PDF (1.7 Mb) - Laboratoire de Télécommunications et Télédétection ...

LABORATOIRE DE TÉLÉCOMMUNICATIONS
ET TÉLÉDÉTECTION
B - 1348 Louvain-la-Neuve Belgique
ABOUT MAXIMUM-LIKELIHOOD PHASE ESTIMATION
IN DS-CDMA COMMUNICATION SYSTEMS
Laurent SCHUMACHER
Thèse présentée en vue de l’obtention du grade de
Docteur en Sciences Appliquées
Jury composéde
Luc VANDENDORPE (UCL - FSA/ELEC/TELE) - Promoteur
Michel GEVERS (UCL - FSA/INMA/AUTO) - Examinateur
Marc MOENECLAEY (Universiteit Gent - FTW/TELIN) - Examinateur
Paul DELOGNE (UCL - FSA/ELEC/TELE) - Examinateur
Marco LUISE (Università di Pisa - DII) - Examinateur
Piotr SOBIESKI (UCL - FSA/ELEC/TELE) - Président
Décembre 1999

Remerciements
Le texte que voici synthétise les résultats de plusieurs années de recherche.
Un tel aboutissement n’est jamais l’œuvre d’une personne seule, mais la
conjugaison, à travers elle, de multiples contributions, parfois modestes,
souvent déterminantes, parfois involontaires, souvent décidées. Nombreuses
sont dès lors les personnes auxquelles je souhaiterais exprimer ici
toute ma gratitude pour leurs précieux conseils et leur soutien de chaque
instant.
Primus inter pares, mes remerciements vont à mon promoteur, le Professeur
Luc Vandendorpe. Son magistère scientifique m’a guidé dans les arcanes
du traitement du signal et son inébranlable confiance a su ranimer
la flamme chaque fois que celle-ci vacillait.
Je tiens aussi à saluer les membres de mon comité d’encadrement, les Professeurs
Michel Gevers et Marc Moeneclaey, ainsi que les autres membres
du jury, les Professeurs Paul Delogne, Marco Luise et Piotr Sobieski, pour
s’être investis dans une tâche qui en rebutait plus d’un. Leurs remarques
pertinentes ont sensiblement contribué à l’amélioration du document final.
Au sortir de mes études universitaires, rien n’indiquait que le diplômé
montois que je suis viendrait arpenter le plateau de Lauzelle. Je sais gré
au Professeur Auguste Laloux de m’avoir ouvert les portes de Louvain-la-
Neuve, me permettant ainsi d’accomplir un rêve d’enfant.
S’il est vrai que le découragement menace souvent le doctorand, les nuages
noirs ne s’attardent pas longtemps dans le ciel de TELE. La convivialité
qui prévaut au laboratoire est l’écrin rêvé pour l’épanouissement des
compétences scientifiques qui s’y développent. Que tous ses membres,
passés et présents, en soient remerciés. Je désire remercier tout spécia-

ii
lement mes condisciples de bureau, Stéphane Pigeon, Laurent Cuvelier,
Benoît Maison et François Deryck, pour l’atmosphère chaleureuse qu’ils y
ont fait régner. Mes pensées vont aussi à ceux qui, dans l’ombre, œuvrent
pour nous assurer le meilleur cadre de travail. J’adresse en outre mes
voeux de succès à Mamoun Guenach, qui met ses pas dans les miens un
peu plus chaque jour.
Ma reconnaissance va également au Fonds National de la Recherche Scientifique
dont le soutien financier m’a permis de mener à bien cette thèse de
doctorat, libéré de toute contingence matérielle.
Un Special Award revient sans conteste à Sarah Zandona, qui s’est astreinte
à plusieurs relectures méticuleuses, en dépit de son absence d’affinités
avec le domaine. Sans elle, ce texte eût été d’une piètre qualité linguistique.
Enfin, je ne peux terminer sans remercier du fond du coeur mes proches,
qui m’accompagnent et me supportent depuis le premier jour.
Laurent Schumacher
Décembre 1999

Contents
1 Introduction 1
1.1 A paradigm shift ......................... 1
1.2 Motivation ............................. 3
1.3 Structure of the thesis ....................... 4
1.4 Notations .............................. 7
2 State of the art 9
2.1 DS-CDMA, a technique whose time has come ......... 9
2.1.1 Spread-spectrum in a nutshell ............. 9
2.1.2 Applications ........................ 16
2.2 Multiuser reception for DS-CDMA systems .......... 24
2.2.1 Detection .......................... 25
2.2.2 Parameter estimation ................... 27
2.2.3 Joint detection and parameter estimation ....... 33
2.3 Phase estimation ......................... 34
2.3.1 Estimation structures ................... 34
2.3.2 Performance characterisation of phase estimators .. 37
2.3.3 Multiuser Phase estimation ............... 40
2.4 Conclusions ............................ 43
3 Tools 45
3.1 System description ........................ 45
3.1.1 System under investigation ............... 45
3.1.2 Definition of Energy-to-Noise ratios .......... 49
3.2 Maximum-Likelihood estimation ................ 51
3.2.1 Maximum A Posteriori and Maximum-Likelihood .. 51
3.2.2 Likelihood function ................... 52
3.2.3 ML condition ....................... 57
3.3 Optimal estimator performance ................. 58

iv CONTENTS
3.3.1 Cramér-Rao Lower Bound ................ 58
3.3.2 ML performance ..................... 60
3.3.3 CRLB for multiuser phase estimation ......... 60
3.4 FF estimation ........................... 62
3.4.1 Closed form of the estimator .............. 62
3.4.2 Variance approximation ................. 66
3.5 Performance evaluation of DD estimators ........... 67
3.5.1 Direct space - Gaussian probability integral ...... 69
3.5.2 Reciprocal space - Characteristic function ....... 71
3.6 Conclusions ............................ 72
4 Data-Aided 75
4.1 Feedback .............................. 76
4.1.1 Open-loop study ..................... 78
4.1.2 Closed-loop study .................... 84
4.2 Feedforward ............................ 94
4.2.1 Pdf of an SU estimator in a multiuser context ..... 94
4.2.2 Linearised multiuser estimator in 2-user system ... 98
4.3 Feedback-Feedforward correspondence ............103
4.4 Conclusions ............................105
5 Decision Directed 109
5.1 Feedback ..............................110
5.1.1 Decisions assumed correct ................112
5.1.2 Actual decisions - Open-loop study ..........116
5.1.3 Actual decisions - Closed-loop study .........141
5.2 Feedforward ............................142
5.2.1 SU DD ML FF estimator .................143
5.2.2 MU DD ML FF estimator ................144
5.2.3 Decisions assumed correct ................147
5.2.4 Actual decisions .....................148
5.3 Conclusions ............................149
6 Conclusions 151
6.1 Achievements ...........................151
6.2 Perspectives ............................152
A Correlation function of the loop noise in a DA recovery loop 157
A.1 BPSK modulation .........................158
A.2 QPSK modulation .........................159

CONTENTS v
B Pdf of Single-User DA ML FF phase estimator 161
B.1 First step: characteristic function ψˆxu,ˆyu (ωr,ωi) ........161
B.2 Second step: pdf Tˆxu,ˆyu (ˆxu, ˆyu) .................162
B.3 Third step: change of variables .................164
B.4 Fourth step: pdf T∆u(∆u) ....................165
B.5 Analytical validation .......................165
C Variance of DA ML FF phase estimators 167
C.1 Multiuser estimator ........................167
C.1.1 BPSK modulation .....................167
C.1.2 QPSK modulation ....................169
C.2 Single-user estimator .......................171
C.2.1 BPSK modulation .....................171
C.2.2 QPSK modulation ....................171
D First order statistics in a linear channel 173
D.1 Expectations of data ¢ decision products ...........174
D.1.1 User ¢ User ........................174
D.1.2 User ¢ Interferer .....................176
D.2 Expectations of decision ¢ decision products .........177
D.2.1 Same I/Q branch .....................177
D.2.2 Cross-talk .........................178
D.3 Conclusion .............................178
E Expectations for DD FB open-loop performance evaluation 179
E.1 BPSK Modulation .........................179
E.1.1 Derivation of E âm u an v ˆ
Φ=0, Φ=∆ .........180
E.1.2 Derivation of E âm u ânv ˆ
Φ=0, Φ=∆ .........182
E.2 QPSK Modulation .........................184
E.2.1 Derivation of E âm u an v ˆ
Φ=0, Φ=∆ .........185
E.2.2 Derivation of E âm u bnv ˆ
Φ=0, Φ=∆ .........187
E.2.3 Derivation of E âm u ân v ˆ
Φ=0, Φ=∆ .........188
E.2.4 Derivation of E âm u ˆb n
v ˆ
Φ=0, Φ=∆ .........189
F Expressions of Uu,DD derived in the reciprocal space 191
F.1 BPSK modulation .........................191
F.2 QPSK modulation .........................194

vi CONTENTS
G COST 207 Channel Models 197
H Curriculum vitae 199
Bibliography 203

List of Figures
1.1 Voice traffic on public networks continues to grow at steady,
predictable rates, while data traffic is growing exponentially
and may surpass voice traffic in many countries by the year
2000 (Source: [3]) ......................... 3
2.1 Block diagram of a digital DS/SS transmitter for radio communications
............................ 10
2.2 Spectrum of a DS/SS signal (Bandwidth expansion factor = 4) 11
2.3 Block diagram of a digital DS/SS receiver for radio communications
.............................. 11
2.4 Spectrum of a FH/SS signal (Bandwidth expansion factor =
4) .................................. 12
2.5 Representation of FDMA, TDMA and CDMA in the timefrequency
plane (Source: [12]) .................. 14
2.6 Illustration of user separation in DS-CDMA systems ..... 15
2.7 UMTS components ........................ 18
2.8 Path diversity (Source: [25]) ................... 21
2.9 Some applications of CDMA nowadays ............ 25
2.10 MUD systems described in [41, 42] ............... 26
2.11 DA estimator ........................... 28
2.12 DD estimator ........................... 28
2.13 NDA estimator .......................... 29
2.14 FB and FF implementations ................... 36
2.15 Hang-up and cycle slip ...................... 39
3.1 Uplink of a coherent CDMA communication system ..... 46
3.2 Sub-domains in the plane (νm u , νn v ) ............. 70
4.1 2-user DA phase recovery loop ................. 77

viii LIST OF FIGURES
4.2 Power spectral density of Additive Noise, Self- and Cross-
Noise ................................ 83
4.3 DA BPSK PLL ........................... 86
4.4 Incidence of the quadratic term of the Taylor-series expansion
at equilibrium of the variance expression ......... 88
4.5 Variance of DA FB estimators in AWGN channel (BPSK) .. 89
4.6 Near-Far effect on DA FB estimators (BPSK) .......... 90
4.7 Variance of DA FB estimators in dispersive channels (BSPK) 91
4.8 Variance of DA FB estimators in AWGN channel (QPSK) .. 92
4.9 Near-Far effect on DA FB estimators (QPSK) ......... 92
4.10 Variance of DA FB estimators in dispersive channels (QSPK) 93
4.11 Pdf of the SU DA ML FF phase estimate in a 2-user, RA
channel context .......................... 96
4.12 Variances of the SU DA ML FF phase estimation error as a
function of the number of user Nu and of the channel type . 97
4.13 Variance of DA FF estimators in an AWGN channel .....102
4.14 Near-Far effect on DA FF estimators ..............103
4.15 Incidence of ISI on DA FF estimators ..............104
4.16 Correspondence between DA FB and FF estimators .....107
5.1 2-user DD phase recovery loop .................111
5.2 Variance of DD ML FB estimators in ISI-free scenario (BPSK) 114
5.3 Variance of DD ML FB estimators in presence of ISI (BPSK) . 115
5.4 Variance of DD ML FB estimators in ISI-free scenario (QPSK) 116
5.5 Variance of DD ML FB estimators in presence of ISI (QPSK) . 117
5.6 S-curves in a 2-user non-dispersive synchronous system, xv,u =
0 ...................................121
5.7 S-surfaces of a 2-user non-dispersive synchronous system,
uncoupled scenario (a: BPSK, b: QPSK) ............122
5.8 S-curves function of ∆u, parametrised on ∆v - 2-user nondispersive
synchronous system, uncoupled scenario (a: BPSK,
b: QPSK) ..............................123
5.9 S-curves function of ∆v, parametrised on ∆u - 2-user nondispersive
synchronous system, uncoupled scenario (a: BPSK,
b: QPSK) ..............................124
5.10 S-surfaces of a 2-user non-dispersive synchronous system,
coupled scenario (a: BPSK, b: QPSK) ..............125
5.11 S-curves function of ∆u, parametrised on ∆v - 2-user nondispersive
synchronous system, coupled scenario (a: BPSK,
b: QPSK) ..............................126

LIST OF FIGURES ix
5.12 S-curves function of ∆v, parametrised on ∆u - 2-user nondispersive
synchronous system, coupled scenario (a: BPSK,
b: QPSK) ..............................127
5.13 S-surfaces of a 2-user non-dispersive synchronous system,
Near-Far scenario (a: BPSK, b: QPSK) .............128
5.14 S-curves function of ∆u, parametrised on ∆v - 2-user nondispersive
synchronous system, Near-Far scenario (a: BPSK,
b: QPSK) ..............................129
5.15 S-curves function of ∆v, parametrised on ∆u - 2-user nondispersive
synchronous system, Near-Far scenario (a: BPSK,
b: QPSK) ..............................130
5.16 U BPSK
u,DD (0) as a function of δv,u (- simulation, ¢ computation
direct space, Æ computation reciprocal space) ......133
5.17 Phasor contributions of user u and interferer v to matched
filter output ym u for BSPK-modulated data symbols ......134
QP SK
5.18 Uu,DD (0) as a function of δv,u (- simulation, ¢ computation
direct space, Æ computation reciprocal space) ......136
5.19 Phasor contributions of user u and interferer v to matched
filter output y m u
5.20 U BPSK
u,DD
5.21 U BPSK
u,DD
for QSPK-modulated data symbols .....136
=30dB ....139
where user u is the strongest and Eb
N0
where user u is the weakest and Eb
N0
=10dB .....140
5.22 2-user DD phase recovery loop .................142
5.23 SU DD ML FF estimator - Fastest update implementation . . 143
5.24 SU DD ML FF estimator - Slow update implementation ...144
5.25 2-user parallel MU DD ML FF estimator ............145
5.26 2-user successive MU DD ML FF estimator ..........146
5.27 Variance of ML FF estimators in presence of ISI (BSPK) ...149
G.1 The Rural Area (RA) channel model is made of 6 taps and
its power delay profile spreads over 0.5 µs. ..........197
G.2 The Typical Urban (TU) channel model is made of 12 taps
and its power delay profile spreads over 5 µs .........198
G.3 The Hilly Terrain (HT) channel model is made of 12 taps
and its power delay profile spreads over 20 µs. ........198

List of Tables
2.1 Air interface parameters of IS-95, cdmaOne and WCDMA
(Sources: [4, 5, 19]) ........................ 19
2.2 Comparison of 2nd-generation Globalstar and Ellipso and
3rd-generation Skybridge (Source: [21, 25, 26, 28]) ...... 22
2.3 Analog and digital phase recovery implementations ..... 35
4.1 Asymptotical variance expressions of DA estimators in a 2user
case ..............................106
6.1 Synthetic view of the achievements of the thesis .......151

LIST OF ABBREVIATIONS xiii
List of abbreviations
ACRB Asymptotic CRLB
ATM Asynchronous Transfer Mode
AWGN Additive White Gaussian Noise
BER Bit Error Rate
BPSK Binary Phase Shift Keying
BRAN Broadband Radio Access Network
BS Base Station
CATV Community Area TeleVision
CDG CDMA Development Group
CDMA Code Division Multiple Access
COST European Cooperation in the field of Scientific and
Technical Research
CRLB Cramér-Rao Lower Bound
CSMA/CA Carrier Sense Multiple Access with Collision Avoidance
DA Data Aided
DD Decision Directed
DFE Decision-Feedback Equaliser
DOA Direction Of Arrival
DS Direct Sequence
EKF Extended Kalman Filtering
EM Expectation-Maximisation
ETSI European Telecommunications Standards Institute
EVD Eigenvalue Decomposition
FB Feedback
FC Full-Carrier
FDD Frequency Division Duplex
FDMA Frequency Division Multiple Access
FF Feedforward
FH Frequency Hopping
FPLMTS Future Public Land Mobile Telecommunication System
GEO Geostationary Earth Orbit
GMPCS Global Mobile Personal Communications by Satellite
GSM Global System for Mobile communications
HIPERLAN High Performance Radio Local Area Network
HT Hilly-Terrain
IMT-2000 International Mobile Telecommunications 2000

xiv LIST OF ABBREVIATIONS
IS Intermediate Standard
ISI Inter-Symbol Interference
ISM Industrial, Scientific and Medical
ISO International Organisation for Standardization
ITU International Telecommunication Union
JD Joint Detection
LAN Local Area Network
LEO Low Earth Orbit
LOS Line-Of-Sight
MAI Multiple Access Interference
MAP Maximum A Posteriori
MC-CDMA Multi-Carrier CDMA
MCRB Modified CRLB
MEO Medium Earth Orbit
MF Matched Filter
MIPS Million of Instructions Per Second
ML Maximum-Likelihood
MSE Mean Square Error
MMSE Minimum MSE
MU Multiuser
MUD Multiuser Detection
MUSIC Multiple Signal Classification
NDA Non Data Aided
OFDM Orthogonal Frequency Division Multiplex
OHG Operators Harmonisation Group
OSI Open Systems Interconnection
PDA Personal Digital Assistant
pdf Probability Density Function
PIC Parallel Interference Canceller
PLL Phase Locked Loop
psd Power Spectral Density
QPSK Quaternary Phase Shift Keying
RA Rural Area
RLS Recursive Least Squares
S-CDMA Synchronous-CDMA
SC Suppressed-Carrier
SAGE Space-Alternating Generalised EM
SIC Successive Interference Canceller
SIR Signal-to-Interference Ratio
SMR Signal-to-Multipath Ratio

LIST OF ABBREVIATIONS xv
SNIR Signal-to-Noise-and-Interference Ratio
SNR Signal-to-Noise Ratio
SS Spread Spectrum
SU Single-User
SVD Singular Value Decomposition
TD-CDMA Time/Code Division Multiple Access
TDD Time Division Duplex
TDMA Time Division Multiple Access
UMTS Universal Mobile Telecommunication Service
UTRA UMTS Terrestrial Radio Access
WATM Wireless ATM
WCDMA Wideband CDMA
WDM Wavelength Division Multiplexing
WLAN Wireless LAN
ZF Zero-Forcing

Chapter 1
Introduction
1.1 A paradigm shift
With his Technology Reports [1], George Gilder is a respected but feared observer
of the infocom world. Owners of the techniques he supports never
fail to praise him, but he is as well strongly criticised for his views on technologies
he believes will not be successful. There was thus no surprise
to see the mix of enthusiastic and negative reactions that his article ”Telecosm
and Beyond: Over the Paradigm Cliff” [2] generated when it was
published in February 1997, as he discussed nothing less than a paradigm
shift rooted in the trade-off between power and bandwidth inherent in
Shannon’s law.
According to this law, every engineer willing to perform reliable transmissions
of information over a noisy channel has to make the most efficient
balance between power and bandwidth. In order to reach a certain bit rate
while mitigating the effect of the noise, s/he can either increase the transmitted
power within a limited bandwidth or use a wider bandwidth with
a limited power.
For decades, the latter option has not been considered. In the ”Industrial
Age”, as George Gilder calls it, bandwidth was regarded as scarce,
and power abundant. It may sound strange to consider that the electromagnetic
spectrum might suffer from scarcity since it is physically infinite.
However, the fact that engineers thought they had to deal with a bandwidth
shortage had more to do with the way the use of the spectrum was
constrained by government regulations. Only the lower part of the spec-

2 Introduction
trum was considered at that time and it was then shared on an exclusive
base between all kinds of application. As a result, transmissions were to
occur over narrow noisy channels plagued by interference. Quality of service
was then ensured by exploiting the remaining degrees of freedom,
namely emitting and switching powers.
Indeed, service providers were first inclined to boost up power within the
limited frequency band allocated to their applications in order to overcome
the poor quality of their noisy channel. However, this strategy alone
could not provide enough channels with enough quality to sustain the
increasing demand for communications. In order to accommodate more
services and more users per service within a band-limited environment,
network switches were then set on work. Their aim was to improve the
efficiency of the time-frequency resource sharing mechanism. It ended
up in exclusive sharing techniques such as Frequency Division Multiple
Access (FDMA) and Time Division Multiple Access (TDMA). For George
Gilder, the paradigm of this time period was ”long and strong”: long
wavelengths, i.e. small and low frequency bands, and strong power. Watts
and MIPS helped to overcome the bandwidth shortage.
These multiple access mechanisms have been designed in times when
analog-based voice services were dominant. However, the move from
analog to digital processing led to the development of data services. Yet,
data transmissions patterns do not necessarily match with patterns designed
to fit voice applications. As the share of data applications rises,
and that of voice applications correspondingly declines (Figure 1.1), the
shortcomings of the use of powerful transmitters and switches in order to
overcome the bandwidth scarcity become obvious.
On the other hand, communications have entered an era of bandwidth
abundance on the wired scene as well as on the wireless one. Fiber optics
developments demonstrate ever increasing throughput, while the constraints
that restricted the use of the electro-magnetic spectrum are being
lifted as government regulations loosen their grip. Techniques enabling
to exploit this bandwidth abundance have emerged: Wavelength Division
Multiplexing (WDM) on optical fibers and Code Division Multiple Access
(CDMA) in the wireless world.
Focusing on wireless applications, George Gilder claims that CDMA is the

1.2 Motivation 3
Figure 1.1: Voice traffic on public networks continues to grow at steady,
predictable rates, while data traffic is growing exponentially and may surpass
voice traffic in many countries by the year 2000 (Source: [3])
wireless access technique of the ”Information Age” in that it appropriately
responds to the new balance between abundance and scarcity: abundance
of bandwidth, scarcity of power. Indeed, present and future devices exhibit
more and more stringent constraints on their power requirements,
whether one speaks of portable devices or of on-board switches. Power
is no longer the key factor it used to be. This role has been overtaken by
bandwidth. Communications are eager to be convoyed by weak signals
using wide bandwidths, leading to a new paradigm for the ”Information
Age”, ”wide and weak”.
1.2 Motivation
Although praising George Gilder’s brilliant demonstration, many critics
pointed out its deficiencies. The least of them is not that his demonstration
is too technology-oriented, implicitly underestimating the weight of
business constraints in the success of technical solutions. In the real world,
they said, the success of a technique does not depend only on its own
merits, but also involves commercial issues, which George Gilder has not

4 Introduction
taken into account. Among these issues is the question whether end-users
can sustain the abundant bandwidth which will provide them with ever
increasing data flows. Indeed, another limiting factor now comes in the
picture, besides power and bandwidth: the information processing power
of a human being.
The subject of this thesis, however, is not to sort out the pros and contras
of CDMA in view of all the constraints that define a successful communication
system. Its aim is to gather knowledge about the estimation of an explicit
synchronisation parameter, viz. the phase in the uplink of a wireless
coherent Direct-Sequence CDMA (DS-CDMA) communication system.
Dealing with the uplink leads to a situation where the Base Station (BS) is
at the receiving end. From this point of view it is valuable to design an
estimation structure that will simultaneously encompass all users active
in the system instead of focusing on one user at a time and neglecting the
others as interferers. The main objective of this thesis is to demonstrate
analytically that the information content of the Multiple Access Interference
(MAI) can be used to improve the quality of the estimation.
However, one might argue that this work is only of academic interest,
since reception is not performed coherently in IS-95, the wireless spreadspectrum
communication system currently in commercial use [4, p. 544].
Indeed, while a pilot signal dedicated to each mobile receiver is inserted
in the downlink, such facility is not used in the uplink by fear of interference.
Yet, developers of third-generation systems are considering to
perform coherent reception also in the uplink [5]. This kind of reception
will be made easier by the insertion of time-multiplexed pilot signals. As
a result, efficient estimators of the phase parameter are required.
1.3 Structure of the thesis
This thesis is divided in four main chapters and seven appendices. The
appendices detail the mathematical developments leading to the relations
studied and illustrated in the chapters of this thesis.
Chapter 2 will introduce the issue of phase estimation in multiuser spreadspectrum
context. It consists of three sections, each of them dealing with
an aspect of this problematic.

1.3 Structure of the thesis 5
The first section will present spread-spectrum communication techniques
and focus on the Direct-Sequence Spread-Spectrum (DS/SS) technique.
It will be shown that DS-CDMA is a technique well suited for providing
multiple access to communication resource. Although this work only
deals with mobile applications of DS-CDMA, its possible applications in
several other communication environments, wired and wireless, will be
briefly reviewed.
DS-CDMA has long been regarded as a method for providing multiple
access without having to design multiuser receivers. A simple correlator
can perform users separation thanks to the inherent orthogonality of the
users in DS-CDMA. However, as the technique gained in popularity, it appeared
that this simple correlator had some shortcomings. Asynchronous
transmissions, dispersive channels, highly loaded systems, and power imbalance
between users plague the system with self- and cross-interference,
respectively Inter-Symbol Interference (ISI) and MAI. This has led to a shift
in the design of the receivers. Noticing that interference has an informative
structure, contrary to additive white noise, developers started to regard interference
as a useful contribution whose exploitation might improve the
performance of the receiver. This new approach has been applied for both
symbol detection and parameter estimation, as it will be described in the
second section of Chapter 2.
The scope of this work, however, will be limited to the parameter estimation
issue, and more precisely, to the estimation of the phase parameter.
The third section of Chapter 2 will review the proposed estimation structures
and the means to perform the characterisation of their performance.
None of the three sections of Chapter 2 briefly described here above claims
to be a thorough presentation of the related issue. They are rather broad
overviews aimed at introducing the subject to newcomers and at pointing
to relevant references.
Following Chapter 2, Chapter 3 will first complete the setting of this
work’s background. The communication system in which the phase estimation
issue has been studied will be presented. Notations will be set for the
following chapters. Elements of estimation theory will be given in order
to introduce the chosen method of Maximum-Likelihood (ML) estimation.

6 Introduction
Since the phase jitter variance is the performance benchmark of the following
developments, a lower-bound, the Cramér-Rao Lower Bound (CRLB)
will be introduced. Some problems faced during the study as well as the
tricks considered to alleviate them will be described afterwards.
The background of this thesis having thus been explained in the previous
two chapters, its subject, namely the estimation of the phase parameter
in a multiuser spread-spectrum environment, shall then be tackled. The
central theme of the next chapters will be the relationship between detection
and estimation stages.
In Chapter 4 the parameter estimation will rely on a perfect knowledge
of transmitted symbols. This situation, called Data-Aided (DA) estimation,
occurs in training periods when transmitter and receiver exchange
predefined sequences aimed at helping to characterise the communication
environment. The specificity of this chapter lies in that it starts from the
premise that the detection stage has no incidence on the estimation one. In
this context two different implementations of the ML phase estimator will
be considered: feedback (FB) and feedforward (FF).
Contrary to Chapter 4, Chapter 5 will take into account possible interactions
between detection and estimation stages. Firstly, decisions will be
assumed to be correct and the difference between DA and DD estimators
due to causality will be underlined. Secondly, the assumption of correct
decisions will be lifted and the incidence of decision errors on the openloop
performance of a Decision-Directed (DD) ML FB estimator will be
derived and illustrated. The closed-loop study will be mentioned in order
to illustrate the benefit of multiuser estimation.
Some developments have been made in the field of Non Data Aided (NDA)
estimation while the present thesis was being written. They are nevertheless
much too incomplete to be presented here.
By way of conclusion, the achievements of this thesis will be summarised
and potential future developments will be outlined.

1.4 Notations 7
1.4 Notations
The following typographic conventions are used throughout this work unless
explicitly specified otherwise.
Scalar variables are denoted by normal-faced symbols, while vectors and
arrays are denoted by bold-faced symbols:
x or x (n) denotes a scalar variable
x denotes a vector or an array
Accents are widely used:
ˆx denotes the estimate of variable x
x ⋆ denotes the complex conjugate of variable x
As far as operators are concerned, the following notations are used:
Pr (X >0) denotes the probability that the random variable X
is positive
E (X) denotes the expectation of the random variable X
Finally, sets of variables are denoted by curly braces xk.

Chapter 2
State of the art
2.1 DS-CDMA, a technique whose time has come
2.1.1 Spread-spectrum in a nutshell
Single-user perspective
The main and common characteristic of spread-spectrum techniques [4,
6, 7] is the expansion of the modulated symbol bandwidth from the minimum
required to transmit the information to a wider one. However these
techniques differ in the way they use this wider bandwidth.
DS/SS system When the whole bandwidth is permanently occupied by
the spread signal, one speaks of Direct Sequence Spread-Spectrum (DS-
/SS). The bandwidth expansion is piloted by a periodic code sequence
whose Nc elementary components are called chips. Its rate, the chip rate
1
1
T
, is higher than the symbol rate Tc T , so that the ratio represents the
Tc
bandwidth spreading factor in the frequency domain. Usually one prefers
to mention the processing gain which is the ratio Tb between the chip rate
Tc
and the bit rate [8]. The bandwidth spreading factor is thus the product of
the processing gain by the dimension of the data modulation T 1 . Diffe-
Tb
1 Authors of [8] note that the bandwidth spreading factor and the processing gain are
sometimes confused, both of them being used to designate the ratio between the spread
bandwidth and the original one (see for instance [9, p. 1]). Moreover, calculating the processing
gain as a bandwidth ratio is regarded in [4, chapter 2] as an approximation of the
true processing gain, defined as a measure of the performance improvement achieved by
systems using spread-spectrum techniques with respect to systems not using them.

10 State of the art
rent possibilities exist as regards the ratio between the code sequence N cTc
and the symbol length T . The present work is only concerned with code
sequences whose period is equal to the symbol length (NcTc = T ).
A digital DS/SS transmitter is presented in Figure 2.1. The spreading operation
is performed in two steps. First, the rate of the information sequence
( 1
1
T ) is increased up to the rate of the spreading sequence ( ). This
Tc
increased rate sequence is then passed through a filter whose taps are the
chips of the code sequence. The sequence output by this filter is the spread
sequence, ready to be shaped and transmitted.
1, -1, 1
Nc
Tc
T T
Tc
Chip
shaping
Figure 2.1: Block diagram of a digital DS/SS transmitter for radio communications
Since the time resolution of the code sequence is Nc times higher than the
one of the symbol sequence, the spreading operation provokes a corresponding
bandwidth expansion in the frequency domain (Figure 2.2). As
a result, the spread signal tends to melt down into the background noise.
It becomes thus hardly noticeable for third parties, achieving a low probability
of interception. This ability to hide pertinent information in the
background noise is a first interesting characteristic of spread-spectrum
transmissions.
At the receiving end (Figure 2.3), the spread signal is correlated with a
synchronised version of the periodic code sequence so as to cancel the effect
of spreading and to restore the original narrow symbol bandwidth.
This despreading operation defines the strict requirements with which
a periodic code sequence has to comply. To be suitable for single-user
spread-spectrum transmissions, the periodic code sequence ought to exhibit
Dirac-like auto-correlation properties. Pseudo-random code sequences
meet such requirements [10].

2.1 DS-CDMA, a technique whose time has come 11
2
Tc
Figure 2.2: Spectrum of a DS/SS signal (Bandwidth expansion factor = 4)
Chip-matched
filter
2
T
Nc
1
Nc
1
Figure 2.3: Block diagram of a digital DS/SS receiver for radio communications
1,-1,1

12 State of the art
While symbols are recovered at the receiving end by compacting their energy
back into the symbol bandwidth, the correlation operation is indeed
a spreading operation by the bandwidth expansion factor for unspread
incoming signals like narrowband interferers. This introduces another interesting
feature of spread-spectrum techniques: the resistance to narrowband
interferers.
FH/SS system Instead of permanently using the whole bandwidth as
DS/SS signals do, the frequency resource can be divided into slots as large
as the symbol bandwidth (Figure 2.4).
2
Tc
Figure 2.4: Spectrum of a FH/SS signal (Bandwidth expansion factor = 4)
The code sequence is then used to define the slot to be used for transmission,
commanding jumps from one slot to the other over time. Hence the
name of the technique: Frequency-Hopping Spread-Spectrum (FH/SS).
On the one hand, the hop scheme is governed by the code sequence. In
order to make it hardly predictable, pseudo-random sequences are also
used in FH/SS systems. On the other hand, the jump rate leads to distinguishing
between Slow-Frequency-Hopping (SFH) and Fast-Frequency-
Hopping (FFH). In SFH systems, several symbols are transmitted between
two frequency jumps whereas several jumps occur within a single symbol
duration in FFH transmissions [4, section 2.4].
2
T

2.1 DS-CDMA, a technique whose time has come 13
Whether DS/SS or FH/SS, the bandwidth expansion of the transmitted
signal is a common characteristic of spread-spectrum techniques. As mentioned
earlier, they both exhibit a low probability of interception since
the spread signal is hardly distinguishable from the background noise
(DS/SS), or hard to catch due to the hops (FH/SS). These techniques are
also robust against jammers thanks to the spreading of interferers at the
decorrelating end (DS/SS) or the avoidance of continuous emission within
a jammed band (FS/SS). Those features are pretty interesting for military
applications. That is why spread-spectrum techniques were first applied
in military projects before moving to the civilian world in the last decades.
In the following sections only DS/SS techniques will be considered. Beside
their resistance to narrowband interferers, another interesting property
of DS/SS signals is their inherent frequency diversity. Since they occupy
a large bandwidth, they are subject to frequency-selective fading [7,
chapter 7]. The resulting multipath transmission can be exploited to improve
the reception, using a RAKE receiver which properly combines the
delayed versions of the transmitted signal [4, subsection 8-4.5].
Multiuser perspective
The preceding features of DS/SS communication systems, viz. resistance
to narrowband interferers and frequency diversity, have been considered
from a single-user perspective. Spread-spectrum techniques are also well
suited for organising multiple access to digital transmissions.
While conventional multiple access schemes like FDMA (Figure 2.5a) and
TDMA (Figure 2.5b) share the time/frequency resources exclusively between
active users, DS-CDMA gives them the opportunity to share the
same bandwidth at the same moment. Transmissions are spread over the
time-frequency plane by code multiplication (Figure 2.5c). This last approach
seems more efficient when the resource is spare [11].
To separate users, DS-CDMA communication systems rely on a proper
choice of code sequences whose mutual correlations ought to be as small
as possible, so that despreading with one code sequence the signal spread
with another one produces zeros. In Figure 2.6, the receiver decorrelates
the signal produced by the transmitter shown in Figure 2.1 using a code
sequence other than the one used at the transmitting end. Since these two
code sequences are orthogonal, the detector output is null.

14 State of the art
(a) FDMA: a narrow frequency
band per user
(b) TDMA (with FDMA component):
several time slots per
frequency channel
(c) CDMA: signal spread over
time-frequency plane
Figure 2.5: Representation of FDMA, TDMA and CDMA in the time-frequency plane (Source: [12])

2.1 DS-CDMA, a technique whose time has come 15
Chip-matched
filter
Nc
1
Nc
1
Figure 2.6: Illustration of user separation in DS-CDMA systems
As far as code sequences are concerned, orthogonal ones are the best choice.
However, they require perfectly synchronous and non-dispersive transmissions
to completely cancel MAI. Quasi-orthogonal sequences are thus
often preferred to orthogonal ones due to their ability to still separate users
when neither synchronism nor non-dispersiveness can be ensured.
FDMA, TDMA and CDMA are thus different in the way they distinguish
users. As a result, the limit on their capacity is also of a different nature
[13].
The number of users that FDMA/TDMA systems can accommodate is
limited by the fractioning into user slots of the global time/frequency resource
they can allocate. The exclusive allocation of these slots avoids interference.
To guarantee that interference is avoided, FDMA/TDMA systems
respect guard times/bands, which results in resource wasting. These
techniques are thus said to be resource-limited [11].
On the other hand, in DS-CDMA schemes, interference is unavoidable.
Each new user is spread over the whole bandwidth and, as a consequence,
raises the interference level up to a point where the Signal-to-Noise-and-
Interference Ratio (SNIR) at the receiver is too poor to permit reception.
As a result, DS-CDMA is interference-limited.
Interfering users can be seen as wideband jammers. A conventional singleuser
receiver, relying only on the correlation properties of the code sequence
to recover the signal transmitted by one user, requires stringent
power control so as to keep the interference level plaguing the reception
below the minimum SNIR level required for reception. Should this power
control fail, the single-user receiver would be unable to recover the trans-
0,0,0

16 State of the art
mitted signal. This happens with the Near-Far effect. It refers to a situation
of power imbalance between users which is encountered, for instance, in
cellular communication systems [6] when a user situated far away from
the BS is affected by the wideband interference generated by another user
located near the BS. Without power control to level users’ energies, the
near user masks the far user at the BS. The Near-Far effect has long been
thought to be an inherent limitation of DS-CDMA. In fact, it is due to the
single-user design of the receiver, which postulates that MAI has been cancelled
at the decorrelator [14]. This question will be developed in the following
sections.
Other interesting features of spread-spectrum techniques that can be mentioned
from a multiuser system perspective are the soft handover and the
overlay over any existing system. Being able to combine several unsynchronised
versions of the same signal, CDMA receivers can communicate
with several BSs, and thus softly switch from one to the other when they
change serving area. On the other hand, the wideband spreading of the
signals enables such systems to overlay over existing systems, as long as
the new system appears as a low-power wideband interferer for them.
2.1.2 Applications
Nowadays, spread-spectrum techniques have found their way in the communication
world, mainly for wireless applications. This section will briefly
review the communication systems already using spread-spectrum techniques
or considering to do so in the near future. The description of standards
mentioned in the following section is limited to the physical layer
(Layer 1 of International Organization for Standardization (ISO) [15] Open
Systems Interconnection (OSI) reference model), which is the global scene
of the present work.
Land mobile services
Mobile applications are concerned with wireless connections within a cellular
network, i.e. between a mobile receiver able to move within the
serving area at typical car speeds and base stations interfacing the mobile
receiver with other mobiles and/or a fixed network. Two scenes are
considered here: the terrestrial and the satellite.

2.1 DS-CDMA, a technique whose time has come 17
Terrestrial-based cellular systems The first generation of terrestrial cellular
system was analog. With second-generation systems like IS-95 and
Global System for Mobile Communications (GSM), a switch has been made
from analog to digital. While GSM organises the multiple access according
to a Frequency Division Duplex (FDD)/TDMA scheme, IS-95 standard [4,
section 9-4.1] relies on CDMA. IS-95 describes a single-carrier DS-CDMA
digital cellular system dedicated to voice and data services for rates up to
9.6 to 14.4 kbps (up to 115.2 kbps with IS-95-B revision). The choice for
CDMA has been made in order to enjoy interference rejection and spectral
efficiency promised by this system, as illustrated in [13] which derives capacity
equations of a spread-spectrum terrestrial cellular network by taking
into account the number of users, bit rates, the reuse factor, possible
sectorisation, and inter-cell interference.
Although second-generation networks are still in deployment in many
places, third-generation systems are already under close investigation. The
incentive of these research activities has been the demand for speed connections
over cellular networks higher than what is affordable today with
second-generation systems. The walk towards third-generation systems
takes place in the framework of International Telecommunication Union
(ITU)’s International Mobile Telecommunications-2000 (IMT-2000) 2 which
defines the requirements for next generation services: rates up to 144 kbps
for vehicular applications, 384 kbps for pedestrian services and 2 Mbps
for indoor systems, with improved spectrum efficiency and service flexibility
within 1,885-2,025 and 2,110-2,200 MHz frequency bands [16, 17]. All
over the world, in IS-95 served areas as well as in GSM countries, spreadspectrum
techniques are retained as candidates for implementing multiple
access in those third-generation systems, with different issues according to
the cellular networks that are already installed.
As far as IS-95 is concerned, the evolution is quite obvious. The next step,
initiated by CDMA Development Group (CDG) [18], will be cdma2000 or
Wideband cdmaOne, heading to bit rates up to 2Mbps. A crucial aspect
of its implementation is the backward compatibility with IS-95. To accommodate
higher speeds, nominal bandwidths have been enlarged from 1.25
MHz in IS-95 to 5 MHz in Wideband cdmaOne. However, a multicarrier
scheme is designed so as to enable overlay of Wideband cdmaOne over
2
formerly known until 1996 as Future Public Land Mobile Telecommunication System
(FPLMTS)

18 State of the art
IS-95. Other enhancements to the physical layer are considered, among
which coherent demodulation in the uplink, faster power control, use of
turbo-codes, and optional Multiuser Detection (MUD) [19].
On the other hand, the evolution from GSM to spread-spectrum techniques
is not as natural as the move from IS-95 to Wideband cdmaOne,
since the nature of the multiple access scheme has to change. In Europe,
where GSM was developed and where IMT-2000 translates into Universal
Mobile Telecommunication Service (UMTS), the European Telecommunications
Standards Institute (ETSI) [20] chose in early 1998 two terrestrial air
interfaces (UTRA): Wideband CDMA (WCDMA) for paired FDD bands
(1,920-1,980 and 2,110-2,170 MHz), and Time/Code Division Multiple Access
(TD-CDMA) for unpaired Time Division Duplex (TDD) bands (1,900-
1,920 and 2,010-2,025 MHz). The main issue is to ensure backward compatibility
with GSM despite the differences in multiple access schemes.
This needs to be done by deriving the clock rates of the third-generation
systems from the GSM clock rate (13 MHz or 26 MHz) and by placing carriers
over a common frequency grid with 200 kHz-spacing.
UMTS
Terrestrial component (UTRA)
WCDMA TD-CDMA
Satellite component (S-UMTS)
Figure 2.7: UMTS components
Table 2.1 synthesises the main parameters of the air interfaces for secondgeneration
IS-95 and third-generation Wideband cdmaOne and WCDMA 3 .
3 An agreement on a globally harmonised third-generation CDMA radio standard was
reached by the Operators Harmonisation Group (OHG) in May 1999 and later endorsed by
all other concerned standardisation bodies. There should be three modes in the harmonised
3G CDMA standard: a FDD single-carrier DS mode for WCDMA, a FDD multi-carrier

Generation 2nd 3rd
System IS-95 Wideband cdmaOne WCDMA
RF channel bandwidth [MHz] 1.25 1.25/5/10/15/20 5/10/20
Downlink RF channel structure
Chip rate [Mcps] 1.2288
Direct spread
1.2288/3.6864
Multicarrier
n¢ 1.2288 (n =1,3,
Direct spread
4.096/8.192/16.384
/7.3728/11.0593
/14.7456
6, 9, 12)
Frame length [ms] 20 20 (data and control)/5 (control information) 10/20 (optional)
Data Downlink BPSK QPSK
modulation Uplink 64-ary
gonalortho-
BPSK
Coherent de- Downlink Common Common pilot channel + auxiliary pilots Common pitection
pilot channel
lot channel +
Uplink - Time-multiplexed pilot
user-dedicated
time-multiplexed
pilots
User-dedicated
time-multiplexed
pilots
Data rates 9.6-14.4 kbps
(IS-95-A)
9.6-115.2 kbps
(IS-95-B)
9.6 kbps - 2 Mbps 128 kbps - 2 Mbps
Table 2.1: Air interface parameters of IS-95, cdmaOne and WCDMA (Sources: [4, 5, 19])
2.1 DS-CDMA, a technique whose time has come 19

20 State of the art
Satellite-based cellular systems The success of cellular systems leads
to a point where users are no longer asking for mobility but for ubiquity.
Land-based cellular networks are unfortunately not available everywhere.
From this point of view, satellite-based cellular systems appear as complementary
to land-based networks, backing them up or even substituting
them in non served areas.
Satellite-based communication services have been available for many years
now, mainly for broadcasting, using Geostationary Earth Orbit (GEO) systems.
However, the GEO orbit (35,860 km above the Earth) is not the most
appropriate for the mobile applications targeted nowadays due to latency
(250 ms round-trip propagation delay) and small link margins [21]. Lower
orbits, like Low Earth Orbit (LEO, 160-480 km) and Medium Earth Orbit
(MEO, 9,660-19,110 km), solve these issues but also introduce new problems.
Indeed, such low orbit satellites move quickly in the sky above the
ground user, which means that handover and correction of large Dopplershifts
(up to 60 kHz for a satellite altitude of 1,500 km at 2.4 GHz [22]) shall
be dealt with.
Moreover, such satellite-based systems work at frequencies higher than
the land-based ones, from one to the tens GHz. At such frequencies,
electro-magnetic fields do not penetrate buildings and are blocked by obstacles.
As a result, transmission is only viable as long as Line-of-Sight
(LOS) visibility is ensured.
For several years, efforts have been made under ITU’s Global Mobile Personal
Communications by Satellite (GMPCS) label in order to provide voice
and data services to handsets worldwide using LEO and MEO satellites
[23].
Commercial service was opened in 1998 by the Iridium consortium [24].
Iridium offers voice and data services up to 9.6 kbps through a 66 LEO
satellite-based cellular network. Multiple access is based on a FDD/TDMA
scheme. Communications between the user and the satellite occur in the
L-band (1.616-1.6265 GHz), while satellites and gateways use K-bands
(19.4-19.6 GHz and 29.1-29.3 GHz). An innovative feature in the Iridium
system is the direct handling of calls from one satellite to the other without
mode for cdmaOne, and a TDD CDMA mode. First and third modes will operate at 3.84
Mcps chip rate, while the FDD multi-carrier will use 3.6864 Mcps chip rate.

2.1 DS-CDMA, a technique whose time has come 21
ground-based relay.
While Iridium is on the air, other second-generation satellite-based cellular
systems are getting ready for opening services. Among them, Globalstar
[25] and Ellipso [26] projects plan to offer multiple access based
on CDMA. Thanks to spread-spectrum, a soft handover between satellite
footprints is possible, as already mentioned in the case of land-based cellular
systems. Moreover, the receiver enjoys path diversity. With a RAKE
receiver several versions of the same signals, transmitted by the different
satellites in view, can be combined so as to improve reception quality and
to avoid blocking (Figure 2.8).
Figure 2.8: Path diversity (Source: [25])
Similarly to the evolution in terrestrial-based applications, third-generation
systems are already knocking at the door. Teledesic [27] and Skybridge
[28] are among the few projects known so far. The latter has to be
mentioned in the current presentation, since it is based on CDMA. Skybridge
announces the deployment of a 64-LEO satellite-based cellular system
working in Ku-band (12-18 GHz) and offering bit rates up to 60 Mbps
in the downlink, and 2 Mbps in the uplink.

22 State of the art
Generation 2nd 3rd
Project Globalstar Ellipso Skybridge
Operator Motorola MCH Inc., Alcatel
Opening of service 1999
Lockheed,
Harris
2001 2001
Number of satellites 48 + 8 spares 6 equatorial +
8 elliptical + 3
spares
64
Orbit LEO MEO LEO
User link Downlink S-band L-band Ku-band
Uplink L-band
Rates Downlink 9.6 kbps 60 Mbps
Uplink 9.6 kbps 2 Mbps
Table 2.2: Comparison of 2nd-generation Globalstar and Ellipso and 3rdgeneration
Skybridge (Source: [21, 25, 26, 28])
Cordless/portable/WLAN service provision
The domain of cordless/portable communication devices is the intermediate
step from the mobile communication world to the fixed communication
one, trying to combine both advantages, viz. mobility and high-speed
transmissions. On the one hand, such portable devices are giving the user
some freedom of position and/or movement within a restricted serving
area: static connections can be engaged from any point and low-speed
mobility is ensured through network and handover management. On the
other hand, these portable terminals are often regarded as wireless gateways
to backbone networks, first Local Area Networks (LAN) then Asynchronous
Transfer Mode (ATM) networks. They are thus expected to offer
higher bit rates than the ones available through mobile devices.
In the late 1980’s and early 1990’s, the first wave of cordless/portable communication
devices, introduced under the trademark of Wireless LAN
(WLAN), was expected to break through thanks to the ease of deployment.
Indeed, wireless connections enable to set up a communication network
without rewiring the communication scene. In fact, these devices
mainly gained popularity thanks to the connection possibilities they added
to portable devices like Personal Digital Assistants (PDA). Among the
several air interface solutions considered, spread-spectrum was retained
for radio-based WLAN thanks to its ability to coexist with already implemented
services in the band used for transmission. Moreover, promoters

2.1 DS-CDMA, a technique whose time has come 23
of spread-spectrum claimed that it would enable cooperation of products
from different vendors without prior dialogue. However, power control
issues appeared to request such concertation [29]. Nevertheless, spreadspectrum
techniques are implemented in the American IEEE 802.11 WLAN
standard to provide immunity with respect to narrowband interferers in
the Industrial, Scientific and Medical (ISM) band (2.4 GHz). Multiple
access is organised using Carrier Sense Multiple Access with Collision
Avoidance (CSMA/CA) mechanism. The IEEE 802.11 standard provides
bit rates up to 2 Mbps [30]. Its equivalent in Europe is ETSI High Performance
Radio Local Area Network/1 (HIPERLAN/1) which was formalised
in 1997. Unlike IEEE 802.11, multiple access techniques normalised
in HIPERLAN/1 do not rely on CSMA/CA but on a FDMA/TDMA combination.
HIPERLAN/1 addresses bit rates up to 23.529 Mbps in the 5.15-
5.30 GHz-band [31, 32].
Nowadays, the next wave of cordless/portable applications, called Wireless
ATM (WATM), is targeting bit rates much higher than the 2 Mbps
rate of IEEE 802.11. For applications offering such high bit rates, spreadspectrum
techniques have been disregarded. Indeed, with a bandwidth
expansion factor as small as 15, spreading data symbols produced at 155
Mbps requires a prohibitive bandwidth of 2 GHz. Moreover, synchronisation
issues become troublesome [33]. Other modulation schemes offering
orthogonality between users and immunity to multipath, like Orthogonal
Frequency Division Multiplex (OFDM), have then been considered in the
European Broadband Radio Access Network (BRAN) project [34]. This
project paves the way beyond HIPERLAN/1 in order to meet demands for
high bit rates transmission. Under the BRAN project, three different standards
(HIPERLAN/2, HIPERACCESS and HIPERLINK) are developed for
broadband cordless/portable communications. These standards target different
bit rates (25-155 Mbps) and environments (static/mobile, indoor-
/outdoor).
Nevertheless spread-spectrum techniques have not completely disappeared
from the cordless/portable communication scene. The parallel transmission
implemented in the OFDM scheme provides a means of spreading
at chip rates lower than the ones requested by DS/SS techniques used
alone. A mix of OFDM and DS/SS, named Multi-Carrier CDMA (MC-
CDMA), is a solution under investigation [33].

24 State of the art
Fixed service provision
The last application to be mentioned in this section is delivery of highspeed
data services over cable TV coaxial networks. While the applications
of spread-spectrum techniques for providing communication services
described so far are all wireless, CDMA has also found its way in
the wired world in order to help Community Area Television (CATV) operators
to turn into multimedia service providers. Since the demand for
such value-added services have been identified, these operators have invested
much time and money to adapt their networks so that they could
provide these services. From this point of view, CDMA has received much
attention as a modulation technique that helps to sustain impairments
encountered on the cable, namely narrowband interference, ingress, and
impulse noise, while enabling to deliver data services without requiring
much change to the infrastructure of a two-way pure coaxial network [35].
Synchronous-CDMA (S-CDMA) systems [36] have demonstrated their ability
to work robustly within the unused low frequency bands of the CATV
medium (5-42 MHz in the United States). Implementation of CDMA communication
systems on CATV networks is the object of ongoing standardisation
work within IEEE 802.14 group [37].
Figure 2.9 illustrates the applications reviewed here above.
2.2 Multiuser reception for DS-CDMA systems
The implementation of DS-CDMA as a multiple access technique has revealed
the inherent limitation of the single-user correlating receiver. Out
of ideal conditions (orthogonal code sequences, synchronous transmissions,
perfect power-control), efficient reception can no longer rely only
on the code correlation properties to separate users. A MAI component
plagues the receiving end, degrading performance, particularly when the
power of the users is not balanced (Near-Far effect).
The design of efficient receivers, in order to work in DS-CDMA systems,
has to take this MAI component into account so as to exploit its information
to improve reception. In a word, the receiver ought to deal with all
active users. Clearly, this increases the complexity of the receiver, that is
why multiuser reception is usually only considered in the uplink. Indeed,
the receiving end, where signals from several users converge, is a network

2.2 Multiuser reception for DS-CDMA systems 25
CATV
cdmaOne
UTRA
Globalstar
Skybridge
IEEE 802.14 IEEE 802.11
Figure 2.9: Some applications of CDMA nowadays
point which should deal with all of them. Hence the obvious benefit of
multiuser reception. However, this advantage has a cost: namely complexity.
A trade-off between performance and complexity needs thus to
be made in order to keep receivers affordable.
The following sections will describe the advances of multiuser reception.
MUD has been the subject of much interest, while the question of multiuser
parameter estimation has less often been studied. After a short introduction
to MUD, multiuser parameter estimation will be presented. Combined
MUD and multiuser parameter estimation will close this section.
2.2.1 Detection
The optimum detector for multiuser DS-CDMA transmissions, a maximum-likelihood
sequence detector, has been described in [38]. Considering
that the activity of the Nu users results in a signal which is similar to
a single-user transmission over a dispersive channel, this detector applies
the Viterbi algorithm to the outputs of a bank of conventional single-user
matched filters. However, the exponential complexity of the Viterbi algorithm
in the number of users Nu makes it hardly practicable. Neverthe-

26 State of the art
less, the way to MUD has been opened. In the sequel of [38], the linear decorrelating
detector has been introduced in [14]. Many other sub-optimal
structures have been proposed afterwards. Their performance, in terms of
Near-Far resistance [39] and asymptotic efficiency [40], is similar to that
achieved by the Viterbi algorithm but with a complexity only linear in Nu.
A review of them can be found in [41, 42]. It is sketched on Figure 2.10.
The issue of MUD in frequency-selective environments, shortly tackled
in [41], is discussed in a more detailed and more up-to-date way in [43].
ZF
MMSE
PIC
Conventional
Matched filter
Optimum
Bank of matched filters
+ Viterbi algorithm
Suboptimum
Linear
DFE
SIC
Multipath fading ?
No
Yes
Conventional
RAKE receiver
Optimum
Bank of RAKE receivers
+ Viterbi algorithm
Figure 2.10: MUD systems described in [41, 42]
To alleviate the Near-Far effect, most of proposed MUD schemes require
knowledge beyond that assumed by the conventional receiver, namely the
channel impulse responses and the users’ signature waveforms. This information
is often collected by using training sequences. To avoid this
loss of throughput while maintaining performance, blind techniques relying
on the same information as the conventional receiver but exhibiting
optimum Near-Far resistance have recently been proposed [43, 44].

2.2 Multiuser reception for DS-CDMA systems 27
2.2.2 Parameter estimation
The ultimate objective of any receiver is to recover transmitted information.
However, this requires most often to recover synchronisation prior to
performing detection. Synchronisation, to be understood here in a broad
sense, concerns recovery of all parameters of the link: timing, frequency,
phase, amplitude, channel response, user’s waveform, etc. A thorough review
of the synchronisation issues in spread-spectrum systems is presented
in [8]. However, it does not really deal with multiuser aspects, assimilating
MAI to a supplementary Gaussian noise contribution (Gaussian approximation,
Section 2.3.3). The present work develops the opposite view.
It does regard MAI as an informative contribution which can be exploited
in order to improve the performance of the parameter estimator. Among
others, this is the main and most innovative contribution of this work.
At this point, a remark should be made with respect to the interaction
between detection and parameter estimation stages. To derive the estimates
of the parameters, some structures rely on transmitted symbols
while others do not. Moreover, the symbols used in the estimation process
can be either the true symbols or the decisions produced by the detector.
This leads to the distinction between Data-Aided (DA), Decision-Directed
(DD), and Non Data-Aided (NDA) estimation.
At least at start-up, and maybe also periodically during transmission if
the parameters to estimate are time-varying, systems transmit training sequences
aimed at helping receivers to collect information about the context
of the transmission. These sequences, known by both transmitter and receiver,
convey no information. Parameter estimators use them in order to
perform estimation, minimising a cost function (I,θ) which depends on
the training sequences and on the parameters (Figure 2.11). Since in this
case symbols are known, the estimation is said to be DA.
Obviously, DA estimation cannot be performed permanently since this
would suppress any information throughput. When parameters have been
acquired thanks to the training sequences, estimators can switch from
training sequences to detector’s outputs. The cost function to minimise
no longer depends on the true symbols I but on the decisions Î (Figure
2.12). In such case one speaks of DD estimation. Provided that the decisions
are mostly correct, DD estimation performs as well as DA, with
the advantage over DA that these are now informative bits and no longer

28 State of the art
r (t)
Phase estimator
maxθ (I,θ)
Training sequence
Figure 2.11: DA estimator
training ones which are transmitted over the channel.
r (t)
Phase estimator
maxθ Î,θ
Figure 2.12: DD estimator
Of course, any decision error degrades the DD estimation process, which
in turn degrades the decision process since the detector often needs good
parameter estimates in order to perform properly. This coupled influence
can lead to a complete collapse of the receiver’s performance. A solution
to such failure is to make the estimation process independent of the detector.
NDA 4 structures are designed in this perspective. They are also
required in short burst transmissions when one cannot afford to waste
throughput with training sequences [45]. The cost function they minimise
has been made independent of the symbols, for instance by averaging it
over their pdf (Figure 2.13). Since NDA estimators do not exploit all the
information available at the receiver, their performance is not as good as
DA or DD structures. On the other hand, they still can work in situations
where decision errors multiply.
Whether DA, DD or NDA, single-user estimation methods are degraded
by MAI [46, 47]. Optimum estimators can be derived, but similarly to
the situation of MUD, their complexity plays against them. To perform
multiuser parameter estimations two options are possible: splitting the
4 NDA estimators are sometimes called ”blind” by analogy with blind detectors [45].
Î

2.2 Multiuser reception for DS-CDMA systems 29
r (t)
Phase estimator
maxθ EI [ (I,θ)]
Figure 2.13: NDA estimator
multiuser estimation problem into single-user ones ([48], Approximate
Maximum-Likelihood in [49]), or performing joint estimation over all active
users. The latter option will be described in the following paragraphs.
Most of the contributions in the field of multiuser parameter estimation
relate either to Expectation-Maximisation (EM), Singular Value Decomposition
(SVD), or Extended Kalman Filtering (EKF). Each method will be
dealt with in a specific paragraph. A fourth paragraph will briefly encompass
other contributions not based on one of the first three methods.
Expectation-Maximisation EM is a two-step estimation algorithm which
is able to produce the ML estimate of a vector of parameters θ when observations
y are the result of a many-to-one mapping of underlying variables
x, also called ”complete data” [50, 51]. The complete data contains extra
information that would ease the parameter estimation but, unfortunately,
these complete data are usually unobservable. The true ML estimation
would request to maximise the log-likelihood function ΛL (x θ) of the underlying
variables with respect to the vector parameter. However, since
these variables are not available, estimation is performed recursively using
the EM algorithm. It is made of two steps: E-step (Expectation) and
M-step (Maximisation), iterated successively until convergence is reached.
In the E-step, the likelihood function to maximise ¯ ΛL is defined as the expectation
over the underlying variables x of their log-likelihood function
ΛL assuming the observations y and the estimate of the vector parameter
θk produced at previous iteration k:
¯ΛL (x θ) =Ex [ΛL (x θ) y,θk] . (2.1)
Following the E-step, the M-step updates the estimate by maximising the
averaged log-likelihood function (2.1) over the vector parameter θ:
ˆθk+1 =argmax
θ
¯ΛL (x θ) . (2.2)

30 State of the art
The algorithm iterates between E- and M-steps until convergence to the
ML estimate, or at least a local extremum is reached, since the likelihood
function does not decrease during iterations [51].
The EM algorithm is well-suited to estimate parameters from the received
signal of a multiuser DS-CDMA system. Indeed, it is a composite signal
built from contributions of Nu users transmitting over possibly multipath
(Np-path) channels. Direct access to the complete data is not possible since
the Nu¢Np signal contributions and noise are mixed together into each reception
filter output. At first sight, finding ML estimates of corresponding
Nu ¢ Np sets of synchronisation parameters would require a joint maximisation
over all parameters, involving all Nu ¢ Np contributions. However,
applying the EM algorithm enables to split this joint problem into
Nu single ones and to define a likelihood function relying on one user at a
time, assuming the other ones.
Furthermore, an evolution of the EM algorithm, the Space-Alternating
Generalised EM (SAGE) algorithm, exhibits faster convergence and lower
complexity. The evolution is twofold. First, the SAGE algorithm involves
additional iteration loops trying to produce refined estimates of a subset
of the whole vector of parameters to be estimated while keeping the other
parameters fixed. Second, the mapping from complete data to incomplete
data is no longer necessarily deterministic, but might be random. In [52],
the joint estimation of delay, azimuth, Doppler frequency, and complex
amplitude is performed in mobile radio environments using the SAGE algorithm.
Singular Value Decomposition-based estimation methods SVD helps
to build the pseudo-inverse matrix bringing out the minimum-norm solution
of a linear least-squares problem [53, p. 414]. This might be used
to degrade a multiuser problem into single-user ones that are easier to
solve [48], or to perform multiuser parameter estimation using the Multiple
Signal Classification (MUSIC) algorithm [53, p. 452]. The MUSIC
algorithm is known to be a method of estimating frequencies of uncorrelated
complex sinusoids in additive noise or to solve Direction Of Arrival
(DOA) problems. Considering the received signal samples of a sum of
uncorrelated signals, the MUSIC algorithm performs an Eigenvalue De-

2.2 Multiuser reception for DS-CDMA systems 31
composition (EVD) of the sample correlation matrix 5 , so as to distinguish
two subspaces: the signal subspace and the noise subspace. Optimally,
the parameters of the signals are orthogonal to the noise subspace. A reliable
estimate is thus obtained by searching for the parameter values which
minimise the norm of their projection onto the noise subspace.
In [49], a modified version of the MUSIC algorithm is introduced to estimate
the propagation delays, the phases, and the amplitudes for all users
of a DS-CDMA system in an Additive White Gaussian Noise (AWGN)
channel. Besides performance results presented in [49], the performance
of this estimator has also been derived in [54] by applying an alternative
perturbation analysis of the second-order statistics. A similarly modified
algorithm is used in [55] for parameter estimation in static fading channels,
while the estimation of propagation delays in time-varying fading
channels is performed in [56] using the same algorithm than in [49]. Back
to static fading channels, another MUSIC-based algorithm is presented in
[57] for channel estimation. In all these works, estimators are shown to be
Near-Far resistant and not to rely on information from the detector. They
are thus suited for acquisition as well as for tracking.
This separation property of SVD is also used in [58] in order to model
MAI as a coloured Gaussian noise and to derive channel parameters as
ML estimates in coloured Gaussian noise. This requires a preamble (DA
estimation).
Extended Kalman Filtering Kalman filters have received much attention
for their ability to perform adaptive least-squares estimation with a
time-varying gain in the update equation. It ensures faster convergence
[53]. However, Kalman filters are not directly applicable to the problem
of parameter estimation since they only apply to linear systems. Unfortunately,
the received signal is a non-linear function of the synchronisation
parameters. EKF, introduced as an extension of the standard Kalman filter
to non-linear systems [59, section 13.7] [60, p. 386], is thus well suited for
estimation in non-linear systems. Moreover, it performs better than the
Recursive Least Squares (RLS) algorithm because it incorporates a priori
knowledge [61].
5
or, equivalently, a SVD of the received signal samples, since EVD is a particular case
of SVD [53, p. 408, 456]

32 State of the art
EKF is implemented in [62] for the DA estimation of timing and complex
coefficients of a tapped-delay line channel model in a single-user wideband
communication system. The time-varying nature of the parameters
is modelled with first-order auto-regressive processes. Sequels of [62] add
rejection of narrowband interference [63] and estimation of Doppler shift
[64]. In [63] the narrowband interference is modelled as an N-th order
auto-regressive process, estimated by a DD EKF estimator relying on the
decisions provided by a RAKE receiver. The lack of knowledge about the
Doppler velocity is solved in [64] by implementing a bank of EKF estimators
assuming this velocity. Their outputs are then combined according
to their a posteriori probabilities so as to form a weighted estimate of the
timing and channels parameters.
Miscellanea Beside those implementing one of the three algorithms mentioned
here above, some other contributions deal with multiuser parameter
estimation in DS-CDMA systems.
Four estimators of the complex amplitude of the signal are introduced
in [65], depending on whether data are known (DA) or not (NDA) and
whether timing has been recovered previously or not. These four estimators
make use of the outputs of a bank of matched filters. Regarding [65]as
based on second-order stationary statistics (the outputs of the matched filter
bank), an estimator based on the second-order cyclostationary statistics
of the received signal collected at the output of the sampler is presented
in [66]. Using these statistics, complex amplitude and timing are obtained
as NDA least-squares estimates based on the Fourier transform of the cyclic
auto-correlation function.
While most contributions are concerned with steady-state performance,
the joint acquisition of both time delay and Doppler velocity is studied
in [67] using a two-dwell correlator system. Acquisition is also an issue
in [68]. The choice of midamble training codes in burst transmission is
discussed with respect to the performance of DA ML- and Matched Filter
(MF)-channel estimators. Instead of wasting throughput in training
sequences, one could rely on blind parameter estimation. Blind channel
estimators are classified into two main categories [69]: statistics-based and
subspace-based [70, 71].

2.2 Multiuser reception for DS-CDMA systems 33
2.2.3 Joint detection and parameter estimation
In order to design a multiuser receiver, the combination of one of the MUD
algorithms and one of the multiuser parameter estimation methods mentioned
previously can be considered.
The EM algorithm has often been applied to perform the parameter estimation
step in the framework of a Gauss-Seidel scheme. This scheme performs
successively a detection step, using previously produced parameter
estimates, and an estimation step fed with the outputs of the detector and
applying the EM algorithm. Considering timing known, a multistage algorithm
is implemented for detection in [72]. Complex amplitudes are
estimated through the EM algorithm. In [73, 74], timing is embedded into
the synchronisation parameters to be estimated using the EM algorithm,
while multistage detection is performed.
Subspace-based methods are used in [75] for the estimation of the propagation
delays, while Minimum Mean Square Error (MMSE) techniques are
applied for the estimation of the path gains and for the detection of the
data bits. A refinement of [75] is presented in [76] where all the parameters
are estimated using SVD while detection is performed according to the
MMSE criterion.
Finally, a tree-search for detection and an adaptive recursive least-squares
multiuser parameter estimator are associated in [77].
However, the design of multiuser receivers can be modified so as to perform
only one global estimation process, involving both data and parameters.
Indeed, the emergence of estimation structures dealing with different
kind of parameters have led to proposals which regard data symbols
as another parameter so that the distinction between detection and parameter
estimation disappears. The algorithms mentioned here above are
then applied to solve this global estimation problem. In that perspective,
the EKF estimation of data symbols (indeed, hard decisions taken over a
parameter embedding both information and all synchronisation parameters
but timing) and timing is described in [61]. Convergence and Near-Far
resistance are evaluated and compared respectively with RLS and singleuser
EKF. Both comparisons demonstrate how much more efficient the
multiuser EKF is.

34 State of the art
A last word in this review, about blind techniques. Most often, receivers
are made of cooperating detectors and parameters estimators. Since detectors
require estimates while estimation can be made NDA, parameter
estimation might be performed before detection. Blind techniques return
this paradigm. Enabling the detection stage to work autonomously, they
lead to structures where data detection can be initiated without preliminary
parameter estimation. However, a parameter estimator might still be
required at the output of the detector. For instance a Phase Locked Loop
(PLL) is used in [78] to mitigate the phase rotation of the Constant Modulus
Algorithm (CMA).
2.3 Phase estimation
After the review of the literature about multiuser reception presented in
the previous section, the present section shall deal with the estimation of
the phase parameter. As explained in Section 1.2, this is the subject of the
present study.
2.3.1 Estimation structures
Analog implementations
Phase recovery structures have been designed first for analog transmissions
[7, section 4.5]. The major actor in this context is the PLL [79], used to
track the carrier in Full-Carrier (FC) modulations or any other pilot signal.
Analytical study shows that the PLL performs ML estimation of the phase
parameter. The situation looks different when considering Suppressed-
Carrier (SC) modulations. At first sight, there is no frequency component
to track. Still, the PLL can be used to recover the phase information of SC
modulations. In order to do so, it tracks the output of a Mth-power nonlinearity
which contains a pertinent frequency component thanks to the
cyclostationarity of the received signal [80]. The squaring loop is an example
of such a Mth-power loop (M = 2). Its study shows that it performs
NDA ML phase estimation. Being well suited for binary modulations,
this squaring loop is however helpless for higher-order balanced modulations,
since applying them the squaring operation results in the vanishing
of the information content [81, p. 279]. Correspondingly, these modulations
require higher-order non-linearities or a slightly different treatment
that avoids calling upon such high-order operations [82]. Other tracking

2.3 Phase estimation 35
Analog Digital
PLL tracking reference wave Waveform regenerator
Costas loop Trackers
Parameter extractor
Parameter search
Table 2.3: Analog and digital phase recovery implementations
loops exist, like the Costas loop. Such loops do not explicitly track a frequency
component. However, their study shows some equivalence with
the squaring loop.
Digital implementations
Digital phase recovery structures have succeeded to analog implementations.
As far as their analytical study is concerned, it is interesting to note
that the ML estimation theory serves as an unifying theoretical framework
for their analysis [83, chapter 2]. Digital implementations are not just digitalised
versions of previously developed analog structures. Among digital
structures, one can distinguish waveform generators, parameter search,
trackers, and parameter extractors. The major analog and digital phase
estimator types are summarised in Table 2.3.
Waveform regenerators are reminiscent of PLL tracking the phase of a FCmodulated
signal. However, they appear ill-suited for digital implementation
since they require several samples per symbol [83, p. 57]. Parameter
search is a brute force method. The value of the parameter which best fulfills
a performance criterion is searched over the interval of possible values
by testing all of them or a set of uniformly distributed ones over this interval.
The efficiency of the method is obtained at the cost of exhaustive
search. Neither waveform generator nor parameter search will be considered
in the following sections.
The current study will restrict its attention to trackers (FB phase estimators)
and to parameter extractors (FF phase estimators). The performance
criterion can be manipulated so as to produce an error signal which
will drive a recovery loop. Here comes the tracker (Figure 2.14 a). On
the other hand, the parameter extractor is a new form of estimator which
is impossible to design in the analog world. It explicitly calculates the

36 State of the art
closed-form estimate of the parameter according to ML theory. The value
of the estimate is used further in the receiver to correct the phase of the
received samples (Figure 2.14 b). FF structures are preferred to FB ones
in burst transmissions. The relevant information is quickly collected at
the receiver and FF estimators introduce no delay due to acquisition. On
the other hand, FB estimation is more suited for continuous transmissions
since it is able to track variations of the parameters instead of always starting
the estimation process from scratches as FF structures do.
Several digital phase estimators are listed in [83, chapter 2]. This classification
has been systematically organised in [84, section I.2] on the basis
of synergy of the estimation stage with the detection process (DA/DD-
/NDA), the underlying estimation theory (ML, non-linearities [82] among
which squaring loop, etc.), the structure type (FB/FF), and the signal modulation.
r (t)
Phase
estimator
(a) FB
r (t)
Phase
estimator
(b) FF
Figure 2.14: FB and FF implementations
The introduction of parameter extractors is not the only difference between
analog and digital phase recovery structures. Another one is the
fact that digital phase estimators can be implemented after timing recovery,
which is not the case in analog structures [85, p. 233]. In digital receivers,
phase estimation requires only one sample per symbol, while timing
recovery needs oversampling. Then, the information initially provided
to the timing estimator is decimated before entering the phase estimation
process [85, p. 275]. This introduces a dependency of the phase estimation
with respect to the timing recovery which translates into a sensitivity to
timing offset, function of the pulse shape [83, pp. 251-255].
However, phase estimators working independently of timing estimators
can be designed. This means that they rely on non-synchronised samples
taken at the output of a prefilter. As a result, oversampling is requested to

2.3 Phase estimation 37
avoid aliasing [84, section I.2.1.4].
2.3.2 Performance characterisation of phase estimators
FB estimation
A lot of work has been done for the performance characterisation of recovery
loops implementing FB estimators. The properties of these loops are
most often described using their open- and closed-loop transfer functions,
in terms of closed-form expressions, Bode plots, and root loci [79, chapter
2]. These specifications are then used to perform two different kinds of
analysis, linear and non-linear, depending on the kind of performance to
derive, either steady-state or dynamic. The following paragraphs are a
short introduction to linear and non-linear analysis.
Linear analysis The linear analysis is performed under the assumption
of small phase error to derive steady-state performance. It is well known
in the estimation literature [85] that the stable operating points of a recovery
loop are located at the positive zero-crossings of its S-curve. The
S-curve is the plot of the mean of the error signal driving the loop, with
respect to the estimation error, considering open-loop conditions. At the
operating points this mean is equal to zero.
The open-loop configuration is obtained by breaking the feedback path of
the loop. Despite this lack of feedback, the influence of the recovery process
is taken into account by substituting the estimation error of the loop
for the parameter to be recovered.
The sensitivity of the recovery loop with respect to the estimation error is
measured on the S-curve as the slope at equilibrium. This value is used to
build a simplified version of the loop. Using this linear model, the firstorder
moments of the phase estimate (bias and variance) can be derived,
as well as the noise bandwidth [79, section 3.1] or the steady-state error in
tracking, depending on the phase stimulus and the shape of the loop filter
[79, section 4.1].
Non-linear analysis The assumption of small phase error is not always
applicable especially when the loop starts working. A non-linear analysis
is then performed. It is used to characterise dynamic performance of the

38 State of the art
loop either during acquisition or during tracking.
The main tool in that account is the Fokker-Planck method for solving
stochastic differential equations. A complete analysis of an analog firstorder
loop is performed in [86, chapter 4] with the help of the Fokker-
Planck method. It derives the steady-state pdf of the phase error and characterises
both acquisition and tracking performance with the time to lock
and the cycle slip frequency. Although not applicable strictly speaking to
discrete time problems, the Fokker-Planck technique can been extended to
the treatment of the stochastic difference equations modelling the working
of digital loops [84, p. I-47].
The analysis of the acquisition is aimed at examining the convergence of
the estimate leading to the locking of the loop. This property is measured
by two ranges, the lock-in range and the pull-in range. The lock-in range
defines the span of possible parameter values for which the FB structure
will converge without missing any parameter cycle [79, p. 68]. On the
other hand, the pull-in range, wider than the lock-in range, guarantees
convergence but with possibly missing cycles. The error signal slowly
drives the loop into its lock-in range where convergence is achieved [79,
p. 72].
A phenomenon that needs to be kept in view when studying acquisition
is the hang-up problem. Due to the periodicity of the mean error signal
with respect to the estimation error, illustrated on the S-curve [83, p. 221],
the structure exhibits unstable tracking points. These points are the zerocrossings
of the S-curve with negative slope (Figure 2.15). At these points,
the loop wrongly appears to have reached equilibrium since the error signal
is null. However, this is an unstable situation. Even a small change of
the parameter value leads to a change of operating point. Measures will
be taken during acquisition so as to avoid being trapped in such a position
[79, p. 68]. Unless the acquisition time is prolonged due to the small value
of the error signal until a change of operating point occurs [80].
Moving to tracking performance, two kinds of incident are to be considered,
namely cycle slips and loss of lock.
Cycle slips occur when the tracked phase parameter exhibits a variation
so large that the loop moves its operating point to a neighbouring stable

2.3 Phase estimation 39
Uu (∆)
Linear
working
area
Hang-up
Cycle slip
Figure 2.15: Hang-up and cycle slip
∆
Stable working point
tracking point (Figure 2.15). This provokes decision errors if the modulation
exhibits rotational symmetry. This failure roots in the narrowness
of the loop bandwidth with respect to the spectral density of the phase
noise [81, p. 224], which prevents the loop from efficiently following variations
of the parameter to track. However, this problem is not as easy
to solve as it might appear at first sight. Indeed, enlarging the loop bandwidth
helps to reduce the cycle slip problem but at the expense of a greater
noise contribution to the system. A degradation of steady-state performance
thus occurs. A trade-off has thus to be made [83, p. 63]. This slip
phenomenon is characterised by the time average between slips. These
are rare incidents which are pretty difficult to study using computer simulations
[85, section 6.4].
Finally, situations might occur where the loop is driven out of lock. This
loss of lock is characterised by the Mean-Time to Lock Loss (MTLL) [87,
88].
FF loops
The study of FF implementations have not attracted as much attention as
FB structures have. There are many reasons amounting for this lack of
interest, the main one being the fact that FB and FF implementations produce
the same estimate provided that their loop/averaging bandwidth
coincide [84, section I.3.2]. Steady-state performance derived for FB estimators
are thus directly extended to FF estimators.

40 State of the art
As far as dynamic performance is concerned, FF implementations are not
plagued by some of the effects encountered with FB loops like hangup
[84, p. I-7]. This comes from the fact that there is no periodic behaviour
like the one illustrated by the S-curve of a FB loop. However, this does not
imply that FF loops are more efficient than FB structures in order to follow
parameter dynamics [85, section 6.4.4].
2.3.3 Multiuser Phase estimation
So far, the review of phase estimation has been mainly concerned with
single-user systems. What happens when the system under investigation
exhibits MAI, like in spread-spectrum communication systems ? Considering
that Nu users are active, a rigorous analysis would request to lead
an analytical study in the Nu-dimensional parameter space. There are very
few works addressing this question directly and without approximation.
It is usually preferred to simplify the problem in one way or another.
Gaussian approximation
The first simplification that comes to mind can be implemented at the
modelling step. Bearing in mind the central limit theorem which states
that the sum of a large number of mutually independent random variables
approaches a Gaussian distribution, it is tempting to model the MAI
as an additive Gaussian noise contribution which modifies the reception
performance of the user under consideration, conditioned on the operating
conditions. The averaging over these operating conditions, also called
interference averaging [4, section 9-3.2], can then be performed either on
the pdf of the MAI (Standard Gaussian Approximation) or on the performance
derived using this conditioned pdf (Improved Gaussian Approximation)
[89]. For instance, a Single-User Maximum-Likelihood (SUML) synchroniser
is presented in [90] in which the MAI has been modelled as a
zero-mean Gaussian random variable. It is demonstrated that the loss of
performance due to the deviation with respect to strict ML estimation is
compensated by performance improvement (Near-Far resistance) with respect
to standard synchronisation approaches. However, such Gaussian
approximations are only valid as long as the central limit theorem applies,
that is to say, in the case of large populations and without dominant term
among the contributing variables. The limits of these Gaussian approximations
are illustrated in [89] for a scarcely populated system and in the
case of a dominant interferer.

2.3 Phase estimation 41
Monte-Carlo simulations
On the other hand, the study is not necessarily led in an analytical way.
Rather than deriving their closed-form expressions, performance is measured
through computer simulations of the communication system (Monte-
Carlo simulations [91, section 5.6.1]). Computer tools enable to build a
block diagram representation of the system by simulating its operation.
Operating conditions are specified through system parameters, while timevarying
phenomena (information to transmit, channel behaviour...) are
simulated by filtering the output of built-in pseudo-random generators.
Running the simulation and measuring the outputs of detection and estimation
blocks give an insight into the performance of the whole system
depending on the specified operating conditions. The validity of the results
produced by this method is guaranteed within a certain confidence
interval provided that some conditions are respected as regards the number
of simulations [91]. Measuring a Bit Error Rate (BER) level requests,
for instance, that a significant number of errors have been observed before
its measure can be accepted within the corresponding confidence interval.
Performance degradation of detection due to estimation errors
Another possible simplification comes from the subject of the study itself.
Instead of deriving the performance of the estimation structures, some authors
rather study the influence of parameter estimation errors on the detection
process. There has been an overwhelming number of contributions
in this field. The references mentioned in the following paragraphs address
this question in spread-spectrum systems.
One of the major issues in the receiver is to ensure synchronisation in the
broad sense, and, more particularly, in code tracking. The sensitivity of
the linear decorrelating detector [14, 39] is investigated in both [40, 92]
in presence of a variety of synchronisation errors (timing, phase and frequency)
in AWGN channel. A Gaussian distribution of propagation delay
estimates is assumed in [40] while synchronisation errors are regarded as
uniformly distributed in [92]. The performance degradation in terms of
bit error probability, asymptotic efficiency, and Near-Far resistance is computed
in [93] and compared to those of the conventional detector.
Moving to frequency-selective channels, the performance degradation in
terms of BER, packet throughput, and delay due to the bias of a non-

42 State of the art
coherent early-late correlator with half-chip spacing is studied in [94]. This
tracking loop is nearly optimal in AWGN channels but its behaviour is
severely distorted by ISI in case of fast fading. However, as long as the
chip duration is shorter than the delay spread, ISI mitigation appears at
the output of the receiver thanks to the code correlation properties. As
regards MAI, its influence can be modelled in a similar way to ISI by substituting
Signal-to-Multipath Ratio (SMR) to Signal-to-Interference Ratio
(SIR). Moreover, tracking errors due to MAI appear to be negligible with
respect to ISI.
Sticking to synchronisation in the broad sense, the incidence of channel
estimation errors on the Joint Detection (JD) receiver of a synchronous
CDMA system is given in [68] in terms of Mean Square Error (MSE) by
linearly adding an error estimation noise term to each estimated tap of
the channel impulse responses. This estimation noise is uncorrelated with
either the channel noise or with the estimation noise affecting other users.
The Signal-to-Noise Ratio (SNR) degradation due to noisy channel estimation
is illustrated for different estimators as a function of the length of
the midamble training code. An appropriate choice of midamble codes
appears to limit the SNR degradation even with suboptimal channel estimation.
Besides code tracking, tight power control is requested to afford using conventional
correlating receivers in strong interfering environments (Near-
Far effect). Power control is analysed in [95] without introducing a Gaussian
approximation of the MAI. The degradation of BER and capacity is
measured with respect to the system load if power control is imperfect.
Rigorous estimation performance study
The present work does not aim at modelling the MAI as a Gaussian noise,
nor at relying on Monte-Carlo simulations. Nor is it concerned with the
influence of estimation errors on the detector. The objective of this work
is to derive, as far as possible, performance expressions of ML phase estimators
in DS-CDMA communication systems. Such approach has not
attracted much interest. A contribution [96] addressing this question appeared
only recently. The pdf of the phase estimate produced by a DD
(indeed DA, since perfect decisions are assumed) first-order PLL is derived
in the presence of AWGN, phase noise, and multiuser interference
in coherent asynchronous DS-CDMA, with the help of the Fokker-Planck

2.4 Conclusions 43
method.
2.4 Conclusions
This chapter had several purposes. First, it has described and scanned the
current and future applications of the multiple access scheme considered
in this thesis, viz. DS-CDMA. Limiting the scope of the present thesis to
mobile communication systems, the stress has then been laid on the design
of multiuser receivers. The work performed so far for symbol detection as
well as for parameter estimation has been reviewed. Finally, the last section
of this chapter has focused on the estimation of the phase which is the
parameter at the centre of the present work.
The next chapter will detail the communication system under investigation.
It will also introduce the analytical foundations required for the
performance study to be lead in Chapters 4 and 5.

Chapter 3
Tools
3.1 System description
3.1.1 System under investigation
Consider the uplink of a coherent CDMA communication system accommodating
Nu users (Figure 3.1). The low-pass equivalent signal tk(t) transmitted
by user k writes:
tk(t) = 2Ek
+
m=
I m k dk(t mT ). (3.1)
Im k = am k + jbm k are the modulated data symbols. Angular modulation
M-PSK will be considered in the following sections, with variance σ2 Ik .
Ekσ2 Ik is thus the emitted energy per symbol Es,k = Eb,k log2 M of user k,
as detailed in Section 3.1.2. T stands for the symbol duration and dk(t) is
the spreading waveform for user k
dk(t) =
Nc 1
n=0
v n k u (t nTc) (3.2)
where vn k is the pseudo-random spreading code, Tc is the chip duration,
Nc = T is the processing gain and u(t) is a rectangular pulse of duration
Tc
Tc. This means that signal is not band-limited, which is not a reasonable
assumption for real systems constrained to work in pre-assigned bands.

46 Tools
I m 1
I m 2
I m Nu
Spreading Power
Matched
Symbol
control
filtering sampling
d1 (t)
d2 (t)
dNu (t)
Ô 2 E1
Ô 2 E2
2 ENu
e jφ1
e jφ2
e jφNu
c2 (t)
c1 (t)
cNu (t)
h⋆ 1 ( t)
h ⋆ 2 ( t)
h⋆ ( t) Nu
t = mT
t = mT
t = mT
Figure 3.1: Uplink of a coherent CDMA communication system
y m 1
y m 2
y m Nu
Coherent
demodulation
e j ˆ φ m 1
e j ˆ φ m 2
e j ˆ φ m Nu
e j ˆ φ m 1 y m 1
e j ˆ φ m 2 y m 2
e j ˆ φ m Nu y m Nu

3.1 System description 47
The spreading waveform dk(t) is normalised so that
T
0
dk(t) 2 dt =1. (3.3)
Signals tk (t) are transmitted through channels having impulse responses
ck (t). Defining
hk(t) =dk(t) ª ck(t), (3.4)
the low-pass equivalent received signal at the BS, sum of the contributions
of the Nu active users, may be written as
Nu
r(t) = 2Ek e jφk
k=1
+
m=
I m k hk(t mT )+n(t). (3.5)
At the receiving end, the bandpass signal is downconverted to baseband
using a local oscillator with correct frequency but arbitrary phase. Thus,
φk, the parameter of interest in this thesis, appears in the expression (3.5)
of the low-pass equivalent received signal r (t) as the carrier phase difference
between transmitter’s and receiver’s oscillators. Finally, n(t) is the
low-pass equivalent of an AWGN with two-sided power spectral density
N0
2 .
The low-pass equivalent received signal r (t) is applied to a bank of filters
matched to the complete impulse responses hk (t). In Chapter 5, when
dealing with DD estimation structures, hard decisions will be taken from
the phase-corrected outputs of these matched-filters. This is definitely not
the optimal detector. Such detector is only suited for synchronous transmissions
of DS-CDMA signals using perfectly orthogonal codes in AWGN
environments. Out of these ideal conditions, its decisions are plagued by
ISI and MAI. This deficiency might be corrected using a more appropriate
detector (See Section 2.2.1). However, it will not be the case here, since
the aim of this thesis is to demonstrate that MAI contributions are informative
and that they may be exploited to improve the performance of the
receiver. Nevertheless, the front-end shown in Figure 3.1 fits all detectors,
whether optimal or not, as it provides sufficient statistics for reception.

48 Tools
The outputs of these channel-matched filters are sampled at symbol rate.
stands for the normalised matched filter output
y p
k
y p
k =
1
Ô 2EkT
= e jφk
+
q=
+ Nu
l=1
l=k
+ν p
k
+
e jφl
h ⋆ k
(t pT ) r (t) dt (3.6)
I q q
kxp k,k
El
Ek
+
q=
I q q
l
xp
k,l
Useful term
+ ISI
MAI
Additive noise
(3.7)
p q
where xk,l represents the normalised channel correlation coefficient between
users k and l for a time shift (p q) T
p q 1
xk,l =
T
+
h ⋆ k (t pT ) hl (t qT) dt (3.8)
and ν p
k are zero-mean complex samples of the noise filtered by the matched
filter
ν p
k =
1
Ô 2EkT
+
h ⋆ k
(t pT ) n (t) dt. (3.9)
Noise samples produced by different matched filters at distant time instants
might be correlated up to the value of the normalised channel cor-
p q p q
q
relation coefficient xk,l . ρk,l and ρp
k,l measure the correlation between
Rice components of the filtered noise:
p q
ρk,l = E ν p
q
p
q
k νl = E νk νl = 1
p q
N0 (xk,l )
Ô
2 EkElT
p q
ρk,l = E ν p
q
p
q
p q
N0 1 (xk,l )
k νl = E νk νl = Ô
2 EkElT .
(3.10)
The first term of (3.7) includes the useful symbol I p
k as well as the interfering
ones through ISI. The following term represents the MAI contribution
found at the output of a filter matched to the channel response of user k.
It results from the activity of interfering users within the same frequency

3.1 System description 49
band at the same time. In an ideal case, this interference would be cancelled
thanks to the correlation properties of the spreading codes v n k .
Unfortunately, a complete cancellation cannot be achieved in most cases
because this would require orthogonal codes and synchronous transmissions
over non dispersive channels. Since these conditions are not fulfilled
most of the time, codes are chosen so as to produce as few MAI as possible
[10].
In the following sections, the code sequences v n k and the channel responses
ck(t) will be supposed to be perfectly known at the BS. Timing
will be regarded as perfectly recovered1 . Moreover, the channel responses
will be assumed to be static2 with power delay profiles defined in accordance
with COST 207 models [97]. Finally, the phases φk will be assumed
to be constant.
3.1.2 Definition of Energy-to-Noise ratios
The denominator of the ratio Eb,k
N0
being known from the previous section,
Eb,k, the energy conveyed by each bit transmitted by user k, has now to
be calculated. To do so, the variance of baseband symbols sent by user k
is to be derived. Multiplying this variance by the symbol length T will
define the baseband symbol energy which is two times the bandpass symbol
energy Es,k. Finally, considering angular modulation with M states,
Eb,k will come out of the division of the bandpass symbol energy Es,k by
log 2 M.
Eb,k = Es,k
log 2 M =
1 1
baseband symbol variance T . (3.11)
log2 M 2
The first step is thus to calculate the variance of baseband symbols. Since
the transmitted signal tk (t) is cyclostationary, the mathematical expectation
operation is combined with stationarisation using a random delay
T0 uniformly distributed over [0,T]. Considering independent identically
1
Sampling at the output of the matched filter occurs at the peak of the auto-correlation
function of the shaping pulse.
2
A channel impulse responses can be regarded to be static when its coherence time is
greater than the symbol rate [7, p. 709]. In the frequency domain, it means that the Doppler
spread of the channel is smaller than the loop bandwidth of the estimator.

50 Tools
distributed data symbols I m k
EI,T0 [tk (t T0) t ⋆ k (t T0)]
= 2Ek
T
⎡
⎣
+ +
I m k (In k )⋆
EI
= 2Ek
T
+
m=
m=
= 2Ekσ 2 Ik
T
= 2Ekσ 2 Ik
T
n=
+
n=
+
m=
+
EI [I m k (In k )⋆ ]
δ (m n)
and using definition (3.8), the variance writes
0
T
0
0
T
T
hk (t mT T0) h ⋆ k (t nT T0) dT0⎦
hk (t mT T0) h ⋆ k (t nT T0) dT0
hk (t mT T0) h ⋆ k (t nT T0) dT0
⎤
(3.12)
(3.13)
hk (t τ) 2 dτ (3.14)
= 2Ekσ 2 Ik x0 k,k (3.15)
where δ (m) is a Kronecker delta function.
Introducing (3.15) in(3.11) gives Es,k and Eb,k
Es,k = EkTx 0 k,k σ2 Ik (3.16)
Eb,k = EkTx 0 k,k
σ2 Ik . (3.17)
log2 M
Making use of Eb,k
, the variance of the Rice components of the noise samples
N0
(3.9) writes
σ 2 (νk) = σ2 (νk) = σ2 νk
2
for M-PSK modulation.
= 1
2
N0 x 0 k,k
EkT
= 1
⎡
⎢
σ
⎣
2
2 Ik
x0 2 k,k
log 2 M
Eb,k
N0
⎤
1
⎥
⎦
(3.18)

3.2 Maximum-Likelihood estimation 51
3.2 Maximum-Likelihood estimation
3.2.1 Maximum A Posteriori and Maximum-Likelihood
Parameter estimation theory [59, 98] classifies estimation methods in two
main domains, distinguishing the Bayes approach and the classic approach
(also called Fisher approach).
In the Bayes approach, the vector parameter is regarded to be a random
variable whose behaviour is described by the pdf Tθ (θ) modelling its distribution
over the span of possible values. The information provided by
this pdf is exploited by the estimation process. Maximum A Posteriori
(MAP) is the most famous algorithm operating under the Bayes umbrella.
This algorithm searches for the vector parameter θ which maximises the a
posteriori pdf.
ˆθMAP =argmax
θ T θr (θ r) = arg max
θ
T rθ (r θ) Tθ (θ)
Tr (r)
. (3.19)
If the observations r and the vector parameter θ are jointly Gaussian, MAP
and MMSE methods produce the same estimate [59, p. 485].
However, the characterisation of the behaviour of the parameter through
its pdf is not always available. Estimation methods of the classic approach
have been designed to solve this issue in as much as they do not call upon
the a priori pdf of the parameter. This is made possible by assuming that
the parameter is uniformly distributed. Indeed, the a priori pdf provides
then no extra information and MAP becomes ML 3 . The ML estimation
process tries to maximise the likelihood function T rθ (r θ), which is the
probability of the observation r assuming the vector parameter θ.
ˆθML =argmax
θ T rθ (r θ) . (3.20)
This estimation method presents the interesting feature that, being conditioned
on the parameters to estimate, the likelihood function to maximise
only depends on the noise distribution. In the case of Gaussian noise ML
and Least-Squares (LS) methods produce the same estimate [59, p. 483].
The present work has been done in keeping with the classic approach.
3 The reader ought to notice that some authors introduce classic estimation as an approach
based on the assumption that the parameter to be estimated is a deterministic constant
[7, 59] rather than as a particular case of the Bayes approach for uniform pdf.

52 Tools
3.2.2 Likelihood function
In the classic approach optimum results are asymptotically obtained by
applying ML estimation. The estimate is obtained from the maximisation
of the likelihood function Λ(r θ). This function has thus to be derived. To
do so, a set of observable samples of the received signal (3.5) is required.
Such a set should be sufficient statistics, which means that all the information
of the time-continuous received signal should be contained in this
set of samples. Several approaches can be considered to build such a set.
In the next sections, the general method of Karhunen-Loève series expansion
will first be presented. Next, it will be shown that the prefiltering and
oversampling of the received signal implemented in nowadays digital receivers
[85, p. 227] can be regarded as a particular case of it. Finally, the
averaging of the likelihood-function over the pdf of the data symbols for
NDA estimation will be considered.
Karhunen-Loève series expansion Let fi (t) be a complete orthonormal
set of NKL functions over the observation interval T0 [98, section 3.2]:
(t) dt = δ (k l) . (3.21)
T0
fk (t) f ⋆ l
Defining ri as the set of the projections of the received signal r (t) (3.5)
on the set of the orthonormal functions fi (t)
ri = r (t) f ⋆ i (t) dt (3.22)
=
T0
Nu
k=1
2Ek e jφk
m=
+
I m k
T0
hk(t mT )f ⋆
i (t) dt +
T0
n(t)f ⋆ i (t) dt
(3.23)
the Karhunen-Loève series expansion states that the set of mutually uncorrelated
coefficients ri can represent r (t) in the limit as NKL + [7,
appendix 4A].
r (t) = lim
NKL+
NKL
i=1
rifi (t) . (3.24)
The ri do not only represent r (t), but also help to build the likelihood
function. In the case under study, they form a NKL-dimensional vector.

3.2 Maximum-Likelihood estimation 53
Provided the complete set of orthonormal functions fi (t) is made of the
eigenfunctions of the auto-correlation function of the received signal r (t),
the ri components conditioned on the vector parameter Φ are complexvalued
statistically independent Gaussian random variables with mean
⎡
Nu
E (ri Φ) = E ⎣ 2Ek e jφk
and variance
k=1
σ 2 riΦ
+
m=
⎡
= E ⎣
T0
I m k
T0
hk(t mT )f ⋆ i
(t) dt
Φ
⎤
⎦ (3.25)
n(t)f ⋆
2
i (t) dt
⎤
Φ⎦
. (3.26)
Assuming the data symbols I m k and channel impulse responses hk (t)
to be known, the NKL-dimensional joint pdf assuming Φ relies then only
on the noise distribution. Since n (t) is statistically independent Gaussian
noise, its components in the Karhunen-Loève series expansion are statistically
independent and jointly Gaussian, so that the NKL-dimensional
joint pdf writes
T (rNKL Φ)
rNKLΦ = NKL
i=1
1
Ô 2πσriΦ
exp 1
2σ 2 riΦ
ri Nu
k=1
Ô 2Ek e jφk
+
m=
I m k
T0
hk(t mT )f ⋆ i
(t) dt
2
.
(3.27)
By analogy to (3.24), the likelihood function Λ(r Φ) comes from (3.27) in
the limit as NKL + .
Λ(r Φ) = lim
NKL+ T rN KLΦ
⎡
= Cst exp ⎣
(rNKL Φ) (3.28)
1
r (t) s (t Φ)
2N0
2 ⎤
dt⎦
. (3.29)
T0

54 Tools
The ML estimate is the value of the parameter that maximises the likelihood
function. From (3.29), it can be interpreted as the value of the parameter
which minimises the distance between the received signal r (t) and
the noiseless signal s (t Φ) assuming Φ.
Since it is assumed that an infinite sequence of data symbols is transmitted
(3.5), but that the observation interval is finite and of length T0, it
is more convenient to change integration limits in (3.29) from[0,T0] to
[ , + ] and simultaneously, to change the summation limits in (3.5)
from [ , + ] to [1,N] so that NT = T0 [83, p. 93]. Practical estimators
built on this modified likelihood function are called pseudo-ML estimators
in [99]. This approximation causes end effects [83, p. 93] traced back
in [81, p. 82]. They generate a noise-independent jitter component whose
incidence depends mainly on the width of the observation interval [81,
p. 82] and on the type of modulation [99, p. 1126]. These end effects will
be neglected in the present thesis.
Using this approximation, relation (3.29) then becomes
Λ(r Φ)
⎡
exp ⎣
exp
⎡
⎢
exp ⎣
1
2N0
Nu
T0
EkT
l=1
l=k
⎤
r (t) 2 dt⎦
exp
N
+
Nu
k=1
(I m k )⋆I n n
k xm
k,k
k=1
N0
m=1 n=
Nu Nu
Ô
EkEl T
N0
k=1
e j(φk
N
φl)
m=1 n=
2EkT
N0
+
e jφk
(I m l )⋆ I n k
N
(I m k )⋆y m
k
m=1
n
xm
l,k
⎤
⎥
⎦ . (3.30)
Moving from (3.29) to(3.30) has led to split the distance r (t) s (t Φ) 2
into its components, the energy of the received signal r (t) 2 , the double
product 2 [r (t) s⋆ (t Φ)], and the energy of the noiseless signal assuming
Φ, s (t Φ) 2 . The energy of the received signal stands explicitely in (3.30),
while the double product turns into the second term, involving matched
filter outputs ym k . Finally, the last two terms come from the expansion of
s (t Φ) 2 .
Here lies the major innovation of ML phase estimation in a multiuser context.
So far, in most carrier recovery structures, the distance in (3.29) is

3.2 Maximum-Likelihood estimation 55
expanded so as to keep only the double product, the correlation of the
received signal, and the conditioned one [r (t) s (t Φ)]. Square terms
are disregarded as not depending on the parameter to estimate. This is
no longer the case in multiuser spread-spectrum systems. The energy of
the conditioned signal s (t Φ) 2 depends on the parameters to estimate
through differences φk φl. It can thus not be disregarded in the estimation
process.
Searching for ML estimate is not easy due to the exponential function in
(3.30). However, since the logarithm is a monotonic function of its argument,
the value which maximises f (Φ) also maximises log [f (Φ)]. So,
usually, instead of dealing with the exponential function appearing in the
likelihood function, one prefers to use its logarithm. Taking the logarithm
and developing s (t Φ) by using (3.7) and (3.8) finally gives
ΛL(r Φ)
Nu 2EkT
= Cst + e jφk
Nu
k=1
EkT
N0
k=1
Nu Nu
k=1
l=1
l=k
N0
N
+
m=1 n=
Ô EkElT
N0
N
(I m k )⋆y m
k
m=1
(I m k )⋆I n n
k xm
k,k
e j(φk φl)
N
+
m=1 n=
(I m l )⋆I n n
k xm
l,k . (3.31)
Sampling independently of the transmitter clock The log-likelihood
function derived by the Karhunen-Loève series expansion depends on the
matched filter outputs ym 1
k produced at rate T . These are sufficient statistics
for phase estimation (not for timing [85, p. 257]). Implicitly, it was assumed
that the sampling of the matched filter output occurred at the right
instants, in synchronisation with the transmitter clock. Instead of working
at 1
T ,[85, chapter 4] establishes conditions under which collecting samples
at a higher rate completely independent of transmitter clock 1 1 > Ts T can
produce sufficient statistics. This is specially useful for digital implementations
since this reduces the number of information flows between analog
stages and digital stages. Samples can be produced by a free-running clock
at rate 1 . By interpolation and decimation of the samples another set of
Ts
and at the right sampling instants.
samples is produced at rate 1
T

56 Tools
On the one hand, this sampling provides samples r (mTs) which are sufficient
statistics to represent r (t) provided a generalised anti-aliasing filter
fulfilling conditions [85, p. 243] has been used. This is the sampling the-
orem which can be understood as a special case of the Karhunen-Loève
series expansion using Whitakker basis functions
gonal functions fi (t).
sin x
x as a set of ortho-
On the other hand, noise samples are complex-valued zero-mean Gaussian
random variables. These samples might be separated into their Rice
components nm I and nm N0
Q , each of them exhibiting a variance of . The Ts
joint pdf of N complex-valued noise samples writes
TnI,nQ (nI, nQ)
= Cst exp
= Cst exp
Ts
N
2N0 m=1
N
Ts
2N0
m=1
n m I 2 +
N
m
n
Q
2
m=1
r m I sm I 2 +
m=1
(3.32)
N
m
rQ s m
Q
2
. (3.33)
In the limit as N + , TnI,nQ (nI, nQ) becomes Λ(r Φ) (3.30).
Low SNR approximation for NDA estimation As explained in Section
2.2.2, it might be desirable under certain circumstances to estimate parameters
without relying on either training sequences (DA case) or on decisions
(DD case). From the point of view of ML, the maximisation of a
modified likelihood function in which the influence of the data symbols
has been cleared is a solution to this problem. This is the NDA approach
which substitutes a log-averaged likelihood function to the usual likelihood
function (3.30) to be maximised. The average is performed with
respect to the data symbols. Note that the average operation is applied
before taking the logarithm, since it is not valid to perform the average
through a non-linearity (the logarithm here) [83, p. 227].
Most of the time, the result of the averaging step is a non-linear function
of the sufficient statistics and the Eb ratio [99]. Instead of dealing with
N0
these rather complicated equations, approximations at low and high SNRs
are preferred [83, pp. 226-250]. While previous references present specific
results for each case considered, the whole approach is formalised in
[100] for low SNR. It suggests expanding the exponential function of (3.30)

3.2 Maximum-Likelihood estimation 57
into a Taylor series and to apply the average operation on each of its term
with respect to the data symbols. Only the remaining terms which are
still function of the parameters to estimate are kept. In the case of the low
SNR limit, this decomposition is further limited to the lowest power of Eb
N0 .
Considering the averaged likelihood function produced by this method in
the case of M-PSK modulations, one can notice that a similar expression
would be obtained by applying a non-linear function to data symbols in
order to cancel the influence of the modulation.
However, the higher the order of the constellation is, the higher the power
to be considered in the Taylor series expansion gets. Moreover, taking the
Mth power of M-PSK modulated samples introduces a M-fold ambiguity
in the estimation process. It has been demonstrated in [82] that it is not
necessary to call upon a non-linearity of order proportional to the dimension
of the constellation. With the help of the polar representation of the
sufficient statistics, applying a non-linearity of order n

58 Tools
respect to the unknown vector parameter Φ and setting the result equal to
zero [98].
∂ΛL(r Φ)
=0. (3.35)
∂Φ
Φ= ˆ Φ
Due to the multiuser context, (3.35) produces as many equations as phase
parameters to be estimated (Nu in the present case). Obviously, this leads
to the search of an optimum in a Nu-dimensional space. Estimation methods
mentioned in Section 4 lead such search. On the other hand, authors
have proposed suboptimum methods aimed at producing reliable estimates
without having to lead such a time-consuming search. Without forgetting
the practical difficulties of the search for the optimum, the present
study will stick to the equations defining the optimum point in order to get
a better insight into the performance of the multiuser estimation process.
3.3 Optimal estimator performance
Since estimators are built upon samples of observed signals embedded
with random noise, they exhibit themselves a random behaviour. In this
perspective, it is natural to quantify their performance by the derivation
of their moments. This derivation is most often limited to the first- and
second-order moments, the mean and the variance, since these are the only
statistics required to completely characterise a stochastic process
from
a
Gaussian perspective. An estimator whose estimation error E θ ˆθ is
null will be called unbiased. The optimal
estimator should not only be un-
ˆθ
2
biased but also have a variance E E ˆθ as small as possible [59,
chapter 2].
3.3.1 Cramér-Rao Lower Bound
Checking that the estimator is unbiased is a pretty easy operation. The
reference value is clear: a null bias. On the other hand, when moving
to second-order statistics performance, the need of a benchmark appears,
against which the variance of a prospective estimator can be tested. On
that account, the Cramér-Rao Lower Bound (CRLB) is precious in estimation
problems, since it provides a lower bound on the variance of any
unbiased estimator.

3.3 Optimal estimator performance 59
The CRLB can be derived from the diagonal elements of the inverse of
the Fisher information matrix. In the case of the estimation of a vector
parameter θ, the elements of the Fisher information matrix I (θ) k,l are the
second derivatives of the log-likelihood function with respect to the parameters:
∂2ΛL (r θ)
I (θ) k,l = E
(3.36)
∂θk ∂θl
∂ΛL (r Φ) ∂ΛL (r θ)
= E
. (3.37)
∂θk ∂θl
The diagonal elements of the Fisher information matrix (3.36) can be interpreted
as a measure of the curvature of the log-likelihood function. The
sharper the function is, the greater the curvature, the lower the variance,
and the better the estimator are [59, p. 29]. Another interpretation of the
CRLB is given in [101], where the bound is derived from the parameter
power spectral density (psd). This spectrum is derived from the psd of
the transmitted signal tk (t) filtered by the channel ck (t), so that it represents
the localisation of the available information about the parameter in
the frequency domain.
Since the CRLB only depends on the Fisher information matrix which itself
in turn only relies on the likelihood function, it is a global benchmark.
The CRLB is derived for a specific problem but irrespective of the estimator
to be tested against that benchmark. It provides a fundamental lower
limit on the variance of any estimator of all the parameters of the problem.
However, some of them are sometimes not to be estimated. Called unwanted
parameters [98, section 2.5], they have yet to be taken into account
in the derivation of the CRLB. This can then appear quite intricate. Some
modified bounds easier to derive have been introduced to tackle this issue.
First, the Modified CRLB (MCRB) was introduced in [102] for cases where
wanted θ and unwanted u parameters coexist. Instead of working as usual
with the pdf T rθ (r θ) of the received signal assuming the wanted parameters
θ, MCRB deals with the pdf T rθ,u (r θ, u) assuming both wanted
and unwanted u parameters. It ends in a looser variance bound than the
CRLB. Nevertheless, approximate equality of CRLB and MCRB is found
to occur in several cases among which phase recovery when all other parameters
and data are known [102, section IV], which will be the case in DA
estimation (Chapter 4).

60 Tools
Besides MCRB, Asymptotic CRLB (ACRB) [103] is another means of getting
a lower bound on the variance while avoiding heavy computations.
ACRB is derived as the high SNR asymptote of the CRLB. It has been
shown in [103] that it equals CRLB when the Fisher information matrix
does not depend on unwanted parameters. As far as the MCRB is concerned,
the ACRB lies above it. However, ACRB and MCRB join when
the unwanted parameters are discrete or when they are continuous but
decoupled from the wanted parameter(s).
3.3.2 ML performance
Now that the CRLB has been introduced as the variance benchmark for
any unbiased estimator the reasons of the optimality of ML estimation can
be listed.
ML estimation provides the optimal estimator in the classic approach because
its estimator is asymptotically consistent and asymptotically efficient.
Consistency means that the estimator is unbiased, while efficiency
relates to the fact that its variance reaches the CRLB. However, these properties
are only valid asymptotically in the case of the ML estimator, that is
to say, in the limit when the number of observed samples N + .
Bearing in mind the Fisher information matrix I (θ), these two properties
can be summed into one statement stating that the ML estimator is asymptotically
distributed according to a Gaussian distribution Æ θ, I 1 (θ) .
3.3.3 CRLB for multiuser phase estimation
Before deriving the CRLB in the case of multiuser phase estimation, a regularity
condition has to be fulfilled [98]
∂ΛL (r Φ)
E
=0 k [1,Nu] . (3.38)
∂φk
For the system under investigation, this condition writes

3.3 Optimal estimator performance 61
∂ΛL (r Φ)
E
∂φu
since data symbols Im u
m n
=
=
2EuT
N0
2EuT
N0
N
+
m=1 n=
N
+
m=1 n=
E [I n u (I m u ) ⋆ m n
] xu,u σ 2 m n
I δ (m n) xu,u
(3.39)
(3.40)
= 0 (3.41)
are independent identically distributed random
variables and xu,u δ (m n) =x0u,u is real.
Having checked the regularity condition, the Fisher information matrix
can be calculated. It will be diagonal since off-diagonal elements
I (Φ) u,v
v=u
∂2ΛL (r Φ)
= E
∂φu ∂φv
=
2 Ô EuEv T
N0
j(φv φu)
e
N
+
m=1 n=
E [I n v (I m u ) ⋆ m n
] xu,v
(3.42)
(3.43)
= 0 (3.44)
are null due to the fact that data symbols I m u and In v
are uncorrelated. Diagonal
elements only remain which, with the help of (3.17), write
∂2
ΛL (r Φ)
I (Φ) u,u = E
(3.45)
= 2N EuTx0 u,uσ2 Iu
(3.46)
N0
= 2N Es,u
. (3.47)
N0
The CRLB is defined as the inverse of the Fisher information matrix. Since
the non-zero elements of the latter lie on the diagonal, the former is given
by the inverse of the corresponding diagonal elements. In a feedforward
implementation using a N-sample window, the CRLB for user u writes [85,
p. 331]
CRLBFF,u = 1
2N
∂φ 2 u
Es,u
N0
1
. (3.48)

62 Tools
If a feedback implementation with closed-loop frequency response Hu (z)
is preferred to a feedforward structure, the one-sided loop bandwidth
BN,u
2BN,u =
1
2T
1
2T
Hu(e 2jπfT
)
2
df (3.49)
is substituted for the size of the observation window N in relation (3.48)
under the condition that either the loop noise is white or the one-sided
loop bandwidth BN,u is narrow [85, p. 349]. The CRLB then becomes
Es,u
CRLBFB,u = BN,uT
N0
1
. (3.50)
In line with the frequency domain interpretation of [101], it is noticed
in [83, chapter 4] that these bounds are the same as those valid for phase
recovery in the case of a pure unmodulated carrier. Things happen as if
the optimal estimator aiming at achieving the CRLB compacted the signal
power into a spectral line to be tracked by a PLL.
3.4 FF estimation
This section will deal with two aspects of the study of FF estimators, namely
the derivation of a closed-form suitable for performance evaluation and
the calculation of the variance.
3.4.1 Closed form of the estimator
In the search for a closed-form expression of the ML FF estimator, the loglikelihood
function (3.31) can be handled in two different ways in order to
extract the parameter of interest. The first possibility consists in estimating
the phase parameter φ or real functions of it (cos φ, sin φ). The second one
would rather deal with the complex phasor e jφ . Both possibilities will be
investigated in the following paragraphs.
Estimation of the phase parameter
Applying (3.35) to(3.31) leads to the following expression
ˆφu = tan
1 (Cu)
(Cu)
(3.51)

3.4 FF estimation 63
where
Cu =
m=1
N
(I m u )⋆ y m u
Nu
Ek
k=1
k=u
Eu
e j ˆ φk
N
+
m=1 n=
(I m u )⋆ I n k
n
xm
u,k . (3.52)
Somehow, this expression of the ML FF phase estimator is similar to classic
ones [7, p. 326]. Indeed, the estimator takes the argument of a complex
number Cu partly built from the product between the matched filter outputs
y m u and the data symbols I m u . Yet, due to the multiuser context, (3.52)
exhibits an additional contribution. The comparison of (3.7) with (3.52)
shows that this additional contribution tends to cancel the influence of the
interfering users coming from the matched filter outputs.
However, relation (3.52) is not appropriate for performance evaluation,
since the parameter estimate of one user is an implicit function of the
parameter estimates of the interfering users. In order to be able to solve
these equations with respect to the phase parameter, a linear relationship
between phase estimates has been searched by applying linearisation to
different factors.
Linearisation of the first derivative about optimum The first linearisation
attempt is a truncated Taylor series expansion of the first derivative of
the log-likelihood function around the optimal value ˆ Φ [85, pp. 343-344].
This is to exploit the fact that the first derivative equals zero at this point.
∂ΛL (r)
∂ΛL (r)
∂2ΛL (r)
= 0 =
+
∂Φ Φ=ˆΦ
∂Φ
∂Φ Φ=Φ0
2
ˆΦ Φ0
Φ=Φ0
(3.53)
where Φ0 is the true value. Solving (3.53) for Φ gives
ˆΦ Φ0 =
∂ 2 ΛL (r)
∂Φ 2
1
Φ=Φ0
∂ΛL (r)
∂Φ
Φ=Φ0
. (3.54)
Considering that the observation window is large enough, the Fisher information
matrix can be substituted for the second derivative of the loglikelihood
function in (3.54)
ˆΦ Φ0 = I (Φ) 1
∂ΛL (r)
. (3.55)
∂Φ
Φ=Φ0

64 Tools
Unfortunately, (3.55) does not produce the wished linear relationship. Indeed,
each phase estimate finally writes as a quotient of functions of the
complex phasors. Considering that such a result is not suited for performance
evaluation, it has been disregarded.
Linearisation of complex exponential A difficulty in dealing with phase
parameters of interfering users in (3.52) is that they appear as arguments of
a non-linear function, namely the exponential function. Instead of linearising
the first-derivative of the log-likelihood function, a second attempt
involves linearising this exponential function. The Taylor series expansion
is limited to the first order under the hypothesis of small estimation error.
e j ˆ
φk = jφk e + ˆφk φk je jφk (3.56)
= e jφk
1+j ˆφk φk . (3.57)
Applying this linearisation to (3.31) leads to the following condition
⎧
e
⎪⎨
jφu
1 j ˆφu φu
⎧
N
(I
⎪⎨
m=1
m u )⋆ ym u
Nu
1+j ˆφk φk
⎫
⎫
⎪⎬
⎪⎬ =0. (3.58)
⎪⎩
⎪⎩
Solving (3.58) gives
where
Du =
N
m=1
Nu
k=1
k=1
N
m=1 n=
(I m u ) ⋆ y m u
Ek
Eu ejφk
+
Ek
Eu ejφk
(I m u ) ⋆ I n k
xm n
u,k
ˆφu φu = e jφu
Du
(e jφu Du)
N
1+j ˆφk φk
⎪⎭
+
m=1 n=
⎪⎭
(I m u )⋆ I n k
(3.59)
xm n
u,k .
(3.60)
Thanks to the linearisation, the tan 1 non-linearity in (3.52) has disappeared.
By regarding some terms of (3.52) as negligible (See Section 4.2.2),

3.4 FF estimation 65
the phase estimation error ˆ φu φu can become linearly dependent of other
phase estimation errors. As it will be shown in the next chapter, this paves
the way for the performance analysis of the ML FF estimator.
Rectangular representation Still estimating the real phase parameter φ,
another strategy might be to try to recover cos φ and sin φ, since φ always
appears as argument of a complex exponential e jφ =cosφ + j sin φ. This
strategy might be applied in two different ways, either by directly estimating
cos φ and sin φ, or by estimating φ and implementing the estimator so
as to track cos φ and sin φ [83, pp. 216-226]. However, as far as the former
case is concerned, derivating with respect to φ or to (cos φ, sin φ) does not
bring out a significantly new estimator, since all these parameters are tied
together as follows
This leads to
dΛ =
∂Λ
∂φ =
=
∂Λ
∂φ =0
∂Λ
∂Λ
d cos φ + d sin φ
∂ cos φ ∂ sin φ
(3.61)
∂Λ ∂ cos φ ∂Λ ∂ sin φ
+
∂ cos φ ∂φ ∂ sin φ ∂φ
(3.62)
∂Λ
∂Λ
sin φ + cos φ.
∂ cos φ ∂ sin φ
(3.63)
∂Λ ∂Λ
=
∂ sin φ ∂ cos φ
tan φ. (3.64)
The latter is a question of implementation rather than a way to obtain a
new expression of the estimator. It does not modify the analytical performance
evaluation to be presented in the following chapters, since the
estimator is still derived from (3.52). This is the reason why it will not be
studied here.
Estimation of the phasor
Having considered the phase estimation in the real space, through the
phase parameter φ as well as through functions of it (cos φ, sin φ), it is
time to move to the complex space.
Planar filtering In (3.52), the phase estimate ˆ φu appears to be obtained
as the argument of the complex number Cu. Instead of tracking the argument,
another possible approach is to track the phasor itself. This tech-

66 Tools
nique is called planar filtering [85, p. 312]. However, it is rather a postprocessing
technique [104], in the sense that the optimum is still defined
with respect to the phase. Only the tracking takes the complex specificity
of the phasor into account. Planar filtering improves the tracking [85,
p. 414] but, as far as estimation is concerned, it does not define another
estimate than the one obtained through phase estimation.
Complex derivation Regarding planar filtering as a post-processing improvement
of a structure based on phase estimation, one could try to directly
estimate the complex exponential e jφ . In the ML approach, the first
derivative to set to zero is then taken with respect to a complex number,
that is to say that the first derivative of a real function with respect to
a complex variable is to be computed. For this derivative to exist, the
Cauchy-Rieman conditions are to be fulfilled.
In the most general case, setting ejφ = x + jywhere x and y are respectively
the real and imaginary parts of ejφ , these conditions state that
∂ [f (x, y)]
=
∂x
∂ [f (x, y)]
(3.65)
∂y
∂ [f (x, y)]
∂x
= ∂ [f (x, y)]
. (3.66)
∂y
However, the likelihood function Λ is a real function of (x, y)
Λ(x, y) =
Ae jφ
which does not fulfill Cauchy-Rieman conditions
(3.67)
= [(Ax + jAy)(x + jy)] (3.68)
= Ax x Ay y (3.69)
∂[f(x,y)]
∂[f(x,y)]
∂x = Ax; ∂y = 0
∂[f(x,y)]
∂x = 0; ∂[f(x,y)]
(3.70)
∂y = Ay.
Derivating the likelihood function with respect to the phasor is thus not
possible.
3.4.2 Variance approximation
The closed-form expressions of the ML FF estimator involve a quotient.
This mathematical relationship is not easy to handle at the time of computing
the variance. Two options appeared in the literature to circumvent

3.5 Performance evaluation of DD estimators 67
this difficulty.
From (3.54), the reader can notice that the phase estimation error is equal
to a ratio between first- and second-derivative of the log-likelihood function.
In [105, 106], the statistical fluctuations of the second derivative are
assumed to be small with respect to its mean value, so as to substitute the
second derivative for its mean. In terms of variance, this finally gives
σ 2
Φ ˆΦ =
∂2ΛL (r)
E
∂Φ2
Φ=Φ0
2
E
∂ΛL (r)
∂Φ
2
Φ=Φ0
. (3.71)
On the other hand, starting from (3.52), the expectation of the square of
the argument may turn into [104]
under the hypotheses that
E [arg (Cu)] 2
E [ (Cu)]
=
2
E [ (Cu)] 2
(3.72)
E [ (Cu)] = 0 (3.73)
E (Cu) E [ (Cu)] 2
E [ (Cu)] 2
(3.74)
E (Cu) E [ (Cu)] 2
E [ (Cu)] 2 . (3.75)
Only the first option (3.71) will be applied in the next chapters. However,
it was worth mentioning the second one (3.72) for review purposes.
3.5 Performance evaluation of DD estimators
The performance of DD estimators is not an easy issue. It is a coupled
problem, since decision errors are eager to cause estimation errors which
in turn will affect the decision process and so on. Few contributions really
tackle the problem. Most of the time, decisions are assumed to be correct,
which, as far as performance is concerned, brings back to DA analysis. A
less rude approach to take into account the possible faulty outcome of the
decision process is to weight the performance derived in the case of DA
structures by the probability of error PE [84, p. II-7].

68 Tools
No such approximation is made in [87]. The performance of carrier phase
recovery systems for single-user transmissions over AWGN channels is
calculated using
analytical
expressions of products between data symbols
⋆
and decisions E Îm k Im
k . Generalising this approach to multiuser systems
working over non ideal channels is pretty intricate. A look at (3.31)
reveals first that the expectations derived in [87], which involve symbols
and decisions from the same user at the same time instant, are now to span
over all users, as a result of the non-orthogonality between users, and over
the whole observation window due the time
dispersiveness
of the chan-
⋆
nel. In the most general case, expectations E Îm k In
l will be computed
for any pair (k, l) and (m, n).
Moreover, (3.31) contains contributions whose positive effect is to mitigate
interference. These terms involve products of decisions, possibly related
to different users or different time instants, which leads to the computation
of
⋆Î
m n
E Îk l
= E (â m k ânl )+E
ˆb mˆn k bl + j E â m k ˆb n
l
E ˆb m
k â n
l . (3.76)
The derivation of such expectations is the subject of the current section.
Considering M-PSK constellations with unit variance (σ 2 I
tions hereafter will be used
p q
Rk,l p q
Ik,l =
=
⎧
⎨
⎩
⎧
⎨
⎩
El
Ek
Ô
2 El
2 Ek
El
Ek
Ô
2 El
2 Ek
e j(φl ˆ
φk) p q
xk,l
e j(φl ˆ
φk) p q
xk,l e j(φl ˆ
φk) p q
xk,l
e j(φl ˆ
φk) p q
xk,l Similarly, for the noise samples
ν m k =
e j ˆ ν
φk m
νk m k =
e j ˆ φk m
νk
(BPSK)
(QPSK)
(BPSK)
(QPSK).
=1), the nota-
(3.77)
(3.78)
(3.79)
. (3.80)
For the sake of simplicity, the following developments will be limited to
conventional hard decisions, although optimal and suboptimal detection

3.5 Performance evaluation of DD estimators 69
strategies have been mentioned in Section 2.2.1.
where
â m k
ˆ b m k
=
=
A p
k
= Āp
k
=
sign (Am k ) = sign Ām k + νm
Ô k
2
2 sign (Am Ô
2
k ) = 2 sign Ām k + νm
k
sign (Bm k ) = sign B¯ m
k + νm
Ô k
2
2 sign (Bm Ô
2
k ) = 2 sign B¯ m
k + νm
k
+ νp
k
+
q=
B p
k
= ¯ B p
=
k + νp
k
+
q=
a q p
kRq k,k bq
p
kIq k,k
a q q
kIp k,k
q
+ bq
kRp k,k
Nu
+
l=1
l=k
Nu
+
l=1
l=k
+
q=
+
q=
a q p
l
Rq
a q q
l
Ip
k,l
k,l b q
l
3.5.1 Direct space - Gaussian probability integral
(BPSK)
(QPSK)
(BPSK)
(QPSK)
(3.81)
(3.82)
p
Iq
k,l + ν p
k
(3.83)
q
+ bq
l
Rp
k,l + ν p
k .
(3.84)
In direct space, the mathematical expectations of (3.76) turns into a double
integral over the noise samples alleviating the arguments (3.83) and (3.84).
In the most general case, the calculation of these expectations ends in a
non separable double integral, since their limits are functions of the same
data symbols. For instance, here is the expression of the expectation of the
product between the decision on the I-branch for user u at instant m and
the decision on the Q-branch for user v at instant n, for QPSK-modulated

70 Tools
symbols since there is no information on the Q-branch in BPSK
E â m u ˆb n
v
= 1
2 EI,ν
m
sign Āu + ν m
u sign ¯B n
v + ν n v
(3.85)
= 1
2 EI
⎧
⎨
+ +
sign
⎩
Ām u + νm
u sign ¯B n
v + νn
v
Tνm u ,νn v (νm u , νn v ) dνm u dνn ⎫
⎬
⎭
v
(3.86)
⎧
⎪⎨
+
= EI
⎪⎩ +
+
Tν m u ,ν n v (νm u , ν n v ) dν m u dν n v
Āmu ¯ Bn v
Ām u
¯ Bn v
Tνm u ,νn v (νm u , νn v ) dνm u dνn v
⎫
⎪⎬
⎪⎭
1
2 .
(3.87)
The decisions are the results of sign functions whose arguments are the
sum of a linear combination of random variables related to the data and
of noise samples. Developing the expectation over the noise brings from
relation (3.85) to(3.86). However, integrating the product of these sign
functions over the plane (ν m u , ν n v ) is equivalent to integrate it over subdomains
where it takes values ¦1 (Figure 3.2).
¯ B n v
ν n v
Ām u
ν m u
Figure 3.2: Sub-domains in the plane (ν m u , ν n v )

3.5 Performance evaluation of DD estimators 71
Exploiting the Gaussian characteristic of the noise samples, one finally
gets relation (3.87). Similar relations can be written for the other expectations
of (3.76).
The fact that the integration limits in (3.87) depend on data symbols complicates
the averaging operation. Recently, an alternative form of the Gaussian
probability integral has been presented [107] and applied to the derivation
of the error probability for various communication systems [108].
Preliminary studies show that this powerful transformation seems very
promising for performance evaluation of estimators. Unfortunately, time
lacked to lead a thorough analysis of this new method and to apply it to
the problem under investigation in the present work.
Coming back to the subject of relation (3.87), the reader notices that it is
suited for numerical computation. However, if the goal is to get an analytical
solution, moving to the reciprocal space and using the characteristic
function has appeared to be an elegant move.
3.5.2 Reciprocal space - Characteristic function
Several methods relying on the characteristic function have been proposed
in order to compute the error probability in presence of noise and interference.
The first to be mentioned here is developed in [89, 109]. The Gaussian
probability integral of the noise samples, whose limits are function of the
interference like in (3.87), is solved assuming the interference. As a result,
the solution has the form of the Gaussian probability function Q (x)
Q (x) = 1
2 erfc
xÔ2
= 1
xÔ2
1 erf
2
= 1
2π
+
x
exp
u2
2
(3.88)
du (3.89)
whose argument depends on the structure of the interference. The calculation
of the error probability would then require to average the obtained
result over the interference pdf. At first sight, this is a very consuming
operation, since it involves a number of computations which is exponential
in the number of symbols contributing to the interference. However,

72 Tools
developing the error function into its Fourier series expansion introduces
a set of basis exponential functions which, combined to the interference
pdf and the integration, makes appear the characteristic function of the
interference. The interesting point is that taking into account the interference
through its characteristic function requires a number of computations
which is only linear in the number of involved symbols.
Another method for deriving the error probability is introduced in [110]
for systems plagued with ISI. A generalisation to systems suffering from
different kinds of interference is presented in [111]. This method embeds
all contributions, signal, interference, and noise into one random variable.
The moment-generating function of this global random variable is
obtained by Fourier transform. It is well-known that the inverse Fourier
transform of the moment-generating function gives the pdf of the random
variable. Computing the error probability turns into a definite integral
of this pdf. As in [89, 109], the use of the characteristic function avoids
exponential computations in favour of linear complexity computations.
Moreover, switching the probability integral and the inverse Fourier transform
integral brings another simplification. This method which relies on
the Fourier transform has also been applied using the Laplace transform
[112].
The main advantage of these techniques is the possibility to perform the
averaging operation over the interference with a computational effort linear
in the number of included symbols. This has been a motivation for
applying the second one to the analysis of DD FB estimators. The details
of the derivation are presented in Appendix E and their exploitation for
the derivation of the performance of DD FB structures will be shown in
Chapter 5.
3.6 Conclusions
The uplink segment of the mobile DS-CDMA communication system under
investigation has been presented in this chapter. Aiming at performing
coherent reception, the carrier phase has to be recovered. Under the
assumption of uniformly distributed random variables, ML has been introduced
as the optimal estimation method. ML estimators are known to
be asymptotically unbiased and their variance is bounded by the CRLB,
which has been derived.

3.6 Conclusions 73
On the other hand, some analytical issues have been dealt with in this
chapter. First, the means to derive a closed-form expression of a ML FF estimate
and to compute its performance have been presented. Next, several
methods for studying the working of DD structures have been explained.
Using the material exposed in the first two chapters, ML phase estimators
will be derived in the following chapters, first in a DA mode (Chapter 4),
then in a DD mode (Chapter 5).

Chapter 4
Data-Aided
This chapter deals with ML estimation of phase parameters in DA context.
In such situation, the receiver has a perfect knowledge of the symbols I p
k
transmitted by user k. This happens during acquisition sessions on the
link between the transmitter and the receiver, when the transmitter emits
predefined symbol sequences used at the receiver to estimate the parameters
of the link.
As mentioned in Section 3.2.3, a necessary but not sufficient condition for
deriving the ML estimate is to set to zero the first derivative of the loglikelihood
function with respect to the parameter of interest. Calculating
the first derivative of (3.31) with respect to Φ and setting the result equal
to zero leads to a set of Nu conditions of the type
∂ΛL(Φ)
∂φu
Φ=ˆΦ
⎡
N
⎤
= 2EuT
N0
⎢
e
⎢
⎢
⎣
j ˆ φu
Nu
k=1
k=u
m=1
(I m u )⋆ y m u
Ek
Eu ej( ˆ φk ˆ φu) N
+
m=1 n=
(Im u )⋆I n n
k
xm
u,k
= 0. (4.1)
The fact that this estimation process works in DA mode appears through
the dependency upon true data symbols I m u and not upon estimates Îm u .
Relation (4.1) can describe feedback as well as feedforward phase recovery
implementations. The former is built as a locked loop tracking the phase
⎥
⎦

76 Data-Aided
according to an error signal u m u,DA
∂ΛL(Φ)
∂φu
Φ= ˆ Φ
= 2EuT
N0
N
m=1
u m u,DA
=0 (4.2)
while the latter explicitly computes a closed-form estimate of the phase
parameter ˆ φu. Both implementations will be studied in the following sections.
4.1 Feedback
The subject of the present section is the Multiuser (MU) DA ML FB phase
estimator that can be derived from (4.1). This section will deal with feedback
structures working in tracking mode. In this mode, the recovery loop
is tracking the variations of the parameter, starting from a rough estimate
obtained during the acquisition mode. Provided a proper design of the
loop, the estimate in tracking loop exhibits small fluctuations around the
true value of the parameter.
The multiuser recovery loop is shown in Figure 4.1 for a 2-user case. Without
the signal flows between the two main branches, it would appear as
two Single-User (SU) recovery loops working in parallel. The exchange of
information between them turns the structure into an MU estimator. This
loop is driven by the error signal u m u,DA
u m ⎡
e
⎢
u,DA = ⎣
j ˆ φm u (Im u ) ⋆ ym u
Nu
Ek
Eu ej( ˆ φm k ˆ φm u ) +
n=
k=1
k=u
(Im u ) ⋆ In n
k
xm
u,k
⎤
⎥
⎦ . (4.3)
It results from two contributions. Besides the classic one, relying on matched
filter outputs ym u [7, 83], the reader notices a second term which depends
only on interfering users. This term comes out from the fact that the
log-likelihood function Λ(r Φ) takes into account the multiuser context of
the transmission. The derivation of a log-likelihood function missing this
aspect leads to the SU DA ML FB estimator, only depending on the term
involving the output ym u of the filter matched to the equivalent channel
model of user u.

(t)
h ⋆ u ( t)
h ⋆ v ( t)
y m u
e j ˆ φ m u
e j ˆ φ m v
y m v
(.) ⋆
NCO
NCO
e j ˆ φ m u y m u
e j ˆ φ m v y m v
u m u
u m v
+
-
(.) ⋆
Figure 4.1: 2-user DA phase recovery loop
-
+
(.) ⋆
(.) ⋆
(.) ⋆
I m u
x m u,v
I m v
4.1 Feedback 77

78 Data-Aided
The introduction of (3.7) into (4.3) gives a better insight into the workings
of the MU FB loop.
u m ⎡
⎢ e
⎢
u,DA = ⎢
⎣
j(φu ˆ φm u ) +
n=
+ Nu
e
k=1
k=u
j(φk ˆ φm u ) Ek
Eu
Nu
e
k=1
k=u
j( ˆ φm k ˆ φm u ) Ek
Eu
+ e j ˆ φm u Im u νm u,DA
(Im u )⋆I n u xm
n
u,u
n=
+
n=
+
(Im u ) ⋆ In n
k
xm
u,k
(Im u )⋆I n n
k
xm
u,k
⎤
⎥ . (4.4)
⎥
⎦
The second term in (4.4) is the MAI contribution which entered the loop
through the matched filter output (3.7). This contribution depends on the
difference between the phases of the interfering users φk and the current
phase estimate for the user of interest ˆ φm u , as also noticed in [96]. That interference
is counterbalanced by the third term, introducing a correction
derived from the log-likelihood function (3.31).
The performance of recovery loops driven by the error signal u m u,DA (4.4)
will be described in the next sections. The jitter variance will serve as performance
measure. Unlike what will be done for FF estimators in the next
section, the pdf of the FB phase estimation error has not been derived in
the present work. As mentioned in Section 2.3, such pdf has been derived
in a single-user context in [86, p. 90] and in a multiuser context in [96].
Notice, however, that the work presented in [96] is performed in the Bayes
approach, while the present thesis follows the Fisher approach.
4.1.1 Open-loop study
Operating point
As described in Section 2.3.2, the first step in the study of a recovery loop
in tracking mode is to determine the operating point of the loop, that is,
the position for which the error signal driving the loop will be null in the
mean. The expressions of the mean of the error signal to be established in
the following will serve for this purpose. They will also help to build a
linear version of the loop, which is more suited for closed-loop investigations.

4.1 Feedback 79
BPSK modulation The data symbols I p
k in (4.4) reduce to their real part.
then writes
U BPSK
u,DA
U BPSK
u,DA
= E u m u,DAˆΦ
=0, Φ=∆
= e j∆u
+
E a m u anu
ˆΦ =0, Φ=∆
⎡
⎢
+ ⎣
n=
Nu
Ek
k=1
k=u
⎡
⎢Nu
⎣
k=1
k=u
Eu
Ek
Eu
e j(δk,u+∆u) +
n=
e j(δk,u+∆u ∆k) +
n=
m n
xu,u
E a m u ank
ˆΦ =0, Φ=∆ x
m n
u,k
E a m u ank
ˆΦ =0, Φ=∆ x
⎤
⎥
⎦
(4.5)
m n
u,k
+E (a m u ν m u ) (4.6)
= K BPSK
D,u sin ∆u (4.7)
where ∆k = φk ˆ φk and δk,l = φk φl. K BPSK
D,u = σ 2 Iu x0 u,u is the phase
detector gain.
In (4.7), U BPSK
u,DA depends on ∆u through a sinusoidal function. This means
that driving the error signal of the loop to zero (U BPSK
u,DA =0) is equivalent
to have ∆u =0, which is an unbiased operating point. Also worth noticing
is the fact that the MAI contribution (second term) is cancelled by the
correcting term (third term) as soon as the phase error is recovered on the
interfering loops (∆k =0 k = u). Indeed, it is not surprising to find the
same final result as for SU estimators. In the DA context, the MAI can be
cancelled thanks to the perfect knowledge of interfering users’ messages.
Multiuser relations are thus equivalent to their single-user counterparts.
Finally, notice that DA estimation introduces no phase ambiguity. Thus
exhibits a 2π-periodicity [83, p. 204].
U BPSK
u,DA
U BP SK
u,DA
Drawing
KBP SK with respect to ∆u produces a S-curve of sinusoidal shape.
D,u
Around the operating point, this sinusoidal curve presents an interesting
linear area in that it produces a mean error signal Uu directly proportional
to the phase estimation error ∆u. On the other hand, it is well-known that
the slope of the S-curve at the operating point can be reduced due to de-
⎤
⎥
⎦

80 Data-Aided
cision errors [83, p. 207]. This will be illustrated in the next chapter, which
deals with DD structures. In the DA case, however, there is no decision
error. The slope is thus equal to 1.
QPSK modulation Moving to QPSK modulation, both real and imagin-
QP SK
ary parts of data symbols will be taken into account in (4.4). Uu,DA writes
QP SK
Uu,DA
= E u m
u,DAˆΦ =0, Φ=∆
⎧
⎨
=
⎩ ej∆u
⎡
+
⎣
n=
E
am u an
u ˆΦ =0, Φ=∆
+E bm u bnu ⎤ ⎫
⎬
⎦ m n
xu,u ˆΦ =0, Φ=∆ ⎭
⎧
⎨
+
⎩ ej∆u
⎡
+
⎣
n=
E
am u bnu
ˆΦ =0, Φ=∆
+E bm u an ⎤ ⎫
⎬
⎦ m n
xu,u u ˆΦ =0, Φ=∆ ⎭
⎧
⎫
⎪⎨
+
⎪⎩
⎧
⎪⎨
+
⎪⎩
⎧
⎪⎨
⎪⎩
⎧
⎪⎨
⎪⎩
Nu
k=1
k=u
n=
+
Nu
k=1
k=u
n=
+
Nu
k=1
k=u
n=
+
Nu
k=1
k=u
n=
+
Ek
Eu ej(δk,u+∆u)
⎡
⎣ E
am u an k
ˆΦ =0, Φ=∆
+E bm u bn k
ˆΦ =0, Φ=∆
Ek
Eu ej(δk,u+∆u)
⎡
⎣ E
am u bn k
ˆΦ =0, Φ=∆
+E bm u an k
ˆΦ =0, Φ=∆
Ek
Eu ej(δk,u+∆u ∆k)
⎡
⎣ E
am u an k
ˆΦ =0, Φ=∆
+E bm u bn k
ˆΦ =0, Φ=∆
Ek
Eu ej(δk,u+∆u ∆k)
⎡
⎣ E
am u bn k
ˆΦ =0, Φ=∆
+E bm u an k
ˆΦ =0, Φ=∆
⎤
⎦ m n
xu,k ⎤
⎦ m n
xu,k ⎤
⎦ m n
xu,k ⎤
⎦ m n
xu,k ⎪⎬
⎪⎭
⎫
⎪⎬
⎪⎭
⎫
⎪⎬
⎪⎭
⎫
⎪⎬
⎪⎭
(4.8)

4.1 Feedback 81
QP SK
where K
D,u = σ 2 Iu x0 u,u
+E (a m u νm u ) E (bmu νm u ) (4.9)
=
QP SK
KD,u sin ∆u (4.10)
. The operating point, the S-curve shape, and
periodicity are thus identical to those derived for BPSK-modulated symbols.
Loop noise
The derivation of the mean of the error signal u m u,DA
is a first step towards
the building of a linearised version of the loop. This linearised loop is used
to perform the closed-loop performance study in the presence of noise and
interference. The variance of the phase jitter is evaluated as the variance
of the loop noise filtered by the linearised loop [79, section 3.1]. The loop
noise embeds additive noise, self-noise (due to the random nature of the
signal ¢ signal terms related to a single user), and cross-noise (due to the
random nature of the signal ¢ signal terms related to a pair of interfering
users). Its characterisation is the subject of the following developments.
BPSK modulation Using (4.6) and (4.7), um u,DA can be split into its mean
value U BPSK
u,DA and the loop noise ξm u which is the sum of the additive noise
and the self- and cross-noise.
⎡
⎢ e
⎢
⎢
⎣
j(φu ˆ φm u ) +
n=
+ Nu
e
k=1
k=u
j(φk ˆ φm u ) + Ek
Eu
n=
Nu
e j( ˆ φm k ˆ φm u ) + Ek
Eu
k=1
k=u
Im u In u xm n
u,u
e j ˆ φm u I m u νm
u,DA
n=
Im u In n
k
xm
u,k
Im u In n
k
xm
u,k
⎤
⎥
⎦
K BPSK
D,u
(4.11)
sin φu ˆ φ m
u .
(4.12)
The psd of the loop noise Sξu (z,∆) is given by the z-transform of its auto-

82 Data-Aided
correlation function C m u,u (∆).
Sξu
(z,∆) =
m=
+
C m u,u (∆) z m . (4.13)
Both psd and auto-correlation function depend on the phase estimation
error ∆. However, in the following paragraphs, the developments will be
limited to the study at equilibrium (∆ =0). The auto-correlation function
of the loop noise is then obtained from the general expression given in
Appendix A by setting u = v, n =0and ∆=0
⎧ +
⎫
C m ⎪⎨
u,u (0) = δ(m)
⎪⎩
p=
[ (x p u,u)] 2
+ N0x0 u,u
2EuT
+ Nu Ek
Eu
k=1 p=
k=u
Nu Ek
Eu
k=1 p=
k=u
+
+
2 ejδk,ux p
u,k
2 ejδk,ux p
u,k
⎪⎬
⎪⎭
x m 2 u,u .
(4.14)
The third and fourth terms of (4.14) are identical, but their cancellation is
only obtained in the case of the MU estimator. The SU estimator does not
exhibit the fourth term. It can thus not compensate the effect of the MAI
present in the matched filter output, which gives rise to the third term of
(4.14).
Introducing (4.14) in(4.13) leads to the expression of Sξu (z,0). For an SU
estimator working in presence of MAI, this psd is made of three contributions:
additive noise, self-, and cross-noise. They are shown in Figure
4.2. The additive noise is the translation in the frequency domain of the
second term of (4.14). It mainly depends on the ratio Eu
. The combination
N0
of the first and fifth terms gives birth to the self-noise which is shaped by
the auto-correlation function of user u’s channel impulse response hu (t)
(xm u,u factors). Finally, the cross-noise comes from the third term, due to
MAI. The structure of this term is given by the cross-correlation function
between channel impulse responses of users u and k (xm u,k factors) with
k = u. Notice however that there is no cross-noise contribution to be
found in Sξu (z,0) in the case of an MU estimator, thanks to the cancellation
mentioned here above. From (4.14), it appears that the additive noise

4.1 Feedback 83
Power Spectral Density
10 −2
10 −3
10 −4
2−user system − 31−chip Gold codes − BPSK modulation − TU channel − E s /N 0 = 20 dB
Additive Noise
Self−Noise
Cross−Noise
10
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
−5
f
Figure 4.2: Power spectral density of Additive Noise, Self- and Cross-
Noise
and the cross-noise contribute to the auto-correlation function only at zero
time-shift (m =0). This leads to flat power spectral densities. On the other
hand, the self-noise contributes to the whole auto-correlation function, up
to the value of the normalised channel correlation coefficient x m u,u. In the
frequency domain, the spectrum of the self-noise vanishes at f =0.[84,
p. II-5] explains this high-pass behaviour as a result of the even symmetry
of the pulse at the matched filter output.
QPSK modulation Similarly, um QP SK
u,DA can be split into its expectation Uu,DA and the loop noise ξm u . As mentioned previously, the loop noise is at least
the sum of the additive noise and the self-noise, but it might also include

84 Data-Aided
a cross-noise contribution when dealing with an SU estimator facing MAI.
u m u,DA
QP SK
= K
D,u
⎪⎩
sin φu ˆ φm
u
+ (Im u )⋆ νm
u,DA
⎧ ⎡
⎢ e
⎢
⎪⎨ ⎢
⎢
+ ⎢
⎣
j(φu ˆ φm u ) +
p=
+ Nu
Ek
Eu
k=1
k=u
ej(φk ˆ φm u ) +
p=
Nu
Ek
Eu ej( ˆ φm k
K
k=1
k=u
QP SK
D,u
sin
φu ˆ φm
u
(Im u ) ⋆ I p m p
uxu,u ˆ φm u ) +
p=
(Im u )⋆ I p p
kxm u,k
(Im u ) ⋆ I p p
kxm u,k
⎤ ⎫
⎥ ⎪⎬ ⎥
⎦
⎪⎭
QP SK
Uu,DA Additive
noise
Self-
+ Cross-
Noise
(4.15)
Following the same procedure than with BPSK-modulated data symbols,
the auto-correlation of the loop noise at equilibrium is derived from (A.2).
Cm u,u (∆ = 0) writes
C m 1
u,u (0) =
2
⎧
⎡
⎢
⎪⎨
⎢
δ (m) ⎢
⎣
⎪⎩
+
p=
x p u,u 2
+ N0x0 u,u
EuT
+ Nu Ek
Eu
k=1 p=
k=u
Nu
k=1
k=u
Ek
Eu
+
+
p=
x p
x p
u,k
2
u,k
2
⎤
⎥
⎦
x m
u,u
2
⎫
⎪⎬
.
⎪⎭
(4.16)
As far as the third and fourth terms are concerned, the same remark as in
the previous paragraph can be made with respect to (4.16). The psd of the
loop-noise is finally obtained by introducing (4.16) into (4.13).
4.1.2 Closed-loop study
In the tracking mode, the estimate exhibits small fluctuations around the
true value of the parameter. In the linear portion of the S-curve, these

4.1 Feedback 85
small fluctuations translate into proportional fluctuations of the error signal.
This enables us to design a linearised model of the recovery loop at
equilibrium. Using this linear model, the jitter variance is obtained as the
variance of the loop noise filtered by the linearised closed-loop transfer
function.
Linear model of the recovery loop
In the general case, the error signal u m l
driving the loop is function of the
phase estimation error ∆ through its mean Ul and through the loop noise.
The working equation of the closed loop writes
ˆφ m+1
k = ˆ φ m k + K0,kFk(z) u m k ,k [1,Nu] (4.17)
where K0,k and Fk(z) represent respectively the gain of NCO and the filter
applied to the error signal u m k in the loop updating ˆ φk.
In the linearised closed-loop derived at equilibrium, Uk is replaced by the
linear term of its Taylor-series expansion. This introduces a linear dependency
of the error signal with respect to the phase estimation error whose
proportionality coefficient is the slope of the S-curve at equilibrium:
u m k = Uk + ξ m k = ∂Uk
∂∆
∆=0
∆+ξ m k ,k [1,Nu] . (4.18)
BPSK modulation U BPSK
u,DA is replaced by its linear expansion around the
operating point ∆=0
⎡
U
⎢
⎣
BPSK
1,DA
U BPSK
2,DA
...
U BPSK
⎤
⎥
⎦
Nu,DA
⎡
∂UBP SK
1,DA
⎢ ∂∆1
⎢ ∆=0
⎢ ∂UBP SK
⎢ 2,DA
= ⎢ ∂∆1
⎢
∆=0
⎢ ...
⎣ ∂UBP SK
∂UBP SK
1,DA
∂∆2
∆=0
∂UBP SK
2,DA
∂∆2
∆=0
...
∂UBP SK
...
...
...
...
∂UBP SK
1,DA
∂∆Nu
∆=0
∂UBP SK
2,DA
∂∆Nu
∆=0
...
∂UBP SK
Nu,DA
⎤
⎥ ⎡
⎥ ∆1 ⎥ ⎢ ∆2 ⎥ ⎢
⎥ ⎣
⎥ ...
⎥ ∆Nu ⎦
⎤
⎥
⎦ .
Nu,DA
∂∆1
∆=0
Nu,DA
∂∆2
∆=0
∂∆Nu
∆=0
(4.19)

86 Data-Aided
In (4.19), the Nu ¢ Nu square matrix of the first derivative of U BPSK
Nu,DA is
the Fisher information matrix, if not some multiplicative terms related to
Es and the loop bandwidth. From that point of view, the off-diagonal
N0
elements characterise the coupling due to the MAI. However, examination
of (4.7) reveals that they are null.
⎡
⎢
⎣
⎤
⎥
⎦ =
⎡
KD,1
⎢ 0
⎣ ...
...
KD,2
...
...
...
...
0
0
...
⎤ ⎡
⎥ ⎢
⎥ ⎢
⎦ ⎣
∆1
∆2
...
⎤
⎥
⎦ . (4.20)
0 0 ... KD,Nu ∆Nu
U BPSK
1,DA
U BPSK
2,DA
...
U BPSK
Nu,DA
There is thus no coupling between recovery processes of different users.
This is a benefit of the DA estimation process.
Using (4.20), (4.17) becomes a set of Nu equations of the type
ˆφ m+1
k = ˆ φ m k
+ KkFk(z)
φk ˆ φ m k
+ K0,kFk(z) ξ m k ,k [1,Nu] (4.21)
where Kk = KD,kK0,k is the loop gain. This is the equation of the loop
shown at Figure 4.3. It illustrates the decoupling between phase recovery
φ m u
φ m v
+
+
ˆφ m u
ˆφ m v
-
-
∆ m u
∆ m v
KD,u
0
K0,u (z 1) 1
KD,v
0
K0,v (z 1) 1
ξ m u
ξ m v
Figure 4.3: DA BPSK PLL
Fu (z)
Fv (z)
processes thanks to the knowledge of interfering users’ symbol sequences
(DA context).

4.1 Feedback 87
QP SK
QPSK modulation Since U
u,DA (4.10) is similar to U BPSK
u,DA (4.7), the linearised
model built in the case of QPSK-modulated data symbols does
not differ significantly from the one obtained with BPSK-modulated data
symbols.
Jitter variance
It was mentioned earlier that the jitter variance σ 2 ˆ φu
would serve as per-
formance measure. Equation (4.21) enables to derive it as the variance of
the loop noise ξm u filtered by the closed loop [85]
σ 2 ˆ φu =
T
2 Uu,DA∆=0 1
2T
1
2T
S ˆ φu
e 2jπfT
df (4.22)
where S ˆ φu (z,∆) is the spectral density of the filtered loop noise.
Ku
Sφu ˆ (z,∆) =
z
1 Fu,u(z)
1
Ku
z 1 Fu,u(z)
1 2
Sξu (z,∆) . (4.23)
Considering a narrow noise bandwidth BN,u and a first-order loop, the
phase jitter variance σ2 φu ˆ can be given by a Taylor-series expansion in the
variable 2 BN,uT of the general variance expression (4.22) at equilibrium
[84, p. II-4].
σ 2 ˆ φu
2 BN,uT
Sξu
∂Uu,DA 2
∂∆u ∆=0
(1, 0) 2
(2 BN,uT ) 2
+
2 ∂Uu,DA
m=
∂∆u ∆=0
m C m u,u (0) .
(4.24)
This expansion is limited to the second order. The quadratic term has to
be taken into account due to the shape of the self-noise spectrum. Since it
vanishes at f =0, it does not contribute to σ2 φu ˆ through the first term which
is linear in BN,uT . The self-noise contribution to the variance comes thus
from the term quadratic in σ2 φu ˆ . Figure 4.4 shows the jitter variance computed
with and without self-noise in the case of BPSK-modulated data
symbols. The importance of the quadratic term of (4.24) appears on the
variance of the MU estimator. Its incidence on the SU estimator is less visible,
the performance of this estimator being already degraded by MAI.

88 Data-Aided
Regardless of the context, notice that the variance of the estimate is lowerand
upper-bounded. The lower bound on the variance is the CRLB, introduced
in Section 3.3.1. Its upper-bound is the variance of an uniformlydistributed
random variable whose span would be the same than the parameter
under investigation, in the present case [0, 2π].
Variance [rad 2 ]
10 1
10 0
10 −1
10 −2
10 −3
10 −4
10 −5
2−user system − 31−chip Gold codes − BPSK modulation − TU channel
Single−user
Multiuser
Without Self−Noise
With Self−Noise
Uniform distribution
CRLB
10
0 5 10 15 20 25 30 35 40
−6
E /N [dB]
s 0
Figure 4.4: Incidence of the quadratic term of the Taylor-series expansion
at equilibrium of the variance expression
Using (4.24), the jitter variance can now be computed considering successively
BPSK- and QPSK-modulated data symbols. The computational
results will be presented in the following figures. Each of these will show
the jitter variance of a single phase estimate, namely the phase estimate of
user 1, exhibited in a different scenario by SU and MU estimators. Such
results have always been obtained by averaging the computations over
1,000 iterations, each one corresponding to a specific snapshot scenario so
as to get results that are independent of the parameters of the scenario
(code sequences, channel responses, etc.).
BPSK modulation Figure 4.5 illustrates the influence of the correlation
properties of the codes and of the load of the system in an AWGN channel,
that is to say, in a situation where the MAI is the only interference. With

4.1 Feedback 89
orthogonal Hadamard codes, there is no MAI. The variance of both SU and
MU estimators are thus equal. Moving to quasi-orthogonal Gold codes,
an irreducible variance floor appears on the SU curve. This floor rises
along with the load. A similar degradation has been shown in [46] which
established the performance of an SU ML chip synchroniser in DS-CDMA
communication systems. However, the MU estimator presented in the
current work goes a step further, in that it mitigates the effect of the MAI
so that the variance sticks to the CRLB even with quasi-orthogonal codes
or with a high system load.
Variance [rad 2 ]
10 1
10 0
10 −1
10 −2
10 −3
10 −4
10 −5
Hadamard
Gold
Single−user
Multiuser
N = 2
u
N = 20
u
BPSK modulation − AWGN channel − 2 B N T = 0.1
Uniform distribution
CRLB
10
0 5 10 15 20 25 30 35 40
−6
E /N [dB]
s 0
Figure 4.5: Variance of DA FB estimators in AWGN channel (BPSK)
Considering a 2-user system over an AWGN channel, the variance curves
of both SU and MU estimators are drawn in Figure 4.6 for different values
of the Near-Far ratio. The performance of the SU estimator being degraded
by the MAI, it is not surprising to see that the irreducible variance
floor rises as the Near-Far ratio grows. On the other hand, the MU estimator
benefits from the MAI mitigation. Its variance still reaches the CRLB,
whichever Near-Far ratios are considered.
Finally, the effect of the frequency selectivity of the channel is illustrated
in Figure 4.7. The variances have been computed for two different baud

90 Data-Aided
Variance [rad 2 ]
10 1
10 0
10 −1
10 −2
10 −3
10 −4
10 −5
2−user system − 31−chip Gold codes − BPSK modulation − AWGN channel − 2 B N T = 0.1
Single−user
Multiuser
Near−Far= 0 dB
Near−Far= 3 dB
Near−Far= 6 dB
Near−Far= 9 dB
Uniform distribution
CRLB
10
0 5 10 15 20 25 30 35 40
−6
E /N [dB]
s 0
Figure 4.6: Near-Far effect on DA FB estimators (BPSK)
rates in an Hilly Terrain (HT) channel 1 . Both SU and MU estimators suffer
from ISI. The variance of the MU estimator is lower than the one of the SU
because the latter also suffers from MAI. For both estimators, the lower
the baud rate is, the longer the symbol becomes. Thus the incidence of the
ISI is also lower. Indeed, neither the SU nor the MU estimator have been
designed to face ISI. While in detection studies ISI and MAI are dealt with
simultaneously by regarding MAI as a time-varying version of ISI [38], the
phase estimation structure handles them separately. Inspection of (3.31)
reveals that the ISI influence vanishes when taking the first derivative of
the log-likelihood function with respect to the phase parameter since it
does not depend on this parameter. The incidence of ISI on recovery loops
has been studied in [113, 114]. Estimation structures taking into account
ISI have been presented in [113, 115]. Without such design, computing the
phase jitter variance in dispersive environments makes it clear that, while
the SU estimator is the only one affected by MAI, both suffer from ISI so
that none of them ever reaches CRLB at high Es
N0 .
1 See Appendix G

4.1 Feedback 91
Variance [rad 2 ]
10 1
10 0
10 −1
10 −2
10 −3
10 −4
10 −5
2−user system − 31−chip Gold codes − BPSK modulation − HT channel − 2 B N T = 0.1
Single−user
Multiuser
R = 1e4 Bauds
R = 1e5 Bauds
Uniform distribution
CRLB
10
0 5 10 15 20 25 30 35 40
−6
E /N [dB]
s 0
Figure 4.7: Variance of DA FB estimators in dispersive channels (BSPK)
QPSK modulation Following the same procedure as in the BPSK case,
(4.16) can be used to derive first the psd of the loop noise (4.13) and then,
the jitter variance (4.24) of the phase recovery loop operating on QPSKmodulated
symbols.
The conclusions drawn in the previous paragraph with BPSK-modulated
data symbols are still valid using QPSK modulation. In an AWGN channel,
the MU estimator has a variance which reaches the CRLB, with orthogonal
as well as with quasi-orthogonal codes, irrespective of the load
enabled by the resolution of the code thanks to the MAI mitigation (Figure
4.8). On the other hand, the SU estimator exhibits an irreducible variance
floor as soon as the orthogonality of the codes is lost. The level of this
floor depends on the load of the system. Facing a Near-Far effect (Figure
4.9), the MU estimator appears to be Near-Far resistant, while the variance
floor limiting the performance of the SU estimator rises with the Near-Far
ratio. Finally, in dispersive channels, the fact that neither the SU nor the
MU estimators take into account the incidence of ISI leads to an irreducible
variance floor on both of them which depends on the level of ISI
degrading their performance.

92 Data-Aided
Variance [rad 2 ]
10 1
10 0
10 −1
10 −2
10 −3
10 −4
10 −5
Hadamard
Gold
Single−user
Multiuser
N u = 2
N u = 20
QPSK modulation − AWGN channel − 2 B N T = 0.1
Uniform distribution
CRLB
10
0 5 10 15 20 25 30 35 40
−6
E /N [dB]
s 0
Figure 4.8: Variance of DA FB estimators in AWGN channel (QPSK)
Variance [rad 2 ]
10 1
10 0
10 −1
10 −2
10 −3
10 −4
10 −5
2−user system − 31−chip Gold codes − QPSK modulation − AWGN channel − 2 B N T = 0.1
Single−user
Multiuser
Near−Far= 0 dB
Near−Far= 3 dB
Near−Far= 6 dB
Near−Far= 9 dB
Uniform distribution
CRLB
10
0 5 10 15 20 25 30 35 40
−6
E /N [dB]
s 0
Figure 4.9: Near-Far effect on DA FB estimators (QPSK)

4.1 Feedback 93
Variance [rad 2 ]
10 1
10 0
10 −1
10 −2
10 −3
10 −4
10 −5
2−user system − 31−chip Gold codes − QPSK modulation − HT channel − 2 B N T = 0.1
Single−user
Multiuser
R = 1e4 Bauds
R = 1e5 Bauds
Uniform distribution
CRLB
10
0 5 10 15 20 25 30 35 40
−6
E /N [dB]
s 0
Figure 4.10: Variance of DA FB estimators in dispersive channels (QSPK)

94 Data-Aided
4.2 Feedforward
Beside the FB estimator, solving (4.1) can also lead to the FF estimator
given by relation (3.52)
where
Cu =
m=1
N
(I m u )⋆ y m u
ˆφu = tan
Nu
Ek
k=1
k=u
1 (Cu)
(Cu) = arg (Cu) (4.25)
Eu
e j ˆ φk
N
+
m=1 n=
(I m u )⋆ I n n
k xm
u,k
(4.26)
as already mentioned in Section 3.4.1. This is the expression of the MU
DA ML FF phase estimator. Before studying the performance of its simplified
version (3.59) obtained by linearising the complex exponential, the
degradation of the performance of the SU estimator working in a multiuser
context will be established in the next section.
4.2.1 Pdf of an SU estimator in a multiuser context
The SU DA ML FF phase estimator is given by [7, p. 326]
N
ˆφu = arg (I m u )⋆ y m
u . (4.27)
m=1
It is the same kind of estimator (argument of a complex number) as the MU
one (4.26) but it lacks the MAI mitigation term. It should thus be stressed
that the SU estimator (4.27) is not the optimal one for either frequencyselective
and/or multiuser contexts due to the presence of interference,
ISI, and/or MAI. Indeed, this estimator has been derived from a log-likelihood
function which does not take into account any interference at all. As
a result, its performance will be degraded by interference. The purpose of
the following calculations is to quantify this degradation on the variance
of the estimation error ∆u = φu ˆ φu. The variance will be calculated from
the pdf of ∆u.
Analytical derivation of the pdf
To derive the pdf of the phase estimation error ∆u, the complex number
whose argument is ∆u is considered:

4.2 Feedforward 95
ˆrue j∆u
= ˆxu + j ˆyu (4.28)
=
N +
(Im u )⋆ In u xm
n
u,u
Ek
Eu
Useful term
+ ISI
ej(φk φu) N +
(Im u ) ⋆ In n
k
xm
u,k MAI
m=1 n=
+ Nu
k=1
k=u
+ e jφu
N
m=1
(I m u )⋆ ν m u,DA
m=1 n=
Additive noise
(4.29)
The characteristic function ψˆxu,ˆyu (ωr,ωi) can be calculated, so that its inverse
Fourier transform gives the joint pdf T ˆxu,ˆyu (ˆxu, ˆyu). Then, a change
of variable from cartesian (x, y) to polar (r, ∆) coordinates and an integration
of Tˆru,∆u (ˆru, ∆u) over the range of ru yields the pdf of the phase
estimation error ∆u. The calculation in the case of BPSK-modulated data
symbols, using a single-tap averaging window (N =1), is detailed in Appendix
B. The pdf finally writes (B.18)
T∆u(∆u)
1
=
2 (NuSx) π⎧
2 (NuSx 2)
k=1
⎪⎨
⎪⎩
exp
1
+
with Sx, cu, f ¦ u and g¦ u
Computational results
exp
1+
g u
4cu
π f u
cu 2 exp
2
(f u ) f u 1 erf
4cu
2 Ô
cu
g + u
4cu
π f
cu
+ u
2 exp
+ 2
(f u )
4cu
1 erf
defined in (B.5), (B.9) and (B.14-B.17).
f + u
2 Ô
cu
⎫
⎪⎬
⎪⎭
(4.30)
Relation (4.30) has been computed in different scenarii. They might appear
simplistic but this is to avoid the computational complexity problem
mentioned in Section B.2. For every scenario, the pdf has been averaged

96 Data-Aided
over 1,000 computations, each one being characterised by a specific random
choice of the users’ phases and codes, and of the channel impulse
responses. This strategy has been chosen in order to produce results not
sensitive to the choice of a set of simulation parameters. Such an averaging
operation is avoided in [96] at the price of the use of stochastic models for
the interference.
Figure 4.11 presents the pdf obtained in a 2-user system using 7-chip Gold
codes in channels whose delay profiles are given by COST 207 Rural Area
(RA) model 2 . Thanks to the DA structure, even an SU ML estimator remains
unbiased in interfering contexts. However, this interference is a
source of degradation. This appears when looking at the variance of the
estimator. Obviously, with respect to the dotted pdf which is obtained in
T ( Δ )
Δ1 1
1.5
1
0.5
2−user system − 7−chip Gold codes − R = 1e5 Bauds − N = 1
AWGN, N u = 1
RA, N u = 2
E s /N 0 = 0 dB
E s /N 0 = 4 dB
E s /N 0 = 8 dB
0
−4 −3 −2 −1 0
Δ [rad]
1
1 2 3 4
Figure 4.11: Pdf of the SU DA ML FF phase estimate in a 2-user, RA channel
context
an SU scenario with AWGN channel, the considered pdf exhibits a larger
variance as a result of two interfering effects. First, the use of quasiorthogonal
codes introduces MAI. Second, the fact that the channel is
frequency-selective causes ISI to also degrade the performance of the estimator.
2 See Appendix G

4.2 Feedforward 97
These influences are illustrated more clearly in Figure 4.12 which shows
the variance of an SU DA ML FF estimator as a function of Es in several
N0
contexts. Due to the heavy computations it would have led to, the variance
has not been computed from (4.30), but rather measured from the pdf derived
from this equation and illustrated in Figure 4.11. The shortcomings
of this method appear in Figure 4.12 at low Es
ratios, where the variance
N0
does not exactly match the CRLB. Nevertheless, the reader distinguishes
the incidence of MAI and of ISI on the variance curves obtained in 1- and
2-user systems considering either AWGN or HT channel.
Variance [rad 2 ]
10 1
10 0
10 −1
10 −2
10 −3
2−user system − 7−chip Gold codes − BPSK modulation − R = 1e5 Bauds − N = 1
AWGN
HT
N u = 1
N u = 2
Uniform distribution
CRLB
10
0 5 10 15 20 25 30
−4
E /N [dB]
s 0
Figure 4.12: Variances of the SU DA ML FF phase estimation error as a
function of the number of user Nu and of the channel type
In a situation where there is neither MAI (Nu =1) nor ISI (AWGN channel),
the variance of the pdf shown in Figure 4.12 equals the CRLB (but
at low Es
ratios, as explained here above). However, due to the interfe-
N0
rence, either MAI or ISI, a variance floor appears which is not depending
on Es
, causing the variance curve to rise away from the CRLB. The level of
N0
this floor depends on how much interference enters the system and thus
on the interfering conditions. The more users there are, the more MAI
plagues the system. The more dispersive the channel is, the more ISI there
is.
MAI
ISI

98 Data-Aided
Another aspect should be taken into account, namely the Near-Far effect.
The study of its influence on the variance of the DA ML FF phase estimator
is deferred to the next section, where the variance of the pdf computed in
the present section will serve as a benchmark for the variance derived from
the closed-form of the estimate.
4.2.2 Linearised multiuser estimator in 2-user system
The previous section was concerned with the derivation of the pdf of an
SU DA ML FF phase estimator in a multiuser context. It was shown that
the performance were not optimal due to the fact that the incidence of
interference was disregarded. The present section will take MAI into account.
In Section 3.4.1, a modified version of the ML FF phase estimator given
by equation (3.52) has been introduced after linearisation of the complex
exponentials. The motivation of this linearisation is to ease the following
derivation of a closed-form expression of the DA ML FF phase estimator
which is suitable for performance evaluation.
Closed-form expression of the phase estimation error
Multiuser case In order to evaluate the performance of the MU DA ML
FF phase estimator, the matched filter outputs y m u are expanded in the analytical
expression (3.59) of the derived estimator. Limiting the developments
to a 2-user case, Du becomes
Du
= ejφu ⎡
⎣ N
Im u 2 x0 u,u + N
m=1
m=1
Ev
+j ∆v Eu ejφv
N +
m=1 n=
+ N
m=1
(I m u ) ⋆ ν m u,DA
+
n=
n=m
(I m u ) ⋆ I n u x
(Im u ) ⋆ In v xm n
u,v
m n
u,u
⎤
⎦
Useful term
+ ISI
MAI
Additive noise
(4.31)
The MAI introduced by the matched filter output is partly cancelled by the
mitigation term. Only a small MAI contribution is left in (4.31), weighted

4.2 Feedforward 99
by j∆v. This weighting term reduces the influence of the MAI proportionally
to the reduction of the phase estimation error on user v. In the case
of an SU estimator there is no such weighting. In this case, the MAI fully
disturbs the estimation process.
Using (4.31) in(3.59) enables to solve the system for the phase estimation
error. Willing to reach a closed form solution, MAI and noise contributions
in the denominator of (3.59) are regarded as negligible with respect to the
direct (x0 u,u) and ISI (xm n
u,u ,m = n and n [ , + ]) terms. Furthermore,
noticing that direct terms are real, these contributions vanish from
the numerator of (3.59). The MU phase estimation error finally writes
∆u
(ISIu + Noiseu) (Directv + ISIv)
(MAIv,u) (ISIv + Noisev)
= (4.32)
(Directu + ISIu) (Directv + ISIv)
where
MAIu,v =
(MAIv,u) (MAIu,v)
Directu =
ISIu =
Ev
Eu
N
m=1
N
I m u 2 x 0 u,u
m=1
+
n=
n=m
j(φv φu)
e
Noiseu = e jφu
(I m u )⋆ I n n
u xm u,u
N
+
m=1 n=
N
m=1
(I m u )⋆ I n n
v xm u,v
(4.33)
(4.34)
(4.35)
(I m u )⋆ ν m u,DA . (4.36)
Single-user case In the same context with the same hypotheses, the SU
phase estimation error writes
∆u = (MAIu,v + ISIu + Noiseu)
. (4.37)
(Directu + ISIu)

100 Data-Aided
Mean
Comparing (4.32) and (4.37), it appears that in a noiseless (Noisek 0 k)
and non dispersive (ISIk 0 k) environment, that is to say in a situation
where MAI is the only interference, the numerator of (4.32) is driven
to zero. By itself, without any more processing, the estimation error of the
MU estimator is small. On the contrary, the corresponding performance of
the SU estimator is plagued by an MAI contribution which requires filtering
(averaging) to disappear. However, if the channel becomes dispersive
(ISIk = 0 k), both estimators exhibit a sensitivity to the ISI contribution.
Nevertheless, it is clear from (4.34) that its incidence can be reduced
by appropriate filtering (enlarging the width N of the averaging window).
From a statistical point of view, an approximation of the expectations of
these quotients can be performed if the denominator is substituted by its
mathematical expectation, as suggested in Section 3.4.2. Then, as already
shown in Figure 4.11, both SU and MU estimators are unbiased since the
expectation operator set their numerators equal to zero thanks to the independence
between data symbols and noise, and between data symbols
from different users (MAI issue) or from the same user but taken at different
time instants (ISI issue).
Variance
The expressions of the variance of several DA ML FF estimators are given
in Appendix C, depending on the structure of the estimator (MU vs SU)
and on the modulation of the data symbols (BPSK vs QPSK).
The reader can notice that all variances present a contribution linear in Es
N0
and a contribution independent of this ratio. The former comes from signal
¢ noise terms while the latter is produced by signal ¢ signal terms [84,
p. II-3]. However, the origin of the latter differs according to the type of
estimator.
In the case of MU estimators, variance expressions (C.4 and C.8) involve
a term linear in Es
, which is not surprising for DA estimators relying on
N0
a perfect knowledge of the channel behaviour [105, 106]. Beside this term
comes a contribution not depending on the Es ratio. Its presence is due to
N0
the dispersiveness of the channel since it reflects the incidence of ISI. Interestingly,
its structure, detailed in (C.6) for the BPSK case, shows that the

4.2 Feedforward 101
variance contribution due to ISI is a function of the mismatching between
the total spread of coefficients xn m
u,u and the width N of the averaging
window:
⎛
N
BPSK 2 ⎜
σISIu = f ⎝
+
n m
x
u,u
2
N N
n m
x
u,u
2
⎞
⎟
⎠ (4.38)
m=1
n=
n=m
m=1
n=1
n=m
When this window becomes wider, the second term of (4.38) mitigates the
first one. The ISI contribution to the variance then reduces, asymptotically
becoming null when N + . It will be shown in the following computational
results that the ISI contribution leads to an irreducible variance
floor plaguing MU estimators in the case of dispersive channels.
On the other hand, beside the contribution linearly dependent on the Es
N0
ratio, variances of SU estimators (C.12 and C.13) also exhibit an irreducible
variance floor. However, contrary to the case of MU estimators, it results
not only from ISI but also from MAI. Analytically, the incidence of the
MAI appears through the third term of the SU variance expressions (C.12)
and (C.13). Graphically, its effect is illustrated in Figure 4.13. Since these
curves are obtained in an AWGN channel, there is no ISI. As a result, the
variance floor, which obviously only appears on the variance curves of the
SU estimator, is due to MAI. Figure 4.13 shows that the variance floor is
reduced by improving the orthogonality properties of the codes, moving
from 7-chip to 31-chip quasi-orthogonal Gold codes, then to 8-chip orthogonal
Hadamard codes. Indeed, choosing better codes reduces the MAI,
and thus leads to lower variance floors. Of course, another way to reduce
the variance is to enlarge the span N of the averaging window, as illustrated
in Figure 4.13 where a window of size N =10is used instead of
one of size N =1.
A perfect power control scenario has been assumed so far. Considering
now power imbalance between users, the behaviour of the estimators in
the presence of a Near-Far effect is clear from both expressions detailed in
Appendix C and from Figure 4.14. In a non-dispersive environment, relations
(C.4) and (C.8) are not explicitly dependent of power ratios between
users, which indicates Near-Far resistance. Figure 4.14 confirms this statement:
the variance curves of the MU estimator are superimposed, showing
no sensitivity to the Near-Far ratio. On the other hand, the SU estimator
appears to be sensitive to the level of power imbalance, as its variance

102 Data-Aided
Variance [rad 2 ]
10 1
10 0
10 −1
10 −2
10 −3
10 −4
2−user system − BPSK modulation − R = 1e5 Bauds − AWGN channel
Single−user
Multiuser
N = 1
N = 10
8−chip Hadamard
7−chip Gold
31−chip Gold
MAI reduction
Uniform distribution
MAI reduction
CRLB, N=1
CRLB, N=10
10
0 5 10 15 20 25 30
−5
E /N [dB]
s 0
Figure 4.13: Variance of DA FF estimators in an AWGN channel
floor rises when the Near-Far ratio increases.
Figure 4.14 is also an opportunity to check the match between the variance
of the SU estimator calculated from the pdf of Section 4.2.1 and
the expressions shown in Appendix C. There is a close match in a perfect
power control scenario (Near-Far ratio = 0 dB). When the Near-Far ratio
increases, the variance derived from linearisation slightly underestimates
the true variance given by the pdf. This underestimation might be due to
the simplification performed in order to reach closed-form expressions of
the estimation error (MAI neglected in front of direct contributions in the
denominator of (3.59)).
Finally, the incidence of dispersive channels is illustrated in Figure 4.15. It
shows the results obtained in an ideal AWGN channel and in a frequencyselective
HT channel. As far as the MU estimator is concerned, the reader
notices the expected variance floor due to ISI. Considering now the SU
estimator, it appears that its variance floor encompasses both MAI and ISI
effects. Indeed, the variance floor already plagues the SU estimator in the
AWGN channel, due to MAI. In the case of the dispersive channel, ISI adds
upon MAI and raises the variance floor. Finally, a slight underestimation

4.3 Feedback-Feedforward correspondence 103
Variance [rad 2 ]
10 1
10 0
10 −1
10 −2
10 −3
2−user system − 7−chip Gold codes − BPSK modulation − R = 1e4 Bauds − AWGN channel − N = 1
pdf
FF
Single−user
Multiuser
Near−Far = 0 dB
Near−Far = 3 dB
Near−Far = 6 dB
Near−Far = 9 dB
Uniform distribution
CRLB
10
0 5 10 15 20 25 30
−4
E /N [dB]
s 0
Figure 4.14: Near-Far effect on DA FF estimators
of the variance is noticed again comparing lienarised and true curves.
Similar curves can be drawn in the case of QPSK-modulated data symbols.
Since they do not bring significant new conclusions, and since it would not
have been possible to check them with results from Section 4.2.1 for the
pdf has been derived in the case of BPSK-modulated data symbols, they
are not shown here.
4.3 Feedback-Feedforward correspondence
Having derived analytical expressions for the variance of several DA estimators,
it is worthwhile to check the performance correspondence between
FB and FF implementations. It is well known that they are related
in as much as their bandwidth correspond [85, p. 349]. This is illustrated
by their respective CRLB expressions (3.50) and (3.48) from which the correspondence
condition is derived :
CRLBu = BN,uT
1
Es,u
N0
= 1
2N
1
Es,u
N0
2 BN,uT = 1
. (4.39)
N

104 Data-Aided
Variance [rad 2 ]
10 1
10 0
10 −1
10 −2
10 −3
2−user system − 7−chip Gold codes − R = 1e5 Bauds − BPSK modulation − N = 1
pdf
FF
Single−user
Multiuser
AWGN
HT
Uniform distribution
CRLB
10
0 5 10 15 20 25 30
−4
E /N [dB]
s 0
Figure 4.15: Incidence of ISI on DA FF estimators
The correspondence can be checked between the pairs of FB/FF estimators
derived in this chapter, either SU or MU, for either BPSK- or QPSKmodulated
data symbols. This study was performed under the hypothesis
of small loop bandwidth (2 BN,uT 0, N + ).
In the case of FB estimators, (4.24) then reduces to
where
∂Uu,DA 2
∂∆u ∆=0
σ 2 ˆ φu
2 BN,uT
Sξu
∂Uu,DA 2
∂∆u ∆=0
(1, 0) (4.40)
= KD,u = σ2 Iux0u,u and Sξu (1, 0) is given by (4.13)
Sξu
(1, 0) =
m=
+
C m u,u (0) (4.41)
with C m u,u (0) written as (4.14) or(4.16) depending on the modulation.
Thanks to the infinite sum in (4.41) the ISI contribution from (4.14) and
(4.16) vanishes. Only noise and MAI terms are thus left in the simplified
expressions listed in Table 4.1. These expressions are derived assuming a
2-user case in order to enable comparison with the relations obtained in

4.4 Conclusions 105
Section 4.2 for FB estimators in such a scenario.
Looking now at FF variance expressions in the case of small loop bandwidths
(N + ), ISI contributions asymptotically disappear in relations
presented in Appendix C, as explained in Section 4.2.2. Among the
remaining terms, only the highest powers of N will be considered. Using
the fact that
n m
x
u,v
2
= e jδv,u
n m
xu,v 2
= e jδv,u 2
n m
xu,v + e jδv,u 2 n m
xu,v = e jδv,u
2
n m
x +2 e jδv,u 2 n m
x (4.42)
u,v
simplified expressions are obtained. They are gathered in Table 4.1. Applying
the bandwidth equivalence expression (4.39), there is indeed correspondence
between FB and FF relations. Moreover, the reader can verify
that, in the absence of MAI, SU estimators exhibit a MAI contribution to
their variance, while the variance of MU estimators only depend on the
Es
N0 ratio.
A graphical validation of the correspondence between FB and FF estimators
is provided in Figure 4.16. It is a reminder of Figure 4.14, completed
with the variances of FB estimators operating in the same scenario. Figure
4.16 shows a close matching between curves of FB and FF estimators.
4.4 Conclusions
Two different implementations of DA estimators, namely FB and FF, have
been considered in this chapter. Their performance in terms of jitter variance
have been derived analytically for MU as well as for SU estimators
and computed in several scenarii. These results have been cross-checked
by comparing asymptotical FB and FF variance expressions. The latter
expression has also been compared to the variance computed using the
analytical expression of the pdf of the SU estimate.
In FB implementations, the main advantage of DA estimation has appeared
to be the decoupling between recovery loops. From the point of
view of the jitter variance, conclusions were the same for FB as well as for
u,v

106 Data-Aided
BPSK SU FB (4.24) with C m u,u (0)
given by (4.14)
Original expression Asymptotical expression
FF (C.12) 1
2 N
MU FB (4.24) with C m u,u (0)
given by (4.14)
FF (C.4) 1
2 N
QPSK SU FB (4.24) with C m u,u (0)
given by (4.16)
FF (C.13) 1
2 N
MU FB (4.24) with C m u,u (0)
given by (4.16)
FF (C.8) 1
2N
1
Es,u
BN,uT
N0
1
Es,u
N0
1
Es,u
BN,uT
N0
1
Es,u
N0
1
Es,u
BN,uT
N0
1
Es,u
N0
1
Es,u
BN,uT
N0
1
Es,u
N0
+ 2 BN,uT
(x0 u,u) 2
Ev
Eu
+
+
p=
+
1
N(x0 u,u) 2
Ev
Eu
p=
+ BN,uT
(x0 u,u) 2
Ev
Eu
+
p=
+
+
1
2 N(x0 u,u) 2
Ev
Eu
p=
jδv,u p 2 e xu,v
jδv,u p 2 e xu,v x p u,v 2
x p u,v 2
Table 4.1: Asymptotical variance expressions of DA estimators in a 2-user case

4.4 Conclusions 107
Variance [rad 2 ]
10 1
2−user system − 7−chip Gold codes − BPSK modulation − R = 1e4 Bauds − AWGN channel − 2 B T = N = 1
N
10 0
10 −1
10 −2
10 −3
FB
FF
pdf
Single−user
Multiuser
Near−Far = 0 dB
Near−Far = 3 dB
Near−Far = 6 dB
Near−Far = 9 dB
Uniform distribution
CRLB
10
0 5 10 15 20 25 30
−4
E /N [dB]
s 0
Figure 4.16: Correspondence between DA FB and FF estimators
FF estimators: the MU estimator is not affected by MAI while its SU counterpart
exhibits a variance floor depending on the level of MAI entering
the system. However, a similar variance floor limits the performance of
both estimators in dispersive environments as a result of ISI.
The estimators studied in this chapter rely on the knowledge of the transmitted
symbols. The next chapter will deal with structures using the fedback
decisions instead of the true symbols.

Chapter 5
Decision Directed
Similarly to what was done in the previous chapter, the first derivative of
the log-likelihood function (3.31) with respect to the phase parameter is
used to derive the ML estimator. In the DD context, it writes
∂ΛL(Φ)
∂φu
Φ= Φˆ
= 2EuT
⎡
⎢
e
⎢
⎢
N0 ⎣
j ˆ N ⋆ φu Îm u y
m=1
m u
Nu
Ek
Eu ej( ˆ φk ˆ φu) N
⎤
⎥
+
⎥
⋆Î ⎥
Îm n m n
u k x ⎦
u,k
.
k=1
k=u
m=1 n=
(5.1)
The main difference between (5.1) and its DA counterpart (4.1) is the use of
Îm u instead of Im u . While, in DA structures, the data information used in the
estimation process are obtained through the transmission of predefined
training sequences, DD phase estimators get this information from the decision
stage of the receiver. Thus, not only do DD estimators, like DA
ones, exhibit coupling between users in that the estimation of the phase
parameter φu of user u also depends on the estimation of φk, k = u, but
the use of decisions within the parameter estimation process introduces
another kind of coupling: between detection and estimation stages.
Again, setting (5.1) equal to zero is a necessary but not sufficient condition
to derive the ML estimate. It can give birth to two different kinds of estimators.
On the one hand, (5.1) can be used as error signal um u,DD driving a

110 Decision Directed
phase recovery loop
∂ΛL(Φ)
∂φu
Φ= ˆ Φ
= 2EuT
N0
N
m=1
u m u,DD
=0. (5.2)
On the other hand, solving (5.1) for Φ produces a DD ML FF phase estimator.
Both implementations, FB and FF, will be dealt with in the following
sections.
5.1 Feedback
The DD recovery loop, shown in Figure 5.1 in a 2-user case, is driven by
the error signal u m u,DD
u m u,DD
⎡
⎢
= ⎢
⎣
e j ˆ φ m u
Nu
k=1
k=u
Î m u
⋆
y m u
Ek
Eu ej( ˆ φ m k ˆ φ m l ) +
n=
Î m u
As mentioned earlier, the use of the decisions Îm u
⋆Î n m n
k xu,k ⎤
⎥
⎦ . (5.3)
to provide the information
requested by the estimation process introduces a coupling between
estimation and decision stages which does not appear in the DA phase recovery
loop since this one can rely on predefined training sequences.
Expanding the matched filter output y m u in (5.3), the error signal derived
from the ML phase estimation of user u writes
u m ⎡
⎢ e
⎢
u,DD = ⎢
⎣
j(φu ˆ φm u ) + ⋆ Îm u I
n=
n u
+ Nu
e
k=1
k=u
j(φv ˆ φm u ) + Ev
Eu
n=
Nu
e
k=1
k=u
j( ˆ φm v ˆ φm u ) + Ev
Eu
n=
+ e j ˆ φm ⋆ u Îm u νm u
xm n
u,u
Î m u
Î m u
⋆
I n v
xm n
u,v
⋆Î n
v xm n
u,v
⎤
⎥ . (5.4)
⎥
⎦
The contributions appearing in (5.4) have the same significance as in the
DA case (4.4). As far as the MAI is concerned, the matched filter output
introduces the interference (second term) and the MU estimator tries to

(t)
h ⋆ u ( t)
h ⋆ v ( t)
y m u
e j ˆ φ m u
e j ˆ φ m v
y m v
(.) ⋆
NCO
NCO
e j ˆ φ m u y m u
e j ˆ φ m v y m v
u m u
u m v
+
-
(.) ⋆
Figure 5.1: 2-user DD phase recovery loop
-
+
(.) ⋆
(.) ⋆
(.) ⋆
Î m u
Î m v
x m u,v
5.1 Feedback 111

112 Decision Directed
mitigate it (third term). However, DD estimators exhibit two restrictions
with respect to their DA counterparts.
First of all, while in DA structures the mitigation term reduces the influence
of the MAI as soon as the phase estimation error related to the
interfering users is small, the success of the mitigation in DD structures
requests also that the detection stage provides correct decisions. Indeed,
the MAI disturbance term and the MAI mitigation term in (4.4) only differ
in their phases since they both rely on predefined training sequences.
This is no longer the case with DD estimators (5.4). In such structures, the
mitigation term cancels the interference provided that two conditions are
fulfilled, namely that the phase estimation error is small and that the decisions
are correct.
The second difference between DA and DD implementations regards the
aforementioned decisions. Using training sequences, DA estimators can
exploit the entire transmitted sequence in order to perform parameter estimation.
On the other hand, DD estimators rely on decisions provided by
the detector. This imposes on them a causal working mode in which the
interference related to undetected symbols cannot be mitigated. As a result,
the MAI mitigation term in (5.4) can be built including, at most, past
and present detected symbols.
In the following paragraphs, the study of DD ML FB estimators will be
split into two main parts. The first part will assume that the decisions are
correct up to the present time. Indeed, it will lead to reinterpret relations
presented in the previous chapter for DA estimators in the light of causality.
On the other hand, the second part will make no assumption regarding
the quality of the decisions, thus including possible faulty outcomes.
A new and original open-loop study will be performed and some aspects
related to the closed-loop performance study will also be introduced. Notice
that, for the ease of treatment, the restriction of causality mentioned
here above will be relaxed in the second part.
5.1.1 Decisions assumed correct
Firstly decisions are assumed to be correct. In this respect, the results
presented in the previous chapter in the case of DA estimators can be used
here to illustrate the performance of DD structures, provided that the estimators
are made causal. This treatment is to be presented hereafter for

5.1 Feedback 113
BPSK- and QPSK-modulated data symbols.
BPSK modulation
Considering first BPSK modulation, the developments return at relation
(4.14), which gives the auto-correlation function of the loop noise at equilibrium.
Limiting the MAI mitigation term to causal contributions, (4.14)
becomes
⎧ +
⎫
C m ⎪⎨
u,u (0) = δ(m)
⎪⎩
p=
[ (x p u,u)] 2
+ N0x0 u,u
2EuT
+ Nu Ek
Eu
k=1 p=
k=u
Nu Ek
Eu
k=1 p=
k=u
+
0
2 ejδk,ux p
u,k
2 ejδk,ux p
u,k
⎪⎬
⎪⎭
x m 2 u,u .
(5.5)
It has been stressed in Chapter 4 dealing with DA estimators that the advantage
of MU estimators with respect to SU ones lies in the mitigation
of the MAI entering the system (third term of (5.5)) by the last term of the
relation. This advantage is partly lost in DD estimators, in as much as only
the causal part of the MAI (p 0) is mitigated. Thus, the DD estimator is
plagued by the anti-causal part of the MAI .
As a result, the variance of the DD estimator is greater than, or at best
equal to the one of the DA estimator. Indeed, under some conditions (nondispersive
channels for instance), the MAI incidence is condensed in the
x 0 u,v
coefficient. Then, if this one is involved in the mitigation term for
DD estimators as it is for DA ones, both estimators exhibit the same variance.
On the other hand, strictly limiting the mitigation term to p

114 Decision Directed
Variance [rad 2 ]
10 1
2−user system − 31−chip Gold codes − BPSK modulation − R = 1e4 Bauds − HT channel − 2 B T = 0.1
N
10 0
10 −1
10 −2
10 −3
10 −4
10 −5
DA
DD with
DD without
Single−user
Multiuser
Uniform distribution
CRLB
10
0 5 10 15 20 25 30 35 40
−6
E /N [dB]
s 0
Figure 5.2: Variance of DD ML FB estimators in ISI-free scenario (BPSK)
Nevertheless, in dispersive environments, the MAI incidence is spread
over a span of coefficients x m u,v, m =0, 1,.... Those among them which
contribute to the anti-causal part of the interference provoke an increase
of the variance. This is illustrated in Figure 5.3, where the variances of
DA and DD estimators are compared in a dispersive HT channel. Again,
curves are shown for two DD estimators, one including the x 0 u,v contribution,
the other not. The variance floor that limits the performance of the
DA estimator as a result of ISI stands below the variance floor related to
the DD estimator. The difference between them is due to the imperfect
MAI mitigation. However, the reader can notice that despite missing the
MAI due to the present symbol the second MU DD estimator (curve ”DD
without”) still performs better than the SU one thanks to the mitigation of
MAI due to past symbols.
QPSK modulation
Moving to QPSK modulation, the auto-correlation function of the loop
noise at equilibrium is given by (4.16). Again, limiting its MAI mitigation

5.1 Feedback 115
Variance [rad 2 ]
10 1
2−user system − 31−chip Gold codes − BPSK modulation − R = 1e5 Bauds − HT channel − 2 B T = 0.1
N
10 0
10 −1
10 −2
10 −3
10 −4
10 −5
DA
DD with
DD without
Single−user
Multiuser
Uniform distribution
CRLB
10
0 5 10 15 20 25 30 35 40
−6
E /N [dB]
s 0
Figure 5.3: Variance of DD ML FB estimators in presence of ISI (BPSK)
term to causal contributions, it turns into
C m 1
u,u (0) =
2
⎧
⎡
⎢
⎪⎨
⎢
δ (m) ⎢
⎣
⎪⎩
+
p=
x p u,u 2
+ N0x0 u,u
EuT
+ Nu Ek
Eu
k=1 p=
k=u
Nu Ek
Eu
k=1 p=
k=u
+
0
x p
x p
u,k
2
u,k
2
⎤
⎥
⎦
⎫
m
x
u,u
2
⎪⎬
.
⎪⎭
(5.6)
The same conclusions apply in the QPSK case as in the BPSK one: due to
partial MAI mitigation, the variance of the DD estimator is greater than
the DA one. In an ISI-free scenario (Figure 5.4), it can lead to an MU DD
estimator exhibiting the same variance than its SU counterpart if the x 0 u,v
contribution is not taken into account (curve ”DD without”). However, the
performance of the DD estimator is better than the SU one in dispersive
channels (Figure 5.5) when MAI coming from past symbols is cancelled.

116 Decision Directed
Variance [rad 2 ]
10 1
2−user system − 31−chip Gold codes − QPSK modulation − R = 1e4 Bauds − HT channel − 2 B T = 0.1
N
10 0
10 −1
10 −2
10 −3
10 −4
10 −5
DA
DD with
DD without
Single−user
Multiuser
Uniform distribution
CRLB
10
0 5 10 15 20 25 30 35 40
−6
E /N [dB]
s 0
Figure 5.4: Variance of DD ML FB estimators in ISI-free scenario (QPSK)
In the following paragraphs of this section dedicated to the FB implementation
of an MU DD ML phase estimator, no assumption is to be made regarding
the correctness of the decisions used in the detection process. As
a result, new and original relations will be derived in which the incidence
of detection errors will appear. Another difference with the current paragraph
is that the causal restriction will be lifted.
5.1.2 Actual decisions - Open-loop study
Similarly to the treatment presented in Chapter 4, the study of the recovery
loop splits into open-loop and closed-loop studies. The present paragraph
presents the first one.
Direct-space - Brute-force development in a simplified context
The analytical expression of Uu,DD, the mean of the error signal u m u,DD ,is
to be derived as a function of the phase estimation error ∆. This expression
illustrates the working of the multiuser phase estimator through the
drawing of S-hypersurfaces. S-hypersurfaces are multi-dimensional extensions
of S-curves. This multi-dimensional aspect comes from the fact
that Uu,DD depends not only on the phase estimation error of user u but

5.1 Feedback 117
Variance [rad 2 ]
10 1
2−user system − 31−chip Gold codes − QPSK modulation − R = 1e5 Bauds − HT channel − 2 B T = 0.1
N
10 0
10 −1
10 −2
10 −3
10 −4
10 −5
DA
DD with
DD without
Single−user
Multiuser
Uniform distribution
CRLB
10
0 5 10 15 20 25 30 35 40
−6
E /N [dB]
s 0
Figure 5.5: Variance of DD ML FB estimators in presence of ISI (QPSK)
also on the estimation errors related to the interfering users.
For reasons which will be explained later, the study will be limited to a
simplified context, namely a 2-user non frequency-selective synchronous
system. In this case, the only interference to be considered is the MAI
due to the second user. Starting from (5.4), these simplifications will be
introduced progressively along with the following developments. In this
2-user system, S-hypersurfaces will degenerate into S-surfaces.
BPSK modulation Considering BPSK-modulated data symbols, the mathematical
expectation of (5.4) in open-loop conditions assuming Φ is given
by
U BPSK
u,DD
(∆) = E u m
ˆΦ u,DD =0, Φ=∆
(5.7)

118 Decision Directed
U BPSK
u,DD (∆)
= e j∆u
+
+
n=
Ev
Eu
Ev
Eu
E â m u anu
ˆΦ =0, Φ=∆
e j(δv,u+∆u)
n=
+
j(δv,u+∆u ∆v)
e
n=
m n
xu,u
E â m u a n
v ˆΦ =0, Φ=∆
+
m n
xu,v
E â m u â n
v ˆΦ =0, Φ=∆
m n
xu,v +E (â m u ν m u ) (5.8)
since the data symbols I p
k and decisions Îp
k are real-only (BPSK modulation).
Due to the signal constellation symmetry, the additive noise contribution
to (5.8) disappears [87]. As a result, the average error signal in the
multiuser context is made of three main contributions, the first two from
the expansion of the matched filter output, embedding a useful contribution
(âm u am u ), the ISI (âm u an u,m = n), and the MAI (âm u an v ), and a last one
being the MAI mitigation introduced by the multiuser estimation process
(âm u ânv ).
Detailed expressions of the first order statistics used in (5.8) are presented
in Appendix D, considering synchronous transmissions over a non dispersive
channel. These are the result of a study called ”brute-force”, in
the sense that it has been performed in the direct space by averaging the
performance over all possible realisations of the data symbols regarded as
random variables. Such exhaustive treatment explains the applied simplifications
(2-user non frequency-selective synchronous system). Without
them, the analytical study would have been unrealistic, at least in the direct
space, due to the exponential complexity of the computations in the
number of users and in the delay spread.

5.1 Feedback 119
QPSK modulation With information spread on I- and Q-branches, the
mean of (5.4) expands into
QP SK
Uu,DD (∆)
= E
⎧
⎨
=
u m u,DD
⎩ ej∆u
⎧
⎨
+
⎪⎨
+
⎪⎩
⎧
⎪⎨
+
⎪⎩
⎧
⎩ ej∆u
n=
⎧
Ev
Eu ej(δv,u+∆u)
⎪⎨
⎪⎩
⎧
⎪⎨
⎪⎩
ˆΦ =0, Φ=∆
⎡
+
⎣
n=
(5.9)
E
âm u anu
ˆΦ =0, Φ=∆
+E ˆb m
u bn
u
ˆΦ=0,
⎤ ⎫
⎬
⎦ m n
xu,u Φ=∆ ⎭
⎡
+
⎣ E
âm u bn
u ˆΦ =0, Φ=∆
+E ˆb m
u an
u
ˆΦ=0,
⎤ ⎫
⎬
⎦ m n
xu,u Φ=∆ ⎭
⎡
+
⎣ E
âm u an
v ˆΦ =0, Φ=∆
+E ˆb m
u bn
v ˆΦ=0,
⎫
⎤ ⎪⎬
⎦ xm n
u,v ⎪⎭
Φ=∆
⎡
+
⎣ E
âm u bnv
ˆΦ =0, Φ=∆
+E ˆb m
u an
v ˆΦ=0,
⎫
⎤ ⎪⎬
⎦ xm n
u,v ⎪⎭
Φ=∆
+
⎣ E
+E ˆb mˆ u bn v ˆΦ=0,
⎫
⎤ ⎪⎬
⎦ xm n
u,v ⎪⎭
Φ=∆
⎡
+
⎣ E
âm u ˆb n
v ˆΦ=0,
Φ=∆
+E ˆb m
u ân
v ˆΦ=0,
⎫
⎤ ⎪⎬
⎦ xm n
u,v ⎪⎭
Φ=∆
) E ˆb m
u ν m
u . (5.10)
n=
Ev
Eu ej(δv,u+∆u)
n=
Ev
∆v)
ej(δv,u+∆u
Eu ⎡
âm u ânv ˆΦ =0, Φ=∆
n=
Ev
∆v)
ej(δv,u+∆u
Eu
n=
+E (â m u νm u
As with BPSK-modulated data symbols, the noise contribution in (5.10)
QP SK
vanishes thanks to the constellation symmetry. Uu,DD is then the result
of three contributions which involve signals on the Q-branch and mixed
product of signals from both I- and Q-branches. The first-order statistics
used in (5.10) are also detailed in Appendix D.

120 Decision Directed
Computational results Introducing results of Appendix D into (5.8) and
(5.10), the S-surfaces have been drawn in three different scenarii, differing
from each other by the level of global coupling (cross-correlation value +
Near-Far ratio) between users.
In an uncoupled context (xv,u =0), S-surfaces degenerate into S-curves.
Figure 5.6 compares the S-curves representing the mean Uu,DD of the error
signal u m u,DD with respect to the phase estimation error ∆u, paramet-
rised on the modulation (BPSK or QPSK) and on the Es
N0
ratio in such an
uncoupled context. Several remarks can be made. Firstly, these curves present
a 2π
M -periodicity due to the phase ambiguity inherent to the decision
process [83, p. 206]. Indeed, without any side information, the receiver
makes ambiguous decisions up to a shift of a multiple of 2π
M . Secondly,
the slopes of both S-curves rise up to 1 with Es
[84, p. II-16]. In the un-
N0
coupled situation, decision errors are only due to the noisy environment.
The higher the Es
ratio is, the less numerous decision errors are, and the
N0
closer the S-curve becomes to its DA counterpart which was shown to exhibit
a unit slope. Moreover, the vulnerability of QPSK to decision errors
due to additive noise with respect to BPSK increasingly turns into a lower
value of the slope at the same Es ratio. However, both BPSK and QPSK
N0
S-curves converge to the unit slope.
Sticking to the uncoupled context, a broader view of the situation is gained
by looking at Figure 5.7 which shows the S-surface Uu,DD (∆u, ∆v) for both
BPSK and QPSK cases. Of course, since the users are considered to be orthogonal
(xv,u =0), the S-surface exhibits no sensitivity to the interfering
estimation process ∆v. This is obvious on Figures 5.9a and 5.9b which
show the traces of the S-surface intersected by planes perpendicular to the
∆u-axis. These traces are thus S-curves Uu,DD (∆v) ∆u function of ∆v and
parametrised on ∆u. These S-curves are flat since there is no sensitivity to
the interfering phase estimation error ∆v. On the other hand, their counterpart
Uu,DD (∆u) ∆v (Figures 5.8a and 5.8b) illustrate the incidence of the
useful phase estimation error on the mean of the error signal. In fact, the
S-curves presented in Figure 5.6 are similar cuts made in the S-surface.
Introducing some coupling in the system modifies the S-surface in a way
that shows the influence of the interfering estimation process on the useful
one. It appears on Figures 5.11a and 5.11b through a broadening of the
conglomerate traces, while Figures 5.12a and 5.12b more clearly present a

5.1 Feedback 121
U u,DD
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
2−user system − x = 0 − AWGN channel
v,u
BPSK
QPSK
E s /N 0 = 0 dB
E s /N 0 = 5 dB
E s /N 0 = 10 dB
E s /N 0 = 20 dB
−1
−2 −1.5 −1 −0.5 0
Δ [rad]
u
0.5 1 1.5 2
Figure 5.6: S-curves in a 2-user non-dispersive synchronous system, xv,u =
0
sensitivity to ∆v.
Going one step further, Figure 5.13 shows the situation in presence of a
Near-Far effect. It exacerbates the results observed with moderate coupling
in Figure 5.10. Conglomerate S-curves (Figures 5.14a and 5.14b) present
a wide broadening while the traces obtained at ∆u constant (Figures
5.15a and 5.15b) have now the shape of S-curves with respect to the phase
estimation error ∆v of the interfering user. The point where both phase
estimation errors ∆u and ∆v get to zero is a stable operating point for the
MU DD phase recovery loop.

122 Decision Directed
Phase Error Detector
Phase Error Detector
1
0.5
0
−0.5
−1
−4
0.4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
−4
0
2−user system − x = 0 − δv,u = 0.1745 rad − Near−Far ratio = 0 dB − E /N = 10 dB
v,u
s 0
−2
−2
0
Δ u [rad]
2
4
4
2
0
Δ v [rad]
(a) S-surface U BPSK
u,DD (∆u, ∆v)
0
2−user system − x = 0 − δv,u = 0.1745 rad − Near−Far ratio = 0 dB − E /N = 10 dB
v,u
s 0
0
Δ u [rad]
2
4
4
2
0
Δ v [rad]
QP SK
(b) S-surface Uu,DD (∆u, ∆v)
Figure 5.7: S-surfaces of a 2-user non-dispersive synchronous system, uncoupled
scenario (a: BPSK, b: QPSK)
−2
−2
−4
−4

5.1 Feedback 123
Phase Error Detector
Phase Error Detector
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
2−user system − x = 0 − δv,u = 0.1745 rad − Near−Far ratio = 0 dB − E /N = 10 dB
v,u
s 0
−1
−4 −3 −2 −1 0
Δ [rad]
u
1 2 3 4
0.4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
(a) S-curve U BPSK
u,DD (∆u) ∆v
0
2−user system − x = 0 − δv,u = 0.1745 rad − Near−Far ratio = 0 dB − E /N = 10 dB
v,u
s 0
−0.4
−4 −3 −2 −1 0
Δ [rad]
u
1 2 3 4
QP SK
(b) S-curve U
u,DD (∆u)
∆v
Figure 5.8: S-curves function of ∆u, parametrised on ∆v - 2-user nondispersive
synchronous system, uncoupled scenario (a: BPSK, b: QPSK)

124 Decision Directed
Phase Error Detector
Phase Error Detector
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
2−user system − x = 0 − δv,u = 0.1745 rad − Near−Far ratio = 0 dB − E /N = 10 dB
v,u
s 0
−1
−4 −3 −2 −1 0
Δ [rad]
v
1 2 3 4
0.4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
(a) S-curve U BPSK
u,DD (∆v) ∆u
0
2−user system − x = 0 − δv,u = 0.1745 rad − Near−Far ratio = 0 dB − E /N = 10 dB
v,u
s 0
−0.4
−4 −3 −2 −1 0
Δ [rad]
v
1 2 3 4
QP SK
(b) S-curve U
u,DD (∆v)
∆u
Figure 5.9: S-curves function of ∆v, parametrised on ∆u - 2-user nondispersive
synchronous system, uncoupled scenario (a: BPSK, b: QPSK)

5.1 Feedback 125
Phase Error Detector
Phase Error Detector
1
0.5
0
−0.5
−1
−4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
−4
0
2−user system − x = 0.2 − δv,u = 0.1745 rad − Near−Far ratio = 0 dB − E /N = 10 dB
v,u
s 0
−2
−2
0
Δ u [rad]
2
4
4
2
0
Δ v [rad]
(a) S-surface U BPSK
u,DD (∆u, ∆v)
0
2−user system − x = 0.2 − δv,u = 0.1745 rad − Near−Far ratio = 0 dB − E /N = 10 dB
v,u
s 0
0
Δ u [rad]
2
4
4
2
0
Δ v [rad]
QP SK
(b) S-surface Uu,DD (∆u, ∆v)
Figure 5.10: S-surfaces of a 2-user non-dispersive synchronous system,
coupled scenario (a: BPSK, b: QPSK)
−2
−2
−4
−4

126 Decision Directed
Phase Error Detector
Phase Error Detector
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
2−user system − x = 0.2 − δv,u = 0.1745 rad − Near−Far ratio = 0 dB − E /N = 10 dB
v,u
s 0
−1
−4 −3 −2 −1 0
Δ [rad]
u
1 2 3 4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
(a) S-curve U BPSK
u,DD (∆u) ∆v
0
2−user system − x = 0.2 − δv,u = 0.1745 rad − Near−Far ratio = 0 dB − E /N = 10 dB
v,u
s 0
−0.4
−4 −3 −2 −1 0
Δ [rad]
u
1 2 3 4
QP SK
(b) S-curve U
u,DD (∆u)
∆v
Figure 5.11: S-curves function of ∆u, parametrised on ∆v - 2-user nondispersive
synchronous system, coupled scenario (a: BPSK, b: QPSK)

5.1 Feedback 127
Phase Error Detector
Phase Error Detector
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
2−user system − x = 0.2 − δv,u = 0.1745 rad − Near−Far ratio = 0 dB − E /N = 10 dB
v,u
s 0
−1
−4 −3 −2 −1 0
Δ [rad]
v
1 2 3 4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
(a) S-curve U BPSK
u,DD (∆v) ∆u
0
2−user system − x = 0.2 − δv,u = 0.1745 rad − Near−Far ratio = 0 dB − E /N = 10 dB
v,u
s 0
−0.4
−4 −3 −2 −1 0
Δ [rad]
v
1 2 3 4
QP SK
(b) S-curve U
u,DD (∆v)
∆u
Figure 5.12: S-curves function of ∆v, parametrised on ∆u - 2-user nondispersive
synchronous system, coupled scenario (a: BPSK, b: QPSK)

128 Decision Directed
Phase Error Detector
Phase Error Detector
1
0.5
0
−0.5
−1
−4
0.3
0.2
0.1
0
−0.1
−0.2
−0.3
−0.4
−4
0
2−user system − x = 0.2 − δv,u = 0.1745 rad − Near−Far ratio = 10 dB − E /N = 10 dB
v,u
s 0
−2
−2
0
Δ u [rad]
2
4
4
2
0
Δ v [rad]
(a) S-surface U BPSK
u,DD (∆u, ∆v)
0
2−user system − x = 0.2 − δv,u = 0.1745 rad − Near−Far ratio = 4 dB − E /N = 10 dB
v,u
s 0
0
Δ u [rad]
2
4
4
2
0
Δ v [rad]
QP SK
(b) S-surface Uu,DD (∆u, ∆v)
Figure 5.13: S-surfaces of a 2-user non-dispersive synchronous system,
Near-Far scenario (a: BPSK, b: QPSK)
−2
−2
−4
−4

5.1 Feedback 129
Phase Error Detector
Phase Error Detector
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
2−user system − x = 0.2 − δv,u = 0.1745 rad − Near−Far ratio = 10 dB − E /N = 10 dB
v,u
s 0
−1
−4 −3 −2 −1 0
Δ [rad]
u
1 2 3 4
0.25
0.2
0.15
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
(a) S-curve U BPSK
u,DD (∆u) ∆v
0
2−user system − x = 0.2 − δv,u = 0.1745 rad − Near−Far ratio = 4 dB − E /N = 10 dB
v,u
s 0
−0.25
−4 −3 −2 −1 0
Δ [rad]
u
1 2 3 4
QP SK
(b) S-curve U
u,DD (∆u)
∆v
Figure 5.14: S-curves function of ∆u, parametrised on ∆v - 2-user nondispersive
synchronous system, Near-Far scenario (a: BPSK, b: QPSK)

130 Decision Directed
Phase Error Detector
Phase Error Detector
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
0
2−user system − x = 0.2 − δv,u = 0.1745 rad − Near−Far ratio = 10 dB − E /N = 10 dB
v,u
s 0
−1
−4 −3 −2 −1 0
Δ [rad]
v
1 2 3 4
0.25
0.2
0.15
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
(a) S-curve U BPSK
u,DD (∆v) ∆u
0
2−user system − x = 0.2 − δv,u = 0.1745 rad − Near−Far ratio = 4 dB − E /N = 10 dB
v,u
s 0
−0.25
−4 −3 −2 −1 0
Δ [rad]
v
1 2 3 4
QP SK
(b) S-curve U
u,DD (∆v)
∆u
Figure 5.15: S-curves function of ∆v, parametrised on ∆u - 2-user nondispersive
synchronous system, Near-Far scenario (a: BPSK, b: QPSK)

5.1 Feedback 131
Reciprocal space - Characteristic function
The appreciation of the developments in direct space as presented in the
previous section, is mixed. To their advantage, one notices that they illustrate
the specificity of the multiuser phase recovery loop in a 2-user system.
Indeed, it shows the incidence of the interfering estimation process
on the useful one. However, the fact that these developments are limited
to a 2-user system due to the heavy derivations that would be required
in the case of more complex systems belongs to their shortcomings. Yet,
would it be possible to develop a performance model that is valid for systems
accommodating more users ?
A way to answer this question is to apply more efficient calculation methods,
such as the one described in Section 3.5.2. The general expression of
the mean of the error signal will be derived in the following paragraphs
by using this method. This expression will be illustrated in several cases.
General expressions The global expressions of the mean Uu,DD of the
estimation error um u,DD have been derived for BPSK- and QPSK-modulated
data symbols.
In the case of an MU DD phase recovery loop dealing with BPSK-modulated
data symbols, the mean of the estimation error is given by relation
(5.8). The expressions of the first-order statistics shown in Appendix E.1
which have been derived in reciprocal space with the help of the characteristic
function, were used in (5.8) instead of their counterparts shown
in Appendix D, and which were obtained in direct space. This led to the
general expression of the mean of the phase estimation error in the case of
BPSK modulation. It is given by relation (F.1).
Similarly, moving to QPSK modulation, the use of the first-order statistics
detailed in Appendix E.2 turns the mean of the estimation error given by
(5.10) into (F.3).
Computational results in a simplified context In order to get some insight
into (F.1) and (F.3), these expressions have been derived at equilibrium
(∆ =0) in a 2-user ideal (neither AWGN nor ISI) system. The means
have then been computed and plotted as a function of δv,u, the true phase
difference between users. On the other hand, the working of the open-loop

132 Decision Directed
configuration has been simulated, enabling to compare analytical and simulation
results. Moreover, they have also been compared to the value at
equilibrium of the S-surfaces parametrised on δv,u obtained in direct space
(Section 5.1.2).
¯ BPSK modulation
With the help of [116], (F.1) becomes at equilibrium in the ideal situation
U BPSK
u,DD (0)
Nu
=2
N0
=0
p q
xk,l =0 p = q
= 1
2 I0
0
u,v sign Ru,u + R 0
0
u,v sign Ru,u R 0
u,v
1
2 I0
sign R0 u,u + R
u,v
0
u,v sign R0 v,u + R0
v,v
+sign R0 u,u R0
u,v sign R0 v,u R0
.(5.11)
v,v
(5.11) was computed with respect to the phase difference between
users u and v, δv,u = φv φu and illustrated in Figure 5.16. Indeed,
this figure is threefold. First, it illustrates the result of the computations
(circles) with and without the mitigating term (second term
of 5.11). Including the mitigating term simulates the MU estimator
while not including it simulates the SU one. Second, it shows simulation
results, drawn as continuous lines. Simulated values of
U BPSK
u,DD (0) were obtained through Monte-Carlo simulations of the
open-loop configuration. The phase was supposed to be perfectly
estimated so as to illustrate the incidence of MAI. Finally, analytical
results derived in direct space (Section 5.1.2) are shown as crosses.
The reader can notice the match between analytical results in direct
and in reciprocal spaces, and also between analytical and simulation
results.
Since Figure 5.16 shows the mean U BPSK
u,DD (0) at equilibrium (∆ =0)
as a function of δv,u, the ideal situation is to have this mean null
everywhere. This is the case for the multiuser curve, not for the
single-user one. Indeed, a bias affects periodically the SU estimation
process around δv,u values such that arg x0
u,v + δv,u = kπ. This
bias comes from the fact that the incidence of the MAI depends on
the phase difference between users. This is described analytically

5.1 Feedback 133
BPSK
U ( 0 )
u,DD
1
0.5
0
−0.5
−1
−1.5
0
2−user system − x = 0.5 − Near−Far ratio = 10 dB − Es /N = 50 dB
v,u
0
Single−user
Multiuser
−3 −2 −1 0
δ [rad]
v,u
1 2 3
Figure 5.16: U BPSK
u,DD (0) as a function of δv,u (- simulation, ¢ computation
direct space, Æ computation reciprocal space)
by the first term of (5.11), which depends on I 0 u,v. Bearing in mind
that φl
ˆ φk = δl,k +∆k, definitions (3.77) and (3.78) explain how
the phase difference between users influence the performance of the
single-user estimator.
To explain this more physically, a detection error occurs as soon as
the interference phasor brings the resulting phasor of the matched
filter output out of the right detection zone (Figure 5.17). This occurs
when the contribution of the interferer is stronger than the one of
the user. In such a case, the output of the matched filter is driven by
the interferer and any parameter estimation process relying only on
this output goes wrong. This is the case of the SU phase estimator,
leading thus to a bias. Nevertheless, this bias disappears in the multiuser
structure thanks to the correcting term in (5.3) that mitigates
the influence of the MAI on the matched filter output.
¯ QPSK modulation
In the 2-user non-dispersive synchronous system, (F.3) writes at equi-

134 Decision Directed
arg x0
0
v,u + δv,u x 0 u,u
Ev
Eu
0 x
v,u
Figure 5.17: Phasor contributions of user u and interferer v to matched
filter output y m u for BSPK-modulated data symbols
librium
Nu
=2
N0
=0
p q
xk,l =0 p = q
= 1
4 sign R 0 u,u + R 0 u,v + I 0 0
u,v Ru,v I 0
u,v
+ 1
4 sign R 0 u,u + R 0 u,v I0 0
u,v Ru,v + I 0
u,v
+ 1
4 sign R 0 u,u R 0 u,v + I0
0
u,v Ru,v I 0
u,v
1
4 sign R 0 u,u R 0 u,v I 0 0
u,v Ru,v + I 0
u,v
+ 1
8 I0
sign R0 u,u
u,v
R0 u,v + I0
u,v
+sign R0 u,u R0 u,v I0
u,v
sign R0 v,v R0 v,u + I0
v,u
+sign R0 v,v R0 v,u I0
v,u
1
8 I0
sign R0 u,u +
u,v
R0 u,v + I0
u,v
+sign R0 u,u + R0 u,v I0
u,v
sign R0 v,v + R0 v,u + I0
v,u
+sign R0 v,v + R0 v,u I0
v,u
QP SK
Uu,DD (0)

5.1 Feedback 135
+ 1
8 R0 u,v
1
8 R0 u,v
sign R0 u,u + R0 u,v + I0
u,v
+sign R0 u,u R0 u,v I0
u,v
sign R0 v,v R0 v,u + I0
v,u
+sign R0 v,v + R0 v,u I0
v,u
sign R0 u,u + R0 u,v I0
u,v
+sign R0 u,u R0 u,v + I0
u,v
sign R0 v,v + R0 v,u + I0
v,u
+sign R0 v,v R0 v,u I0
.
v,u
(5.12)
QP SK
The reader can notice that Uu,DD given by (5.12) becomes equal to
U BPSK
u,DD (5.11) if(5.12) is only made of the terms weighting I 0 v,u while
deleting I 0 v,u in the argument of the sign functions. The reason for
keeping only terms weighting I 0 v,u
is that these are the ones present
in BPSK since the error signal takes the imaginary part of the phasecorrected
matched filter output. On the other hand, deleting I 0 v,u
in sign functions comes from the fact that BPSK hard-decisions rely
only on real parts of the phase-corrected matched filter output.
Figure 5.18 presents the result of the computation of (5.12) in the
chosen context. As in Figure 5.16, continuous curves are the result of
Monte-Carlo simulations of the open-loop process with perfect estimate
of the phase. Circles show the computational results of the
QP SK
expression of Uu,DD derived in reciprocal space, while the crosses
illustrate the expression derived in direct space. Again, the matching
between computations and simulations is good. However, the
simulation results have sometimes slightly diverged from the computational
ones due to numerical inaccuracies.
QP SK
Similarly to the BPSK case, a bias due to MAI affects Uu,DD . This
bias appears when the MAI contribution in the matched filter output
y p
k is stronger than the one from the user of interest, so that the matched
filter output is driven by MAI (Figure 5.19). Then the chosen
detection process produces wrong estimates of the data and an estimation
process relying only on y p
k is biased, as shown in Figure 5.18.
This bias appears around values of δv,u such as arg x0
u,v +δv,u = k π
2 .
Nevertheless, thanks to the introduction of a correcting term, this
bias is cancelled by the MU estimator.

136 Decision Directed
QPSK
U ( 0 )
u,DD
0.2
0.1
0
−0.1
−0.2
−0.3
0
2−user system − x = 0.5 − Near−Far ratio = 4 dB − Es /N = 50 dB
v,u
0
Single−user
Multiuser
−3 −2 −1 0
δ [rad]
v,u
1 2 3
QP SK
Figure 5.18: Uu,DD (0) as a function of δv,u (- simulation, ¢ computation
direct space, Æ computation reciprocal space)
arg x0
v,u + δv,u
x 0 u,u
Ev
Eu
0 x
v,u
Figure 5.19: Phasor contributions of user u and interferer v to matched
filter output y m u for QSPK-modulated data symbols

5.1 Feedback 137
In the remainder of this chapter, only the case of BPSK modulation will be
considered.
Numerical integration To fully exploit the expression (F.1), numerical
integration (Romberg method [117]) has been applied. In such approach,
the fact that the integrand is divided by the integration variable in all
terms of (F.1) is rather inconvenient. It can be solved with a classic change
of variable (Ω =lnω). This leads to (F.2) which is more appropriate for
numerical integration.
The concordance between (5.11) and (F.2) has been tested in different 2user
snapshot scenarii considering either the strongest or the weakest user
with Eb =10or 30 dB. The results are shown in Figures 5.20 and 5.21 for
N0
an SU estimator in a single-user system (Reference curve), and for singleuser
(SU curve) and multiuser (MU curve) estimators in the 2-user system.
In each case, the first subfigure represents the considered snapshot
scenario by drawing the phasors of the two users. The second subfigure
illustrates U BPSK
u,DD (0) computed according to (5.11) at equilibrium with respect
to the phase offset δu,v between the two users. This phase offset is
the one generated by the phase oscillators. That is why this offset does not
match with the offset shown in the first subfigure which also includes the
influence of the channel. The cross in the second subfigure indicates the
value of U BPSK
u,DD (0) given by numerical integration of relation (F.2) at the
offset value δu,v of the snapshot scenario. Finally, the third subfigure illustrates
U BPSK
u,DD (∆u, 0) numerically integrated from (F.2) at the offset δu,v of
the snapshot scenario. The value of U BPSK
u,DD (0) derived analytically with
(5.11) is shown by a circle. The match between the analytical derivation
(5.11) and the numerical integration (F.2) is thus measured by the concordance
between crosses and circles.
In Figure 5.20, the strongest user of the system is considered. The second
subfigure shows that no estimation process exhibits a bias at equilibrium,
whatever the true phase difference between users is, thanks to the fact that
the MAI introduced by the interfering user is small with regard to the useful
contribution of the user. It has thus no influence on the decision stage.
As a result, U BPSK
1,DD (∆1) shown in the third subfigure has the form of a
classic S-curve.
The situation is quite different in the scenario where user 1 is the weak-

138 Decision Directed
est (Figure 5.21). The MAI contribution of the interfering user provokes
decision errors, which introduce a bias affecting the SU estimator (second
subfigure). This bias also appears on the drawing of U BPSK
1,DD (∆1).

150
210
120
240
90
270
3.7124
2.4749
1.2375
4.9499
180 0
v
(a) Polar representation of the
snapshot scenario
60
300
u
30
330
Bias
Figure 5.20: U BPSK
u,DD
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
Δ u = Δ v = 0
SU
MU
−1
−2 −1 0
δ [rad]
u,v
1 2
15
10
5
0
−5
−10
Δ v = 0, δ u,v = 0.88714 rad
Reference
SU
MU
−15
−2 −1 0
Δ [rad]
u
1 2
(b) Comparison of biases computed by (5.11) and derived from the
numerical integration of (F.2)
where user u is the strongest and Eb
N0 =30dB
5.1 Feedback 139

140 Decision Directed
150
210
120
240
90
270
2.3839
1.1919
3.5758
180
u
0
(a) Polar representation of the
snapshot scenario
v
60
300
30
330
Bias
Figure 5.21: U BPSK
u,DD
0.03
0.02
0.01
0
−0.01
−0.02
Δ u = Δ v = 0
SU
MU
−0.03
−2 −1 0
δ [rad]
u,v
1 2
0.015
0.01
0.005
0
−0.005
−0.01
Δ v = 0, δ u,v = 0.55436 rad
Reference
SU
MU
−0.015
−2 −1 0
Δ [rad]
u
1 2
(b) Comparison of biases computed by (5.11) and derived from the
numerical integration of (F.2)
where user u is the weakest and Eb
N0 =10dB

5.1 Feedback 141
5.1.3 Actual decisions - Closed-loop study
Expressions (F.1) and (F.2) are too complex to be used for a closed-loop
study as the one lead in the DA case. Nevertheless, by simplifying them
to a 2-user environment without ISI nor AWGN, one can write
U BPSK
u,DD (∆u, ∆v) = U BPSK
u,DD
where U BPSK
u,DD
writes
∂U BPSK
u,DD
∂∆u
= 1
while
∆=0
2 x0u,u
Ev
+ 1
2
⎧
⎪⎨
⎪⎩
∂UBP SK
u,DD
∂∆v
∂U BPSK
u,DD
∂UBPSK
u,DD
(0, 0) +
∂∆u
∆=0
∆u + ∂UBPSK
u,DD
∂∆v
is given by (5.11). After some calculations,
∆=0
0
sign Ru,u + R 0 0
u,v +signRu,u R 0
u,v
Eu
R 0 u,v
∆=0
∂UBP SK
u,DD
∂∆u
∆v
(5.13)
∆=0
sign R0 u,u + R0
u,v 1 sign R0 v,v + R0
v,u
+sign R0 u,u R0
u,v 1 sign R0 v,v R0
v,u
2 I0
2 δ R0 u,u + R
u,v
0
u,v 1 sign R0 v,v + R0
v,u
+δ R0 u,u R0
u,v 1 sign R0 v,v R0 ⎫
⎪⎬
⎪⎭
v,u
(5.14)
becomes
∆=0
∂∆v
∆=0
= 1
⎧
⎪⎨
Ev
2 Eu ⎪⎩
sign R0 u,u + R0
u,v sign R0 v,u + R0
u,v
sign R0 u,u R0
u,v sign R0 v,u R0
u,v
2 I0
2 sign R0 u,u + R
u,v
0
u,v δ R0 v,v + R0
u,v
+sign R0 u,u R0
u,v δ R0 v,v R0 ⎫
⎪⎬
.
⎪⎭
u,v
(5.15)
R 0 u,v
The reader can notice that the explicit sensitivity of U BPSK (∆) with re-
u,DD
spect to the interfering user only appears in the case of the MU phase
estimator which contains an MAI mitigation term in its error signal um u,DD .

142 Decision Directed
This is shown by relation (5.15), which is not null if and only if MU estimation
is applied.
Using the previous equations, the phase recovery loop described by these
expressions can be drawn. It is shown in a 2-user system in Figure 5.22.
φ m u
φ m v
+
ˆφ m u
+
ˆφ m v
-
∆ m u
- ∆ m v
∂UBP SK
u,DD
∂∆m u
∂UBP SK
v,DD
∂∆m u
K0,u (z 1) 1
∂UBP SK
v,DD
∂∆m v
∂UBP SK
u,DD
∂∆m v
K0,v (z 1) 1
Figure 5.22: 2-user DD phase recovery loop
ξ m u
ξ m u
Fu (z)
Fv (z)
Unfortunately, due to the complexity of the developments, the study of
the recovery loop has been stopped here.
5.2 Feedforward
Before studying the performance of DD ML FF estimators, some implementation
issues are worth a discussion. These issues are raised by the
fact that the estimators no longer rely on training sequences but on decisions.

5.2 Feedforward 143
5.2.1 SU DD ML FF estimator
The causal restriction introduced in the previous section dealing with DD
ML FB estimators applies more forcibly in the FF case, for both SU and
MU estimators. Indeed, since the DD ML FF estimator is based on fedback
decisions produced downstream in the communication chain after phase
correction (see Figure 2.14 b), its estimate is, at first sight, constrained to
use past decisions only. Hence, the SU DD ML FF phase estimator writes
ˆφ m u
= arg
m 1
n=m N
⋆ n
Îu y n
u . (5.16)
Obviously, an initial phase estimate obtained in DA mode is requested in
order to start the whole process. With the help of this initial estimate, the
phase of matched filter outputs is corrected and decisions based on these
corrected outputs are obtained. These decisions are then exploited by the
DD phase estimator to compute the first DD estimate.
Several options are possible as far as the update rhythm is concerned. In
the fastest mode (Figure 5.23), a new phase estimate is computed as soon
as a new decision is produced. This new decision is combined with N 1
past ones in order to compute the updated estimate according to (5.16).
On the other hand, slower modes can be considered, in which the phase
e j ˆ φ 0 u
m N yu m N Îu m N+1 yu m N+1 Îu m 1 yu m 1 Îu ˆφ
e j ˆ φ m u
y m u
Î m u
ˆφ
y m+1
u
e j ˆ φ m+1
u
Î m+1
u
m+N 1 yu m+N 1 Îu Figure 5.23: SU DD ML FF estimator - Fastest update implementation
estimate is applied to several (at most N) successive matched filter outputs
(Figure 5.24). Once the decisions based on these phase-corrected matched
filter outputs are produced, an updated estimate of the phase is computed.
With respect to the previous phase estimate, the slowly updated one is
mostly based on brand new decisions, while the fastest estimator described
here above recycles N 1 past decisions. Notice also that the slow-

144 Decision Directed
e j ˆ φ 0 u
m N yu m N Îu m N+1 yu m N+1 Îu m 1 yu m 1 Îu ˆφ
e j ˆ φ m u
y m u
Î m u
y m+1
u
Î m+1
u
m+N 1 yu m+N 1 Îu ˆφ e j ˆ φ m+N
u
Figure 5.24: SU DD ML FF estimator - Slow update implementation
est estimation mode nicely applies to a burst transmission scenario. In
fact, the choice of an update mode is a trade-off between the computational
load and the adaptability of the phase estimate. The slowest solution
is definitely the best one from the point of view of the computational
load, since it requires up to N times less operations than the fastest one.
However, it assumes that the phase to be estimated does not significantly
change over the time span of the treated symbol block. If it does, wrong
decisions will be produced due to an erroneous phase estimate, and the
whole reception process will collapse. The fastest estimator makes no such
assumptions but implies much more computation.
5.2.2 MU DD ML FF estimator
From a multiuser perspective, the constraint to rely on available decisions
applies not only to the single-user part of the estimate, as studied in the
previous paragraph, but also to the MAI mitigation term introduced in
(4.26). Moreover, since the mitigation part uses phase estimates related to
the interfering users, the estimator is also constrained to rely on the latest
phase estimates. Hence, these restrictions can lead to two different implementations.
A first option consists in using past decisions and past estimates for all
users. The MU estimator writes
ˆφ m u = tan 1 (Cm u )
(Cm u ) = arg (Cm u ) (5.17)

5.2 Feedforward 145
where
C m u
=
m 1
n=m N
Î n u
⋆
y n u
Nu
Ek
k=1
k=u
Eu
e j ˆm 1
φk m 1
n=m N p=
m 1
Î n u
⋆ p p
Îk xn
u,k
(5.18)
This is the parallel implementation, since all users are simultaneously dealt
with. A 2-user version is shown in Figure 5.25. With respect to the DA
case, the MAI mitigation is incomplete, because it is strictly limited to its
causal part (p

146 Decision Directed
related to weaker users. (4.26) writes then
C m u
=
m 1
Î n u
⋆
y n u
n=m N
Nu
Ek
e
Eu
k=1
ku
j ˆ m
m 1 1
m 1
φl n=m N p=
e j ˆ φ 0 u ˆ φ ˆ φ
e j ˆ φ 0 v
m N yu m N Îu m N Îv m N yv m N+1 yu m N+1 Îu m N+1 Îv m N+1 yv m 1 yu m 1 Îu m 1 Îv m 1 yv ˆφ
y m u
Î m u
e j ˆ φ m+1
u
Î m v
y m v
ˆφ
e j ˆ φ m+1
u
Î n u
Î n u
y m+1
u
Î m+1
u
Î m+1
v
e j ˆ φ m+1
v
y m+1
v
⋆ p p
Îk xn
u,k
⋆ Î p
l
xn p
u,l
Figure 5.26: 2-user successive MU DD ML FF estimator
m+N 1 yu m+N 1 Îu m+N 1 Îv m+N 1 yv (5.19)
The successive implementation uses the most up-to-date information in its
reception process. As a result, the MAI mitigation is slightly better than in
the parallel implementation in as much as the MAI related to the current
symbols (p = m) of the more powerful users is also mitigated. The price
to pay is a delay on the reception of weak users. This delay is as great as
the user is weak, due to the fact that the estimator waits for new decisions
and updated estimates from more powerful users.

5.2 Feedforward 147
Finally, notice that the design option discussed in the SU case (fast vs. slow
update) is also applicable in the MU case.
5.2.3 Decisions assumed correct
Computational results are presented in the following pages. They have
been obtained under the assumption that decisions and phase estimates
were correct. In that situation, the only difference between DA and DD
implementations lies in the fact that the mitigation performed in MU DD
structures is limited to the causal part of the message, as mentioned in the
previous paragraph. With this restriction, the following results are just a
reinterpretation of DA results of Chapter 4.
Introducing the causal restriction in (4.31) leads to the following MU DD
ML FF estimator
∆u =
ISIu + MAIa
v,u + Noiseu (Directv + ISIv)
MAIc
v,u (ISIv + Noisev)
(Directu + ISIu) (Directv + ISIv)
MAIc
v,u MAIc u,v
(5.20)
Comparing to its DA counterpart (4.32), the reader can notice that the MAI
contribution in (4.32) is now split into its causal MAI c v,u
MAI c u,v =
Ev
Eu
and anti-causal MAI a v,u parts
MAI a v,u =
Ev
Eu
j(φv φu)
e
j(φv φu)
e
N
m=1 n=
N
m
+
m=1 n=m+1
(I m u ) ⋆ I n m n
v xu,v (I m u ) ⋆ I n n m
v xv,u (5.21)
(5.22)
with respect to time index m. The anti-causal part of the MAI contributes
to the biasing term (second numerator term of (5.20)) already mentioned
in Section 4.2.2 in the case of ISI.
From the point of view of the variance, the split of the MAI into its causal
and anti-causal part adds new terms to the expressions presented in Appendix
C. A simplified version is obtained by limiting the study to the

148 Decision Directed
most important one, MAIa
v,u (Directv). Its incidence in the variance
expressions of Appendix C is a supplementary term whose numerator
writes, depending on the modulation,
¯ BPSK modulation
N
0 2
Nxu,u 4 Ev
Eu
¯ QPSK modulation
+
m=1 n=m+1
4 Ev
Eu
x
n m
v,u
0 2 N
Nxu,u 2
+
m=1 n=m+1
e 2jδu,v x
n m
v,u
2
(5.23)
n m
x 2 . (5.24)
Its denominator is given either by (C.7) for BPSK- or by (C.11) for QPSKmodulated
symbols.
Both expressions (5.23) and (5.24) induce a variance increase independent
of the Es ratio whose importance depends on the relative weight of the
N0
unmitigated anti-causal part of the MAI. This variance increase graphically
translates into a raise of the variance floor (See Figure 5.27, curve ”DD
with”). The variance is even greater if the zero-shift contribution x0 u,v is not
taken into account (curve ”DD without”), as already explained in Section
5.1.1. However, in dispersive environments such as the one considered in
Figure 5.27, a (slight) improvement with respect to the SU estimator is still
noticeable thanks to the mitigation of the MAI due to past symbols.
5.2.4 Actual decisions
No study similar to the one performed in DA phase estimation (Section
4.2) has been made for its DD counterpart. Such analysis would have
been pretty difficult, since it would have required to deal with the two
different kinds of coupling, namely the coupling between users and the
coupling between estimation and detection stages. Since the resolution of
the coupling between users in the DA context already leads to simplifications,
it could be expected that the study of the DD context would require
stronger approximations.
v,u

5.3 Conclusions 149
Variance [rad 2 ]
10 1
2−user system − 31−chip Gold codes − BPSK modulation − R = 1e5 Bauds − HT channel − 2 B T = 0.1
N
10 0
10 −1
10 −2
10 −3
10 −4
10 −5
DA
DD with
DD without
Single−user
Multiuser
Uniform distribution
CRLB
10
0 5 10 15 20 25 30 35 40
−6
E /N [dB]
s 0
Figure 5.27: Variance of ML FF estimators in presence of ISI (BSPK)
On the other hand, as mentioned in Section 2.3.2, the study of DD ML FF
phase estimators is usually not performed as is in the estimation literature.
One relies on a correspondence between FF and FB estimators [84, p. II-
3] to extend the performance derived for FB estimators to FF estimators.
Such extension is only valid as long as some conditions regarding the error
detector and the closed-loop impulse response are respected [85, p. 344].
5.3 Conclusions
The study of MU DD estimators has been performed in this chapter by
following two different tracks.
The first one was based on the assumptions of correct decisions. Introducing
the causal restriction of DD estimators into DA relations derived in
Chapter 4, it has led to a reinterpretation of these relations in a DD perspective.
It has been shown that the variance of the MU DD estimator is
greater than, or at best equal to the one of the MU DA estimator, but lower
than, or at worst equal to the one of the SU estimator. The quality of the
estimator mainly depends on the way it deals with the interference related
to the present symbol.

150 Decision Directed
The second track made no assumption with respect to the correctness of
the decisions. It has focused on the open-loop study of a MU DD phase
recovery loop. Several methods for deriving the mean of its error signal
have been presented, namely a bidimensional brute-force method in the
direct space and a multidimensional one in the reciprocal space. Using
the first one, a graphical representation of the mean has been illustrated.
In the considered 2-user system, it stands as the bidimensional extension
of well-known S-curves, therefore called S-surfaces. For systems encompassing
more than 2 users, one speaks of S-hypersurfaces.
On the other hand, the closed-loop study of the MU DD recovery loop has
been initiated. It has been demonstrated that its structure could be seen as
a set of loops whose error signals are function of all estimation errors.

Chapter 6
Conclusions
6.1 Achievements
This thesis has tackled the parameter estimation issue in a multiuser spread-spectrum
communication system. The following table summarises
the options considered as far as the interaction between estimator and detector
(DA/DD/NDA) and the structure of the estimator (FB/FF) are concerned.
The degree of achievement within each option is also mentioned.
Estimator Performance study
DA:
minθ Î,θ
DD:
Λ(I,θ)
Λ Î,θ
FB
FF
FB
Open-loop Closed-loop
Closed-form solution
Decisions assumed correct
minθ (I,θ) Open-loop
FF Decisions assumed correct
NDA: ¯ Λ(θ) FB
minθ EI [ (I,θ)] FF
Table 6.1: Synthetic view of the achievements of the thesis
Regarding the parameter to be estimated as an uniformly distributed random
variable, the log-likelihood function has been derived in Chapter 3 as
the cost function to be minimised by the optimal estimator, whether DA,
DD, or NDA. In the case of DA and DD estimations, it has been shown
that taking into account the conditional signal energy in the writing of the
log-likelihood function leads to the addition of a supplementary term in

152 Conclusions
the expression of the estimators, beside the well-known contribution depending
on the correlation between symbols and phase-corrected matched
filter outputs. Usually, the conditional signal energy is disregarded in the
literature as being independent of the parameter to estimate. This is valid
in SU environments but not in MU ones. This observation has been the
starting point of this thesis. As a result, the supplementary contribution
appearing in the expressions of the estimators has been shown afterwards
to mitigate the incidence of the MAI entering the system through the matched
filter outputs.
Moving to implementations, two MU DA ML phase estimators, one FB
and one FF, have been first described in Chapter 4. Their performance
have been derived analytically and compared to those of their SU counterparts
working in the same multiuser environment. The mitigation effect
mentioned here above leads to performance improvement with respect to
SU estimators: MU DA ML phase estimators exhibit less jitter variance
than their SU counterparts. Furthermore, they reach the CRLB and appear
to be Near-Far resistant. However, they suffer from ISI.
Following the analysis of DA estimators, the study of DD structures presented
in Chapter 5 has been twofold. On the one hand, assuming correct
decisions, the developments led for DA estimators have been reinterpreted
in a DD perspective. On the other hand, relaxing assumptions
on the quality of the decisions, this work focused on FB implementations
(recovery loops). The investigations have been limited to the open-loop
study. S-hypersurfaces have been introduced as multi-dimensional extensions
of well-known S-curves. Moreover, the results presented in [87] for
a single-user case and AWGN channels have been generalised for a multiuser
case and possibly frequency-selective channels.
For all the investigated estimators, the performance study has been performed
analytically. To the knowledge of the author, this is the first analytical
demonstration of the efficiency of multiuser parameter estimation
in spread-spectrum environments.
6.2 Perspectives
This thesis does not claim, in any way, to have covered the subject in its
entirety. There remains several issues that can be used as starting points

6.2 Perspectives 153
and/or central themes for future studies. They shall be the subject of this
concluding section.
Phase model
The first action point is related to the parameter at the centre of this thesis.
In the present work, the phase has been modelled as constant during the
estimation. However, it might be modelled as a slowly changing parameter
whose fluctuations are characterised by the phase noise. The filtering
of this noise through the estimation device affects the performance of
the estimator [84, part IV]. The incidence of the phase noise should thus
be taken into account in future developments.
Close to the concern of phase noise, a specific fluctuation of the phase
would be of great interest, namely its continuous raise due to the integration
over time of a frequency offset. This issue is of crucial importance for
coherent transmissions.
Validity of working hypotheses
The estimators presented in this work have been derived under the assumption
that all other parameters of the communication system (timing,
power, channel impulse response, etc.) either were known or had been
perfectly recovered. There is very little knowledge in as much stringent
these hypotheses are. As reviewed in Section 2.3.3, there are numerous
references regarding the incidence of estimation errors on the detection
performance. However, the issue of imperfect recovery of one parameter
on the estimation of another one has not received much attention. The
sensibility of the proposed estimators to incorrect values of those parameters
ought thus to be studied in order to get a better view on the working
of the proposed structures.
Generalisation to other parameters - Joint 2 estimation
The previous action point stressed the importance of the other parameters
of the link. Indeed, beyond the sensibility study considered, the scope of
the present work might be broadened so as to encompass all those parameters.
The phase is definitely not the only parameter to estimate. From
an analytical point of view, and interestingly enough, it is characterised by
the fact that it stands explicitly through a phasor in relations defining for

154 Conclusions
instance the matched filter output. On the other hand, other parameters,
such as the timing, are implicit. They appear in these relations through
non-linear functions, which makes their estimation more complex. However,
recovering them is of crucial interest, as for instance timing in the
case of spread-spectrum receivers. Hence, investigating the multiuser estimation
of other parameters is most probably the next step ahead.
Furthermore, it is obvious that the estimation of any parameter is connected
to the recovery of the other ones. It might thus be more straightforward
to tackle the problem in its globality from the very beginning, that
is to say design a parameter estimator which simultaneously addresses all
parameters of the link for all active users. This would lead to a ”joint 2 ” estimation
process, the exponent indicating that the joint characteristic has a
double meaning: for each parameter, this global estimator would take all
active users into account and, for each user, it would attempt to recover all
the parameters of this link.
Coupled structures for DD estimation
Figure 5.22 can be seen both as an achievement and as a starting point.
On the one hand, it is an achievement in that the means of coupling in an
MU DD ML FB phase estimator have been made obvious. Yet, it is also a
starting point, as all elements for further study are available because the
expression of U BPSK
u,DD has been established. Further investigations of such
MU DD phase recovery loops will definitely lead to the study of strongly
coupled feedback systems.
NDA estimation
The closed-loop study of DD estimators mentioned here above would conclude
the treatment of DD structures. However, even with this addendum,
the review of possible estimation structures would still miss NDA estimators.
This lack ought to be filled through a dedicated analysis of NDA structures,
whose main property is to be independent of the detection stage.
Monte-Carlo simulations for validating calculations and investigating
dynamic phenomena
The analytical developments presented in this thesis are interesting in that
they give a precise insight in the ins and outs of the parameter estima-

6.2 Perspectives 155
tion process in a multiuser spread-spectrum environment. However, the
confirmation of the calculations through Monte-Carlo simulations would
ground more strongly the conclusions drawn in the previous section. Moreover,
Monte-Carlo simulations are not just a tool for validating calculations.
They also enable to investigate dynamic behaviours, such as the
estimator’s one, and to deal with time-varying channels.
As far as the estimator is concerned, the investigation performed during
this thesis has focused on its steady-state performance. Yet, as was
stressed in Section 2.3.2, the dynamic behaviour (acquisition, cycle slips,
etc.) of the MU estimator also deserves much attention. It ought thus to
be investigated in order to get a really complete view of its performance.
On the other hand, the channels used throughout this work have been
regarded as static, although the thesis took place in a mobile radio environment.
The time-varying characteristic of the channels have thus not
been taken into account. Hence, it would be more realistic to investigate
the behaviour of the presented estimators in such channels. Monte-Carlo
simulations are a powerful tool in that perspective.
Gaussian issues
Section 2.3.3 stressed that the performance study of devices operating in
MAI-plagued environments often relied on a Gaussian model of this interference.
This thesis has taken the exact opposite view in that it dealt with
the actual MAI throughout all developments. However, it might be worth
raising the question of the efficiency of this approach. Indeed, the use of
the Gaussian approximation which models the effect of a large number of
independent random sources, implies that the load of the system, i.e. the
number of active users, is large. As a result, the exhaustive approach followed
in this thesis should apply for small to moderate system loads but
might be outperformed by the Gaussian approximation from the point of
view of calculation complexity when the load becomes heavy. An interesting
issue is the limit of the system load at which it becomes more efficient
to change of MAI model. Some preliminary results were already presented
in [89]. However, it would deserve deeper investigation.
The other issue related to Gaussian matters refers to the developments
presented in [107]. It is clear from Appendix E that analytical developments
related to DD structures often involve the calculation of a probab-

156 Conclusions
ility of the type Pr (A >0), where A is a random variable. Indeed, A is a
linear combination of several random variables and of Gaussian noise. Assuming
the other variables, the random nature of A only comes from the
noise. As a result, the probability mentioned here above can be obtained
assuming the other random variables, as a Q function whose argument
depends on the assumed variables. Completing the calculation requires to
average this expression over the joint pdf of the assumed variables. However,
the Q function that has appeared due to the Gaussian random variable
is not really well suited for such a derivation. Yet, another writing of
the Q function has been proposed [107] (See Section 3.5.1), which might
help to solve this calculation issue.
MC-CDMA
The question of coherent reception is the last key-issue for future works to
mention. Now topical for third-generation developments, it is also of crucial
importance for another kind of air interface, viz. OFDM systems. In
the last few years, an hybrid air interface, MC-CDMA, combining OFDM
and CDMA, has attracted much interest. It would be worth extending the
study led in this thesis for single-carrier CDMA to MC-CDMA systems.

Appendix A
Correlation function of the
loop noise in a DA recovery
loop
General expressions of the cross-correlation function C m n
u,v (∆) of two loop
noise samples ξ m u and ξ n v are given in this appendix as a function of the
vector estimation error ∆=Φ ˆΦ.

158 Correlation function of the loop noise in a DA recovery loop
A.1 BPSK modulation
After long but easy calculations, Cm n
u,v (∆) writes, in the case of BPSKmodulated
data symbols
m n
Cu,v (∆)
= E [ξ m u ξn v ]
= δ (u
⎧
v)
⎪⎨
⎧
⎪⎨
δ (m n)
+
p=
+ N0x0 u,u
2EuT
+ Nu Ek
Eu
k=1 p=
k=u
+ Nu Ek
Eu
k=1 p=
k=u
j∆u p 2 e xu,u +
+
e j(δk,u+∆u)
2 p
xu,k
e j(δk,u+∆u
2 ∆k) p
xu,k ⎧
⎨
e j(δk,u+∆u)
p
x
u,k
e j(δk,u+∆u
∆k) p
xu,k ⎪⎩
2
⎪⎩
Nu + Ek
Eu
k=1 p= ⎩
k=u
+ ej∆u
xm n j∆u
u,u e xn m
u,u
+[1 δ (u v)]
⎧
⎪⎨
⎪⎩
ej(δv,u+∆u)
xm n
u,v ej(δu,v+∆v) xn m
v,u
ej(δv,u+∆u)
xm n j(δu,v+∆v
u,v e ∆u) xn m
v,u
ej(δu,v+∆v)
xn m j(δv,u+∆u
v,u e ∆v) xm n
u,v
ej(δv,u+∆u ∆v) ⎫
⎪⎬
.
xm n 2
⎪⎭
u,v
⎫
⎬
⎭
⎫
⎪⎬
⎪⎭
⎫
⎪⎬
⎪⎭
(A.1)

A.2 QPSK modulation 159
A.2 QPSK modulation
Likewise, with QPSK-modulated data symbols,
m n
Cu,v (∆)
= E [ξ m u (ξn v )⋆ ]
⎧
⎡
⎢
⎪⎨
⎢
δ (m n) ⎢
= δ (u v)
⎢
⎣
1
2
p=
+
x p u,u 2
+ N0x0 u,u
2EuT
+ 1
Nu + Ek
2 Eu
k=1 p=
k=u
x p
u,k
2
+ 1
Nu + Ek
2 Eu
k=1 p=
k=u
x p
u,k
2
Nu Ek
Eu
k=1
k=u
+
cos ∆k
p=
⎪⎩ 1
2 xm n
u,u
2 cos 2∆u
+ 1
⎡
xm n
u,v
⎢
[1 δ (u v)] ⎢
2 ⎣
2 cos (∆u +∆v)
+
xm n
u,v
2 cos ∆u
+
xm n
u,v
2 cos ∆v
xm n2
u,v
x p
u,k
2
⎤ ⎫
⎥ ⎪⎬
⎥
⎦
⎪⎭
⎤
⎥ . (A.2)
⎦

Appendix B
Pdf of Single-User DA ML FF
phase estimator
In this appendix, the pdf of the Single-User DA ML FF phase estimator
given by [7, p. 326] is derived analytically in a multiuser context. Calculations
are simplified by restricting the study to BPSK modulated data
symbols and by using a single-tap averaging window (N =1). For detection
[38] as well as for parameter estimation, such one-shot approach ends
in a sub-optimal implementation in the case of asynchronous signals.
B.1 First step: characteristic function ψˆxu,ˆyu (ωr,ωi)
Using (3.77) and (3.78), the characteristic function writes
ψˆxu,ˆyu (ωr,ωi)
⎧ ⎧ ⎧ ⎡
Nu
N
+ Ek
ωr ⎣ Eu
⎪⎨ ⎪⎨ ⎪⎨
k=1 n=
m=1
+I
= E exp j
⎪⎩ ⎪⎩ ⎪⎩
m u νm u e jφu
⎡
Nu
N
+ Ek
+ωi ⎣ Eu
k=1 n=
m=1
+Im u νm u e jφu
Im u In n
k
Rm
u,k
Im u In n
k
Im
u,k
⎤
⎦
⎤
⎦
⎫⎫⎫
⎪⎬ ⎪⎬ ⎪⎬
⎪⎭ ⎪⎭ ⎪⎭
(B.1)
Developing the expectation in (B.1) is pretty intricate due to the ISI terms
E (Im u In u ). As a result of their presence, calculating the expectation over
all data symbols progressively, assuming one data symbol after the other,

162 Pdf of Single-User DA ML FF phase estimator
generates nested functions which are not easy to handle. However, a simplifying
hypothesis, setting the span N equal to 1, enables to derive these
expectations without too much effort. Such narrow averaging window
is unfortunately not realistic for practical implementations. The present
choice should then be seen as a trade-off between the will to reach an analytical
result and the tolerable complexity of the derivations.
With this hypothesis, (B.1) becomes
ψˆxu,ˆyu (ωr,ωi)
= exp
+
n=
n=0
N0x 0 u,u
4EuT
2
ωr + ω 2 0
i + jxu,uωr cos R n u,uωr + I n u,uωi Nu
k=1
k=u
n=
+
cos R n u,kωr + I n u,kωi
.
(B.2)
On the other hand, neglecting the ISI terms enables to avoid the nested
functions issue mentioned here above. It is thus possible to write the characteristic
function for widths N of the averaging window greater than 1.
It can easily be shown that the characteristic function obtained in a case
where N > 1 is the N-th power of the characteristic function obtained
assuming N =1.
B.2 Second step: pdf Tˆxu,ˆyu (ˆxu, ˆyu)
The product of cosines functions in (B.2) does not facilitate the search for
an analytical solution of the inverse Fourier transform. However, such
closed form solution is within reach if this product can be written as a
sum, like in the following
where
N
cos (dkω) =
k=1
Dl =
q=1
1
2 (N 1)
2 (N 1)
l=1
N
(l 1) mod 2
1 2
[cos (Dlω) sin (Dlω)] (B.3)
2 (N q)
(N+1 q)
dq
(B.4)

B.2 Second step: pdf Tˆxu,ˆyu (ˆxu, ˆyu) 163
with x being the greatest integer value lower than or equal to x.
This trick is very helpful to turn the integration of a product into a sum of
integrations. However, it has a major drawback. It was stressed in Section
3.5.2 that the derivation in the reciprocal space, using the characteristic
function, helped to reduce the number of computations from exponential
down to linear complexity. Expanding the product as in (B.3) cancels this
advantage, since the final result now exhibits a sum whose span enlarges
exponentially with the number of symbols contributing to the interference.
Indeed, this sum performs the averaging operation over all possible outcomes.
Despite this loss of performance, the derivation can and will go on.
Defining Sx as the time span of the normalised channel correlation coefficient,
that is to say the number of non-zero coefficients x q
k,l for a pair of
users (k, l), dq in (B.4), will be in the present case the elements of Nu ¢ Sx
vectors ¯ R p u and Īp u so that
¯R p u =
p %Sx
Ru, p
Sx
Ī p u =
p %Sx
Iu, p
Sx
(B.5)
(B.6)
where % stands for the modulo operator and p [1,NuSx] . Using (B.3)
and [118, p. 15, relation (11)], Tˆxu,ˆyu (ˆxu, ˆyu) writes
Tˆxu,ˆyu (ˆxu, ˆyu)
+ 2
1
=
2π
=
+
1
2 (NuSx+1) cuπ
⎧
2 (NuSx 2)
k=1
⎪⎨
⎪⎩
exp
ψˆxu,ˆyu (ωr,ωi) e j(ωrxu+ωiyu) dωrdωi
⎧
⎪⎨
+exp
⎪⎩ 1
4cu
⎧
⎪⎨
⎪⎩ 1
4cu
(B.7)
⎡
x2 u + y
⎢
⎣
2
u +2xu F k
u x0
u,u
+2yuGk u + x0 2 u,u 2F k
u x0 u,u
+ F k ⎤⎫
⎪⎬
⎥
⎦
2
u + Gk 2
⎪⎭
⎡ u
x2 u + y
⎢
⎣
2
u 2xu F k
u + x0
u,u
2yuGk u + x0 2 u,u +2Fk u x0 u,u
+ F k ⎫
⎪⎬
⎤⎫
⎪⎬
⎥
⎦
2
u + Gk 2
⎪⎭ ⎪⎭
u
(B.8)

164 Pdf of Single-User DA ML FF phase estimator
where
F k u =
G k u =
cu = N0x 0 u,u
4EuT =
x0 2
u,u
N
(l 1) mod 2
1 2
q=1
4
2 (N q)
N
(l 1) mod 2
1 2
q=1
2 (N q)
1
Es
N0
(N+1 q)
(N+1 q)
B.3 Third step: change of variables
¯R q u
(B.9)
(B.10)
Ī q u . (B.11)
Switching from the cartesian (x, y) to the polar (r, ∆) coordinate system
according to ˆxu =ˆru cos ∆u and ˆyu =ˆru sin ∆u enables us, with the help of
[118, p. 146, relation (31)], to write the joint pdf Tˆru,∆u (ˆru, ∆u) as follows
where
Tˆru,∆u (ˆru, ∆u)
= Tˆxu,ˆyu (ˆxu,
ˆyu)
∂ (ˆxu, ˆyu)
∂
(ˆru, ∆u)
1
=
2 (NuSx+1) cuπ
⎧
2 (NuSx 2)
⎨
g u exp ˆru exp
4cu
⎩ +exp ˆru exp
k=1
g + u
4cu
2f u
4au ˆru
1 2
(ˆru) 4cu
1
4cu (ˆru) 2 + 2f + u
4cu ˆru
⎫
⎬
⎭
(B.12)
(B.13)
f +
u = F k u + x 0
u,u cos ∆u + G k u sin ∆u
(B.14)
f u =
F k u x0
u,u cos ∆u + G k u sin ∆u
(B.15)
g + u = x 0 2 k
u,u +2Fux 0
u,u + F k 2
u + G k 2 u
(B.16)
g u = x 0 2 k
u,u 2Fu x 0 u,u +
F k 2
u + G k 2 u . (B.17)

B.4 Fourth step: pdf T∆u(∆u) 165
B.4 Fourth step: pdf T∆u (∆u)
Integrating ˆru out produces the marginal density of ∆u which is the desired
pdf
=
1
2 (NuSx) π⎧
2 (NuSx 2)
k=1
⎪⎨
⎪⎩
exp
1
+exp
1+
T∆u(∆u)
+
= Tˆru,∆u (ˆru, ∆u)dˆru
0
g u
4cu
π
cu
f u
2 exp
2
(f u ) f u 1 erf
4cu
2 Ô
cu
g + u
4cu
π
cu
B.5 Analytical validation
f + u
2 exp
(f + u ) 2
4cu
1 erf
f + u
2 Ô
cu
⎫
⎪⎬
.
⎪⎭
(B.18)
In order to validate (B.18), an AWGN scenario was considered. In such
environment, F k u = Gku =0, leading to f u = f + u and g u = g+ then becomes
u . T∆u(∆u)
T∆u(∆u)
= 1
2π exp
1+
Eb
N0
π Eb
cos ∆u exp
N0
Eb
cos
N0
2 ∆u
1+erf
Eb
cos ∆u
N0
.
(B.19)
which is the expression presented in [7, p. 262, relation (4.2.103)]. This
expression has been derived in a non frequency-selective environment.
However, it was shown in [119] that it could be extended to dispersive
environments by including ISI in the SNR.

Appendix C
Variance of DA ML FF phase
estimators
This appendix presents the variance expressions of the closed-form DA
ML FF phase estimators derived in Section 4.2.2. These expressions differ
from BPSK- to QPSK-modulated data symbols, since
E (I m k )2
= E I m k 2
(C.1)
= σ 2 Ik (C.2)
in the case of BPSK, while, in the case of QPSK
E (I m k )2 =0. (C.3)
C.1 Multiuser estimator
C.1.1 BPSK modulation
Considering first BPSK-modulated data symbols, the variance of the multiuser
phase estimator given by (4.32) can be derived according to the
method described in Section 3.4.2. It finally writes
BPSK 2
σ =
∆u
1
Es,u
x
N0
0 u,u
Num BPSK
u
Den BPSK
u
+
σBPSK 2
ISIu
Den BPSK
u
(C.4)

168 Variance of DA ML FF phase estimators
where the numerator Num BPSK
u
Num BPSK
u
= N 3 x0
u,u x0 2
v,v
+N x0 ⎡
u,u
N
⎣ N
x
m=1
N
n=1
n=m
Nx0 v,v
m=1 n=1
N x expands into
m n
v,v
m n
u,v
2 + +
2 +
n=
n=m
x m n
v,v
2
⎤
⎦
e2j(φu φv) x
m n
u,v
2
Nx0
N N
v,v xn m
v,u x0 u,v + x
m=1 n=1
0 2j(φu
v,u e φv)
N N N
xn m p m n p
v,v xu,v xv,u +2 xn m m p
v,v xv,u x
m=1
N
m=1
⎡
n=1
n=m
+
n=
n=m
p=1
N
N
p=1
N
xn m
x0 u,v
m=1 n=1 p=1
⎣e2j(φu φv) x0 v,u
p n
v,v xu,v x
N
x
N
n m
v,u
N
m p
v,u
p n
xv,v + x
N
x
m=1 n=1 p=1
p=n
n m
v,u
n p
v,v
p n
xv,v + x
n p
v,v
N + N
p m x
+
x m n
v,v v,u
m=1 n= p=1
2 + e2j(φu φv) x
N + N
m p x
+
x n m
v,v v,u
m=1 n= p=1
2 + e2j(φu φv) x
N N N
+
xn m p
v,u xm u,v xn m m
v,v + xp v,v
m=1 n=1 p=1
N N N
2j(φu + e φv) xn m p m p m
v,u xv,u xv,v + x
m=1 n=1 p=1
N N N
+
xn m m p
v,u xu,v xm n m p
v,v + xv,v m=1 n=1 p=1
N N N
2j(φu + e φv) xn m m m p
xp v,u xv,v + x
m=1 n=1 p=1
while the variance of the ISI contribution is given by
v,u
n p
v,u
⎤
⎦
m n
v,u
n m
v,u
n m
v,v
m n
v,v
2
2
(C.5)

C.1 Multiuser estimator 169
BPSK 2
σISIu = 4 Nx 0 u,u
+2
2
2 Ev
Eu
⎧
⎪⎨
⎪⎩
⎧
⎪⎨
⎪⎩
⎧
⎪⎨
⎪⎩
N
m=1
N
m=1
N
m=1
+
n=
n=m
N
N
n=1
n=m
+
n=
n=m
m=1 n=1
n=m
N
m=1
+ N
m=1
+
x
n m
N
n=
n=m
N
n=1
n=m
u,u
2
x
n m2
x
u,u
x
n m
u,u
2
x
n m2
x
u,u
x
n m
v,v
2 +
x
n m2
+ x
On the other hand, the denominator Den BPSK
u
Den BPSK
u
⎧
⎨
=2
⎩
N 2 x 0 u,u x0 v,v
1
2
N
m=1 n=1
C.1.2 QPSK modulation
N x
n m
v,u
v,v
2 +
x
n m 2
u,u
n m
u,u
writes
2
x
n m 2
u,u
n m
u,u
2
x
n m 2
v,v
n m
v,v
2
e2j(φu φv) x
n m
v,u
⎫
⎪⎬
⎪⎭
⎫
⎪⎬
⎪⎭
⎫
⎪⎬
.
⎪⎭
(C.6)
⎫2
⎬
2 .
⎭
(C.7)
Moving to QPSK-modulated data symbols, the variance expression does
not apparently differ from the corresponding expression (C.4) obtained
considering BPSK-modulated symbols
σ
2 QP SK
∆u
=
1
Es,u
x
N0
0 u,u
QP SK
Numu QP SK
Denu 2 QP SK
ISIu
QP SK
Denu
σ
+
. (C.8)
Indeed, the differences lie in the definitions of the numerator Num QP SK
u

170 Variance of DA ML FF phase estimators
QP SK
Numu = N 3 x0
u,u x0 2
v,v
+N x0 N +
u,u
m=1 n=
n=m
N
Nx0 v,v
m=1 n=1
N
m=1
N
+
N
+
+
n=
n=m
N
N x N
x
p=1
N
xm n 2
v,v
m n
u,v
n m n
v,v xp
N
x
x0 u,v
m=1 n=1 p=1
+
m=1 n=
N
N
2 + x0
u,vxn m
v,u
u,v x
N
xm n
p=1
the variance of the ISI contribution σ
σ
2 QP SK
ISIu
= 4 Nx 0 u,u
+2
2
2 Ev
Eu
v,u
m p
v,u
n m p
v,u xn v,v
2
xn m m p
v,u xu,v m=1 n=1 p=1
2 QP SK
ISIu
p m
xv,v + x
p m
xv,v + x
m p
v,v
m n
v,v
,
⎧
⎪⎨ N +
n m
x
u,u
⎪⎩ m=1 n=
n=m
2
N N
n m
x
u,u
m=1 n=1
n=m
2
⎫
⎪⎬
⎪⎭
⎧
⎨ N +
xn m
u,u
⎩m=1
n=
n=m
2 N N
xn m
u,u
m=1 n=1
n=m
2
⎫
⎬
⎭
⎧
⎨ N +
xn m
v,v
⎩
2 + N N
xn m
v,v
2
⎫
⎬
⎭ ,
m=1
QP SK
and the denominator Denu QP SK
Denu =2
n=
n=m
N 2 x 0 u,u x 0 v,v
1
2
N
m=1 n=1
m=1
n=1
n=m
N
n m
x
v,u
2
2
(C.9)
(C.10)
. (C.11)
Comparing (C.11) to(C.7), the reader can notice that the terms weighted
by I2
k have disappeared due to (C.3).

C.2 Single-user estimator 171
C.2 Single-user estimator
After deriving the variance of the Multiuser DA ML FF phase estimator
(4.32), the variance of the Single-User estimator (4.37) is to be derived.
C.2.1 BPSK modulation
Applying the same procedure as the one applied for obtaining (C.4), the
variance of a Single-User DA ML FF estimator writes in the case of BPSKmodulated
data symbols
BPSK 2
σ =
∆u
1
2 N
⎧
+
⎪⎨
⎪⎩
+ Ev
Eu
1
Es,u
N
N0
m=1
N
m=1
N
+
n=
n=m
N
n=1
n=m
m=1 n=1
C.2.2 QPSK modulation
x
n m
x
N x u,u
n m
u,u
2
2 x
2 N 2 x0 2 u,u
n m
v,u
2
x
n m 2
u,u
n m
u,u
2
e2j(φu φv) x
2 N 2 x0 2 u,u
⎫
⎪⎬
⎪⎭
n m
v,u
2
.
(C.12)
Finally, the variance of a Single-User DA ML FF phase estimator writes for
QPSK-modulated data symbols

172 Variance of DA ML FF phase estimators
σ
2 QP SK
∆u
=
1
2 N
N
+
m=1
+ Ev
Eu
1
Es,u
N0
+
n=
n=m
N
m=1 n=1
xn m
u,u
2
N
2 N 2 x0 u,u
N
xn m2
v,u
N
m=1 n=1
n=m
2
xn m
u,u
2 N 2 x0 2 . (C.13)
u,u
Again, the terms weighted by I 2 k
disappear when switching from BPSKto
QPSK-modulated data symbols.
2

Appendix D
First order statistics in a linear
channel
The purpose of this appendix is to develop the expressions of the first order
statistics of products involving data symbols and decisions present
in (5.8) and (5.8), considering a 2-user synchronous communication systems
transmitting over a linear channel. The following developments will
be limited to Signal ¢ Signal first order statistics since these are the only
statistics required to compute (5.8) and (5.8). Indeed, products involving
Noise disappear from these relations thanks to the constellation symmetry
[87].
Using the Rice component of the additive noise samples ν m k and νm k
given by (3.80) and with the following definitions
¯ BPSK
¯ QPSK
X ¦ = x 0 u,u cos ∆u ¦
Y ¦ =
Xk = x 0 u,u cos
Eu
Ev
π
4
k 1
+( 1) 2
Ev
Eu
x 0 u,v cos (δv,u +∆u) (D.1)
x 0 u,v cos (δu,v +∆v) ¦ x 0 v,v cos ∆v (D.2)
k 1
+( 1) 4 ∆u
Ev
Eu
x 0 u,v cos
π
4 +( 1)k 1
(δv,u +∆u)
(D.3)

174 First order statistics in a linear channel
Yk =
where k =1...8.
Eu
Ev
x 0 v,u cos
π
4
k 1
+( 1) 2 x 0 v,v cos
k 1
+( 1) 4 (δu,v +∆v)
π
4 +( 1)k 1 ∆v
(D.4)
the analytical expressions of the expectations appearing in (5.8) and (5.10)
can be derived. But before writing any expression, notice that the linearity
of the channel and the synchronous transmission modify the expression
of the matched filter output (3.7) in such a way that it only depends now
on the current symbols and is thus subject to MAI only
y m u = ejφuI m u x0u,u +e
Useful term, no ISI
jφv
Ev
Eu Im v x0u,v MAI
Additive noise
+ν m u
(D.5)
This writing, combined with the statistical independence between the data
flows of the users, drives to zero most of the expectations in (5.8) and
(5.10), namely those for which temporal indexes n and m differ (n m = 0).
As a result, only terms for which n = m are non zero. Their computation
follows.
D.1 Expectations of data ¢ decision products
Data ¢ decision products come from the expansion of the matched filter
output. Two different kinds of products can be considered. The first one
involves only signals from the user of interest and is thus the useful contribution.
On the other hand, the second one depends on User ¢ Interferer
products as a result of the MAI introduced through the matched filter output.
D.1.1 User ¢ User
Same I/Q branch
Consider first the expectation of products involving data symbols and decisions
related to the same user over the same branch.

D.1 Expectations of data ¢ decision products 175
BPSK
QPSK
Cross-talk
E â m u an u
ˆΦ
n=m
=0, Φ=∆
= 1 P ν m u X+ P ν m u
E â m u a n
u ˆΦ
n=m
=0, Φ=∆
= 1
16
X
(D.6)
8
[1 2P (ν m u Xk)] (D.7)
k=1
E ˆb m
u b n
u
ˆΦ=0, Φ=∆
= 1
16
n=m
8
[1 2P (ν m u Xk)] . (D.8)
k=1
Moving to products involving contributions from different I/Q branches,
the derived expectations quantify the incidence of cross-talk. Obviously,
there are no contribution to take into account in the case of BPSK-modulated
data symbols. On the contrary, in the QPSK case
E â m u bn u
ˆΦ
n=m
=0, Φ=∆
= 1
8
4
[P (ν m u Xk+4) P (ν m u Xk)] (D.9)
k=1
E ˆb m
u a n
u
ˆΦ=0, Φ=∆
= 1
8
n=m
4
[P (ν m u Xk) P (ν m u Xk+4)] . (D.10)
k=1
Having dealt so far with the useful User ¢ User products only, it is time to
consider User ¢ Interferer contributions due to MAI.

176 First order statistics in a linear channel
D.1.2 User ¢ Interferer
Similarly to the previous section, several combinations will be considered,
either on the same branch or with cross-talk.
Same I/Q branch
On the same branch, the expectations write
BPSK
QPSK
E
E â m u anv
ˆΦ
n=m
=0, Φ=∆
= P ν m u X P ν m u
â m u a n
v ˆΦ
n=m
=0, Φ=∆
= 1
8
4
k=1
P ν m u X k 1
k+2 2
E ˆb m
u b n
v ˆΦ=0, Φ=∆
= 1
8
4
k=1
n=m
P ν m u X k 1
k+2 2
P
P
X+
ν m u X k+2 k 1
2
ν m u X k 1
k+2 2
where k represents the least integral value strictly greater than k
Cross-talk
(D.11)
(D.12)
(D.13)
Considering cross-talk, the QPSK case needs again to be studied alone.
The expectations of User ¢ Interferer products are given by
E â m u bnv
ˆΦ
n=m
=0, Φ=∆
= 1
8
4
k=1
P ν m u Xk+2 k 1 1 2
P
ν m u X 3
k 1
+2( 1)k
2
(D.14)

D.2 Expectations of decision ¢ decision products 177
E ˆb m
u a n
v ˆΦ=0, Φ=∆
= 1
8
4
k=1
n=m
P ν m
u X k 1 3 +2( 1)k
2
P ν m
u X k 1
k+2 1 .
2
(D.15)
These were the expectations quantifying the incidence of the MAI introduced
in the system through the matched filter output. On the other hand,
the incidence of the MAI mitigation has now to be computed. It depends
on decision ¢ decision products.
D.2 Expectations of decision ¢ decision products
Dealing with MAI mitigation terms, the current section has only to consider
User ¢ Interferer products, first on the same branch, then with crosstalk,
and always in both BPSK and QPSK cases if applicable.
D.2.1 Same I/Q branch
The expectations of decision ¢ decision products related to a unique branch
write
BPSK
E â m u ânv ˆΦ =0, Φ=∆
n=m
= 2P (νm u X + ) P (νm u X + ) P (νm u X + ) P (νm u X + )
2P (νm u X ) P (νm u X ) P (νm u X ) P (νm u X )
+1
(D.16)
QPSK
E â m u â n
v ˆΦ
n=m
=0, Φ=∆
= 1
16
8
k=1
4P (ν m u Xk) P (ν m v Yk)
2P (ν m u Xk) 2P (ν m v Yk)+1
(D.17)

178 First order statistics in a linear channel
E ˆb mˆn u bv ˆΦ=0,
n=m
Φ=∆
= 1
8
4P (νm u Xk) P (ν
16
m v Yk)
2P (νm u Xk) 2P (νm v Yk)+1
k=1
D.2.2 Cross-talk
. (D.18)
Considering cross-talk, the corresponding expectations become
E â m u ˆb n
v ˆΦ=0,
n=m
Φ=∆
= 1
⎧
4
⎪⎨
4P (νm u Xk) P
16
k=1 ⎪⎩
νm
v Y5+(k%4) 2P (νm u Xk) 2P νm
v Y5+(k%4) +1
4P νm u X5+(k%4) P (νm v Yk)
2P νm ⎫
⎪⎬
⎪⎭
u X5+(k%4) 2P (νm v Yk)+1
(D.19)
E ˆb m
u â n
v ˆΦ=0,
n=m
Φ=∆
= 1
⎧
4
⎪⎨
4P (νm v Yk) P
16
k=1 ⎪⎩
νm v X
5+(k%4)
2P (νm v Yk) 2P νm u X
5+(k%4) +1
4P νm v Y5+(k%4) P (νm u Xk)
2P νm v Y ⎫
⎪⎬
⎪⎭
5+(k%4) 2P (νm u Xk)+1
(D.20)
where k%4 is the rest of the division of k by 4.
D.3 Conclusion
Comparing the expectations of data ¢ decision products (D.6-D.15) to
the ones of decision ¢ decision products (D.16-D.20), it appears that the
former only depends on X while the latter also depends on Y . As
far as open-loop performance is concerned, contributions from the matched
filter output are thus only a function of the phase estimation error
related to the user of interest, while MAI mitigation terms are sensitive to
estimation errors related to both user and interferer(s).

Appendix E
Expectations for DD FB
open-loop performance
evaluation
The present appendix describes the calculation of the expectations of products
between true data symbols and decisions. Such expectations are
encountered in the derivation of the open-loop performance of DD FB estimators.
They depend on the data symbols in such a way that, at first
sight, their computation would require to consider all hypotheses regarding
the messages sent. Fortunately, the use of the characteristic functions
helps to avoid this time-consuming approach.
E.1 BPSK Modulation
⋆ E Îm u In
v ˆ ⋆Î
Φ=0, Φ=∆ and E Îm n u v ˆ
Φ=0, Φ=∆ will be derived
in this section for BPSK modulated data symbols. Since the modulation
is binary, there is neither conjugate in the argument of the expectation
nor quadrature component in data symbols and hard decisions. Thus,
⋆ m
E Îu I n
v ˆ
Φ=0, Φ=∆ = E â m u anv ˆ
Φ=0, Φ=∆ (E.1)
⋆Î
m n E Îu v ˆ
Φ=0, Φ=∆ = E â m u â n v ˆ
Φ=0, Φ=∆ . (E.2)

180 Expectations for DD FB open-loop performance evaluation
E.1.1 Derivation of E âm u an v ˆ
Φ=0, Φ=∆
Using A p
k introduced in (3.83) which, in the current BPSK context (no information
on the Q-branch), writes
A p
k =
+
q=
a q q
kRp k,k +
Nu
l=1
l=k
+
the expectation E âm u an v ˆ
Φ=0, Φ=∆ becomes
E
â m u anv ˆ
Φ=0, Φ=∆
q=
a q
q
l
Rp
k,l + e j ˆ φk p
νk
= E sign (A m u ) a n v ˆ
Φ=0, Φ=∆
= 1
⎧
Pr A
⎪⎨
2
⎪⎩
m u > 0 an v =1, ˆ
Φ=0, Φ=∆
Pr Am u < 0 anv =1, ˆ
Φ=0, Φ=∆
Pr Am u > 0 anv = 1, ˆ
Φ=0, Φ=∆
+Pr Am u < 0 an v = 1, ˆ ⎫
⎪⎬
⎪⎭
Φ=0, Φ=∆
(E.3)
(E.4)
(E.5)
where an 1
v = ¦1 with equal probability 2 . In order to derive the probabilities
in (E.5), the pdf of Am u is requested. It will then be integrated over
[ , 0] and [0, + ]. This pdf can be derived as the inverse Fourier transform
of the characteristic function
ψA m u (ωu) =E e jAm u ωu . (E.6)
Switching the inverse Fourier transform and the integration over [ , 0]
and [0, + ] and using the fact that
+
0
0
e jEω dE = 1
+ πδ (ω) (E.7)
jω
e jEω dE = πδ (ω)
1
jω
(E.8)

E.1 BPSK Modulation 181
leads to a new writing of (E.4)
E â m u a n v ˆ
Φ=0, Φ=∆
=
1
⎡
+
⎢ ψA
⎢
2jπ ⎣
m
ωu a u
n v =1, ˆ
dωu
Φ=0, Φ=∆ ωu
+
ωu an v = 1, ˆ Φ=0, Φ=∆
ψA m u
dωu
ωu
⎤
⎥ . (E.9)
⎦
Introducing (E.3) into (E.6) enables to write the characteristic function in
open-loop conditions as follows
ψAm u (ωu a n v = ¦1, ˆ Φ=0, Φ=∆)
⎧ ⎧ ⎡ +
⎪⎨ ⎪⎨ ⎢
a
⎢ p=
= E exp jωu ⎢
⎣
⎪⎩ ⎪⎩
p m p
uRu,u + Nu +
= exp
⎡
⎢
⎣
+
p=
p=n
σ2 (νu) ω2 u
2
cos ωuR
k=1
k=u
p=
m n
¦ jωuRu,v m p
u,v
⎤ ⎡
Nu ⎥
⎢
⎦ ⎣
k=1
k=u
where, in open-loop, (3.77) turns into
p q
Rk,l =
El
Ek
a p p
kRm u,k + (νm u )
q=
+
cos ωuR
m q
u,k
e j(δk,l+∆k) x p q
k,l
⎤⎫
⎥⎪⎬
⎥
⎥
⎦
⎪⎭
Finally, inserting (E.11) into (E.9) produces the expectation
E â m u a n v ˆ
Φ=0, Φ=∆
= 1
π
+
exp
⎡
⎣ +
p=
p=n
σ2
(νu) ω2
u
2
cos ωuR
m n
u,v
sin ωuR
⎤ ⎡
⎦ ⎣ Nu
m p
u,v
k=1
k=v
q=
+
a n v
⎫
⎪⎬
= ¦1
⎪⎭
(E.10)
⎤
⎥
⎦ (E.11)
. (E.12)
cos ωuR
m q
u,k
⎤
⎦ dωu
ωu .
(E.13)

182 Expectations for DD FB open-loop performance evaluation
(E.13) applies to the multiuser context under investigation. To validate it,
one can move to the situation studied in [87], i.e. a single user transmission
(u = v, El =0 l = u) over an AWGN channel (xm n
u,u = x0u,uδ(m n)).
This produces
E â m u amu ˆ
Φ=0, Φ=∆
= 1
π
+
exp
σ2 (νu) ω2 u
2
sin ωu cos ∆ux 0 dωu
u,u . (E.14)
Using [120, p. 123], (E.14) finally turns into the same result as in [87]
E â m u a m u ˆ
Φ=0, Φ=∆
=1 2Q
⎛
⎝ cos ∆ux 0 u,u
σ 2 (νu)
with Q (x) defined in (3.89). The reader can also notice that
E â m u amu ˆ
Φ=0, Φ=∆
ωu
⎞
⎠ (E.15)
= (1 PE)+( 1) PE (E.16)
= 1 2 PE (E.17)
where PE represents the BPSK error probability in AWGN channels as a
function of the phase misalignment ∆u. If∆u =0, PE becomes [121, p. 94]
PE = Q σ 1
(ν) = Q
2Eb
. (E.18)
N0
E.1.2 Derivation of E âm u ân v ˆ
Φ=0, Φ=∆
The derivation of E âm u ân v ˆ
Φ=0, Φ=∆ is a little more intricate but fol-
lows the same procedure as in the previous section. Am u and Anv being the
arguments of the discriminating functions producing hard decisions âm u

E.1 BPSK Modulation 183
and ân
v , the expectation E âm u ân v ˆ
Φ=0, Φ=∆ becomes
E â m u â n v ˆ
Φ=0, Φ=∆
= E sign (A m u )sign(A n v ) ˆ
Φ=0, Φ=∆
(E.19)
= Pr A m u > 0,Anv > 0 ˆ
Φ=0, Φ=∆
Pr A m u > 0,Anv < 0 ˆ
Φ=0, Φ=∆
Pr A m u < 0,Anv > 0 ˆ
Φ=0, Φ=∆
+Pr A m u < 0,A n v < 0 ˆ
Φ=0, Φ=∆ . (E.20)
Once again, the characteristic function ψA m u ,A n v (ωu,ωv) is of great help to
derive the probabilities in (E.20). Following the steps which lead from
(E.5) to(E.9), one gets
E
â m u â n v ˆ
Φ=0, Φ=∆
= 1
π 2
+
+
ψA m u ,An v
+ (ν n v )
ωu,ωv ˆ
dωu dωv
Φ=0, Φ=∆
ωu ωv
(E.21)
where the characteristic function writes in open-loop conditions
ψAm u ,An
ωu,ωv v
ˆ
Φ=0, Φ=∆
= E e j(Amu ωu+An
v ωv)
ˆ
Φ=0, Φ=∆
(E.22)
=
⎧ ⎧ ⎡
+
⎢ a
jωu ⎣ p=
⎪⎨ ⎪⎨
E exp
⎪⎩ ⎪⎩
p m p
uRu,u + Nu +
k=1 p=
k=u
a p
+ (ν
p
kRm u,k
m u )
⎡
+
⎢ a
+jωv ⎣ q=
⎤
⎥
⎦
q n q
vRv,v + Nu +
l=1 q=
l=v
a q
⎫⎫
⎪⎬ ⎪⎬
⎤
q
l
Rn
v,l ⎥
⎦
⎪⎭ ⎪⎭
= exp
Nu
k=1 p=
1
2
cos ωuR
+
σ 2 (νu) ω2 m n
u +2ρu,v ωuωv + σ 2 (νv) ω2 v
m p
u,k
(E.23)
n p
+ ωvRv,k
. (E.24)

184 Expectations for DD FB open-loop performance evaluation
Using (E.24) in(E.21) finally gives the expectation
E â m u â n v ˆ
Φ=0, Φ=∆
= 1
π 2
+
+
exp
Nu
k=1 q=
1
2 σ2 (νu) ω2 u
+
E.2 QPSK Modulation
cos ωuR
+2ρm n
u,v ωuωv + σ2 (νv) ω2
v
m q
u,k
+ ωvR n q
v,k
dωu
ωu
dωv
ωv .
(E.25)
Dealing now with QPSK modulated data symbols, both in-phase and quadrature
components have to be taken into account so that
⋆ m
E Îu I n v ˆ
Φ=0, Φ=∆
= E â m u a n v ˆ
Φ=0, Φ=∆ + E ˆb m
u b n
v ˆ
Φ=0, Φ=∆
+j E â m u b n v ˆ
Φ=0, Φ=∆ E ˆb m
u a n
v ˆ
Φ=0, Φ=∆
(E.26)
⋆Î m n
E Îu v ˆ
Φ=0, Φ=∆
= E â m u â n v ˆ
Φ=0, Φ=∆ + E ˆb mˆn u bv ˆ
Φ=0, Φ=∆
+j E â m u ˆb n
v ˆ
Φ=0, Φ=∆ E ˆb m
u â n
v ˆ
Φ=0, Φ=∆ .
(E.27)
It can be shown that the computation of the eight expectations listed here
above is unnecessary. Thanks to the symmetry of the QPSK constellation,
the following relationships apply
E
E
â m u a n v ˆ
Φ=0, Φ=∆
â m u b n v ˆ Φ=0, Φ=∆
= E ˆb m
u b n
v ˆ
Φ=0, Φ=∆ (E.28)
= E ˆb m
u a n
v ˆ
Φ=0, Φ=∆ . (E.29)
Analogous relationships may be written regarding the expectation of the
product of decisions. Thus, only four expectations will be derived in the
next sections.

E.2 QPSK Modulation 185
E.2.1 Derivation of E âm u an v ˆ
Φ=0, Φ=∆
The derivation of E âm u an v ˆ
Φ=0, Φ=∆ in a QPSK environment is very
similar to the one lead in Section E.1.1 for BPSK. The main difference lies in
the argument A p
k (3.83) of the decision function âp
k =
Ô
2
2 sign A p
k . Unsurprisingly,
cross-talk from quadrature components into in-phase decisions
of the same user appear due to the phase misalignment of oscillators.
Moreover, MAI comes out from both in-phase and quadrature components.
Thus, the characteristic function in open-loop conditions now writes
ωu a n Ô
2
v = ¦
2 , ˆ
Φ=0, Φ=∆
⎧ ⎧ ⎡ +
a
⎪⎨ ⎪⎨
⎢ p=
⎢
= E exp jωu ⎢
⎣
⎪⎩ ⎪⎩
p m p
uRu,u f p uI
+ Nu +
a
k=1 p=
k=u
p p
kRm ψA m u
= exp
⎡
Nu
⎢
⎣
k=1
k=v
q=
σ2 (νu) ω2 u
+
2
cos ωuR
+ (ν m u )
m n
¦ jωuRu,v m q
u,k
⎡
⎢
⎣
cos ωuI
m p
u,u
u,k b p
p
kIm u,k
+
p=
p=n
m q
u,k
cos ωuR
m p
u,v
⎤⎫
⎥
⎥⎪⎬
⎥
⎥ a
⎥
⎦
⎪⎭
n ⎫
Ô ⎪⎬
2
v = ¦
2
⎪⎭
cos ωuI
(E.30)
⎤
⎥
⎦
m p
u,v
⎤
⎥
⎦ . (E.31)
Comparing (E.11) and (E.31), the reader can notice the dependency of the
p q
characteristic function on cosines functions of Ik,l introduced in (3.78). In
p q
open-loop conditions, considering QPSK, R
p q
Rk,l p q
Ik,l =
=
Ô
2
2
Ô
2
2
El
Ek
El
Ek
k,l
and Ip q
k,l write
e j(δk,l+∆k) x p q
k,l
e j(δk,l+∆k) x p q
k,l
(E.32)
. (E.33)

186 Expectations for DD FB open-loop performance evaluation
Finally, the expectation writes
E â m u a n v ˆ
Φ=0, Φ=∆
= 1
2π
+
exp
⎡
⎣ +
p=
p=n
⎡
⎣ Nu
k=1 q=
k=v
σ2
(νu) ω2
u
2
cos ωuR
+
sin
ωuRm n
u,v cos ωuI
m p
cos ωuI ⎤
⎦
m p
u,v
cos ωuR
m q
u,k
u,v
cos ωuI
m q
u,k
m n
u,v
⎤
⎦ dωu
ωu .
(E.34)
(E.34) was validated in the same way as done previously with (E.13). In
the context described in Section E.1.1, the expectation reduces to
E
â m u am u ˆ Φ=0, Φ=∆
= 1
2π
+
exp
cos
σ2 (νu) ω2 Ô
u
2
2 sin
Ô 2
2 sin ∆ux 0 u,uωu
2 cos ∆ux 0 u,uωu
dωu
which, after integration [120, p. 123], leads to
E â m u amu ˆ
Φ=0, Φ=∆
= 1
⎧
⎨
2 ⎩ Q
⎡
⎣ x0u,u cos ∆u + 3π
⎤ ⎡
4
⎦ Q ⎣ x0u,u cos ∆u + π
⎤⎫
⎬
4
⎦
⎭ .
σ 2 (νu)
ωu
σ 2 (νu)
(E.35)
(E.36)
This is exactly the relation (36) presented in [87]. Moreover, noticing that
E â m u a m u ˆ
Φ=0, Φ=∆
=
Ô 2
2
2
(1 PE)
Ô 2
2
2
PE
=
(E.37)
1
2 (1 2PE) (E.38)

E.2 QPSK Modulation 187
setting ∆u =0turns (E.36) into (E.18), the BPSK error probability which is
to be understood here as the bit error probability affecting each branch of
a QPSK constellation in an AWGN channel.
E â m u a m u ˆ
Φ=0, Φ=∆
1
2
= 1
1 2Q 2σ
2
2 (ν) (E.39)
= 1
2Eb
1 2Q
(E.40)
2
N0
2Eb
PE = Q . (E.41)
N0
E.2.2 Derivation of E âm u bnv ˆ
Φ=0, Φ=∆
Since the decision with which it has been dealt in this section is the same
as the one used in the previous section, the characteristic function used
to derive the expectation is still ψA m u (ωu). However, its inverse Fourier
transform is now conditioned on b n v , leading to
E
=
â m u bnv ˆ
Φ=0, Φ=∆
1
4jπ
⎡
+
⎢
⎣ +
ψA m u
ψA m u
ωu bn Ô
2
v = 2 , ˆ
dωu
Φ=0, Φ=∆ ωu
ωu b n v =
Ô 2
2 , ˆ Φ=0, Φ=∆
dωu
ωu
⎤
⎥ . (E.42)
⎦
Inserting (E.31) in(E.42) produces a slightly modified version of (E.34)
E â m u b n v ˆ
Φ=0, Φ=∆
= 1
2π
+
exp
⎡
⎣ +
p=
p=n
⎡
⎣ Nu
k=1 q=
k=v
σ2 (νu) ω2
u
2
cos ωuR
+
sin
ωuI m n
u,v cos ωuR
m p
cos ωuI ⎤
⎦
m p
u,v
cos ωuR
m q
u,k
u,v
cos ωuI
m q
u,k
m n
u,v
⎤
⎦ dωu
ωu .
(E.43)

188 Expectations for DD FB open-loop performance evaluation
which, adapted to the conditions of the validation test, brings out
E â m u b m u ˆ
Φ=0, Φ=∆
= 1
⎧ ⎡
⎨
1 Q ⎣
2 ⎩ x0u,u cos ∆u + 3π
⎤ ⎡
4
⎦ Q ⎣ x0u,u cos ∆u + π
⎤⎫
⎬
4
⎦
⎭ .
σ 2 (νu)
σ 2 (νu)
(E.44)
Again, it is exactly the same relation as the one presented in [87] under
reference (37). Moreover, the reader can notice that (E.44) is equal to zero
if there is no phase error (∆u =0). In this case there is no cross-talk between
in-phase and quadrature components. Decisions made on one Rice
component no longer depend on the other component.
E.2.3 Derivation of E âm u ânv ˆ
Φ=0, Φ=∆
Like in Section E.1.2, the arguments of two decisions functions of the type
(3.83) need to be taken into account. Building up the expectation according
to (E.20), the characteristic function ψA m u ,A n v (ωu,ωv) is derived by applying
the same procedure as before
ψAm u ,An v (ωu,ωv ˆ Φ=0, Φ=∆)
= E e j(Amu ωu+An
v ωv)
ˆ
Φ=0, Φ=∆
⎧ ⎧ ⎡ +
a
⎢ p=
⎢
jωu ⎢
⎣
⎪⎨ ⎪⎨
= E exp
⎪⎩ ⎪⎩
p m p
uRu,u f p uI
+ Nu +
a
k=1 p=
k=u
p p
kRm + (νm u )
⎡ +
a
⎢ q=
⎢
+jωv ⎢
⎣
q n q
vRv,v b q vI
+ Nu +
a
l=1 q=
l=v
q q
l
Rn
+ (ν n v )
m p
u,u
u,k b p
p
kIm u,k
n q
v,v
v,l b q
q
l
In
v,l
(E.45)
⎤ ⎫⎫
⎥
⎦
⎪⎬ ⎪⎬
⎤ (E.46)
⎥
⎦
⎪⎭ ⎪⎭

E.2 QPSK Modulation 189
= exp
Nu
k=1 p=
1
σ
2
2 (νu) ω2 n
u +2ρm u,v ωuωv + σ 2 (νv) ω2
v
+
cos ωuR
m p
u,k
Using (E.47) in(E.21) finally gives
E
â m u ân v ˆ Φ=0, Φ=∆
=
1
2π 2
+
+
exp
⎡
⎣ Nu
k=1 q=
n p
+ ωvRv,k cos ωuI
1
2 σ2 (νu) ω2 u +2ρm n
m q
cos ωuRu,k +
cos ωuI
m p
u,k
E.2.4 Derivation of E âm u ˆb n
v ˆ
Φ=0, Φ=∆
m p
u,k
n p
+ ωvIv,k
.
u,v ωuωv + σ 2 (νv) ω2 v
n q
+ ωvRv,k n p
+ ωvIv,k Not only are the arguments of two decision functions, A p
k
⎤
⎦ dωu
ωu
(E.47)
dωv
ωv .
(E.48)
and Bp
k ,tobe
considered in this section, but the second one is related to the quadrature
component. This has been defined in (3.84). The characteristic function of
interest is thus ψA m u ,Bn v (ωu,ωv) which writes in open-loop conditions
ψAm u ,Bn v (ωu,ωv ˆ Φ=0, Φ=∆)
= E e j(Amu ωu+Bn
v ωv)
ˆ
Φ=0, Φ=∆
⎧ ⎧ ⎡ +
a
⎢ p=
⎢
jωu ⎢
⎣
⎪⎨ ⎪⎨
= E exp
⎪⎩ ⎪⎩
p m p
uRu,u f p
uI
+ Nu +
a
k=1 p=
k=u
p p
kRm + (νm u )
⎡ +
a
⎢ q=
⎢
+jωv ⎢
⎣
q n q
vIv,v + b p
n q
vRv,v + Nu +
a
l=1 q=
l=v
q
q
q
l
In
v,l + bq
l
Rn
v,l
+ (νn v )
m p
u,u
u,k b p
p
kIm u,k
(E.49)
⎤ ⎫⎫
⎥
⎦
⎪⎬ ⎪⎬
⎤ (E.50)
⎥
⎦
⎪⎭ ⎪⎭

190 Expectations for DD FB open-loop performance evaluation
= exp
Nu
k=1 p=
1
σ
2
2 (νu) ω2 n
u +2ρm u,v ωuωv + σ 2 (νv) ω2
v
+
cos ωuI
m p
u,k
Using (E.51) in(E.21) finally gives
E â m u ˆb n
v ˆ
Φ=0, Φ=∆
=
1
2π2 + +
exp 1
⎡ 2
⎣ Nu + cos
k=1 q= cos
n p
m p
+ ωvRv,k cos ωuRu,k ωvI
n p
v,k
σ2 (νu) ω2 u +2ρm n
u,v ωuωv + σ2 (νv) ω2 v
⎤
m q n q
ωuRu,k + ωvI
v,k ⎦
m p n p
ωuIu,k ωvRv,k dωu
ωu
.
(E.51)
dωv
ωv .
(E.52)

Appendix F
Expressions of U u,DD derived
in the reciprocal space
The general expressions of the mean Uu,DD of the error signal u m u,DD driving
a multiuser DD phase recovery loop, obtained in the reciprocal space,
are presented in this appendix for both BPSK- and QPSK-modulated data
symbols.
F.1 BPSK modulation
Inserting the results of Appendix E into (5.8) leads to
U BPSK
u,DD (∆)
= 1
π
+
p=
ej∆ux p
u,u
+
N0x0 u,u
4EuT ω2
a sin (ωaR p u,u)
exp
+
cos (ωaR q u,u)
q=
q=p
Nu
l=1
l=u
+
q=
cos
ωaR q
u,l
dωa
ωa

192 Expressions of Uu,DD derived in the reciprocal space
+ 1
π
+ 1
π 2
Nu
l=1
l=u
Nu
l=1
l=u
+
p=
+
p=
El
Eu
e j(δl,u+∆u)
p
xu,l +
exp
N0x0 u,u
4EuT ω2
a sin ωaR p
u,l
+
cos ωaR q
u,l
q=
q=p
Nu
k=1
k=l
+
q=
cos ωaR q
dωa
u,k ωa
El
Eu
e j(δl,u+∆u ∆l) p
x
+
+
⎧
⎪⎨
exp
⎪⎩
Nu
k=1 q=
N0
4T
+
⎡
⎢
⎣
cos
u,l
x0 u,u
Eu ω2 a
+2 (xpu,v) Ô ωaωb
EuEl
+ x0 l,l
El ω2 b
ωR q
u,k
⎤⎫
⎥⎪⎬
⎥
⎦
⎪⎭
p q
+ ωbRl,k dωa
ωa
dωb
ωb
.
(F.1)
The writing of (F.1) is not appropriate for numerical integration due to the
presence of the integration variable in the denominator of the integrand.
Applying a classic change of variable (Ω =lnω), another writing of (F.1)is
obtained, avoiding this shortcoming
U BPSK
u,DD (∆)
= 2
π
+
p=
ej∆u
m p
xu,u +
exp
N0x0 u,u
4EuT e2Ωa
sin eΩa
ej∆ux +
cos eΩa
ej∆u
m q
x
q=
q=p
Nu
l=1
l=u
+
q=
u,u
cos eΩa
El
Eu
m p
u,u
e j(δl,u+∆u) x m q
u,l
dΩa

F.1 BPSK modulation 193
+ 2
π
+ 2
π 2
2
π 2
Nu
l=1
l=u
Nu
l=1
l=u
Nu
l=1
l=u
p=
+
+
p=
+
p=
El
Eu
ej(δl,u+∆u) x
+
exp
N0x0 u,u
sin eΩa
El
+
q=
q=p
Nu
k=1
k=l
cos
+
q=
El
Eu
+
+
El
Eu
+
+
m p
u,l
4EuT e2Ωa
Eu
e Ωa
ej(δl,u+∆u)
m p
xu,l
ej(δl,u+∆u) x
El
Eu
cos eΩa
Ek
Eu
ej(δl,u+∆u ∆l) m p
x
⎧ u,l
x0 u,u
Eu e2Ωa
Nu
k=1 q=
⎪⎨
exp
⎪⎩
+
⎪⎨
exp
⎪⎩
N0
4T
m q
u,l
e j(δk,u+∆u) x m q
u,k
⎡
⎢
p
⎢ (xm
u,l )
⎢ +2 Ô e
⎣ EuEl
(Ωa+Ωb)
⎧
⎪⎨
cos
+ e
⎪⎩
Ωb
ej(δl,u+∆u ∆l) m p
x
⎧ u,l
x0 u,u
Eu e2Ωa
Nu
+
k=1 q=
N0
4T
⎧
⎡
⎢
⎣
⎪⎨
cos
⎪⎩
+ x0 l,l
El e2Ωb
eΩa
Ek
Eu
e j(δk,u+∆u) m q
xu,k
Ek
El
e j(δk,l+∆l) p q
xl,k 2
p
(xm
u,l )
Ô e
EuEl
(Ωa+Ωb)
+ x0 l,l
El e2Ωb
eΩa
Ek
Eu
e j(δk,u+∆u) m q
xu,k eΩb
Ek
El
e j(δk,l+∆l) p q
xl,k
dΩa
⎤⎫
⎥⎪⎬
⎥
⎦
⎪⎭
⎫
⎪⎬
⎪⎭
dΩadΩb
⎤⎫
⎥⎪⎬
⎥
⎦
⎪⎭
⎫
⎪⎬
⎪⎭
dΩadΩb.
(F.2)

194 Expressions of Uu,DD derived in the reciprocal space
F.2 QPSK modulation
Similarly, inserting the results of Appendix E into (5.10) leads to
QP SK
Uu,DD (∆)
= 1
π
+
p=
+ 1
π
+ 1
π 2
Nu
l=1
l=u
Nu
l=1
l=u
ej∆ux p
u,u
+
+
p=
+
p=
N0x0 u,u
4EuT ω2
a sin (ωaR p u,u)cos(ωaI p u,u)
exp
+
cos (ωaR q u,u)cos(ωaI q u,u)
q=
q=p
Nu
+
l=1 q=
l=u
El
Eu
+
cos ωaR q
u,l cos ωaI q
dωa
u,l ωa
e j(δl,u+∆u)
p
xu,l
exp
N0x0 u,u
4EuT ω2
a sin ωaR p
u,l cos
+
cos ωaR q
u,l cos ωaI q
u,l
q=
q=p
Nu
k=1
k=l
+
q=
El
Eu
e j(δl,u+∆u ∆l) p
x
+
+
⎧
⎪⎨
exp
⎪⎩
Nu
k=1 q=
ωaI p
u,l
cos ωaR q
u,k cos ωaI q
dωa
u,k ωa
+
N0
4T
u,l
x0 u,u
Eu ω2 a
+2 (xp
Ô u,l)
ωaωb
EuEl
⎡
⎢
⎣
+ x0 l,l
El ω2 ⎤⎫
⎥⎪⎬
⎥
⎦
⎪⎭
b
cos ωR q
p q
u,k + ωbRl,k
cos ωI q
p q
u,k + ωbIl,k dωa
ωa
dωb
ωb

F.2 QPSK modulation 195
1
π 2
Nu
l=1
l=u
+
p=
El
Eu
e j(δl,u+∆u ∆l) p
x
+
+
⎧
⎪⎨
exp
⎪⎩
Nu
+
k=1 q=
N0
4T
⎡ u,l
x
⎢
⎣
0 u,u
Eu ω2 a
2 (xp
Ô u,l)
ωaωb
EuEl
+ x0 l,l
El ω2 ⎤⎫
⎥⎪⎬
⎥
⎦
⎪⎭
b
cos ωI q
p q
u,k + ωbR
l,k
cos ωI q
p q
u,k ωbIl,k dωb
ωb .
dωa
ωa
(F.3)

Appendix G
COST 207 Channel Models
The COST 207 channel models are tapped delay lines modelling the behaviour
of a mobile radio channel in several typical environments. Usually,
one distinguishes the Rural Area (RA), the Typical Urban (TU), and
the Hilly Terrain (HT) environments. Values and spacing of their taps are
given in [97, Section 2.4.4]. Their impulse responses are illustrated hereafter.
Power [W]
1
0.8
0.6
0.4
0.2
COST 207 Rural Area (RA) channel impulse response
−1 0 1 2 3 4 5 6
x 10 −7
0
Time [s]
Figure G.1: The Rural Area (RA) channel model is made of 6 taps and its
power delay profile spreads over 0.5 µs.

198 COST 207 Channel Models
Power [W]
1
0.8
0.6
0.4
0.2
COST 207 Typical Urban (TU) channel impulse response
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10 −6
0
Time [s]
Figure G.2: The Typical Urban (TU) channel model is made of 12 taps and
its power delay profile spreads over 5 µs
Power [W]
1
0.8
0.6
0.4
0.2
COST 207 Hilly Terrain (HT) channel impulse response
0 1 2
x 10 −5
0
Time [s]
Figure G.3: The Hilly Terrain (HT) channel model is made of 12 taps and
its power delay profile spreads over 20 µs.

Appendix H
Curriculum vitae
Laurent Schumacher
Born in Mons, Belgium, on April 28th, 1971.
Education
1994 – 1999 Ph.D. in Applied Sciences – Université catholique de
Louvain
Thesis: About Maximum-Likelihood Phase Estimation in
DS-CDMA Communication Systems
Supervisor: Prof. L. Vandendorpe.
1988 – 1993 Electrical Engineer – Faculté Polytechnique de Mons,
Belgium
Orientation: Telecommunications
Masters thesis: Segmentation of cursive writing
Supervisor: Prof. H. Leich.
Professional experience
October 1995 –
September 1999
October 1994 –
September 1995
Degree candidate, Fonds National de la Recherche
Scientifique, Université catholique de Louvain,
Communication and Remote Sensing Laboratory.
Scholar, Fonds National de la Recherche Scientifique,
Université catholique de Louvain, Communication
and Remote Sensing Laboratory.

200 Curriculum vitae
August 1993 –
September 1994
July 1992 –
September 1992
August 1991 –
September 1991
Publications
Refereed conference papers
Research assistant, Université catholique de
Louvain, Communication and Remote Sensing
Laboratory.
Trainee, IBM Danmark A/S.
Trainee, Generale Bank, Human Resources and IT
Departments.
L. Schumacher and L. Vandendorpe, ”Maximum likelihood joint phase estimators
in CDMA communications systems”, Proc. IEEE Third Symposium
on Communications and Vehicular Technology in the Benelux, Eindhoven, The
Netherlands, October 1995, pp. 76-82.
L. Schumacher and L. Vandendorpe, ”Open loop analysis of maximumlikelihood
decision-directed phase estimation in CDMA communication
systems with QPSK modulation”, Proc. IEEE Fourth Symposium on Communication
and Vehicular Technology in the Benelux, Ghent, Belgium, October
1996, pp. 114-121.
L. Vandendorpe and L. Schumacher, ”Maximum likelihood data-aided
phase estimators in CDMA communication systems with QPSK modulation”,
Proc. IEEE Globecom’96 Communication Theory - Mini-Conference,
London, United Kingdom, November 17-22, 1996, pp. 219-223.
L. Schumacher and L. Vandendorpe, ”MAI Mitigation in DA ML Carrier
Phase Recovery Loops for DS-CDMA Systems”, Proc. IEEE Vehicular Technology
Conference VTC 1999-Fall, Amsterdam, The Netherlands, September
19-22, 1999, pp. 1850-1854.
Submitted papers
L. Schumacher and L. Vandendorpe, ”MAI Mitigation in DA ML Carrier
Phase Recovery Loops for DS-CDMA Systems”, submitted to IEEE Transactions
on Communications, April 1999.

201
L. Schumacher and L. Vandendorpe, ”Performance study of DD ML Phase
Estimators for DS-CDMA Communications Systems”, submitted to EU-
SIPCO 2000 - European Signal Processing Conference, Tampere, Finland, September
4-9, 2000.

Bibliography
[1] Gilder Technology Group Inc. Gilder: The Gilder Technology Report.
[online]. No Date [Cited 6 August 1999]. Available from:
.
[2] George Gilder. Telecosm and Beyond: Over the Paradigm Cliff. Forbes
ASAP, February 1997. [Cited 9 August 1999] Available from:
.
[3] Bernard Guidon. Next-generation telecommunication networks
take shape. Telecommunications News, September
1998. [Cited 9 August 1999] Available from:
.
[4] Roger L. Peterson, Rodger E. Ziemer, and David E. Borth. Introduction
to Spread Spectrum Communications. Prentice Hall, 1995.
[5] Tero Ojanperä and Ramjee Prasad. An Overview of Air Interface
Multiple Access for IMT-2000/UMTS. IEEE Communications
Magazine, 36(9):82–95, September 1998.
[6] Raymond L. Pickholtz, Laurence B. Milstein, and Donald L.
Schilling. Spread Spectrum for Mobile Communications. IEEE
Transactions on Vehicular Technology, 40(2):313–322, 1991.
[7] John Proakis. Digital Communications. Mac Graw Hill, 1992.
[8] Ricardo De Gaudenzi, Filippo Gianetti, and Marco Luise. Signal
Synchronization for Direct-Sequence Code-Division Multiple Access
Radio Modems. European Transactions on Telecommunications,
9(1):73–89, February 1998.
[9] Andrew J. Viterbi. CDMA - Principles of Spread Spectrum Communication.
Addison-Wesley, 1995.

204 BIBLIOGRAPHY
[10] Dilip V. Sarwate and Michael B. Pursley. Crosscorrelation Properties
of Pseudorandom and Related Sequences. Proceedings of the IEEE,
68(5):593–619, May 1980.
[11] William C. Y. Lee. Overview of Cellular CDMA. IEEE Transactions
on Vehicular Technology, 40(2):291–302, May 1991.
[12] Eric Lawrey. The suitability of OFDM as a modulation technique for
wireless telecommunications, with a CDMA comparison. Master’s
thesis, James Cook University of North Queensland, Australia, 1997.
[online]. 18 October 1997 [Cited 13 August 1999]. Available from:
.
[13] Klein S. Gilhousen, Irwin M. Jacobs, Roberto Pavodani, Andrew J.
Viterbi, Lindsay A. Weaver, Jr., and Charles E. Wheatley III. On the
Capacity of a Cellular CDMA System. IEEE Transactions on Vehicular
Technology, 40:303–312, May 1991.
[14] Ruxandra Lupas and Sergio Verdú. Linear Multiuser Detectors for
Synchronous Code-Division Multiple-Access Channels. IEEE Transactions
on Information Theory, 35(1):123–136, January 1989.
[15] ISO. ISO homepage. [online]. No date [Cited 3 August 1999]. Available
from: .
[16] ITU. International Mobile Telecommunications IMT-2000. [online].
28 July 1999 [Cited 3 August 1999]. Available from:
.
[17] Ermanno Berruto, Mikael Gudmundson, Raffaele Menolascino,
Werner Mohr, and Marta Pizarroso. Research Activities on UMTS
Radio Interface, Network Architectures, and Planning. IEEE Communications
Magazine, 46(2):82–95, February 1998.
[18] CDG. CDMA Development Group. [online]. No date [Cited 3 August
1999]. Available from: .
[19] Douglas N. Knisely, Sarath Kumar, Subhasis Laha, and Sanjiv
Nanda. Evolution of Wireless Data Services: IS-95 to cdma2000.
IEEE Communications Magazine, 36(10):140–149, October 1998.
[20] ETSI. ETSI - Standardizing Telecommunications Products and Services.
[online]. No date [Cited 3 August 1999]. Available from:
.

BIBLIOGRAPHY 205
[21] Barry Miller. Satellites Free the Mobile Phone. IEEE Spectrum,
35(3):26–35, March 1998.
[22] A. Kajiwara. Mobile Satellite CDMA System Robust to Doppler
Shift. IEEE Transactions on Vehicular Technology, 44(3):480–486, August
1995.
[23] Arild Flystveit and Arvid Bertheau Johannessen. Global Mobile Personal
Communications by Satellite. Telektronikk, 94(2):22–33, 1998.
[24] Iridium LLC. :: Welcome to Iridium ::. [online].
No date [Cited 3 August 1999]. Available from:
.
[25] Globalstar. Globalstar. [online]. No date [Cited 6 April 1999]. Available
from: .
[26] Ellipso. Ellipso. [online]. No date [Cited 3 August 1999]. Available
from: .
[27] Teledesic LLC. Teledesic. [online]. July 1999 [Cited 3 August 1999].
Available from: .
[28] Skybridge. SkyBridge. [online]. No date [Cited 3 August 1999].
Available from: .
[29] Kaveh Pahlavan, Thomas H. Probert, and Mitchell E. Chase.
Trends in Local Wireless Networks. IEEE Communications Magazine,
33(3):88–95, March 1995.
[30] Kaveh Pahlavan and Prashant Krishnamurthy. Wideband Local
Access: Wireless LAN and Wireless ATM. IEEE Communications
Magazine, 35(11):34–40, November 1997.
[31] Richard O. LaMaire, Arvind Krishna, Pravin Bhagwat, and James
Panian. Wireless LANs and Mobile Networking: Standards and Future
Directions. IEEE Communications Magazine, 34(8):86–94, August
1996.
[32] Luis Correia and Ramjee Prasad. An Overview of Wireless Broadband
Communications. IEEE Communications Magazine, 35(1):28–33,
January 1997.

206 BIBLIOGRAPHY
[33] Norihiko Morinaga, Masao Nakagawa, and Ryuji Kohno. New Concepts
and Technologies for Achieving Highly Reliable and High-
Capacity Multimedia Wireless Communications Systems. IEEE
Communications Magazine, 35(1):34–40, January 1997.
[34] ETSI. BRAN Homepage. [online]. 2 July 1999 [Cited 3 August 1999].
Available from: .
[35] Dan Spoelman and Gary Law. Mapping Out A Modem Strategy.
CED Magazine, 24(4):72–92, April 1998.
[36] Terayon. S-CDMA, The Upstream Advantage. [online].
1998 [Cited 3 August 1999]. Available from:
.
[37] Matt Brandt. IEEE 802.14 Information. [online].
1998 [Cited 3 August 1999]. Available from:
.
[38] Sergio Verdú. Minimum Probability of Error for Asynchronous
Gaussian Multiple-Access Channels. IEEE Transactions on Information
Theory, 32(1):85–96, January 1986.
[39] Ruxandra Lupas and Sergio Verdú. Near-Far Resistance of Multiuser
Detectors in Asynchronous Channels. IEEE Transactions on
Communications, 38(4):496–508, April 1990.
[40] Stefan Parkvall, Erik G. Ström, and Björn Ottersten. The Impact of
Timing Errors on the Performance of Linear DS-CDMA Receivers.
IEEE Journal on Selected Areas in Communications, 14(8):1660–1668,
October 1996.
[41] Alexandra Duel-Hallen, Jack Holtzman, and Zoran Zvonar. Multiuser
Detection for CDMA Systems. IEEE Personal Communications,
2(2):46–58, April 1995.
[42] Shimon Moshavi. Multi-User Detection for DS-CDMA Communications.
IEEE Communications Magazine, pages 124–136, October 1996.
[43] Geert Leus and Marc Moonen. Multi-User Detection in Frequency-
Selective Fading Channels. In Seminar on Digital Signal Processing and
Wireless Communications, 28 May 1999, Leuven, Belgium, May 1999.

BIBLIOGRAPHY 207
[44] Michael Honig, Upamanyu Madhow, and Sergio Verdú. Blind Adaptive
Multiuser Detection. IEEE Transactions on Information Theory,
41(4):944–960, July 1995.
[45] Franco Mazzenga and Giovanni Emanuele Corazza. Blind Least-
Squares Estimation of Carrier Phase, Doppler Shift, and Doppler
Rate for m-PSK Burst Transmission. IEEE Communication Letters,
2(3):73–75, March 1998.
[46] Kaj Goethals. DA Chip Synchronizers for Bandlimited DS/SS M-
PSK Signals Using CDMA on Mobile Satellite Communications
Channels. In IEEE International Conference on Communications ICC’94,
1-5 May 1994, New-Orleans, United States of America, volume 2, pages
1150–1154, May 1994.
[47] Kaj Goethals and Marc Moeneclaey. NDA Chip Synchronizers for
Bandlimited DS/SS Signals Using CDMA on Nonselective Fading
Channels. In 3rd European Conference on Satellite Communications, 2-4
November 1993, Manchester, United Kingdom, pages 336–340, November
1993.
[48] Stephen E. Bensley and Behnaam Aazhang. Subspace-Based Estimation
of Multipath Channel Parameters for CDMA Communication
Systems. In Proceedings IEEE Telecommunications Conference, Communication
theory Mini-Conference Record, pages 154–158, 1994.
[49] Eric G. Ström, Stefan Parkvall, Scott L. Miller, and Björn E. Ottersten.
Propagation Delay Estimation in Asynchronous Direct-
Sequence Code-Division Multiple Access Systems. IEEE Transactions
on Communications, 44(1):84–93, January 1996.
[50] Meir Feder and Josko A. Catipovic. Algorithms For Joint Channel
Estimation and Data Recovery - Application to Equalization in
Underwater Communications. IEEE Journal of Oceanic Engineering,
16(1):42–55, January 1991.
[51] Todd K. Moon. The Expectation-Maximization Algorithm. IEEE
Signal Processing Magazine, pages 47–60, November 1996.
[52] Bernard H. Fleury, Martin Tschudin, Ralf Heddergott, Dirk Dalhaus,
and Klaus Ingeman Perdesen. Channel Parameter Estimation in Mobile
Radio Environments Using the SAGE Algorithm. IEEE Journal
on Selected Areas in Communications, 17(3):434–450, March 1999.

208 BIBLIOGRAPHY
[53] Simon Haykin. Adaptive Filter Theory. Prentice Hall International,
1991.
[54] Li-Chung Chu and Urbashi Mitra. Analysis of MUSIC-Based Delay
Estimators for Direct-Sequence Code-Division Multiple-Access Systems.
IEEE Transactions on Communications, 147(1):133–138, January
1999.
[55] Stephen E. Bensley and Behnaam Aazhang. Subspace-Based Channel
Estimation for Code Division Multiple Access Communication
Systems. IEEE Transactions on Communications, 44(8):1009–1020, August
1996.
[56] Eric G. Ström, Stefan Parkvall, Scott L. Miller, and Björn E. Ottersten.
DS-CDMA Synchronization in Time-Varying Fading Channels.
IEEE Journal on Selected Areas in Communications, 14(8):1636–
1642, August 1996.
[57] Hui Liu and Guanghan Xu. A Subspace-Method for Signature
Waveform Estimation in Synchronous CDMA Systems. IEEE Transactions
on Communications, 44(10):1346–1354, October 1996.
[58] Stephen E. Bensley and Behnaam Aazhang. Maximum-Likelihood
Synchronization of a Single User for Code-Division Multiple-Acces
Communication Systems. IEEE Transactions on Communications,
46(3):392–399, March 1998.
[59] Steven M. Kay. Fundamentals of Statistical Signal Processing. Prentice
Hall International, 1993.
[60] Jerry M. Mendel. Lessons in Estimation Theory for Signal Processing
Communications and Control. Prentice Hall, 1995.
[61] Teng Joon Lim and Lars K. Rasmussen. Adaptive Symbol and Parameter
Estimation in Synchronous Multiuser CDMA Detectors. IEEE
Transactions on Communications, 45(2):213–220, February 1997.
[62] Ronald A. Iltis. Joint Estimation of PN Code Delay and Multipath
Using the Extended Kalman Filter. IEEE Transactions on Communications,
38(10):1677–1685, October 1990.
[63] Ronald A. Iltis. An EKF-Based Joint Estimator for Interference, Multipath,
and Code Delay in a DS Spread-Spectrum Receiver. IEEE
Transactions on Communications, 42(2/3/4):1288–1299, April 1994.

BIBLIOGRAPHY 209
[64] Alfred W. Fuxjaeger and Roland A. Iltis. Adaptive Parameter Estimation
using Parallel Kalman Filtering for Spread Spectrum Code and
Doppler Tracking. IEEE Transactions on Communications, 42(6):2227–
2230, June 1994.
[65] Todd K. Moon, Zhenhua Xie, Craig K. Rushforth, and Robert T.
Short. Parameter Estimation in a Multi-User Communication System.
IEEE Transactions on Communications, 42(8):2553–2560, August
1994.
[66] F. Mazzenga and F. Valataro. Parameter Estimation in CDMA Multiuser
Detection Using Cyclostationnary Statistics. Electronic Letters,
32(3):179–181, February 1996.
[67] Alfred W. Fuxjaeger and Roland A. Iltis. Acquisition of Timing and
Doppler-Shift in a Direct-Sequence Spread-Spectrum System. IEEE
Transactions on Communications, 42(10):2870–2880, October 1994.
[68] Bernd Steiner and Peter Jung. Optimum and Suboptimum Channel
Estimation for the Uplink of CDMA Mobile Radio Systems with
Joint Detection. European Transactions on Telecommunications, 5(1):39–
50, January 1994.
[69] Hui Liu, Guanghan Xu, Lang Tong, and Thomas Kailath. Recent
developments in blind channel equalization: From cyclostationarity
to subspaces. Signal Processing, 50(1-2):83–99, April 1996.
[70] Xiaodong Wang and H. Vincent Poor. Blind Equalization and Multiuser
Detection in Dispersive CDMA Channels. IEEE Transactions on
Communications, 46(1):91–103, January 1998.
[71] Emre Akta and Urbashi Mitra. Blind Channel Estimation for Multiuser
CDMA Systems. In ICC ’98 Conference Record, volume 2, pages
1064–1068, June 1998.
[72] Urs Fawer and Behnaam Aazhang. A Multiuser Receiver for Code
Division Multiple Access Communications over Multipath Channels.
IEEE Transactions on Communications, 43(2/3/4):1556–1565,
1995.
[73] Andrej Radovic. An Iterative Near-Far Resistant Algorithm for
Joint Parameter Estimation in Asynchronous CDMA Systems. In
PIMRC’94, pages 199–203, 1994.

210 BIBLIOGRAPHY
[74] Dirk Dalhaus, Bernard H. Fleury, and Andrej Radović. A Sequential
Algorithm for Joint Parameter Estimation and Multiuser Detection
in DS/CDMA Systems with Multipath Propagation. Wireless Personal
Communications, pages 161–178, 1998.
[75] Stefan Parkvall and Erik G. Ström. Parameter Estimation and Detection
of DS-CDMA Signal subject to Multipath Propagation. In
Proceedings of IEEE/IEE Workshop on Signal Processing in Multipath Environments,
Glasgow, Scotland, 1995.
[76] Erik G. Ström and Stefan Parkvall. Joint Parameter Estimation and
Detection of DS-CDMA Signals in Fading Channels. In Proceedings
IEEE Global Telecommunications Conference, volume 2, pages 1109–
1113, 1995.
[77] Zhenhua Xie, Craig K. Rushforth, Robert T. Short, and Todd K.
Moon. Joint Signal Detection and Parameter Estimation in Multiuser
Communications. IEEE Transactions on Communications, 41(7):1208–
1216, August 1993.
[78] John R. Barry and Anuj Batra. A Multidimensionnal Phase-Locked
Loop for Blind Equalization of Multi-Input Multi-Output Channels.
In ICC ’96 Conference Record, volume 3, pages 1307–1312, 1996.
[79] Floyd M. Gardner. Phaselock Techniques. John Wiley & Sons, Inc.,
1979.
[80] L. E. Franks. Carrier and Bit Synchronization in Data Communication
- A Tutorial Review. IEEE Transactions on Communications,
28(8):1107–1120, August 1980.
[81] Umberto Mengali and Aldo N. D’Andrea. Synchronization Techniques
for Digital Receivers. Plenum press, New York, 1997.
[82] Andrew J. Viterbi and Audrey M. Viterbi. Nonlinear Estimation
of PSK-Modulated Carrier Phase with Application to Burst Digital
Transmission. IEEE Transactions on Information Theory, 29(4):543–551,
July 1983.
[83] Floyd Gardner. Demodulator Reference Recovery Techniques
Suited for Digital Implementation. ESTEC Contract No.
6487/86/NL/DG, European Space Agency, August 1988.

BIBLIOGRAPHY 211
[84] Thierry Jesupret, Marc Moeneclaey, and Gerd Ascheid. Digital Demodulator
Synchronization. ESTEC Contract No. 8437/89/NL/RE,
European Space Agency, June 1991.
[85] Heinrich Meyr, Marc Moeneclaey, and Stefan A. Fechtel. Digital
Communication Receivers - Synchronization, Channel Estimation, and
Signal Processing. Wiley Series in Telecommunications and Signal
Processing. John Wiley & Sons, Inc., 1998.
[86] Andrew J. Viterbi. Principles of Coherent Communications. McGraw-
Hill Book Company, 1966.
[87] Ricardo De Gaudenzi, Tobias Garde, and Vieri Vanghi. Performance
Analysis of Decision-Directed Maximum-Likelihood Phase Estimators
for M-PSK Modulated Signals. IEEE Transactions on Communications,
43(12):3090–3100, December 1995.
[88] Dae Sun Oh, Won Gi Jeon, Yong Soo Cho, Hyung Woon Park, and
Ki Ho Kim. Convergence Analysis of a PLL for a Digital Recording
Channel with an Adaptive Partial Response Equalizer. In IEEE
Globecom ’96, pages 979–983, November 1996.
[89] M. O˘guz Sunay and Peter J. McLane. Calculating Error Probabilities
for DS CDMA Systems: When Not to Use the Gaussian Approximation.
In IEEE Globecom ’96, pages 1744–1748, November 1996.
[90] Erik G. Ström and Fredrik Malmsten. Maximum Likelihood Synchronization
of DS-CDMA Signals Transmitted over Multipath
Channels. In ICC ’98 Conference Record, volume 3, pages 1546–1550,
June 1998.
[91] Michel B. Jeruchim, Philip Balaban, and K. Sam Shanmugan. Simulation
of Communication Systems. Plenum Press, 1992.
[92] Fu-Chun Zheng and Stephen K. Barton. On the Performance of
Near-Far Resistant CDMA Detectors in the Presence of Synchronization
Errors. IEEE Transactions on Communications, 43(12):3037–3045,
December 1995.
[93] Sergio Verdú. Optimum Multiuser Asymptotic Efficiency. IEEE
Transactions on Communications, 34(9):890–897, September 1986.

212 BIBLIOGRAPHY
[94] Bas W’t Hart, Richard D. J. Van Nee, and Ramjee Prasad. Performance
Degradation Due to Code Tracking Errors in Spread-Spectrum
Code-Division Multiple-Access Systems. IEEE Journal on Selected
Areas in Communications, 14(8):1669–1679, October 1996.
[95] Rick Cameron and Brian Woerner. Performance Analysis of CDMA
with Imperfect Power Control. IEEE Transactions on Communications,
44(7):777–808, July 1996.
[96] Wei Huang, Ivan Andonovic, and Masao Nakagawa. PLL Performance
of DS-CDMA Systems in the Presence of Phase Noise, Multiuser
Interference, and Additive Gaussian Noise. IEEE Transactions
on Communications, 46(11):1507–1515, November 1998.
[97] COST 207 Management Committee. COST 207 Digital Land Mobile
Radio Communications - Final Report. Office for Official Publications
of the European Communities, 1989.
[98] Harry L. Van Trees. Detection, Estimation and Modulation Theory.
Wiley, 1968.
[99] M. H. Meyers and L. E. Franks. Joint Carrier Phase and Symbol
Timing Recovery for PAM Systems. IEEE Transactions on Communications,
28(8):1121–1129, August 1980.
[100] Marc Moeneclaey and Geert de Jonghe. Tracking Performance Comparison
of Two Feedforward ML-Oriented Carrier-Independent
NDA Symbol Synchronizers. IEEE Transactions on Communications,
40:1423–1425, September 1992.
[101] Thomas Alberty. Frequency Domain Interpretation of the Cramér-
Rao Bound for Carrier and Clock Synchronization. IEEE Transactions
on Communications, 43(2/3/4):1185–1191, April 1995.
[102] Aldo N. D’Andrea, Umberto Mengali, and Ruggero Reggiannini.
The Modified Cramér-Rao Bound and Its Application to
Synchronization Problems. IEEE Transactions on Communications,
42(2/3/4):1391–1399, March 1994.
[103] Marc Moeneclaey. On the True and the Modified Cramér-Rao
Bounds for the Estimation of a Scalar Parameter in the Presence
of Nuisance Parameters. IEEE Transactions on Communications,
46(11):1536–1544, November 1998.

BIBLIOGRAPHY 213
[104] Martin Oerder and Heinrich Meyr. Digital Filter and Square Timing
Recovery. IEEE Transactions on Communications, 36(5):605–612, May
1988.
[105] Kaj Goethals. ML-Oriented Symbol Synchronizers for Mobile Satellite
Communications Using Narrowband M-PSK. In 1st IEEE Symposium
on Communications and Vehicular Technology in the Benelux
SCVT’93, October 1993, Louvain-la-Neuve, Belgium, pages 1.3.1–1.3.8,
October 1993.
[106] Kaj Goethals and Marc Moeneclaey. ML-Oriented DA Symbol Synchronization
for Nonselective Fading Channels Using Imperfect
Channel Gain Estimates. In 4th International Workshop on Digital Signal
Processing Techniques, 26-28 September 1994, London, United Kingdom,
pages 2–8, September 1994.
[107] Marvin K. Simon and Dariush Divsalar. Some New Twists to Problems
Involving the Gaussian Probability Integral. IEEE Transactions
on Communications, 46(2):200–210, February 1998.
[108] Marvin K. Simon and Mohamed-Slim Alouini. A Unified Approach
to the Performance Analysis of Digital Communications over Generalized
Fading Channels. Proceedings of the IEEE, 86(9):1860–1877,
September 1998.
[109] Norman C. Beaulieu. The Evalutation of Error Probabilities for Intersymbol
and Cochannel Interference. IEEE Transactions on Communications,
39(12):1740–1749, December 1991.
[110] O. Shimbo and M. I. Celebiler. The Probability Of Error Due To Intersymbol
Interference And Gaussian Noise In Digital Communication
Systems. IEEE Transactions on Communications, 19(4):115–119, April
1971.
[111] Luc Vandendorpe and Olivier van de Wiel. Performance Analysis of
Linear Joint Equalization and Multiple Access Interference Cancellation
for Multitone CDMA. Wireless Personal Communications, 3(1-
2):17–36, February 1996.
[112] Carl W. Helstrom. Calculating Error Probabilities for Intersymbol
and Cochannel Interference. IEEE Transactions on Communications,
34(5):430–435, May 1986.

214 BIBLIOGRAPHY
[113] Saïd Moridi and Hikmet Sari. Analysis of Four Decision-Feedback
Carrier Recovery Loops in the Presence of Intersymbol Interference.
IEEE Transactions on Communications, 33(6):543–550, June 1985.
[114] Sami Hinedi and William C. Lindsey. Intersymbol Interference Effects
on BPSK and QPSK Carrier Tracking Loops. IEEE Transactions
on Communications, 38(10):1670–1676, October 1990.
[115] Sami M. Hinedi and Marvin K. Simon. Suppressed Carrier
Synchronizers for ISI Channels. In Communication Theory Mini-
Conference on behalf of IEEE Globecom ’96, pages 62–66, 1996.
[116] I.S. Gradshteyn. Table of Integrals and Products. Academic Press, 1965.
[117] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. Numerical
Recipes in C - The Art of Scientific Computing. Cambridge Univesity
Press, 1992.
[118] A. Erdélyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi. Tables
of Integral Transforms, volume 1. McGraw-Hill Book Company Inc.,
1954.
[119] Sirikiat Ariyavisitakul. Equalization of a Hard-Limited Slowly-
Fading Multipath Signal Using a Phase Equalizer with Time-
Reversal Structure. IEEE Journal on Selected Areas in Communications,
10(3):589–596, April 1992.
[120] F. Oberhettinger. Tabellen zur Fourier Transformation. Springer, Berlin,
1957.
[121] Richard D. Gitlin, Jeremiah F. Hayes, and Stephen B. Weinstein. Data
Communications Principles. Plenum Press, New York, 1992.