discriminator.

5.2 High Resolution Correlator

The High Resolution Correlator (HRC) was introduced in [22] for BPSK signals. The HRC uses

multiple correlator outputs, from a conventional GNSS receiver (Figure 3.2), to yield an approximation

to W1 Code Correlation Reference Waveform, that will be analysed in Sub Sub Section 5.3.2.

The structure of the HRC receiver is similar to the one illustrated in Figure 3.2, but with the difference

that the HRC receiver requires eight correlators: Very-Early-in-phase VEI , Early-in-phase EI , Late-in-

phase LI , Very-Late-in-phase VLI , Very-Early-quadrature VEQ , Early-quadrature EQ , Late-quadrature

LQ , Very-Late-quadrature VLQ .

If the input RF signal is given by (4.1) and the receiver is coherent, then the correlators outputs of the

HRC will be given by (4.2) and

( )

( )

( )

( ) VLqeeXXeeXXeVL

VEqeeXXeeXXeVE

VLieeXXeeXXeVL

VEieeXXeeXXeVE

NTfRTfRTfADQ

NTfRTfRTfADQ

NTfRTfRTfADI

NTfRTfRTfADQ

,~~

,~~

,~~

,~~

)sin(2

)sin(2

)(sinc

)sin(2

)sin(2

)(sinc

)cos(2

)cos(2

)(sinc

)cos(2

)sin(2

)(sinc

+

++

−

∆+++

∆+=

+

++

−

∆−++

∆−=

+

++

−

∆+++

∆+=

+

++

−

∆−++

∆−=

φθπτλεαθπλεε




(5.2)

where ∆ is the early-late spacing, ∆λ is the very-early-very-late spacing, with 1>λ , typically 2=λ ,

and VEiN , VLiN , VEqN , VLqN , are zero-mean Gaussian random variables, with variances 2σ and cross-

30

correlations

{ } { } ( )

{ } { } ( )

{ } ( )

{ } ( )

{ } ( )∆=

∆=

∆=

∆+==

∆−==

XZqYi

XLxEx

XVLxVEx

XVLxExLxVEx

XVLxLxExVEx

RNNE

RNNE

RNNE

RNNENNE

RNNENNE

2

,,

2

,,

2

,,

2

,,,,

2

,,,,

21

21

σ

σ

λσ

λσ

λσ

(5.3)

with the x one of VE , E , L or VL , and with Y and Z one of i or q .

The response of the HRC discriminator is given by

( ) ( ) ( )[ ] ( ) ( )[ ]VLVEVLVELELE QQIIQQIId −+−−−+−= λελ, (5.4)

Figure 5.5 shows the HRC code discriminator responses for BPSK signals with unlimited input

bandwidth, E-L spacing cT1.0=∆ and several λ values. The linear area of the code discriminator

around the stable lock point at 0=ε remains similar for different λ values, but the areas where the

code discriminator response is different from zero are narrower.

Figure 5.5: HRC code discriminator response for BPSK signals for several λ values.

Figure 5.6 shows the code discriminator output and the multipath error envelope for the HRC with

unlimited bandwidth. While for BPSK signals the code discriminator has one stable lock point at 0=ε ,

for BOC(1,1) signals the code discriminator has three stable lock points: one at 0=ε and two false-

lock points at cT±=ε . For CBOC(6,1,1/11) signals the situation is even worse, as the HRC code

31

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.06

-0.04

-0.02

0

0.02

0.04

0.06


disc

rimin

ator

out

put,

d ( ε

)

∆=0.1Tc

λ=1.5

λ=2

λ=2.5

discriminator presents even more false-lock points than for the BOC(1,1) signals. The multipath error

envelope shows that the HRC discriminator is not well adapted for use with CBOC(6,1,1/11) signals,

as it can be seen than the tracking error will be different from zero for a larger set of delays of the

reflected ray, than for the case of the BOC(1,1) signal. From the three signals and the HRC

discriminator, the BPSK signal is the one that offers the best performance in the presence of multipath.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025


disc

rimin

ator

out

put,

d ( ε )

∆=0.05Tc

BW

Tc=∞

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.015

-0.01

-0.005

0

0.005

0.01

0.015


trac

king

err

or, ε

/ T C

∆=0.05Tc

BW

Tc=∞

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.6: Code discriminator outputs (left) and multipath error envelope (right) for HRC, withunlimited pre-correlation bandwidth and λ = 2.

Figure 5.7 illustrates the effects of the different input bandwidths on the multipath error envelopes. The

multipath error envelopes for input bandwidth 12=cW TB have similar envelopes as the case of

unlimited pre-correlation bandwidth; but, while for unlimited bandwidth the shapes of multipath error

envelope were sharp, for the 12=cW TB bandwidth these shapes are smooth. For input bandwidth

5=cW TB it is important to refer that the behaviour the CBOC(6,1,1/11) signal closes up BOC(1,1)

signals, as it happens for the NELP code discriminator.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025


trac

king

err

or, ε

/ T C

∆=0.05Tc

BW

Tc=5

α=0.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.015

-0.01

-0.005

0

0.005

0.01

0.015


trac

king

err

or, ε

/ T C

∆=0.05Tc

BW

Tc=12

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.7: Multipath error envelopes for the HRC, with normalized bandwidth BWTc = 5 (left) andBWTc = 12 (right).

For weak noise conditions and linearising (5.4) around 0=ε , from (B.3) and (B.5), the steady-state

32

normalized code error variance, ( )2/ cTε , is given by:

( )( )

( )[ ] ( ) ( )( ) ( )( )[ ][ ]( )2

~~

2

0

2

)2()2(2

2/12/1211

/

2

cXXXX

XXXXPLLc

TRR

RRRR

ΝC

BT

∆′−∆′

∆−−∆++∆−∆−+⋅≈

λλ

λλλλλε (5.5)

Figure 5.8 plots the code error variance, ( )2/ cTε , versus the pre-correlation bandwidth, WB , and the

E-L spacing, ∆ , for dB/Hz400 =NC , integration period s004.0=T , 2=λ and code loop bandwidth

Hz1=PLLB . In general, the code error variance will decrease for lower E-L spacings and higher pre-

correlation bandwidths. However, for each pre-correlation value there is a E-L spacing that minimizes

the code error variance. The selected E-L spacing must provide both good multipath mitigation (see

Figure 5.8) and low noise propagation. The E-L spacings cT08.0=∆ and cT15.0=∆ were considered

suitable for 5=cW TB and 12=cWTB , respectively, with 2=λ .

[23]-, [24], [25], [26]

33


code

trac

king

erro

r, ε

/ Tc

BWTc=∞

λ=2

0 ∆/2 0.5-∆/2 0.5 0.5+∆/2

-α∆/4

-α∆/8

0

-α∆/8

α∆/4

BPSK

BOC(1,1)

Figure 5.9: Multipath error envelopes as function of E-L spacing and attenuation coefficient, for theHRC discriminator with λ=2.

5.3 Code Correlation Reference Waveform

The concept of Code Correlation Reference Waveform (CCRW) was presented in [23] to describe

different code correlation techniques used by some major GPS receiver manufacturers. These

techniques had in common the fact that, they were specially designed to mitigate multipath, and

instead of using a replica of the navigation signal, use a reference waveform, the CCRW. This

technique has also been named strobe correlator or gating correlator [23]-[26].

CCRW was first developed to improve the multipath performance of BPSK signals. More recently the

same technique was extended to BOC signals [27], [28], and proved to be able to mitigate the

multipath effects for this modulation also. In this section, the CCRW concept will be applied to BPSK,

BOC and a MBOC signal, the CBOC(6,1,1/11).

Figure 5.10 shows the receiver's structure considered for use CCRW. Let RF signal be given by (4.1),

then, the correlators PI and PQ outputs are given by (4.2) and WI and WQ are given by

( ) ( ) ( ){ }

( ) ( ) ( ){ }WqeeWXeeWXeW

WieeWXeeWXeW

NTfRTfRTfADQ

NTfRTfRTfADI

,~~

,~~

)sin()sin()(sinc

)cos()cos()(sinc

+++−++=

+++−++=


φθπτεαθπεε (5.6)

with )()()(~ εεε hRR XWWX∗= , where { })()()( εε −= tWtXERXW is the cross-correlation between )(tX and

the CCRW )(tW .

The code discriminator considered in this section is given by

)()()()()( εεεεε WPWP QQIId += (5.7)

34

Phase

Rotation

Gating Signal

Generator

I&D

I&D

I&D

I&D

CodeDiscriminator

Local Code

Generator

Signal

Generator

Local Phase

Generator

Phase

Discriminator

d(ε)

IP

IW

QW

QP

C(t-ε)

W(t-ε)

cos (t)φ^

φ^

sin (t)

z (t)i

z (t)q

y (t)q

y (t)i

X(t-ε)

sin(ω t) 0

cos(ω t) 0

r(t)

Figure 5.10: Block diagram of the receiver code and phase loops using the strobe correlator.

The receiver's performance will be strongly dependent of the selected CCRW. In the past several

CCRW have been studied, as a way to improve the code tracking performance [23]-[28]. The CCRW

studied and plotted in Figure 5.11 can be partitioned in two sub-categories: transition-based, if a

CCRW pulse appears only when a signal value transition occurs; or per-chip, if a CCRW pulse occurs

for every chip code. The determination of CCF XWR will depend if the CCRW is transition-based or

per-chip.

● CCF for transition-based CCRW

Let the CCRW and the received BOC(n,m) signal (with unlimited bandwidth) be defined as:

( )

( )∑

∑

−′=

−′−+

= −

k

ck

k

ck

mn

k

kTtXCtX

kTtWCC

tW

)(

2

)1()( 1

2

(5.8)

where )(tW ′ is a CCRW pulse and )(tX ′ depends on the signal modulation. For instance, for BPSK

signals with 2/1/ =mn , ( )cTttX Π=′ )( .

Considering that )( fGXW is the Fourier Transform (FT) of the cross-correlation function between the

received signal and CCRW:

)()( fGR XW

FT

XW →τ (5.9)

35

Then,

( ) ( ) ( ) ( )

( ) ( )

( ) ( )∑

∑∑

∑∑

−+

= =

−−

−−

′′′−+′

=

′′−+

=

′

−+

′=−=

l

lfil

mn

l

k m

kjfij

mn

j

k

jfi

j

j

mn

j

k

kfi

kXW

efWfXCC

efWfXCC

C

efWCC

efXCfWfXfG

π

π

ππ

21

2

1 1

)(21

2

21

2

2

2

)1(

2

)1(

2

)1()(

(5.10)

where ( )fX , ( )fW , ( )fX ′ and ( )fW ′ are the FT of ( )tX , ( )tW , ( )tX and ( )tW ′ respectively, and

≠=′

=′=′ ∑

=− 0,0

;with, 0

lC

NCCCC

l

chip

lk

lkkl .

Thereby,

( ) ( )( )τπ2/)1(1)( imn

chipXW efWfXNfG −+′′= (5.11)

The cross-correlation function can be determined applying the Inverse Fourier Transform (IFT).

)()( τXW

IFT

XW RfG → (5.12)

● CCF for per-chip CCRW

Now let the CCRW and the received BOC(n,m) signal (with unlimited bandwidth) be defined as:

( )

( )∑

∑

−′=

−′=

k

ck

k

ck

kTtXCtX

kTtWCtW

)(

)(

(5.13)

Applying the same steps as in the previous section, it is easy to show that

( ) ( )fWfXNfG chipXW′′=)( . (5.14)

And then determine the cross-correlation function, applying the IFT

)()( τXW

IFT

XW RfG → . (5.15)

36

0 1 2 3 4 5 6

W4

W3

W2

W1

rect

signal

time, Tc

Figure 5.11: CCRW waveforms for a BPSK signal.

5.3.1 Rectangular CCRW

The rectangular pulse )/( LtΠ , defined as 1 for 2/Lt ≤ and 0 otherwise, has been used in the CCRW

concept as alternative to the conventional E-L discriminator for BPSK signals [23]. It is possible to

show that, for BPSK signals the E-L discriminator is equivalent to a CCRW transition-based:

∆

−Π∗=

∆−−

∆+∗=

∆+−

∆− ∑ c

k

kXXXX

kTtsXtXtXXtRtR

~

22

~

22~~ . (5.16)

where ks depends on the signal code and can take one of the values -1, 0 or +1. Because of (5.16),

the here named Rectangular (RECT) CCRW, is also known as narrow CCRW [23].

-L/2 0 L/2

0

1

time

valu

e

Figure 5.12: W1 CCRW pulse.

It is important to state that (5.16) is only valid for BPSK signal. Nevertheless a CCRW may be defined

as

37

−Π=∑

L

kTtstW c

k

k)( (5.17)

and applied to other signals. However for other signals it would not be equivalent to a E-L

discriminator.

Comparing Figures 5.1 and 5.13, it is possible to confirm the similarities between the multipath

performance of the Narrow CCRW and the E-L discriminator for BPSK signals, and the differences for

BOC and CBOC signals.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05


disc

rimin

ator

out

put,

d (

ε )

L/Tc=0.05

BW

Tc=∞

BPSK

BOC(1,1)

CBOC(6,1,1/11)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.03

-0.02

-0.01

0

0.01

0.02

0.03


trac

king

err

or,

ε / T

C

L/Tc=0.05

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.13: Code discriminator outputs (left) and multipath error envelopes (right) for the RECT

CCRW with unlimited pre correlation bandwidth.

For BOC and CBOC signals, the RECT CCRW (Figure 5.13), given by (5.17), offers a worse

performance than the NELP discriminator (Figure 5.13). However by changing the CCRW to a New

RECT CCRW

+−Π=∑

L

TktstW c

k

k

)5.0()( , (5.18)

for BOC and CBOC signals, it was found that the multipath performance will increase. Reflected rays

with delays larger than about half chip duration, i.e. 2cT , will cause no effect on the tracking error

(Figure 5.14).

38

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05


disc

rimin

ator

out

put,

d (

ε )

L/Tc=0.05

BW

Tc=∞

BOC(1,1)

CBOC(6,1,1/11)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04


trac

king

err

or,

ε / T

C

L/Tc=0.05

α=0.5

BW

Tc=5 BOC(1,1)

BW

Tc=5 CBOC(6,1,1/11)

BW

Tc=12 BOC(1,1)

BW

Tc=12 CBOC(6,1,1/11)

Figure 5.14: Code discriminator outputs (left) for the New RECT CCRW with unlimited pre correlationbandwidth. Multipath error envelopes (right) for the New RECT CCRW for different signals and precorrelation bandwidths.

Assuming weak noise and from (B.3) and (B.6), the steady-state normalized code error variance,

22

cTε , is defined as:

[ ]2~0

2

)0(

1

)(WX

cDLL

RNC

TLB

′=ε (5.19)

For each pre-correlation bandwidth, there is a pulse duration, L , that minimize the code error

variance. This can be seen in Figure 5.15, where the normalized code error variance, ( )2/ cTε , versus

the pre-correlation bandwidth, WB , and the E-L spacing, ∆ , is plotted for dB/Hz400 =NC , integration

period s004.0=T , and code loop bandwidth Hz1=PLLB . The pulse durations cTL 05.0= and cTL 1.0=

were considered suitable for the normalized pre-correlation bandwidths 5=cW TB and 12=cWTB ,

respectively.

39


code

trac

king

erro

r, ε

/ Tc

BWTc=∞

-α∆/2

0

α∆/2

0 0.5+∆/2 1+∆/2

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.16: Multipath error envelopes as function of CCRW pulse width and attenuation coefficient,for RECT CCRW with BPSK signals and for New RECT CCRW with BOC and CBOC signals.

40

5.3.2 W1 CCRW

This technique is also known as HRC (for BPSK signals the W1 CCRW is equivalent to the technique

presented in section 5.2) or Double Delta [18], [23].

The W1 CCRW pulse is defined as

Π−

Π=

L

t

L

ttG

22)( (5.20)

where L is the total pulse duration (Figure 5.17).

-L/2 -L/4 0 L/4 L/2

-1

0

1

time

valu

e

Figure 5.17: W1 CCRW pulse

The W1 CCRW is illustrated in Figure 5.11 and is given by

∑ −=k

ck kTtGstW )()( (5.21)

where ks depends on the signal sign in such way that a W1 CCRW pulse only occurs at the beginning

of the chip when a signal sign transition happens.

Figure 5.18 shows the discriminator response and the multipath error envelope for the W1 CCRW and

unlimited input bandwidth. BPSK signals are well suitable to this CCRW, the discriminator response

has only one stable lock point at 0=ε and the multipath error envelope has three protuberances1, the

first at about 0=τ and the last two at around cT=τ . The discriminator response for BOC signals,

compared to the one for BPSK signals, has two false-lock points at 2cTm=ε , that introduce an

ambiguity in the code discriminator response. The multipath error envelope for BOC signals has two

more protuberances for delays around 2cTm=τ , relative to the same plot for BPSK signals. For the

CBOC signals, the discriminator response has even more false-lock points than for BOC signals, and

the multipath error envelope is similar to the BOC one, but with several small protuberances due to the

high frequency component.

1 Areas of the multipath error envelope, where the tracking error is not zero.

41

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025


disc

rimin

ator

out

put,

d (

ε )

L/Tc=0.2

BW

Tc=∞

BPSK

BOC(1,1)

CBOC(6,1,1/11)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025


trac

king

err

or, ε

/ T C

L/Tc=0.2

BW

Tc=∞

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.18: Code discriminator outputs (left) and multipath error envelopes (right) for unlimited precorrelation bandwidth and W1 CCRW.

Figure 5.19 shows the effect of different pre-correlation bandwidth values on the multipath error

envelope. Regarding to the multipath error envelopes illustrated in Figure 5.18, the protuberances on

the multipath error envelopes in Figure 5.19 are smoother. For bandwidth 5=cWTB , the amplitude of

the protuberances is significantly increased and, hence, the multipath performance is not so good

compared to to a normalized pre-correlation bandwidth 12=cWTB .

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025


trac

king

err

or,

ε / T

C

L/Tc=0.2

BW

Tc=5

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02


trac

king

err

or,

ε / T

C

L/Tc=0.2

BW

Tc=12

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.19: Multipath error envelope for several signals for the W1 CCRW and pre correlationbandwidth BWTc = 5 (left) and BWTc = 12 (right).

Similarly to (5.18), a more adequate New W1 CCRW can be written:

( )[ ]∑ +−=k

ck TktGstW 5.0)( (5.22)

Comparing to the the results obtained with W1 CCRW, and plotted in Figures 5.18 and 5.19, the New

W1 CCRW offers a much better performance, as it can be seen in Figure 5.20. For BOC and CBOC

signals and the new CCRW, the multipath error envelope, illustrated in Figure 5.20, is preferable to the

42

one for W1 CCRW, plotted in Figure 5.18: the new CCRW offers a lower tracking error for reflected

rays delayed about 2cT and eliminates the tracking error due to reflected rays delayed about cT .

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05


disc

rimin

ator

out

put,

d (

ε )

L/Tc=0.2

BW

Tc=∞

BOC(1,1)

CBOC(6,1,1/11)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.03

-0.02

-0.01

0

0.01

0.02

0.03


trac

king

err

or,

ε / T

C

L/Tc=0.2

BW

Tc=∞

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.20: Code discriminator outputs (left) and multipath error envelopes (right) for unlimited precorrelation bandwidth and the New W1 CCRW.

Figure 5.22 shows the steady-state normalized code error variance, 22

cTε , given by (5.19), as a

function of the normalized pre-correlation bandwidth and CCRW pulse duration. For both the CCRWs

and for all the three signals, cTL 1.0= and cTL 05.0= for 5=cW TB and 12=cW TB , respectively, were

considered suitable pulse widths (see also Figure 5.21).


code

trac

king

erro

r, ε

/ Tc

BWTc=∞

0 L/2 0.5-L/2 0.5 0.5+L/2

-αL/4

-αL/8

0

-αL/8

αL/4

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.21: Multipath error envelopes as function of CCRW pulse width and attenuation coefficient,for W1 CCRW with BPSK signals and for New W1 CCRW with BOC and CBOC signals.

43

5.3.3 W2 CCRW

The W2 CCRW pulse is equal to the W1 CCRW pulse, given by (5.20). Although the CCRW W2 is

different and written as

∑ −=k

ck kTtGtW )()( (5.23)

where )(tGGk ±= and ± depends on the chip sign. The main difference between W1 CCRW and W2

CCRW is that, in the latter the CCRW pulses will appear at the beginning of every chip, while in the

former CCRW only occur when a signal value transition happens, as shown in Figure 5.11.

For BPSK signals, the discriminator response for W2 CCRW has one stable lock point at 0=ε . For

BOC(1,1) signals, the discriminator function has two stable lock points at 0=ε and at cT=ε . For

CBOC(6,1,1/11) signals, there are other two zero crossing points that can act as lock points. However,

this area has a lower slope and is smaller (the points will be harder to lock) than the stable lock points

44

at 0=ε and at cT=ε .

Compared to the others CCRW seen so far (RECT, W1), the W2 has the best multipath error envelope

for the three signals. The multipath error envelope has only one protuberance at 0≈τ , which means

that the W2 CCRW is only affected by reflected rays delays lower than L . It is important to refer that,

for CBOC(6,1,1/11) the multipath error envelope has a bias error; however this is a deterministic error

that is easily corrected.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05


disc

rimin

ator

out

put,

d (

ε )

L/Tc=0.2

BW

Tc=∞

BPSK

BOC(1,1)

CBOC(6,1,1/11)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025


trac

king

err

or,

ε / T

C

L/Tc=0.2

BW

Tc=∞

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.23: Code discriminators outputs (left) and multipath error envelopes (right) for unlimitedbandwidth and the W2 CCRW.

Assuming that the received signal has a limited input bandwidth, the multipath error envelopes will

change. For bandwidths of 5=cW TB and 12=cW TB the multipath error envelopes will change to the

curves plotted in Figure 5.24. Beside the original and now smoother protuberance, several other and

smaller protuberances appear in the multipath error envelopes. Other important change is that a bias

appears for BOC signals, and the initial bias for CBOC(6,1,1/11) is increased, but, as before, these

bias are deterministic and can be corrected.

Figure 5.25 plots the steady-state normalized error variance, 22

cTε , given by (5.19), versus the

CCRW pulse duration and the normalized pre-correlation bandwidth, for several signals. For the W2

CCRW, considering also the multipath performance (see Figure 5.26), cTL 3.0= and cTL 15.0= for

5=cW TB and 12=cW TB , respectively, were considered suitable pulse widths.

45

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03


trac

king

err

or,

ε / T

C

L/Tc=0.2

BW

Tc=5

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02


trac

king

err

or,

ε / T

C

L/Tc=0.2

BW

Tc=12

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.24: Multipath error envelopes for the W2 CCRW and pre correlation bandwidth BWTc = 5 (left)and BWTc = 12 (right).

46

multipath delay, τ / Tc

code

trac

king

erro

r, ε/

Tc

BWTc=∞

0 L/2

-αL/4

0

αL/4

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.26: Multipath error envelopes as function of CCRW pulse width and attenuation coefficient,for W2 CCRW.

5.3.4 W3 CCRW

The W3 CCRW pulse is sketched in Figure 5.27. As can be seen, while the previous analysed pulses

were symmetric, the W3 CCRW pulse is asymmetric, and is given by

−Π−

Π=

2

2

2

1

2)(

L

Lt

L

ttG . (5.24)

-L/2 0 L/2

-0.5

0

1

time

valu

e

Figure 5.27: W3 CCRW pulse.The W3 CCRW is a per-chip CCRW and is defined by (5.23), with CCRW pulse given by (5.24).

Figure 5.28 illustrates the code discriminator response and the multipath error envelope for the W3

CCRW, different signals and unlimited input bandwidth. Despite the discriminator output be clearly

different, the multipath error envelope is very similar to the one obtained with the W2 CCRW, plotted in

Figure 5.23. As it is possible to see in the multipath error envelopes presented in Figures 5.23 and

5.28, the main difference in the multipath error envelopes from the W2 CCRW tracking error is a

deterministic bias error.

47

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06


disc

rimin

ator

out

put,

d (

ε )

L/Tc=0.2

BW

Tc=∞

BPSK

BOC(1,1)

CBOC(6,1,1/11)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08


trac

king

err

or,

ε / T

C

L/Tc=0.2

BW

Tc=∞

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.28: Code discriminator output for several signals (left) and multipath error envelopes forseveral signals (right) for the W3 CCRW.

Figures 5.29 illustrates the multipath error envelopes for the W3 CCRW and two different pre

correlation bandwidths. For the wider bandwidth 12=cW TB , the multipath error envelopes are very

similar to the ones plotted in 5.28, but for the bandwidth 5=cW TB , several small protuberances appear

in the multipath error envelopes.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08


trac

king

err

or,

ε / T

C

L/Tc=0.2

BW

Tc=5

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.01

0.02

0.03

0.04

0.05

0.06

0.07


trac

king

err

or,

ε / T

C

L/Tc=0.2

BW

Tc=12

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)


The steady-state normalized error variance, 22

cTε , given by (5.19), is plotted in Figure 5.30 as a

function of the normalized pre-correlation bandwidth, cWTB , and of the CCRW pulse duration, L , for

several signals and with Hz1=PLLB and s004.0=T . It is possible to see that W4 CCRW pulses with a

total duration cTL 05.0= and cTL 1.0= are adequate, both in the presence of noise and multipath

(Figure 5.31), for normalized bandwidths 12=cW TB and 5=cW TB , respectively.

48


code

tra

ckin

g er

ror,

ε /

Tc

BWTc=∞

0 3L/4

-αL/4

0

αL/4

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.31: Multipath error envelopes as function of CCRW pulse width and attenuation coefficient,for W3 CCRW.

49

5.3.5 W4 CCRW

The W4 CCRW is a per-chip CCRW given by (5.23), where the waveform elements, plotted in Figure

5.32, are asymmetric pulses defined as

−Π−

Π=

3

2/

32)(

L

Lt

L

ttG . (5.25)

-L/3 0 L/3 2L/3

-1

0

1

time

valu

e

Figure 5.32: W4 CCRW pulse.

The W3 and W4 CCRWs behaviours are quite similar and, as so, their performances are also very

similar. As it happens with W3, W4 CCRW also presents a deterministic tracking error bias. Figures

5.33 and 5.34 (when compared to Figures 5.28 and 5.29) confirm the similarities between the W3 and

W4 CCRWs.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.06

-0.04

-0.02

0

0.02

0.04

0.06


disc

rimin

ator

out

put,

d (

ε )

L/Tc=0.15

BW

Tc=∞

BPSK

BOC(1,1)

CBOC(6,1,1/11)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025


trac

king

err

or,

ε / T

C

L/Tc=0.15

BW

Tc=∞

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.33: Code discriminator output for several signals (left) and multipath error envelopes forseveral signals (right) for the W4 CCRW.

For a input bandwidth 12=cW TB , the effects of the input bandwidth on the multipath error envelope are

small, as it can be seen in Figure 5.34. It is important to refer, that for the pre correlation bandwidth

5=cW TB a bias and several protuberances appear on the multipath tracking error. However, as before,

the bias is deterministic and can be removed.

50

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.03

-0.02

-0.01

0

0.01

0.02

0.03


trac

king

err

or,

ε / T

C

L/Tc=0.15

BW

Tc=5

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02


trac

king

err

or,

ε / T

C

L/Tc=0.15

BW

Tc=12

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)


The steady-state normalized error, 22

cTε , given by (5.19), is represented in Figure 5.34 versus the

normalized pre-correlation bandwidth and the CCRW pulse width, for several signals and with

Hz1=PLLB , s1 nTc = and s004.0=T . The W4 CCRW pulses with a total duration of cTL 1.0= and

cTL 2.0= are suitable, considering also the multipath performance (Figure 5.35), for bandwidths of

12=cWTB and 5=cWTB , respectively.


code

trac

king

erro

r, ε

/ Tc

BWTc=∞

0 2L/3

-αL/3

0

αL/3

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.35: Multipath error envelopes as function of CCRW pulse duration and attenuation coefficient,for W4 CCRW.

51

5.4 Teager-Kaiser Operator

The non-linear quadratic TK operator was first introduced for measuring the real physical energy of a

system [29]. It was found that this non-linear operator is simple, efficient and able to track

instantaneously varying spatial modulation patterns [30]. Since its introduction, several other

applications have been found for TK operator, including estimating the code tracking error of a GNSS

receiver [31], [21].

The discrete-time TK operator for a complex valued signal ][nx is given by [32]

( )]1[]1[]1[]1[2

1][][])[(

*** −+++−−= nxnxnxnxnxnxnxψ . (5.26)

If this operator is applied to the ACF of a BPSK signal, the TK energy of the function will exhibit a peak

at zero lag [31]. If instead of a BPSK signal it is used a BOC(1,1) or a CBOC(6,1,1/11) signal is used,

52

the TK energy of the autocorrelation function will still exhibit a large peak at zero lag, but other lower

peaks will appear at secondary lobes. Figure 5.37 shows the behaviour of the TK output for different

signals using a sample time cs TT 04.0= .

As it was mentioned earlier, in the presence of multipath, the CCF function between the received and

the locally generated signals can be viewed as a superposition of shifted and distorted versions of the

undisturbed ACF. Applying the TK operator, the TK energy of the CCF will exhibit a peak at zero lag

(Figure 5.38). The lower peaks are the result of ACF secondary lobes and received echoes.

In [21] was analysed a deconvolution algorithm that used the TK operator to remove the reflected rays

from the incoming signals. The main assumption for that algorithm was that, as the multipath signals

provides time-aligned peaks located at the multiple rays time of arrival, then it should be possible to

remove the reflected rays from the incoming signal, and hence eliminate the code tracking errors

produced by them. The computation load of this method will vary considering the number of

correlators used. But as it is possible to operated with few correlators, the computation load could be

very low, and thereby adequate for use in low cost, mass-market receivers [21]. The drawback of

using fewer correlators is that method will lose resolution and will present bigger errors estimating the

peak positions.

53

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.4

-0.2

0

0.2

0.4

0.6

0.8

1


norm

aliz

ed T

K op

erat

or,

Ψ (

ε )

CBOC(6,1,1/11)

BOC(1,1)

BPSK

Figure 5.37: TK operator output for BPSK, BOC(1,1) and CBOC(6,1,1/11) signals.

-1 -0.5 0 0.5 1 1.5-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

norm

aliz

ed T

K o

pera

tor,

Ψ

( ε

)


τ=0.2

τ=0.4

τ=0.6

Figure 5.38: TK operator output for BOC(1,1) for a LOS signal and one reflect ray with different delaysrelative the LOS signal.

To keep the complexity as low as possible, while obtaining high resolution, was proposed in [33] a

continuous update technique, in which a discriminator constructed from the TK operator is used to

estimate the code delay.

Similarly to algorithm present in [33], the TK operator can be used to replace the Early and Late

correlators output and used with a classical Early-minus-Late discriminator (Figure 5.39).

( ) LEd ψψε −= (5.27)

where Eψ and Lψ are the result of applying the TK operator to the CCF between the received signal

and the early and the late version of the locally generate signal with ∆= 2sT , respectively, and are

proportional to

∆−

∆++

∆+

∆−−

∆−

∆−∝

∆−

∆++

∆+

∆−−

∆+

∆+∝

*

~~

*

~~

~~

*

~~

*

~~

~~

2

5

2

3

2

3

2

5

2

1

22)(

2

3

2

5

2

5

2

3

2

1

22)(

εεεε

εεεψ

εεεε

εεεψ

XXXXXXXX

XXXXL

XXXXXXXX

XXXXE

RRRR

RR

RRRR

RR

(5.28)

Now, considering a receiver architecture like the one plotted in Figure 3.2, but with six correlators: very

very late (VVL), VL, L, E, VE, very very early (VVE); with E-L spacing ∆ , VE-VL spacing ∆3 and VVE-

VVL spacing ∆5 , Eψ and Lψ can be given as:

54

[ ]

[ ]

[ ]

[ ]VEVVLVEVVLLL

VVLVVLVEVEVEVEVVLVVLLLL

VVEVLVVEVLEE

VVEVLVLVVEVLVVEVVEVLVLVVEVLVVEVVEVLVVEVL

VLVVEVLVVEVLVVEVLVVEVVEVLVVEVLVVEVLVVEVL

VLVLVVEVVEVVEVVEVLVLEEE

QQIIQI

iQIiQIiQIiQIQI

QQIIQI

QiIIiQQiIIiQQQIIQQII

QQIiQQiIIIQQIiQQiIII

iQIiQIiQIiQIQI

+−+=

−++−+−+=

+−+=

−+−++++=

++−+++−=

−++−+−+=

22

22

22

22

))(())((2

1)(

))(())((2

1)(

εψ

εψ

(5.29)

Thereby, the discriminator response for the purposed TK based algorithm is given by:

( ) [ ] [ ]{ }

[ ] [ ] [ ] [ ]VVEVLVVEVLVEVVLVEVVLLLEE

VEVVLVEVVLLLVVEVLVVEVLEE

QQIIQQIIQIQI

QQIIQIQQIIQId

+−+++−+=

+−+−+−+=

2222

2222ε

(5.30)

This method has the advantage of using the well known classic DLL, keeping complexity low and

improving the performance under a multipath environment. As it can be seen in Figure 5.39 (right), the

method significantly improves the performance comparative to the narrow correlator. It has particularly

good performance for BPSK signals. For BOC(1,1) the results are also good, and for CBOC(6,1,1/11)

despite being worse compared to BPSK and BOC(1,1), the performance is still superior to the Narrow

Correlator. On the other hand the method will require at least 6 correlators (if considered a coherent

receiver) for tracking the code delay, while a classic E-L can do the same with only two correlators. But

the complexity is still low.

However, the CCRW, judging the multipath error envelopes, offers a best performance than this

technique based on the TK operator and has a lower complexity, as the CCRW requires only 4

correlators for a non coherent receiver, or 2 correlators if it is considered a coherent receiver.

Figure 5.40 shows the multipath error envelopes for input bandwidth 5=cW TB and 12=cW TB . As it

happened with the other multipath mitigation techniques analysed in this chapter, the code tracking

error increases when the pre correlation bandwidth is narrower.

This TK-based technique uses more correlators than the other techniques. Thus, it is more difficult to

deduce an expression for the steady-state code error variance. For this reason and this technique, the

effect of noise will be only considered in the simulation results, that will be presented in Sub Section

7.3.

55

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025


disc

rimin

ator

out

put,

d (

ε )

∆=0.05Tc

BW

Tc=∞

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.015

-0.01

-0.005

0

0.005

0.01

0.015


trac

king

err

or,

ε / T

C

∆=0.05Tc

BW

Tc=∞

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.39: Code discriminator response (left) and multipath error envelopes (right) for several signalsand unlimited pre-correlation bandwidth.

0 0.5 1 1.5 2-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025


tra

ckin

g e

rro

r, ε/T

C

∆=0.05Tc

BW

Tc=5

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)

0 0.5 1 1.5 2-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02


tra

ckin

g e

rro

r, ε/T

C

∆=0.05Tc

BW

Tc=12

α=0.5

BPSK

BOC(1,1)

CBOC(6,1,1/11)

Figure 5.40: Multipath error envelopes for several signals BWTc = 5 (left) and BWTc = 12 (right).

56

Chapter 6

Simulation Setup

In Chapter 5, the behaviour of some multipath mitigation technique in the presence of multipath was

analysed. Nevertheless, the assumptions made in the previous Chapter were very simple and ideal. It

is necessary to set up a test environment more close to reality. Thereby, a GNSS receiver and a more

realistic multipath model were developed in Simulink with GRANADA FCM Blockset.

6.1 GRANADA FCM Blockset

The GRANADA FCM (Factored Correlator Model) Blockset is a simulation tool that allows to quickly

assess the receiver performance under different scenarios [34].

Compared to hardware GNSS receiver test benches (e.g. FPGA based), the FCM provides more

flexibility and thereby it is more adequate for architecture design and development phase. FCM allows

to analyse and control the internal receiver signals, so new algorithms for acquisition, tracking, lock

detection and other can be easy developed and tested.

If compared to highly realistic software simulators that simulate part or the whole GNSS receiver’s

signal processing chain, the FCM is much faster, because it models the outputs of the high frequency

stages of a GNSS with a low frequency algorithm (as illustrated in Figure 6.1). Thereby, FCM

significantly increases simulation speed, while still weighting numerous effects, like the carrier phase

and frequency errors, code delay errors, and Doppler effects [34].

Therefore, FCM is a good choice to simulate a receiver employing the techniques presented in

Chapter 5. Figure 6.2 shows a possible implementation with code and carrier loops).

57

Figure 6.1: Typical signal processing chain for the tracking of GNSS signals [35].

Figure 6.2: Possible implementation of a multi-channel GNSS receiver's tracking loops [35].

6.1.1 Using FCM with CBOC signals and Strobe Correlator

FCM can simulate a GNSS receiver with conventional correlators for BPSK and BOC signals.

However, it offers no support for MBOC signals nor CCRWs, so, a workaround is required.

First it is necessary understand how the FCM works. FCM assumes that the correlator outputs of the

nth integration can be approximated by [34]

nnnnnn WLREAS +≈ (6.1)

where nA is the post-correlation signal amplitude, nE is the contribution of the complex carrier, nR is

the contribution of the code misalignment and is related with the ACF, nL accounts for the period

expansion or compression due to satellite-receiver relative motion, and nW is the post-correlation

noise.

The factors nA and nE will not be affected by changes in the signal modulation or the use of CCRW.

58

However, both the factors nR and

nL depend of the signal modulation and the correlation technique

used. In this thesis, the period expansion or compression due to satellite-receiver relative motion was

neglected; thus the contribution of the factor nL was also neglected (and its contribution scheduled for

future work).

It is necessary to rewrite the FCM internal function that determines the factor nR . The new function

can generate the ACF of BPSK, BOC and MBOC signals, applying the expressions given in Sub

Section 2.3.1; and the CCF between a signal and a CCRW, using the procedure described in Section

5.3.

6.1.2 FCM with correlated correlators noise

The FCM does not assume that the noise of the correlator outputs, nW , is correlated. However, this

happens, and each correlator output has a noise component, as it was seen in (3.7), that is correlated

with the noise present in the other correlator outputs. So it was necessary to generate Gaussian

random variables, with a predefined covariance matrix.

Let nyy ,,1 L denote the desired Gaussian random variables with covariance matrix [ ]2

ijR σ= , where

{ }jiij yyE=2σ , and let nuu ,,1 L denote a set of independent of independent Gaussian random variables

with unit variance. In [36] was presented a method to generate { }iy from { }iu :

Auy = , (6.2)

The covariance matrix can be written as

{ } { }TTT AAuuEyyER == (6.3)

where { } IuuET = . Any matrix A which satisfies T

AAR = can be used in (6.2) to generate the desired

correlated random variables.

Since R is a real symmetric matrix, it can be written as

T

n

PPR

=

λ

λ

L

MOM

L

0

01

(6.4)

where P is an orthogonal matrix, and iλ are eigenvalues of R . Considering that the iλ are indexed

in decreasing order, the jth column of P , denoted jp is simply an eigenvector of R associated with

iλ , normalized to unit magnitude. Thereby

59

=

n

nppA

λ

λ

0

01

1 OL (6.5)

The random variables generated this way will have a unitary variance. In this situation the noise

power at the output of the correlator NP will be equal to one. To simulate the behaviour of the

correlators' output under different carrier to noise density ( 0/ NC ) it was decided to keep 1=NP and

change the signal power at the correlator output SP .

6.2 Multipath channel model

Apart from the one used in the computation of multipath error envelopes, there are other more

complex multipath models. The most realistic but also the most complex approach to assess the

multipath performance of a given signal/receiver combination is to consider not only the signal

characteristics and the receiver architecture but also different multipath environments and different

elevation angles. Such models have been derived in the past [37], based on extensive measurements

campaigns.

For the following determination of typical multipath errors that are valid for a dedicated environment,

the wide band channel model presented in [37] has been used. This model provides general

distribution of the number of occurring multipath signals, their corresponding path delays and relative

amplitudes as well as associated model parameters for different environments and elevation angles.

Effects like shadowing are also taken into account by modelling the amplitude of the line-of-sight

(LOS) component.

The considered model comprises a direct path, near echoes and far echoes. Each path is described

by its complex amplitude relative to free space propagation, path delay mτ relative to 1st path,

amplitude distribution of ma or instantaneous power distribution 2

mm aP = .

A navigation signal is transmitted from the satellite in many directions. Different reflectors kR cause

echoes with a round-trip detour ks∆ and a delay cskk /∆=∆τ (where c is the velocity of light) with

respect to the propagation delay of an undisturbed signal. All reflected components of the signal

superimpose at the receiver input.

The channel is assumed to be a wide-sense stationary with uncorrelated scatterers (WSSUS),

thereby it can be modelled as a filter structure with delay taps. The complex impulse response of the

satellite wideband channel can then be defined as sum of Mk ,...,1= signal paths with amplitude )(tEk

and delay 1τ and kk tt τττ ∆+= )()( 1 , with Mk ,...,2= :

60

( )( )∑=

=N

k

kk ttEth1

)(),( τδτ (6.6)

where the amplitude of each echo is complex and can be rewritten as

)()()( ti

kkketatE

φ= . (6.7)

For WSSUS channels, the phases )(tkφ are uniformly distributed in [2,0[ π , since there is no

correlation among scatterers with different geometrical distances by definition.

6.2.1 Direct Path

The direct path is defined as the shortest path between the satellite and the receiver. This path can be

either clear; partial obstructed (shadowing) with moving obstacles, vegetation, building or other

obstacles; or total blocked. The model referred in [37] describes the direct path ray for the three

situations.

● Clear

The amplitude of the direct path ray, 1a , is random and can be described by the Rician

distribution:

+−

=

2

2

2022

1exp)(

σσσkkk

kRice

aaI

aapdf , (6.8)

where the Rice-factor ( )22/1 σ=c denotes the carrier-to-multipath (signal-to-multipath ratio

SMR) and )(0 xI is the modified Bessel function of the first kind and 0th order.

● Shadowing

The probability density function of the ray amplitude is 1a a Rayleigh-type with a lognormal-

distribution mean power 0

22 P=σ :

( )( )

−−⋅=

−=

2

2

0

0

0ln

22

2

log10exp

1

10ln2

10)(

2exp)(

σ

µ

σπ

σσ

P

PPpdf

aaapdf kk

kRayl

.

(6.9)

(6.10)

● Blocked

The ray amplitude is 01 =a , or in other words, there is no direct path ray.

6.2.2 Near Echoes

The near echoes will appear in the delay interval em ττ <<0 , NEMm += 1,...,3,2 . The number of the

61

near echoes, NEM , is Poisson distributed with mean NEλ according to

λλ −= eN

NPN

Poisson!

)( . (6.11)

The delay distribution of the near echoes follows an exponential distribution

∆−=∆

bbP k

k

ττ exp

1)(exp (6.12)

The mean power of near echoes ( ) ( )2

kaS ετ = is exponentially decreasing

( ) keSSδττ −= 0 , (6.13)

or in log scaling

( )τ

τ

dB

d

dB

S

dB

S−= 0 (6.14)

with ( ) ( )ed log10/log10 δ= . Given a mean power ( )τS for a fixed delay τ , the amplitude ka of the near

echoes will vary around this mean value according to a Rayleigh distribution, (6.9), with ( )τσ S=22 .

6.2.3 Far Echoes

The number of far echoes, FEM , is Poisson distributed with mean FEλ according to equation (6.11).

The amplitudes of ka of the far echoes follow a Rayleigh distribution (6.9). The delays of the far

echoes are uniformly distributed in [ [maxe ττ , .

6.2.4 Multipath model with FCM

If the received signal may be written as

ki

k

kk exasπφτττ 2

1

11 )()( ∑=

∆−= , (6.15)

then the cross-correlation between the received signal and the locally generated signal will be

ki

k

kxksx eRaRπφτττ 2

1

11 )()( ∑=

∆−= (6.16)

Thereby, a multipath scenario may be set up with FCM, considering that each signal path

ki

kk exaπφττ 2

1 )( ∆− is an independent channel. The sum of all these channels will result in the

correlators' output of the multipath signal given by (6.15). The parameters φτ ,, kka ∆ are determined

according to Section 6.2.

62

6.3 Simulation Plan

Figure 6.3 shows the implemented GNSS receiver simulator in Simulink, using the Granada FCM

blockset. The block FCM+SCM+CCM models the correlator outputs considering correlated noise and

the multipath channel model.

The block normalization estimates the signal power and then normalizes the correlator outputs by the

estimated signal power. The DLL block implements the code discriminator and code low pass filter.

The INS aiding simulates the aiding signal to the Aided PLL. The block Aided PLL, implements the

carrier phase discriminator.

Figure 6.3: Simulink block diagram of the implement receiver simulator.

The developed GNSS receiver was set up with code loop bandwidth HzBDLL 1= , carrier loop

bandwidth HzBPLL 3= , integration time msT 4= and chip duration nsTc 1= . The receiver was tested

with BPSK, BOC(1,1) and CBOC(6,1,1/11) signals. The simulations were performed for two pre

correlation bandwidths 5=cW TB and 12=cW TB . These two input bandwidths were selected, because

they allow to simulate the behaviour of very low cost receivers, that have narrower input bandwidths

and the behaviour of wideband GNSS receivers, that can take advantage of the wideband

components of the new GPS and Galileo signals.

Table 6.1 shows the multipath mitigation techniques selected for the simulation plan. The E-L spacing,

∆ , and the CCRW pulse duration, L , were chosen for each technique and each bandwidth, in a way

that the code discriminator offers a good all-round performance (in the presence of both multipath and

noise). For BOC and CBOC signals, instead of RECT and W1 CCRWs it is used New RECT and New

63

W1 CCRWs.

Bandwidth 5=cW TB Bandwidth 12=cW TB

NELP with 1.0=∆ cT NELP with 05.0=∆ cT

HRC with 15.0,2 =∆= cTλ HRC with 08.0,2 =∆= cTλ

RECT CCRW with 1.0=cTL RECT CCRW with 05.0=cTL

W1 CCRW with 3.0=cTL W1 CCRW with 15.0=cTL




TK with 1.0=∆ cT TK with 1.0=∆ cT

Table 6.1: Multipath mitigation techniques selected for simulation.

The receiver trajectory is illustrated in Figure 6.4. The directions x and y are orthogonal and belong to

a plane tangential to Earth's surface. The selected trajectory simulates the dynamics of a vehicle in an

urban environment and it has the duration of 50s.

0 50 100 150 200 250-20

0

20

40

60

80

100

120

140

y, m

x, m

Figure 6.4: Receiver trajectory.

Figure 6.5 shows the acceleration components, the velocity and the dynamics relative (velocity, vr,, and

acceleration, ar) to the satellite of the receiver versus the simulation time.

64

Eight multipath scenarios were selected to evaluate the performance of the different techniques:

● No Multipath (NMP) – Where the receiver is tested without multipath;

● Suburban (SUB) – Multipath model for a suburban scenario and elevation angle E=35º, [37];

● Urban I (URB1)– Simulates the multipath present in a urban environment for elevation angle

E=45º, [37];

● Urban II (URB2) – The same as the previous scenario, but for elevation angle E=25º [37];

● Urban III (URB3) – This scenario is based on the previous one, but it adds more and stronger

reflected rays.

● Urban IV (URB4) – The same as before, but with even more reflected rays and stronger

multipath amplitudes.

● Urban V (URB5) – This scenario differs from the last one in the number of near echoes.

● Urban VI (URB6) – It is the worst scenario of the set of simulations.

Figure 6.6 illustrates the typical distribution of amplitudes versus the echoes delays for the all

scenarios, obtained from the multipath channel model. There is no plot for the “no multipath” scenario,

because in this scenario there are no echoes. The eight multipath environments selected to test the

techniques studied are defined in Table 6.2.

65

a) Sub-urban b)Urban I

c) Urban II d) Urban III

e) Urban IV and V f) Urban VI

Figure 6.6: Echoes amplitudes versus echoes delays, for the different scenarios.

66

Environment NMP SUB URB1 URB2 URB3 URB4 URB5 URB6

Time Share of Shadowing A 0 0.54 0.56 0.79 0.79 0.79 0.79 0.79

1a , clear, Rice c [dB] - 10.7 8.5 3.2 3.2 3.2 3.2 3.2

1a , shadowed, Rayleigh/LNµ [dB] - -7.6 -3.0 -12.1 -12.1 -12.1 -12.1 -12.1

σ [dB] - 3.2 2.7 6.3 4.3 4.3 4.3 4.3

NEN , Poisson NEλ 0 1.2 3.6 4.0 5.0 8.0 15.0 8

Maximum near echoes delay eτ [ns] - 400 600 600 600 600 600 600

Delay NEτ∆ exp. b [µs] - 0.039 0.081 0.063 0.15 0.15 0.15 0.15

( )tS0S [dB] - -24.4 -23.5 -17.0 -10.0 -7.0 -7.0 -4.0

d [dB] - 23.6 8.5 26.2 20 20 20 20

FEN , Poisson FEλ 0 0.8 0.8 0.8 0.8 0.8 0.8 0.8

ka , Rayleigh σ [dB] - -28.2 -28.2 -28.2 -28.2 -28.2 -28.2 -28.2

Maximum delay eτ [µs] - 5 5 5 5 5 5 5

Table 6.2: Multipath Model Parameters

67

Chapter 7

Results Discussion

This Chapter is subdivided into three sections. First, the multipath mitigation techniques analysed in

Chapter 5 will be compared between them, so it may be possible to have an idea of what to expect

from the simulation results. Next, the benefits of using a aided PLL will be discussed. And finally, the

simulation results will be presented and discussed.

7.1 Comparison between different techniques

Figures 7.1-7.3 show the multipath error envelopes of the considered techniques (see Table 6.1) for

the different signals and bandwidths. Considering the multipath error envelopes, W2, W3 and W4

CCRWs are expected to have the best multipath mitigation capabilities of all the tested techniques.

W1 CCRW and HRC are other techniques expected to have good performance in the presence of

multipath. NELP and RECT CCRW should have the worse performance in the presence of multipath.

Figure 7.1: Multipath error envelopes for BPSK signals, for the different techniques and bandwidths.

68

0 0.2 0.4 0.6 0.8 1 1.2-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04


code

tra

ckin

g er

ror,

ε /

Tc

BWTc=5

α=0.5

NELP

HRCRECT

W1

W2

W3W4

TK

0 0.2 0.4 0.6 0.8 1 1.2-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025


code

tra

ckin

g er

ror,

ε /

Tc

BWTc=12

α =0.5

NELP

HRCRECT

W1

W2

W3W4

TK

Figure 7.2: Multipath error envelopes for BOC(1,1) signals, for the different techniques andbandwidths.

0 0.2 0.4 0.6 0.8 1 1.2-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04


code

tra

ckin

g er

ror,

ε /

Tc

BWTc=5

α=0.5

NELP

HRCRECT

TK

0 0.2 0.4 0.6 0.8 1-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02


code

tra

ckin

g er

ror,

ε /

Tc

BWTc=12

α =0.5

NELP

HRCRECT

TK

0 0.2 0.4 0.6 0.8 1 1.2-0.03

-0.02

-0.01

0

0.01

0.02

0.03


code

tra

ckin

g er

ror,

ε /

Tc

BWTc=5

α=0.5

W1

W2W3

W4

0 0.2 0.4 0.6 0.8 1-0.015

-0.01

-0.005

0

0.005

0.01

0.015


code

tra

ckin

g er

ror,

ε /

Tc

BWTc=12

α=0.5

W1

W2W3

W4

Figure 7.3: Multipath error envelopes for CBOC(6,1,1/11) signals, for the different techniques andbandwidths.

Figures 7.4-7.6 show the normalized steady-state code error variance, ( )2

cTε , for the selected

techniques (Table 6.1) and the different signals. It can be seen that the NELP and the HRC

discriminators have lower steady-state error variance relative to CCRW discriminators. So, these

techniques should offer superior performance in the presence of noise comparative to CCRWs. The

TK based technique was not included in Figures 7.4-7.6 because it was not deduced a formula for

determining the steady-state code error variance.

69

0 0.2 0.4 0.6 0.8 1 1.2-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04


code

tra

ckin

g er

ror,

ε /

Tc

BWTc=5

α =0.5

NELP

HRCRECT

W1

W2

W3W4

TK

0 0.2 0.4 0.6 0.8 1-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025


code

tra

ckin

g er

ror,

ε /

Tc

BWTc=12

α =0.5

NELP

HRCRECT

W1

W2

W3W4

TK

30 35 40 45 5010

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

C/N0, dB

BWTc=5

NELPHRCRECTW1W2W3W4

30 35 40 45 5010

-7

10-6

10-5

10-4

10-3

10-2

10-1

C/N0, dB

Nor

mal

ize

d co

de

err

or

vari

an

ces

BWTc=12

NELPHRCRECTW1W2W3W4

Figure 7.4: Normalized code error variances for BPSK signals,for the different techniques andbandwidths.

30 35 40 45 5010

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

C/N0, dB

BWTc=5

NELPHRCRECTW1W2W3W4

30 35 40 45 5010

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

C/N0, dB

Nor

mal

ize

d co

de

err

or

vari

an

ces

BWTc=12

NELPHRCRECTW1W2W3W4

Figure 7.5: Normalized code error variances for BOC(1,1) signals, for the different techniques andbandwidths.

30 35 40 45 5010

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

C/N0, dB

BWTc=5

NELPHRCRECTW1W2W3W4

30 35 40 45 5010

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

C/N0, dB

Nor

mal

ize

d co

de

err

or

vari

an

ces

BWTc=12

NELPHRCRECTW1W2W3W4

Figure 7.6: Normalized code error variances for CBOC(6,1,1/11) signals for the different techniquesand bandwidths.

70

7.2 Aiding signal

As it was seen on Chapter 3, the receiver tracking loops can be aided: the PLL can be aided with a

signal, that estimates the frequency shift due to the Doppler effect; and the DLL can be aided with the

phase rate estimated by the PLL. The concept of aided loops was applied to the GNSS receiver

developed in Section 6.3. In this Section will be compared the code tracking error of aided tracking

loops with unaided tracking loops.

Two different environments are considered in this section: “no multipath” (NMP) and “urban

E=25º” (URB2). Different receiver settings are also considered (see Table 7.1). The GNSS receiver,

here considered, uses the NELP discriminator with pre-correlation bandwidth 12=cWTB , E-L spacing

cT05.0=∆ , integration time and code period sTTcode 004.0== , number of chips 4092=chipN , and

BOC(1,1) signals.

#1 Unaided Hz18=PLLB , Hz1=DLLB , NMP

#2 Aided Hz3=PLLB , Hz1=DLLB , NMP

#3 Unaided PLL Hz18=PLLB , Hz1=DLLB , URB2

#4 Aided PLL, Hz3=PLLB , Hz1=DLLB , URB2

Table 7.1: Tested conditions.

Figure 7.7 shows the mean (colour bars) and standard deviation (error bars) of the code tracking error

for the conditions described in Table 7.1. The receiver with the Aided PLL offers the best performance

in the presence of noise (conditions #1 and #2): the aided loops (#2) may present a higher code error

mean, but they also have lower code error variance than the unaided loops (#1). In the presence of

multipath, the benefits of using an aided PLL (condition #4) are even more clear: both the code

tracking error mean and variance are much lower than using an unaided PLL (condition #3). So, the

aiding signal by itself can be used to mitigate the multipath effect.

The advantage of using aided loops is that lower PLL bandwidth, PLLB , can be used, because the

71

Figure 7.7: Code tracking error for aided PLL versus not aided PLL without multipath (left), and withmultipath.

aiding signal helps the tracking loops estimating the receiver dynamics. So the estimated phase will

have less noise (as it is possible to see in Figure 7.7, the standard deviation for the aided PLL is lower

than for the unaided PLL). As the phase rate aids the DLL, then the code delay estimation will also be

more accurate and have less noise.

Although the presented results were valid only for BOC signals and the NELP discriminator, the

benefits of aided loops are extendable to all signals and discriminator functions.

7.3 Simulation Results

The simulation plan defined in Section 6.3 consists of testing eight multipath mitigation techniques, two

pre-correlation bandwidths, three signals modulations and eight multipath scenarios. So, the number

of results is large (8x2x3x8=384 simulations) and only some selected plots will be presented here.

Nevertheless, all results are included in Appendix C.

Figure 7.8 shows the mean (colour bars) and standard deviation (error bars) of the code tracking error

for all multipath scenarios, pre-correlation bandwidth BWTc = 5 and different techniques and signals. It is

possible to see how the different techniques behave for each signal modulation. Lower expected

values and variances means that a technique has a better performance.

As expected, the NELP discriminator has the lowest tracking error standard deviation of all tested

techniques, but the code tracking error mean is the second highest.

HRC offers the lowest code tracking error mean for all the three modulations. When compared with the

NELP results, the code error standard deviation is higher than was expected, especially for BPSK

signals. This seems to go against what was seen in Section 7.1. However, it is important to say that

multipath error envelopes and steady-state code tracking error variance only give an idea of the

performance in the presence of multipath only and in the presence of noise only, respectively. On the

other hand, the results obtained with the Simulink model considered both the presence of multipath

and noise, and, thereby, they are more representative of the real behaviour.

RECT CCRW offers a code tracking error mean similar to NELP, but with higher standard deviation.

This fact was expected because, as it was seen in Section 5.3.1, the RECT CCRW can be shown to

be equivalent to a conventional E-L discriminator for BPSK signals, and it is similar for near echoes (

ck T5.0<τ ) and for BOC and CBOC signals.

W1, W2, W3 and W4 CCRWs have a similar behaviour: lower tracking error mean than the NELP and,

generally, lower standard deviation than the HRC. From these four CCRWs, W3 has lower code

tracking error mean, while W4 has lower code tracking error standard deviation; overall W3 has the

best performance of the four. Other important point to refer is that for these CCRWs, there is an

increase in code tracking error standard deviation for BOC and CBOC. This behaviour was expected

as it was seen in Section 7.1.

The proposed implementation of the TK operator, could not match up with the performance of other

72

techniques: TK has the highest code tracking error mean and standard deviation.

For normalized pre-correlation bandwidth BWTc = 5, the HRC and W3 are the techniques that have the

lowest code tracking error.

0

2

4

6

8

10

12

14

16

18

20

NELP HRC RECT W1 W2 W3 W4 TK

co

de t

rac

kin

g e

rro

r, m

.

BPSK

BOC

CBOC

Figure 7.8: Code tracking error mean and variance (average of the eight scenarios) for normalizedbandwidth BWTc = 5, different techniques and modulations.

Figure 7.9 plots the mean (colour bars) and standard deviation (error bars) of the code tracking error

for all multipath scenarios, normalized pre-correlation bandwidth BWTc = 12 and different techniques

and signals. Overall, there is a performance increase (both the code tracking error mean and standard

deviation decrease) relative to the results plotted in Figure 7.8. However, CCRWs (neglecting RECT

CCRW) and HRC are the techniques that best benefit with the use of a larger pre-correlation

bandwidth, offering lower tracking error mean and standard deviation when compared to others.

The TK-based shows a slight performance increase, but, again, its performance is much far away from

the other techniques.

For normalized pre-correlation bandwidth BWTc = 12 the best techniques were HRC and W3 and W4

CCRWs. Based on the results, HRC is by far the best technique for BOC and CBOC signals, while

W3 and W4 CCRWs are superior for BPSK signals.

Until now, the results presented were the average of the tracking error mean and standard deviation

over the eight scenarios. These results can translate the overall performance of the different technique

into different environments: without multipath, weak multipath and heavy multipath. Now, Figures 7.11

and 7.10 plot the average of the code tracking error mean and standard deviation only for the

scenarios with strong multipath: Urban III, Urban IV, Urban V and Urban VI (see Table 6.2 and Figure

6.6). This way it is easier to see the differences between the different algorithms, in the presence of

multipath. Or in other words, the real capability of the different techniques to mitigate the multipath

effect.

73

0

2

4

6

8

10

12

14

16

18

20


code

tra

ckin

g er

ror,

m

.

BPSK

BOC

CBOC

Figure 7.9: Code tracking error mean and variance (average of the eight scenarios) for normalizedbandwidth BWTc = 12, different techniques and modulations.

In general, in Figures 7.10 and 7.11, there is an increase on the error mean and standard deviation,

relative to Figures 7.8 and 7.9. This was expected, because, as it was stated before, the effect of

multipath depends on the intensity of multipath, and for the last two Figures, the scenarios considered

were the four worse of the eight considered environments. Otherwise, in Figures 7.11 and 7.10, there

are not different observations to make relative to Figures 7.8 and 7.9: HRC and W3 CCRW, for both

pre-correlation bandwidths, and W4 CCRW for pre-correlation bandwidth BWTc = 12, continue to be the

best technique.

0

5

10

15

20

25

30

35


code

tra

ckin

g er

ror,

m

.

BPSK

BOC

CBOC

Figure 7.10: Code tracking error mean and variance (average of the four worst scenarios) fornormalized bandwidth BWTc = 5, different techniques and modulations.

74

0

5

10

15

20

25

30

35


cod

e t

rack

ing

err

or,

m

.

BPSK

BOC

CBOC

Figure 7.11: Code tracking error mean and variance (average of the four worst scenarios) fornormalized bandwidth BWTc = 12, different techniques and modulations.

Finally, some words must be addressed regarding the complexity of the different techniques. The

CCRW techniques require four correlators and the discriminator has two multiplications and one

addition. The NELP requires six correlators, four additions and four multiplications. The HRC requires

ten correlators and the discriminator two multiplications and six additions. The TK is by far the most

complex. Table 7.2 compares the complexity of the different techniques considered.

Considering the computational complexity and the code tracking error mean and standard deviation,

W3 CCRW was considered the most adequate technique to use in a low cost GNSS receiver:

● it is one of the techniques with lower code tracking error mean and standard deviation (only

HRC was better);

● it has lower computational complexity than a conventional NELP discriminator and than the

HRC;

● it behaves well for the three tested modulations;

If possible, it is preferable to use wider pre-correlation bandwidth, because less signal power is lost

and, thereby, lower code tracking errors can be achieved. However, larger pre-correlation bandwidths,

require higher sampling rates and more expensive receivers.

75

Technique Correlators Multiplications Additions Overall

NELP 6 4 3 Medium

HRC 10 1 6 Medium/High

RECT

4 2 1 Low

W1

W2

W3

W4

TK 14 8 7 High

Table 7.2: Computational complexity.

76

Chapter 8

Conclusion and Final Remarks

8.1 Summary

This dissertation focused in multipath mitigation techniques suitable for civilian low cost GNSS

receivers.

The basic concepts of GNSS were introduced. GNSS navigation signals were described, namely GPS

L1 and L1C, and Galileo E1 signals. The operation of GNSS receiver tracking loops was described. It

was explained what is multipath, when it happens and what is its effect on the tracking loops.

It was given an overview of current multipath mitigation techniques, with focus on the correlation

based techniques. Several techniques were analysed: NELP, HRC, CCRW with several waveforms

and a TK based technique.

A GNSS receiver simulator was implemented in Simulink using the GRANADA FCM blockset. The

developed receiver simulator required also the implementation of a correlator noise generator, a

stochastic multipath model and some workarounds, so GRANADA FCM blockset could support MBOC

signals and CCRW. Some multipath environments were described and some techniques were picked

up for simulation.

The simulation results were presented and discussed, and the techniques suitable for implementation

on mass market civilian receivers were identified.

8.2 Conclusion

The purpose of this thesis was to study and identify multipath mitigation techniques for urban

scenarios, which are suitable for low cost GNSS receivers. The main research objectives were

accomplished: the GNSS principles, different multipath mitigation techniques giving special attention to

the correlation based techniques were studied and multipath mitigation techniques suitable for low

cost GNSS receiver were identified.

All the techniques analysed operate at the tracking loops level and can be implemented in mass

market receiver. However they behave differently in the presence of multipath and noise:

● NELP is not a bad option: it works well in the presence of weak multipath and its performance

77

is good in the presence of noise. The negative point is that, in the presence of stronger

multipath, the tracking error is large when compared with other techniques. The use of new

signals, as BOC(1,1) and CBOC(6,1,1/11) signals, helps the NELP discriminator to achieve

lower code tracking errors.

● HRC offers good performance. However, it has the disadvantage that, using CCRWs, it is

possible to have similar performance and lower computational complexity.

● The performance of CCRWs are affected by the selected waveform. With exception to the

RECT and New RECT CCRW, the tested CCRW presented good performance, however W3

CCRW offered the best performance among CCRWs.

● TK was the most complex (from the computational point of view), but the worse technique.

The TK operator may be a powerful algorithm, as it was seen in [21], but it fails as it was

shown in Section 5.4. Another disadvantage was the fact that from all the tested algorithms,

this was the one that required a higher number of correlators and, hence, the most

computationally heavier.

From all techniques, W3 CCRW is the one that offers the lowest computational complexity (as the

others CCRWs) and the tracking performance nearest to the HRC. Thereby, from the tested

techniques, this one was considered the most suitable for low cost receivers.

Other important conclusion, is that using larger pre-correlation bandwidths, it is possible to have a

lower tracking error. Nevertheless, wider bandwidth required higher sampling frequencies and so,

expensive hardware. Thus, if the cost does not exceed the receiver budget it is preferable to use

larger bandwidth.

8.3 Future work

The work developed in this dissertation was only “the tip of the iceberg”: many other techniques can

be analysed, special CCRWs. As it was possible to see, the CCRW based techniques can be very

flexible, and their performance can be tuned with different reference waveforms. There are unlimited

possible waveforms to be tested. Also different CCRW can be combined in the same discriminator to

achieve better performance, as it was proposed in [18].

The techniques were tested in a virtual environment: the Matlab Simulink. The next step is to test the

techniques in a real GNSS receiver with real signals (e.g. GPS L1). The virtual environment can give a

very good idea how the different algorithms and signals behaves, but it can not replace the tests in

real environments.

78

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81

Appendix A

Cross-Correlation Function

Let ( )[ ]aa Tttta −Π=)( and ( )[ ]bb Ttttb −Π=)( stand for two unity rectangular pulses of duration aT

and bT , centred in at and bt respectively (Figure A.1).

-Ta/2 0 +Ta/20

+1

-Tb/2 0 +Tb/20

+1

Figure A.1: Signals a(t) (left) and b(t) (right).

The cross-correlation between the two signals will be

−

+−Λ

+−

+

+−Λ=

ba

ba

ba

a

ba

babaAB

TT

ttt

TT

T

TT

tttTTR , (A.1)

where )(tΛ is a triangular pulse zero centred, with 1,0)( >=Λ tt and 1)0( =Λ . Figure A.2 shows the

cross-correlation function ABR , when 0== ba tt .

-(Ta+Tb)/2 -(Ta-Tb)/2 0 +(Ta-Tb)/2 +(Ta+Tb)/20

Ta.Tb

Figure A.2: Cross-correlation function ABR .

82

Appendix B

Noise Analysis

The code discriminator response at time kTt = , where T is the integration time, can be linearised

around 0=ε as [10]

kkk Ngd += ε0 , (B.1)

where 0g is the discriminator gain, kε is the code tracking error at time kTt = , and kN is a zero-

mean Gaussian random variable with variance 2

dσ . For finite bandwidths the discriminator gain

depends on the input signal and the strobe pulse width.

The transfer function of the linearised closed code loop in the z-domain is [10]

)()(1

)()()(

0

0

zVzFg

zVzFgzH

+= (B.2)

where )(zF and )(zV are, respectively, the loop filter and the numerically controlled oscillator transfer

functions. Thereby, it can be shown that the steady-state normalized code error variance is given by

2

0

22 2

g

TB dDLL σε = (B.3)

where DLLB is the closed-loop noise bandwidth in Hz.

Considering weak noise condition (neglecting the contribution of the noise x noise terms), the variance

of kN in (B.1) was determined in [10] as:

● For the Narrow Early-Late Power discriminator:

( )( )[ ]∆−∆≈ XXXd RR

TNC1)2(

4 2~

0

2σ , (B.4)

and the discriminator gain is )2()2(4 ~~0 ∆′∆−=XXXX

RRg .

● For the High Resolution Correlator:

83

( )( )[ ] ( ) ( )( ) ( )( )[ ]{ }2/12/1211

/

1 2

0

2 ∆−−∆++∆−∆−+≈ λλλλλσ XXXXd RRRRTΝC

, (B.5)

with the discriminator gain given by [ ])2()2(2 ~~0 ∆′−∆′=XXXX

RRg λλ .

● For Code Correlation Reference Waveforms:

( )0)(2

1 2~

0

2

XX

cd R

TNC

TL≈σ , (B.6)

and the discriminator gain is given by [ ] )0()0()()( ~~

0

~~0 WXXXWXXXRRRRg ′=

′=

=ε

εε .

84

Appendix C

Tables of Results

NMP SUB URB1 URB2 URB3 URB4 URB5 URB6

ε 2ε ε 2ε ε 2ε ε 2ε ε 2ε ε 2ε ε 2ε ε 2ε

NELP -0.11 0.44 0.02 0.45 0.20 0.42 1.52 0.45 12.07 2.65 20.43 5.08 20.21 5.21 24.12 9.73

HRC -0.12 1.14 -0.17 2.86 0.20 2.07 0.29 11.15 0.30 7.30 -0.79 14.92 1.03 11.12 8.44 139.21

RECT -0.01 0.23 0.14 0.26 0.23 0.25 1.22 0.50 10.66 11.45 19.85 15.43 19.61 20.57 24.70 10.68

W1 0.01 1.20 0.03 1.39 0.10 1.22 0.99 1.37 5.05 5.15 11.46 11.35 10.50 9.01 15.29 9.31

W2 0.04 1.27 0.19 1.19 0.02 1.33 0.76 0.97 5.03 4.93 10.59 10.55 10.35 8.07 14.55 8.03

W3 -0.08 2.81 -0.09 2.51 -0.30 3.00 0.93 2.47 2.54 8.12 6.50 12.28 6.74 12.77 9.28 13.25

W4 -0.16 0.67 0.04 0.65 -0.04 0.71 0.76 0.48 4.93 2.48 8.73 5.40 9.57 8.11 13.74 11.25

TK 0.02 12.59 0.30 14.04 0.43 13.72 3.44 17.21 21.13 43.66 25.97 47.50 27.00 49.73 30.17 49.08

Table C.1: BPSK signals and bandwidth BWTc = 5.



NELP 0.04 0.13 0.12 0.14 0.12 0.15 1.43 0.26 8.65 1.45 15.20 3.28 15.95 5.20 19.85 7.57

HRC -0.06 0.45 0.24 0.84 0.14 0.75 0.15 2.76 0.30 3.62 1.48 12.34 0.48 5.16 1.01 25.80

RECT -0.02 0.23 -0.01 0.29 0.15 0.31 1.13 0.88 7.00 4.02 14.90 10.85 14.81 12.43 18.43 10.93

W1 -0.02 1.26 -0.05 1.39 0.26 1.00 0.95 1.20 4.83 4.03 10.07 9.38 10.40 9.03 14.45 10.39

W2 0.25 4.30 1.15 7.30 0.30 4.80 0.67 4.58 5.50 14.66 11.41 16.15 12.47 19.17 14.80 19.92

W3 0.05 6.72 -0.99 8.20 -0.37 10.89 0.62 7.14 2.69 13.56 7.53 13.59 6.42 20.78 9.17 15.05

W4 -0.12 2.92 0.05 2.85 0.01 2.92 0.88 4.46 4.16 6.25 9.50 14.06 11.99 18.99 15.43 16.18

W5 -0.10 6.23 0.04 6.84 0.24 6.57 3.28 9.73 15.59 31.80 19.71 34.39 21.04 37.73 23.16 42.28

Table C.2: BOC(1,1) signals and bandwidth BWTc = 5.

85



NELP 0.08 0.17 -0.07 0.17 0.21 0.16 1.47 0.24 9.20 1.81 15.55 3.19 15.54 4.28 19.82 5.27

HRC -0.05 0.52 0.13 0.91 0.25 0.74 0.38 2.29 0.89 3.93 3.96 42.22 1.03 20.37 1.99 7.95

RECT -0.01 0.26 0.02 0.28 0.07 0.36 1.14 0.34 7.35 5.24 12.72 10.18 14.81 10.73 18.30 15.01

W1 -0.18 1.15 0.09 1.23 0.02 1.35 0.99 1.87 4.86 2.56 9.52 6.22 10.39 8.77 14.75 9.52

W2 -0.21 5.18 0.47 5.30 -0.46 5.14 1.28 6.63 6.29 11.45 10.13 19.15 11.02 20.77 13.81 27.20

W3 -0.26 7.36 0.13 7.11 -0.51 9.44 -0.10 9.31 2.27 10.64 7.37 19.19 6.02 19.15 9.43 18.16

W4 0.06 3.10 0.41 3.15 0.06 2.62 0.64 4.13 4.37 5.77 10.77 16.23 8.90 10.55 13.07 12.16

TK 0.06 7.46 0.25 8.68 0.19 8.62 3.46 13.01 17.54 40.35 20.96 36.24 20.88 39.04 22.89 42.13

Table C.3: CBOC(6,1,1/11) signals and bandwidth BWTc = 5.



NELP -0.03 0.18 -0.01 0.21 0.12 0.18 1.07 0.20 7.63 1.42 17.10 3.01 17.48 4.57 22.80 5.12

HRC 89.00 0.50 -0.08 1.04 0.01 0.77 0.34 1.79 0.39 4.52 -2.38 15.02 0.84 5.51 3.44 35.62

RECT -0.02 0.05 -0.02 0.07 0.08 0.06 0.84 0.14 6.03 3.32 16.46 14.73 16.28 14.66 22.74 15.87

W1 -0.05 0.15 -0.02 0.29 -0.04 0.19 0.37 0.29 1.92 0.60 4.59 2.82 4.81 2.04 7.03 5.82

W2 -0.08 0.17 0.03 0.33 0.01 0.21 0.36 0.25 2.27 0.85 4.57 1.53 4.02 2.51 7.03 4.44

W3 0.01 0.42 0.06 0.55 -0.09 0.46 0.19 0.69 0.70 0.84 2.08 2.54 1.98 1.65 3.18 4.21

W4 0.00 0.13 -0.04 0.13 -0.01 0.13 0.41 0.14 1.72 0.74 3.77 1.81 3.52 2.50 5.90 4.33

TK -0.05 5.73 0.20 5.63 0.41 5.29 2.01 6.79 18.28 34.45 25.16 36.59 25.71 33.75 29.15 39.34

Table C.4: BPSK signals and bandwidth BWTc = 12.



NELP -0.05 0.06 0.10 0.05 0.10 0.07 0.83 0.11 5.14 0.65 11.01 1.91 12.16 3.29 16.57 5.33

HRC 0.06 0.16 -0.05 0.35 0.13 0.26 0.18 0.60 -0.15 1.58 0.08 2.45 0.36 6.17 -0.84 6.45

RECT -0.02 0.04 0.01 0.04 0.05 0.05 0.69 0.10 3.77 1.07 9.78 6.27 10.11 7.65 16.81 10.50

W1 -0.04 0.25 0.03 0.18 0.04 0.18 0.21 0.24 1.72 0.70 4.55 3.39 3.95 1.60 8.21 3.03

W2 0.04 0.77 -0.09 0.88 -0.05 0.81 0.35 0.47 1.73 1.79 3.94 2.74 5.94 5.10 7.43 9.47

W3 -0.20 2.03 -0.32 2.45 -0.06 2.59 -0.17 4.17 0.50 4.66 2.72 8.40 1.77 5.07 2.80 4.61

W4 -0.02 0.50 -0.15 0.47 0.21 0.63 0.52 0.73 1.68 1.64 3.40 2.55 3.63 1.74 5.27 3.56

W5 -0.02 2.55 0.02 2.60 0.24 2.79 1.64 3.86 12.82 21.11 18.63 27.52 19.37 33.15 21.94 34.06

Table C.5: BOC(1,1) signals and bandwidth BWTc = 12.

86



NELP 0.01 0.03 0.09 0.04 0.13 0.05 0.66 0.05 3.18 0.25 7.13 0.90 7.09 1.09 10.95 1.75

HRC 0.00 0.07 -0.06 0.15 0.04 0.13 0.26 0.45 0.20 1.21 -0.18 1.03 1.28 3.32 -0.18 2.52

RECT 0.00 0.05 0.05 0.07 0.11 0.06 0.44 0.11 4.13 1.52 8.64 4.99 9.34 5.97 17.97 6.54

W1 0.04 0.23 0.10 0.26 0.05 0.24 0.35 0.33 1.80 0.67 4.43 1.91 4.29 1.75 6.66 3.64

W2 0.09 0.78 0.34 0.78 -0.10 0.71 0.17 0.79 1.16 1.08 3.77 2.87 3.85 1.84 5.74 5.18

W3 0.10 2.00 -0.19 2.01 -0.11 2.10 0.06 2.70 0.71 3.75 2.81 7.60 1.89 4.84 3.77 7.79

W4 -0.01 0.44 -0.21 0.48 -0.06 0.62 0.29 0.72 0.90 0.71 2.79 1.59 2.48 1.68 4.50 3.34

TK -0.03 0.56 0.07 0.60 0.17 0.61 1.01 0.93 6.41 8.10 12.86 15.51 13.93 18.59 17.90 25.69

Table C.6: CBOC(6,1,1/11) signals and bandwidth BWTc = 12.

87

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