synchronization and multipath delay estimation …...[p1] r. hamila, j. vesma, t. saramaki and m....

Tampereen teknillinen korkeakouluJulkaisuja 375

Tampere University of TechnologyPublications 375

Tampere 2002

Ridha Hamila

Synchronization and Multipath Delay Estimation Algorithms for Digital Receivers

Tampereen teknillinen korkeakouluJulkaisuja 375

Tampere University of TechnologyPublications 375

Tampere 2002

Ridha Hamila

Synchronization and Multipath Delay Estimation Algorithms for Digital Receivers

Thesis for the degree of Doctor of Technology to be presented with due permission for public examination and criticism in Tietotalo Building, Auditorium TB109, at Tampere University of Technology, on the 19th of June 2002, at 12 o’clock noon.

i

ABSTRACT

This thesis considers the development of synchronization and signal processing techniquesfor digital communication receivers, which is greatly influenced by the digital revolution ofelectronic systems. Eventhough synchronization concepts are well studied and establishedin the literature, there is always a need for new algorithms depending on new system re-quirements and new trends in receiver architecture design. The new trend of using digitalreceivers where the sampling of the baseband signal is performed by a free running oscil-lator reduces the analog components by performing most of the functions digitally, whichincreases the flexibility, configurability, and integrability of the receiver. Also, this new de-sign approach contributes greatly to the software radio (SWR) concept which is the naturalprogression of digital radio receivers towards multimode, multistandard terminals wherethe radio functionalities are defined by software.

The first part of this research work introduces a new technique for jointly estimating thesymbol timing and carrier phase of digital receivers with non-synchronized sampling clockfor both data-aided and non-data-aided systems using a block-based feed-forward architec-ture. This technique is a practical, rapidly converging, fully digitally implemented synchro-nization concept based on a low-order polynomial approximation of the likelihood functionsusing the Farrow-based interpolator. A review of maximum likelihood theory, which is thebasis for coherent theory of synchronization, defines first the criteria and general frameworkfor developing near-optimum synchronization schemes for digital communication systems.Then, efficient Farrow-based polynomial approximations of the typical likelihood func-tions are derived for systematic symbol timing, carrier phase and fine acquisition frequencysynchronization algorithms.

Another important receiver functionality closely related to synchronization is propagationdelay estimation which is the basis of positioning technologies. Mobile phone positioning isbecoming unavoidable after the mandate imposed by communications regulatory bodies onemergency call positioning. The second part of this thesis reviews and develops new tech-niques with subchip resolution capabilities for estimating closely-spaced multipath delays inspread-spectrum CDMA systems. Generally, multipath delays caused by distant reflectorshave relatively large delay spread, with more than one chip interval between different paths,that can be resolved using conventional delay-locked-loop techniques. However, shorterexcess path delays result in overlapped fading multipath components that introduce signifi-cant errors to the line-of-sight path delay and gain estimation. Overlapping fading multipathcomponents are considered as one of the major sources of error that have strong impact onhigh precision mobile positioning solutions, as well as, on mobile applications of dedicatedsystems like the Global Positioning System (GPS). An overview of the most promising ge-olocation positioning techniques for wireless systems that are being standardized is firstprovided with a survey of fundamental concepts and major problems in positioning. Then,the characteristics of different channel models and conventional multipath delay estimationtechniques based on maximum likelihood theory are also discussed. Two new techniqueswith subchip resolution capabilities are proposed for estimating closely-spaced overlappedmultipath components. These techniques are intended to improve the accuracy of locationestimates by estimating correctly the delay of the line-of-sight path.

Preface

Research work for this thesis has been carried out during the years 1997-2002 in the researchproject "Advanced Transceiver Architectures and Implementations for Wireless Commu-nications" at the Institute of Communications Engineering (formerly TelecommunicationsLaboratory) of Tampere University of Technology, Finland. This thesis was financially sup-ported by the Academy of Finland and Tampere Graduate School of Information Scienceand Engineering (TISE), which are gratefully acknowledged.

First, I would like to express my sincere and deep gratitude to my supervisor Profes-sor Markku Renfors for his invaluable guidance, continuous support, and infinite toleranceduring the course of this work.

I would like to thank Associate Professor Peter Handel from the Department of Signals,Sensors and Systems, Royal Institute of Technology, Stockholm, and Professor Timo Laaksofrom the Signal Processing Laboratory, Helsinki University of Technology, for reviewingmy thesis, and for their constructive feedback and comments on the manuscript.

Special thanks are due to Professor Jaakko Astola my former M.Sc. advisor, to ProfessorMoncef Gabbouj for his encouragement as well as for his kind advice, to Professor TapioSaramaki for creating an enthusiastic work atmosphere, to professor Jarmo Harju, Dr. PerttiKoivisto coordinator of TISE, to Dr. Jari Syrjarinne, to Professor fred harris for giving methe opportunity to visit San Diego State University, California, and to Professor James F.Kaiser for sharing with me Teager energy concept and scientific experience.

I am indebted to all my colleagues and friends at the Institute of Communications Engi-neering for the pleasant work environment and for the help I have received during my work.I owe special thanks to Dr. Jussi Vesma and M.Sc. Simona Lohan for their cooperation,suggestions, and for the fruitful technical discussions, as well as, to Dr. Juha Yli-Kaakinenfor the LATEX help. In particular special thanks are due to Tarja Eralaukko, Sari Kinnari andElina Orava.

Special thanks to all my friends in Finland for their support and care. I’m very obliged toMohamed Maala, family Gabbouj, Faouzi Alaya Cheick, Saara Maala, Vesma family, PaulaLinna, Ali Hazmi, family Hammouda, Monaem Lakhzouri, Mejdi Trimeche, Mika Niem-inen, Sari Luhtala, Asheesh family, Abdo family, and to the small Tunisian community inTampere.

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iv PREFACE

Most of all I wish to express my deepest gratitude to Satu Lassila, my mother Fatma, mybrothers and sisters for their love, endless support, encouragement, and understanding allthese years.

Finally, I would like to dedicate this thesis to the memory of my father Salem who didnot have the chance to see the outcome of this work, God bless him.

RIDHA HAMILA

Tampere, May 31, 2002

Contents

Preface iii

List of Publications ix

List of Supplementary Publications xi

List of Symbols and Acronyms xiii

1 Introduction 1

2 Polynomial-Based ML Technique for Synchronization in Digital Receivers 52.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5

2.2 Synchronization and Timing Recovery Techniques : : : : : : : : : : : : : 6

2.3 Maximum Likelihood Estimation : : : : : : : : : : : : : : : : : : : : : : 6

2.3.1 ML Principle for Optimum Receivers : : : : : : : : : : : : : : : : 6

2.3.2 Likelihood Function for Systematic Synchronization Algorithms : : 9

2.3.3 Data-Aided Estimation : : : : : : : : : : : : : : : : : : : : : : : : 11

2.3.4 Non-Data-Aided Estimation : : : : : : : : : : : : : : : : : : : : : 12

2.4 Timing Adjustment Using Polynomial Interpolation : : : : : : : : : : : : : 13

2.4.1 Receiver with Non-Synchronized Sampling : : : : : : : : : : : : : 13

2.4.2 The Farrow Structure : : : : : : : : : : : : : : : : : : : : : : : : : 13

2.5 Polynomial Approximation for the Log-Likelihood Function : : : : : : : : 15

2.5.1 Data-Aided Estimator : : : : : : : : : : : : : : : : : : : : : : : : 15

2.5.2 Non-Data-Aided Estimator : : : : : : : : : : : : : : : : : : : : : : 17

2.6 Summary of Simulation Results : : : : : : : : : : : : : : : : : : : : : : : 18

3 Multipath Delay Estimation for Accurate Positioning 213.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21

3.2 Techniques for Personal Positioning: Principles and Problems : : : : : : : 21

v

vi CONTENTS

3.3 Fundamentals of GPS : : : : : : : : : : : : : : : : : : : : : : : : : : : : 24

3.3.1 Satellite Signals : : : : : : : : : : : : : : : : : : : : : : : : : : : 25

3.3.2 Signal Characteristics : : : : : : : : : : : : : : : : : : : : : : : : 26

3.4 Multipath Delay Estimation Techniques : : : : : : : : : : : : : : : : : : : 27

3.4.1 Channel Models : : : : : : : : : : : : : : : : : : : : : : : : : : : 27

3.4.2 Conventional ML Based Approaches : : : : : : : : : : : : : : : : : 29

3.4.3 Multipath Delay Estimation Using Peak Tracking with PulseSubtraction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 31

3.4.4 Multipath Delay Estimation Based on Teager-Kaiser Operator : : : : 33

4 Summary of Publications 374.1 Receiver Synchronization Studies : : : : : : : : : : : : : : : : : : : : : : 37

4.2 Multipath Delay Estimation Studies : : : : : : : : : : : : : : : : : : : : : 38

4.3 Author’s Contribution to the Publications : : : : : : : : : : : : : : : : : : 39

5 Conclusions 41

References 43

Publications 47

List of Figures

2.1 Timing recovery methods. : : : : : : : : : : : : : : : : : : : : : : : : : 7

2.2 Degradation of symbol error probability due to different timing errors, �� ,using QPSK modulation. : : : : : : : : : : : : : : : : : : : : : : : : : : 8

2.3 Degradation of symbol error probability due to different phase errors, ��,using QPSK modulation. : : : : : : : : : : : : : : : : : : : : : : : : : : 8

2.4 Digital receiver with non-synchronized sampling. : : : : : : : : : : : : : : 11

2.5 Illustrative example of a typical log-likelihood function for �0 = 0:52T and�0 = 0, as well as its first order derivative. : : : : : : : : : : : : : : : : : 11

2.6 The Farrow structure for polynomial-based interpolation filter. : : : : : : : 14

2.7 Example of the input and output samples for the Farrow structure. Here theover-sampling ratio is � = T=Ts = 2. : : : : : : : : : : : : : : : : : : : 14

2.8 DA symbol timing and carrier phase synchronization scheme. : : : : : : : 17

2.9 NDA symbol timing recovery scheme. : : : : : : : : : : : : : : : : : : : 18

2.10 NDA carrier phase estimation scheme for M-PSK modulations. : : : : : : 19

3.1 Principles of E-OTD and A-GPS methods in cellular network. : : : : : : : 23

3.2 GPS signal structure. : : : : : : : : : : : : : : : : : : : : : : : : : : : : 25

3.3 GPS signal spectrum (positive side). : : : : : : : : : : : : : : : : : : : : 26

3.4 User received minimum power levels with respect to the user elevationangle and signal structure. : : : : : : : : : : : : : : : : : : : : : : : : : : 27

3.5 Non-coherent DLL S-curve for a multipath channel. : : : : : : : : : : : : 31

3.6 Effect of multipath propagation on the S-curve of a non-coherent DLLusing different discriminator values � and two paths channel. : : : : : : : 32

vii

List of Publications

[P1] R. Hamila, J. Vesma, T. Saramaki and M. Renfors, "Discrete-Time Simulation ofContinuous-Time Systems Using Generalized Interpolation Techniques," in Proc.1997 the Summer Computer Simulation Conference, SCSC’97, Arlington, Virginia,USA, July 1997, pp. 914–919.

[P2] R. Hamila, J. Vesma, H. Vuolle, and M. Renfors, "Joint Estimation of Carrier Phaseand Symbol Timing Using Polynomial-Based Maximum Likelihood Technique," inProc IEEE 1998 International Conference on Universal Personal Communications,ICUPC’98, Florence, Italy, October 1998, pp. 369–373.

[P3] R. Hamila, J. Vesma, H. Vuolle, and M. Renfors, "NDA Maximum Likelihood Ap-proach for Timing and Phase Adjustment by Polynomial-Based Interpolation," inProc. 6th IEEE International Workshop on Intelligent Signal Processing and Com-munication Systems, ISPACS’98, Melbourne, Australia, Nov. 1998, pp. 248–252.

[P4] R. Hamila, J. Vesma, H. Vuolle, and M. Renfors, "Effect of Frequency Offset on Car-rier Phase and Symbol Timing Recovery in Digital Receivers," in Proc. 1998 URSIInternational Symposium on Signals, Systems, and Electronics, ISSSE’98, Pisa, Italy,September 1998, pp. 247–252.

[P5] R. Hamila, M. Renfors, "New Maximum Likelihood Based Frequency Estimator forDigital Receivers," in Proc. IEEE Wireless Communications and Networking Con-ference, WCNC´99, New Orleans, USA, Sept. 1999, pp. 206–210.

[P6] R. Hamila, S. Lohan, and M. Renfors, "Effect of Correlation Estimation on MultipathDelay Estimation Techniques in DS-CDMA Systems," in Proc. of 4th InternationalSymposium on Wireless Personal Multimedia Communications, WPMC’01, Aalborg,Denmark, September 2001, pp. 331–335.

[P7] R. Hamila, S. Lohan, and M. Renfors, "Novel Technique for Closely-Spaced Mul-tipath Delay Estimation in DS-CDMA Systems," Submitted to Signal Processing,(EURASIP).

[P8] S. Lohan, R. Hamila, and M. Renfors, "Performance Analysis of an Efficient Multi-path Delay Estimation Approach in a CDMA Multiuser Environment," in Proc. of12th IEEE International Symposium on Personal, Indoor and Mobile Radio Commu-nications, PIMRC´01, San Diego, California, USA, Sept. 2001, pp. 6–10.

ix

List of Supplementary Publications

[S1] R. Hamila, J. Vesma and M. Renfors, "Polynomial-Based Maximum Likelihood Tech-nique for Synchronization in Digital Receivers," to be published in the IEEE Trans-actions on Circuit and Systems II: Analog and Digital Signal Processing.

[S2] J. Vesma , R. Hamila, T. Saramäki and M. Renfors, "Design of Polynomial Interpola-tion Filters Based on Taylor Series," in the European Signal Processing Conference,Rhodes, Greece, September 1998, pp. 283–286.

[S3] R. Hamila, J. Astola, F. Alaya Cheikh, M. Gabbouj, and M. Renfors, "Teager Energyand the Ambiguity Function," IEEE Transactions on Signal Processing, Vol. 47, No.1, pp. 260–262, January 1999.

[S4] R. Hamila, F. Alaya Cheikh, J. Vesma, J. Astola, and M. Gabbouj, "Relationshipbetween Wigner-Distribution and the Teager Energy," in Proc. European Signal Pro-cessing Conference, Rhodes, Greece, September 1998, pp. 1857–1860

[S5] R. Hamila, M. Renfors, G. Gunnarsson, M. Alanen, "Data Processing for MobilePhone Positioning," in Proc. Vehicular Technology Conference, Amsterdam, Nether-lands, Sept. 1999, pp. 446–449.

[S6] R. Hamila, S. Lohan, and M. Renfors, "Novel Technique for Multipath Delay Estima-tion in GPS receivers," in Proc. IEEE International Conference on Third GenerationWireless and Beyond, San Francisco, USA, May 2001, pp. 993–998.

[S7] S. Lohan, R. Hamila, and M. Renfors, "Cramer Rao Bound for Multipath Time De-lays in a DS-CDMA System," in Proc. of 4th International Symposium on Wire-less Personal Multimedia Communications, Aalborg, Denmark, September 2001, pp.1043–1046.

[S8] R. Hamila, S. Lohan and M. Renfors, "Subchip Multipath Delay Estimation for Down-link WCDMA System Based on Teager Operator," Submitted to IEEE Communica-tions Letters.

xi

List of Symbols and Acronyms

SYMBOLS

a(n) data symbolsÆ(:) Dirac delta function�t time spacing(�t)c coherence timeBc coherence bandwidthBd Doppler spreadC(t) spreading codecl(:) FIR filter coefficientEb energy of a bitEs energy of a symbolEf:g statistical expectationFs sampling frequencygT (:) transmitter filtergMF (:) receiver matched filter

j imaginary unit (j =p�1)

N block lengthN0 single sided noise PSDP (:) apriori probabilityp(:j:) conditional probability density functionRef:g real partr(t) received signals(t;�) baseband transmitted signal

xiii

xiv List of Symbols and Acronyms

T symbol intervalT0 observation intervalTm delay spreadTs sample interval� synchronization parameter vector~� synchronization parameter trial vector

� estimated synchronization parameter vector� time delay� estimated time delay� phase error

� estimated phase error� 2 [0; 1) fractional interval�(:) likelihood function�(:) log-likelihood function�(:) rectangular function5(:) triangle function(:)� complex conjugationj:j absolute valueb:c integer part

ACRONYMS

3G Third Generation Wireless System3GPP 3rd Generation Partnership ProjectADC Analog to Digital ConverterAF Ambiguity FunctionAFLT Advanced Forward Link TrilaterationA-GPS Assisted Global Positioning SystemANSI American National Standards InstituteARIB Association of Radio Industries and BusinessesBPSK Binary Phase Shift KeyingBS Base StationC/A-Code Coarse Acquisition CodeCDMA Code Division Multiple Access

List of Symbols and Acronyms xv

CI Cell IdentityDA Data AidedDD Decision DirectedDLL Delay Locked LoopDoD Department of DefenseDSP Digital Signal ProcessingDS-SS Direct Sequence - Spread SpectrumE-OTD Enhanced Observed Time DifferenceETSI European Telecommunications Standards InstituteFCC Federal Communications CommissionFIR Finite Impulse ResponseGPS Global Positioning SystemIPDL Idle Period Down LinkGSM Global System for Mobile CommunicationLLF Log Likelihood FunctionLOS Line Of SightMAP Maximum A Posteriori ProbabilityML Maximum LikelihoodMLE Maximum Likelihood EstimationMS Mobile StationNCDLL Non-Coherent Delay Lock LoopNDA Non-Data AidedNLOS Non-Line Of SightOTDOA Observed Time Difference of ArrivalPAM Pulse Amplitude ModulationPLL Phase Locked LoopPPS Precise Position ServicePRN Pseudo Random NoiseQAM Quadrature Amplitude ModulationQPSK Quadrature Phase Shift KeyingRF Radio Frequencysec SecondSMG Standard Mobile GroupSNR Signal-to-Noise RatioSPS Standard Position Service

xvi List of Symbols and Acronyms

SWR Software RadioSVs Space VehiclesTDMA Time Division Multiple AccessTDOA Time Difference Of ArrivalTOA Time Of ArrivalTOT Time Of TransmissionUS United StatesVLSI Very Large Scale Integration

Chapter 1Introduction

The high competition between Americans, Asians, and Europeans has led to the coexistenceof several radio communication systems and promises a very difficult path towards the con-vergence to a unique standard wireless mobile system, despite of the benefits of commonworldwide standard. This competition introduces new challenges in receiver architecturedesign to accommodate the different air interfaces. The natural progression of digital ra-dios towards multimode, multistandard terminals is a potential pragmatic solution whichis known as the Software Radio (SWR) concept. The main idea is that the radio function-alities of user terminals are defined by software which allows them to dynamically adaptto different radio environments. Even though, the software radio was first introduced formilitary applications, the digital revolution of electronic systems has enabled software ra-dio to gradually enter also commercial communication systems. The popularity of digitalsolutions over their analog counterparts resides mainly on their greater flexibility, high inte-gration efficiency, and configurability, as well as on the fast speed at which semiconductortechnologies, digital signal processors, and VLSI circuits are developing.

The rapid evolution of communication systems and the new requirements imposed bythe standardization authorities have brought various consumer electronics device types anda high demand on new communication services, such as mobile phone positioning withhigh accuracy. Even if safety was the primary motivation for mobile positioning, a lotof commercial applications have already emerged in the market. For the public interest,mobile phone positioning in a cellular network with reliable and rather accurate positioninformation has become unavoidable after the Federal Communications Commission man-date, FCC-E911 docket on emergency call positioning in USA, and the coming E112 in theEuropean Union [1]. The basic rule of emergency services requires mobile operators to au-tomatically transfer caller location details to the emergency authorities without regard tovalidation procedures intended to automatic caller identification and localization.

International analysts have also predicted that mobile location services are commerciallypotential and the most compelling value-added services to the forthcoming third generationmobile systems, taking into consideration the huge range of services that can be providedto the mobile users. Recently, several potential technologies enabling the development ofmobile location techniques have emerged and are already competing in the market [2]. A

1

2 INTRODUCTION

large number of commercial location-based services are being provided, such as fleet andresource management, vehicle tracking and person to person location and messaging appli-cations [3], [4], location specific traffic information, yacht position, map-based guidance,and navigation [5]. Location based services will allow mobile users to receive personalizedand lifestyle-oriented services relative to their geographic location, and provide enormousopportunities for companies and organizations to effectively reach their target groups ofstrong purchasing power. However, the successful evolution of the mobile location servicesto a massmarket industry relies mainly on its accuracy, interoperability, end user privacy andavailability of attractive services. In view of the sensitivity of location data to the end userprivacy and security, appropriate technical solutions that satisfy requirements for securityissues must be established to ensure compliance with regulatory rules in this area.

Since the introduction of the software radio concept, several architectures for designingSWR radio platforms have been recently developed through the efforts of commercial andnoncommercial organizations [6], [7], [8], [9], [10], [11]. The ultimate goal of designing asingle-chip, flexible wireless transceiver supporting multimode and multistandard compati-bility introduced new architectures more suitable for integration. The current trend is to pushthe analog/digital interface towards the antenna to simplify the analog parts and allow theimplementation of most receiver functions digitally, increasing therefore the flexibility, con-figurability, and integrability. As one important functionality, these new receivers requireflexible and efficient synchronization algorithms which depend on the system requirementsand receiver architecture design.

In digital communication systems, the continuous-time received signal is sampled andthese samples are used to make the decisions on the transmitted symbols. The sampling ofthe received signal must be synchronized to the incoming data symbols. In conventional re-ceivers, the synchronization is performed using a feedback or feed-forward loop to controlthe phase of the sampling clock which generally requires complicated phase-locked-loopcircuits [12]. The new trend of using digital receivers where the sampling of the basebandsignal is not synchronized to the incoming data symbols, reduces the analog componentssince most of the functions are performed digitally [13], [14], [15]. In this receiver archi-tecture, the sampling of the baseband signal is clocked by a free running oscillator, and thusthe sampling is not synchronized to the incoming symbols. Therefore, timing adjustment,and in practice also the carrier phase estimation must be done by digital methods after sam-pling. One way to perform this is to calculate the value of the signal at the desired timeinstants using interpolation. Maximum likelihood (ML) estimation theory is the basis forcoherent theory of synchronization [13]. It provides a general framework for developingnear-optimum synchronization schemes for digital communication systems. Joint estima-tion of signal parameters using the ML approach yields usually to better estimates withrespect to the lower bound on the variance, known as the Cramer Rao bound, compared toestimates obtained from separate optimization of the likelihood functions [16].

Another essential receiver functionality closely related to synchronization is the propa-gation delay estimation which is the basis for positioning technologies. Multipath effects ofthe mobile channel could degrade the location estimate substantially, and they are regardedas a killer issue in location estimation, and still a challenging topic for research work.

The main goal of the first part of this thesis work is to develop flexible and efficient all-digital synchronization algorithms with good performance and fast convergence. The workfocuses on new algorithms for jointly estimating the symbol timing and carrier phase of

3

digital receivers with non-synchronized sampling clock for both data-aided and non-data-aided systems. To achieve efficient implementation, modest oversampling is used and yetgood timing accuracy is achieved by developing low-order polynomial approximations ofthe likelihood functions. The main emphasis is on the derivation of efficient Farrow-basedpolynomial approximations of the likelihood function for systematic and practical symboltiming, carrier phase, and fine acquisition frequency synchronization algorithms.

The second main topic is to develop new techniques with subchip resolution capabilitiesfor estimating closely-spaced multipath delays in spread-spectrum CDMA systems in orderto estimate correctly the delay of the LOS path. Overlapped closely-spaced multipath com-ponents are considered as one of the major sources of error for accurate mobile terminalpositioning solutions.

This thesis is organized as follows: In Chapter 2, the polynomial-based maximum like-lihood technique for parameter synchronization in digital receivers with non-synchronizedsampling clock is established for both DA and NDA systems. Chapter 3 provides an overviewof the most promising geolocation solutions, and introduces new techniques for estimatingoverlapped closely-spaced multipath components. A summary of the main results of thisthesis and the author’s contribution to the publications are clarified in Chapter 4. The resultsof this work are given in the publications included in the appendices. Finally, conclusionsare drawn in Chapter 5.

Chapter 2Polynomial-Based ML Techniquefor Synchronization in DigitalReceivers

2.1 INTRODUCTION

In this chapter, new symbol timing and carrier phase estimators are introduced based on MLtheory for both data-aided (DA) and non-data-aided (NDA) systems, using a block-basedfeed-forward architecture. This new technique is a practical, fully digitally implementedsynchronization concept using interpolation for jointly estimating the symbol timing andthe carrier phase. The interpolation method used in this context is efficiently implementedusing the so-called Farrow structure [17] which is characterized by its simple and flexiblerealization. The likelihood function is expressed in terms of the polynomial coefficientsobtained from the Farrow-based interpolator. The feed-forward architecture we are consid-ering provides rapid acquisition characteristics, which are very important especially in themobile communication systems where the channel characteristics are rapidly changing, andin the case of TDMA system, also the transmission is bursty. Furthermore, the used interpo-lation approach allows to use modest oversampling (typically twice the symbol rate) whileproviding the temporal resolution of a highly oversampled system in a computationally ef-ficient manner. In [18], [19], a somewhat similar idea was developed for symbol timingestimation through polynomial approximation of the log-likelihood function. Our approachdiffers from this earlier idea in that the polynomial approximation of the log-likelihoodfunction is here derived from the piecewise polynomial approximation of the signal in anefficient way. This allows the use of lower sampling rate (oversampling factor of 2 insteadof 4). For other approaches to all-digital symbol timing recovery we refer to [14], [15], [20].

This chapter is organized as follows. Section 2.2 discusses briefly the importance of syn-chronization function and different timing recovery techniques. Section 2.3 gives a briefoverview of the maximum likelihood principle for optimum receivers, and presents the typ-

5

6 POLYNOMIAL-BASED ML TECHNIQUE FOR SYNCHRONIZATION IN DIGITAL RECEIVERS

ical likelihood functions required for deriving systematic symbol timing and carrier phasesynchronization algorithms. Timing adjustment using polynomial interpolation concept fordigital receiver with non-synchronized sampling is reviewed in Section 2.4. The polynomialapproximation of the likelihood functions for both DA and NDA systems is derived in Sec-tion 2.5. Finally, a summary of the simulation results which are provided in the Publications[P1]-[P5] is presented in Section 2.6.

2.2 SYNCHRONIZATION AND TIMING RECOVERY TECHNIQUES

Synchronization is one of the fundamental functions in communication systems. Its task isto lock the synchronization parameters of the receiver with the received signal. Estimationof synchronization parameters, such as symbol timing, carrier phase and frequency is veryessential for performing demodulation and detection of the data from the transmitted signalwith high reliability. In conventional receivers, the synchronization is performed using afeedback or feed-forward loop to control the phase of the sampling clock as shown in Fig.2.1 (a) and (b). These analog and hybrid methods of synchronization are well establishedin the literature and studied thoroughly [13], [21]. However, their major drawbacks are thelarge amounts of board space, power consumption, and circuit complexity of phase-locked-loops (PLLs) which are used to control the sampling of incoming signals [12]. Anothersynchronization method suitable for digital receiver implementation has been introducedduring the last decade as shown in Fig. 2.1 (c). In this architecture, the received signalis sampled by a free running clock and thus sampling is not synchronized to the incomingdata symbols. Consequently, the synchronization parameters are estimated by digital meth-ods after sampling. This new type of receiver architecture with non-synchronized samplingclock is very relevant to the context of software radio. In fact, existing communication stan-dards are using different data rates, and consequently they employ different master clockrates. In order to reduce the receiver complexity, we can use a single master clock with thisreceiver architecture and generate the different clock rates virtually by means of samplingrate conversion [10].

The synchronization function is generally very critical to the error performance. Typicalexamples shown in Figs. 2.2 and 2.3 illustrate the effect of timing offset and phase error ontheoretical symbol error probability using QPSK modulation [15], [16].

2.3 MAXIMUM LIKELIHOOD ESTIMATION

In this section, the maximum a posteriori (MAP) and maximum likelihood (ML) estimationcriteria are shortly reviewed. Then, the ML criterion is used to device the symbol timingand carrier phase estimators for both DA and NDA systems.

2.3.1 ML Principle for Optimum Receivers

The optimum decision rule to detect the symbol sequence a in a received signal corruptedby noise is based on the maximum a posteriori probability (MAP) estimation criterion, alsoknown as Bayesian estimation [16]. This decision criterion is trying to select the values of

MAXIMUM LIKELIHOOD ESTIMATION 7

(c) Digital Recovery

(b) Hybrid Recovery

(a) Analog Recovery

TIMING CONTROL~

ADC DSPRF/AnalogFront-end

DSP

TIMING CONTROL

ADC

~

RF/AnalogFront-end

TIMING CONTROL

DSPADC

~

RF/AnalogFront-end

Fig. 2.1 Timing recovery methods.

a in each transmitted signal interval based on the observation vector r, such that the set ofposteriori probabilities is maximized. Under the condition that the prior probabilities areall equal (symbols are uniformly distributed), a detector based on the MAP criterion re-duces to the decision criterion based on the maximum of the conditional probability densityfunctions, known as maximum-likelihood (ML) criterion [15].

An optimal ML receiver would perform joint estimation of data a and synchronizationparameters� = f�; �g simultaneously [20]. Here, � and � denote the time delay and carrierphase, respectively. The ML criterion for both data detection and synchronization param-eter estimation is trying to select the set of values fa;�g which maximizes the likelihoodfunction p(rja;�) as follows

(a; �) = arg maxa;�

p(rja;�): (2.1)


2 3 4 5 6 7 810

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bol E

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babi

lity

Es/N

0 (dB)

∆ τ=0∆ τ=0.1/T∆ τ=0.2/T∆ τ=0.3/T∆ τ=0.4/T

Fig. 2.2 Degradation of symbol error probability due to different timing errors, �� , using QPSKmodulation.

0 2 4 6 8 10 12 14 16 18 2010

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babi

lity

Es/N

0 (dB)

∆ φ=0∆ φ=π/32∆ φ=π/16∆ φ=π/8

Fig. 2.3 Degradation of symbol error probability due to different phase errors, ��, using QPSKmodulation.

Unfortunately, the optimal joint detection/estimation approach is rather complicated forpractical receiver implementation [15], [20]. Therefore, separate data detection algorithms,and independent estimates of each synchronization parameter are commonly derived for re-


alizable receiver structures. Depending on the data dependency, there are three main classesof ML synchronizers:

� The ML data-aided (DA) synchronization algorithms use known data sequence a0(preamble, training sequence) at the receiver for the estimation as follows

(�)DA = arg max�

p(rja = a0;�): (2.2)

� The ML decision-directed (DD) synchronization algorithms use detected data se-quence a instead of the true symbol values for the estimation,

(�)DD = arg max�

p(rja = a;�): (2.3)

� The third class is the ML non-data-aided (NDA) synchronization algorithms. Theyare using neither the data symbol values nor their estimates. However, they removethe data dependency by averaging over all prior probabilities as follows

p(rj�) =Xa

p(rja;�)P (a): (2.4)

Then, by maximizing the above likelihood function (2.4) with respect to each/allsynchronization parameter, the ML NDA algorithms are given by

(�)NDA = arg max�

p(rj�): (2.5)

2.3.2 Likelihood Function for Systematic Synchronization Algorithms

A linearly modulated transmitted signal in baseband is given by

s(t;�) = ej�0Xn

a(n)gT (t� nT � �0); (2.6)

where the couple (�0; �0) are the true values of the unknown clock timing and carrier phaseoffsets, respectively, fa(n)g is the sequence of independent equiprobable data symbols,gT (t) is the transmitter filter pulse shape, and 1=T is the symbol rate.

We assume that s(t;�) is completely known at the receiver except for the synchroniza-tion parameter vector�which consists of the time delay � and carrier phase �, and possiblythe data symbols. The parameter vector � is also assumed to remain unchanged over anobservation interval T0 (or sufficiently large number N = T0=T of transmitted symbols).

The linearly modulated received signal is given by

r(t) = s(t;�) + �(t); (2.7)

where �(t) is an independent white stationary Gaussian noise with flat power spectral den-sity N0. Recalling the Gaussian nature of the random process, the likelihood function thatis to be maximized is given by [15], [16]

p(rja;�) = � exp

��

1

�2

ZT0

��r(t)� ~s(t; ~�)��2dt�; (2.8)


where � is a positive constant, independent of the synchronization parameters which can beneglected, and ~s(t; ~�) is the trial local replica generated at the receiver defined by

~s(t; ~�) = ej~�Xn

a(n)gT (t� nT � ~�): (2.9)

Maximum likelihood estimation (MLE) is simply trying to minimize the distance betweenthe corrupted received signal r(t) and the trial local replica ~s(t; ~�). Equivalently, MLEis trying to estimate one or both values from the parameter set � = f� ; �g using the trialvector ~� = f~� ; ~�g in order to maximize the likelihood function p(rja;�).

As mentioned earlier, consistent approximations of the above likelihood function areneeded to derive systematic ML synchronization algorithms. By expanding the integral inEq. (2.8), and observing that the energy quantities resulting from the received noisy signaland the local trial signal can be ignored since they are independent of the synchronizationparameters, it follows [16]

p(rja;�) � � exp

�2

�2

Z T0

0

Renr(t) ~s�(t; ~�)

odt

�: (2.10)

The above approximation is accurate for constant envelope signals (equal symbol energies),e.g., M-PSK, but it is commonly used also for QAM signals.

In the basic system model of Fig. 2.4, after down-conversion to the baseband throughmixing and filtering, the received signal is first applied to a receiver matched filter gMF (t),then sampled with a fixed sampling rate 1=Ts . In practice, the order of sampling and matchedfilter can be reversed. Under the assumptions that the transmitter and receiver matched filterssatisfy the Nyquist properties and the channel is ideal, there is no inter-symbol interferenceprovided that the samples are taken at the correct time instant, i.e., �0 = � . By replacingthe local trial signal replica (2.9) in Eq. (2.10), after some manipulations, the likelihoodfunction or objective function becomes [20]

�(a;�) = � exp

�2

�2Rene�j�

Xn

a�(n)m(n; �)o�

; (2.11)

where m(n; �) denotes samples from the output of the matched filter given by

m(n; �) =

Z T0

0

r(t)gMF (nT + � � t)dt: (2.12)

Taking the logarithm of (2.11) and dropping the resulting constants, we obtain the log-likelihood function (LLF)

�(a;�) = Rene�j�

Xn

a�(n)m(n; �)o: (2.13)

An illustrative example of the likelihood function and its first order derivative is depictedin Fig. 2.5 using QPSK constellation signal and an oversampling factor equal to 50. Sev-eral techniques for searching the maximum location of the likelihood function have beendevised with a complexity mostly related to the data sample rate and used technology. The


Fig. 2.4 Digital receiver with non-synchronized sampling.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.6

1.7

1.8

1.9

2

Timing offset (τ)

Lik

elih

ood

Func

t.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Timing offset (τ)

Firs

t der

ivat

ive

Fig. 2.5 Illustrative example of a typical log-likelihood function for �0 = 0:52T and �0 = 0, aswell as its first order derivative.

straightforward approach of calculating the log-likelihood function from the discrete-timematched filter output samples would require excessive oversampling factor to achieve suf-ficient temporal resolution, especially with high-order modulations. Generally, the moststraightforward approaches are based on the partial derivatives of the LLF. Namely, theiterative search procedure or the error tracking feedback method [20]. However, the maxi-mum search can be facilitated and optimized using the proposed idea of polynomial-basedapproximation of the maximum likelihood function.

2.3.3 Data-Aided Estimation

In this case, the symbols fa(n)g used for synchronization (preamble, training sequence)are known at the receiver. Similarly, the decision-directed synchronization algorithms use


detected data sequence instead of the true symbol values. For jointly estimating the sym-bol timing and carrier phase (� ; �), the log-likelihood function has to be maximized withrespect to the two parameters � and � as follows [15]

(� ; �) = argmax�;�

�(�; �) (2.14)

where the LLF (2.13) can be expressed as

�(�; �) =��X

n

a�(n)m(n; �)��Rene�j��argPn

a�(n)m(n;�)�o

: (2.15)

By observing that the magnitude term in Eq. (2.15) is independent of �, the joint estimationof (� ; �) can be performed by first estimating � by finding the maximum of the magnitudeterm

� = argmax�

��Xn

a�(n)m(n; �)�� (2.16)

and then, the carrier phase estimate � is easily deduced as

� = argXn

a�(n)m(n; � ): (2.17)

2.3.4 Non-Data-Aided Estimation

Under the assumptions of low SNR and independent equiprobable transmitted data symbols,using quadratic Taylor series expansion of the exponential term in the objective function inEq. (2.11) and then removing of the data dependency by averaging, the desired approxima-tion of the marginal likelihood function is given by [20]

�(�; �) =Xn

E�ja(n)j2

��m(n; �)��2 +Re

nXn

E[a2(n)]�m�(n; �)

�2e�2j�

o: (2.18)

Detailed mathematical derivations of the above marginal likelihood function are providedin [20], Chapter 5.

Consequently, for a uniformly distributed phase, the likelihood function of Eq. (2.18) isreduced to

�(�) =Xn

��m(n; �)��2: (2.19)

Hence, the NDA timing estimate for any linear digital modulation is given by [15]

� = argmax�

Xn

��m(n; �)��2: (2.20)

For M-PSK signalling, joint symbol timing and carrier phase estimate is feasible and carrierphase estimate is given by

� =1

Marg

Xn

�m(n; � )

�M: (2.21)

TIMING ADJUSTMENT USING POLYNOMIAL INTERPOLATION 13

2.4 TIMING ADJUSTMENT USING POLYNOMIAL INTERPOLATION

This section reviews and demonstrates how the symbol timing adjustment can be done dig-itally using an interpolator if the sampling of the receiver signal is not synchronized tothe symbol timing instants. It is also discussed how the polynomial-based interpolationmethods can be efficiently implemented using the so-called Farrow structure. In the nextsection, the polynomial-based interpolation and the Farrow structure are utilized to derivethe proposed ML based symbol timing and carrier phase estimation algorithm.

2.4.1 Receiver with Non-Synchronized Sampling

In this work we consider the receiver structure where the sampling is done using a fee-runningoscillator and the symbol timing and carrier phase estimation and correction are done com-pletely in the digital part of the receiver, as shown in Fig. 2.4. Here the received basebandsignal after the matched filter, denoted by m(t), is sampled using the over-sampling ratioof � = T=Ts, where T and Ts are the symbol and sampling intervals, respectively. In thedigital part of the receiver, the timing error (�i) and phase error (�) are estimated (to be dis-cussed in the next section) and the symbol timing adjustment is done first using interpolatorand, after that, the phase error is corrected using a phase rotator.

For convenience, it is assumed that the ADC sampling rate is an integer multiple of thesymbol rate, and the interpolator output sampling rate is equal to the symbol rate. In thiscase, the fractional interval is constant, �i = �, in stationary conditions. The fractionalinterval and timing offset are related as follows:

� =�

Ts�j �

Ts

kwhere � 2 [0; 1): (2.22)

Here, bxc stands for the integer part of x.

2.4.2 The Farrow Structure

It has turned out that the polynomial-based interpolation methods are the most widely usedsolution for performing the symbol timing adjustment in digital receivers. The reason forthe popularity of these methods lies in the fact that interpolator can then be efficiently imple-mented using the Farrow structure [17]. This filter structure, shown in Fig. 2.6, consists ofL+1 parallel FIR filters having the transfer functions Cl(z) for l = 0; 1; � � � ; L. HereL de-notes the degree of the polynomial interpolation. The impulse response of the interpolatorin each Ts-segment with basepoints k = I1 to I2:

h(t) = h(iT ) = h[(k + �i)Ts] =LXl=0

cl(k)�li (2.23)

with cl(k)’s being the fixed coefficients of the FIR filters of length I2 � I1 + 1.The synchronized output samples of this structure are be given by

y(i) =

I2Xk=I1

m(ni � k)h[(k + �i)Ts] =

LXl=0

vl(ni)�li; (2.24)


Fig. 2.6 The Farrow structure for polynomial-based interpolation filter.

Fig. 2.7 Example of the input and output samples for the Farrow structure. Here the over-samplingratio is � = T=Ts = 2.

where

vl(ni) =

I2Xk=I1

m(ni � k)cl(k): (2.25)

The FIR filter coefficients for the Lagrange interpolation are given, e.g., in [14] and for moregeneral interpolation techniques, see [17], [22], [23].

The idea of the Farrow structure is that the output samples of the FIR filters vl(ni) asgiven by Eq. (2.25) form a polynomial approximation for the continuous-time signal m(t)in the interval niTs � t < (ni + 1)Ts. The output sample y(i) is then calculated by eval-uating the value of the polynomial at the time instant determined by �i according to Eq.(2.24). Therefore, the time instant for the output sample y(i) = y(iT ) is determined by(see Fig. 2.7)

iT = (ni + �i)Ts with 0 � �i < 1: (2.26)

The advantage of the Farrow structure is that the filter coefficients cl(k) are fixed and thetime instant for the output sample can be easily controlled by the fractional interval �i.

POLYNOMIAL APPROXIMATION FOR THE LOG-LIKELIHOOD FUNCTION 15

2.5 POLYNOMIAL APPROXIMATION FOR THE LOG-LIKELIHOODFUNCTION

In this section, we derive the polynomial approximation for the log-likelihood function forboth DA and NDA cases. We consider the case of a non-synchronized receiver where thesampling rate is twice the symbol rate (T = 2Ts), consequently, we are going to use � and �in the following to denote the timing error as fractions of the symbol and sampling interval,respectively. The function of the interpolation filter is to calculate the correct output sample,y(i) = y(iT ), at a time using a set of adjacent input samples, m(n) = m(nTs), obtainedfrom the output of the matched filter based on a timing error estimate �. Here n denotes thelargest integer for which nTs � iT .

2.5.1 Data-Aided Estimator

In the following, the polynomial approximation for the log-likelihood function is derivedby utilizing the fact that the output samples of the FIR filters in the Farrow structure, de-noted by vl(n), form an Lth order polynomial approximation for the input signal m(n).This approximation is piecewise polynomial, i.e., there are different polynomial approxima-tions for each sample interval. Because we are using two samples per one symbol interval,there are two different polynomial approximations of the log-likelihood function [P2] cor-responding to the two halves of the symbol interval. Consequently, from expression (2.16)the symbol timing is estimated by

� = argmax�

�j�1(2�)j for 0 � � < T=2j�2(2� � 1)j for T=2 � � < T

(2.27)

where the LLF functions �1(2�) and �2(2� � 1) are generated using odd and even samplesof the matched filter output m(n; �), as follows(

�1(2�) =PN

n=1 a�(n)m(2n � 1; 2�) for 0 � � < T=2

�2(2� � 1) =PN

n=1 a�(n)m(2n; 2� � 1) for T=2 � � < T .

(2.28)

Also, by using the timing estimate � in the corresponding symbol time interval, the carrierphase estimate (Eq. (2.17)) becomes

� =

�arg �1(2� ) for 0 � � < T=2arg �2(2� � 1) for T=2 � � < T :

(2.29)

The above expressions can be rewritten with respect to the sample interval denoted hereby the fractional interval �. The LLF equation (2.28) becomes(

�1(�Ts) =PN

n=1 a�(n)m(2n � 1; �Ts)

�2(�Ts) =PN

n=1 a�(n)m(2n; �Ts):

(2.30)

The values of m(n; �Ts) can be interpolated for different values of � 2 [0; 1) by utilizingthe Farrow structure and the original samples m(n). Thus, the interpolated sample values


expressed as a function of the FIR branch filter outputs in the Farrow structure, vl(n), andthe fractional interval, �, are given by [P2]

(m(2n� 1; �Ts) =

PL

l=0 vl(2n� 1)�l

m(2n; �Ts) =PL

l=0 vl(2n)�l:

(2.31)

where L is the degree of polynomial approximation. Finally, by substituting Eq. (2.31) into(2.30), the polynomial approximation of the log-likelihood function can be written as:

8<:

�1(�Ts) =PL

l=0

�PN

n=1 a�(n)vl(2n� 1)

��l

�2(�Ts) =PL

l=0

�PN

n=1 a�(n)vl(2n)

��l:

(2.32)

Consequently, the symbol timing and the carrier phase estimation can be performed easilywith the aid of the Farrow structure. First, the polynomial coefficients of the log-likelihoodfunctions are calculated by averaging the product of a�(n) and vl(n) over N symbols ac-cording to Eq. (2.32). It has turned out through simulations that the values of N between16 and 64 provides good results. Second, the value of the timing error estimate � that max-imizes either of the log-likelihood functions f�1(�Ts);�2(�Ts)g is computed. Finally, byusing the timing estimate, the carrier phase estimate � is calculated. A formal descriptionof this algorithm is written as follows:

If max j�1(�Ts)j > max j�2(�Ts)j then� = argmax� j�1(�Ts)j

� = arg[�1(�Ts)]else

� = argmax� j�2(�Ts)j

� = arg[�2(�Ts)]end If

The overall scheme of the proposed DA block-based symbol timing and carrier phaserecovery is illustrated in Fig. 2.8. After the parameters � and � are estimated, timing ad-justment is performed by evaluating the value of the polynomial at the given value of �.Subsequently, the carrier phase is corrected by the factor e�j�.

The efficiency of this algorithm is due to the fact that here the block averaging of the like-lihood function is done in the polynomial coefficient domain, and the multiplications with �are needed only for the whole estimation block. The above procedure can be used as such inthe data-aided (DA) systems where the symbol values are known in advance. For example,GSM does have a known synchronization symbol sequence. In case of decision-directed(DD) systems, a reasonably good initial estimate for the sampling phase must be known.This scheme can also be utilized in non-data-aided (NDA) systems with some modificationsas shown in the following section.

POLYNOMIAL APPROXIMATION FOR THE LOG-LIKELIHOOD FUNCTION 17

Fig. 2.8 DA symbol timing and carrier phase synchronization scheme.

2.5.2 Non-Data-Aided Estimator

Similarly to the DA estimators, the symbol timing for the NDA case is derived from expres-sion (2.20) as follows [P3]

� = argmax�

(�1(2�) for 0 � � < T=2�2(2� � 1) for T=2 � � < T

(2.33)

where the likelihood functions �1(2�) and �2(2� � 1) are given by(�1(2�) =

PN

n=1

��m(2n� 1; 2�)��2 for 0 � � < T=2

�2(2� � 1) =PN

n=1

��m(2n; 2� � 1)��2 for T=2 � � < T:

(2.34)

The likelihood functions (2.34) can also be expressed as a function of the fractional intervaland interpolation filter branches as follows;(

�1(�Ts) =PN

n=1

��PL

l=0 vl(2n� 1)�l��2

�2(�Ts) =PN

n=1

��PL

l=0 vl(2n)�l��2: (2.35)

The carrier phase estimate for an M-PSK signal is deduced by using the above timing esti-mate � , as follows

� =

8><>:

1M

argPN

n=1

�PL

l=0 vl(2n� 1)�l�M

1M

argPN

n=1

�PL

l=0 vl(2n)�l�M

:(2.36)

By noticing that the summation inside the parentheses is actually the output sample of theFarrow structure y(i) for both 0 � � < T=2 and T=2 � � < T , Eq. (2.36) can be rewritten


Fig. 2.9 NDA symbol timing recovery scheme.

as

� =1

Marg

NXn=1

�y(i)

�M: (2.37)

Finally, the symbol timing estimate is reduced to the search of the value of � that maximizeseither of the likelihood functions f�1(�Ts);�2(�Ts)g. Then, the timing estimate is usedfor computing the carrier phase estimate �. Alternatively, this algorithm is described asfollows;

If max[�1(�Ts)] > max[�2(�Ts)] then� = argmax�[�1(�Ts)]

� = 1M

argPN

n=1

�y(i)

�Melse

� = argmax�[�2(�Ts)]

� = 1M

argPN

n=1

�y(i)

�Mend If

The overall scheme of the NDA block-based symbol timing and carrier phase recovery isillustrated in Figs. 2.9 and 2.10. Notice that the influence of the data symbols of this NDAfeed-forward phase estimator is removed by the Mth power. Remark also that in this case,the averaging of the likelihood function cannot be done in the polynomial coefficient do-main. Consequently, the algorithm is not as efficient as in the DA case with respect to thecomputational complexity.

2.6 SUMMARY OF SIMULATION RESULTS

Simulations have been performed to analyze the performance of digital receivers with non-synchronized sampling using the proposed polynomial-based ML synchronization algo-rithms for both DA and NDA cases and the results are shown in the publications [P1]-[P5].The proposed schemes are simulated using an AWGN channel model, and different signalconstellation types (i.e., PAM, QPSK, 16-QAM, and 64-QAM) for different block lengths

SUMMARY OF SIMULATION RESULTS 19

Fig. 2.10 NDA carrier phase estimation scheme for M-PSK modulations.

N . The transmitter and receiver pulse-shaping filters are square root raised cosine filterswith excess bandwidth of 35%.

A frequency domain optimized third order interpolation filter (L = 3) of length 8 de-signed by using a minimax synthesis technique [24] is compared in the simulations with theconventional cubic Lagrange interpolator [13]. The length of the FIR filters in the Farrowstructure for the optimized interpolator is twice as long as for the Lagrange interpolator.However, the filter coefficients are symmetrical for this optimized design which makes itscomplexity of the same order with the Lagrange interpolator. The number of the FIR filterbranches is four for both interpolators. Also, the simulations are done using the worst-casetiming offset and some selected values of the phase error.

Several sets of experiments have been conducted for both DA and NDA synchronizationapproaches. Theoretical expectations are compared with simulated worst case (with respectto timing offset) SEP as a function of Eb=N0 using different types of modulations. Also,variance and mean of the timing jitter as well as symbol error probability (SEP) of bothinterpolators are compared. The effect of a relatively small frequency error, compared tothe symbol rate, on the synchronization scheme is analyzed in [P4]. The simulation resultsdemonstrate the efficiency and excellent performance of the proposed block-based feedfor-ward estimators even in the presence of a small frequency error. Also, the results showthe excellent performance of the frequency domain optimized interpolator compared to thetraditional Lagrange interpolator specially if we want to make use of shorter block lengthswhich provide fast acquisition. In [P5], a new frequency estimator for fine acquisition suit-able for this digital reciever type is derived. The simulation results of this work are given inthe publications [P1]-[P5] included in the appendices, and a summary of the main results ispresented in Chapter 4.

Chapter 3Multipath Delay Estimation forAccurate Positioning

3.1 INTRODUCTION

In this chapter, we provide an overview of the most promising geolocation positioning tech-niques for wireless systems that are being standardized, including a survey of fundamentalconcepts and major problems in positioning. Then we briefly review the GPS system. Fi-nally, we discuss and introduce new techniques with subchip resolution capabilities forestimating closely-spaced multipath delays in spread spectrum CDMA systems. Generally,multipath delays caused by distant reflectors have relatively large delay spread, more thanone chip interval, that can be detected using conventional techniques [25], [26]. However,shorter excess path delays result in overlapped fading multipath components that introducesignificant errors to the LOS path time and gain estimation. The proposed algorithms areintended to improve the accuracy of location estimates of GPS, as well as, mobile phonepositioning using the cellular network assisted GPS technology by estimating correctly theLOS path. In the sequel, we will focus our study mainly on the GPS case.

3.2 TECHNIQUES FOR PERSONAL POSITIONING: PRINCIPLES ANDPROBLEMS

Although mobile location is an enhanced feature of cellular systems which are primarilydesigned only for voice and data transmission, there exist some basic parameters avail-able to the network that can be used to generate a rough position estimate. These availablepieces of information, such as serving cell identity (CI), timing advance, and measured sig-nal strengths of the serving and neighboring cells, can provide only a limited precision oflocation which cannot in any case satisfy the FCC requirements [1]. Even though, mobilepositioning is a rather new concept and it is still in its infancy, various geolocation tech-nologies have been recently devised using either cellular network-based, mobile-based,

21

22 MULTIPATH DELAY ESTIMATION FOR ACCURATE POSITIONING

or hybrid approaches [2]. Hereafter, we briefly discuss the most prominent positioningmethods that have been approved for standardization within the 3rd Generation PartnershipProject (3GPP)1, by the sub-committee T1P1.5 of the American T1 Standards Committee,and subsequently in ETSI Technical Committee SMG. The methods that are being standard-ized by T1P1.5 for GSM are Time of Arrival (TOA), Enhanced Observed Time Difference(E-OTD), and Assisted GPS (A-GPS) [27]. In addition, Observed time difference of Ar-rival - Idle Period Down Link (OTDOA-IPDL) and Cell Identity methods have been addedto allow easy migration to the upcoming 3G systems [28].

� Time of Arrival

Time of Arrival positioning method with known time of transmission (TOT), is amultilateral and pure network-based approach where multiple base stations listen tohandover access bursts and triangulate the position of the mobile. Measurements ofthe exact time of arrival of at least three MS to BSs radio links has to be performedwith respect to a synchronized and common reference time clock [53]. The locationof the user equipment is consequently given by the intersection of the three TOA cir-cles. TOA positioning requires full network synchronism, which is not the case forGSM and the forthcoming 3G networks [2].

A more practical and suitable technique for the asynchronous networks is the TimeDifference of Arrival (TDOA) positioning where TOA differences are used in orderto eliminate the unknown TOT from the observations. TDOA technique relies sim-ply on the time difference at which signal arrives at multiple BSs, rather than on theabsolute arrival time like the TOA method [29]. TOA estimates of a serving BS andits neighbor yield one TOA difference which result in a hyperbola that lies betweenthe two BSs. Measurements of TOA from at least four serving neighbor BSs resultin three hyperbolas. The intersection of the three hyperbolas determines the mobiledevice position, and this is known as hyperbolic trilateration [29].

� Enhanced Observed Time Difference

Enhanced Observed Time Difference is basically reversed TOA or mobile terminalbased TOA, where the handset is much more actively involved in the positioning pro-cess. E-OTD is a unilateral approach that involves the mobile station in estimatingthe timing differences between the various base stations [30]. In E-OTD, the mobilestation listens to bursts from several base stations and measures the observed timedifferences, which are then used for trilateration of the mobile position as shown inFigure 3.1.

Unilateral E-OTD principle has been approved for standardization in different cellu-lar systems. For GSM it is called E-OTD, in 3G systems it is OTDOA-IPDL, and inUS-CDMA it is called Advanced Forward Link Trilateration (AFLT). These systemshave different names mainly because of the different network types and measurementprocesses of TOA signals [27].

� Assisted Global Positioning System (A-GPS)

13GPP is a joint activity of the European ETSI, American National Standards Institute, and Japanese ARIP tostandardize next generation wireless communication systems.

TECHNIQUES FOR PERSONAL POSITIONING: PRINCIPLES AND PROBLEMS 23

MLU

Base Station

GPS antennaMeasurementerror margin

MLU

MS equipped with GPS

MLU

Fig. 3.1 Principles of E-OTD and A-GPS methods in cellular network.

Assisted Global Positioning System relies on mobile stations having an integratedGPS receiver aided by the cellular network assistance and support to enhance thepositioning, especially for the non-line-of-sight (NLOS) conditions and indoor en-vironments. Assistance data is transmitted from the network to the MS in order toexpedite the GPS signal acquisition search and therefore shortening the time-to-first-fix from minutes to a second or less [31]. GPS-based positioning appears to be themost prominent and the leading candidate for wireless systems in terms of accuracy,reliability, latency, availability, and continuity of service [2].

The major problem of the above described geolocation solutions, with the exception ofA-GPS, is that they are unable to deliver reliable positioning to the end user with homo-geneous performance in urban, suburban, rural, and indoor environments. Currently, theefforts are aiming at defining hybridised solutions that satisfy all the imposed requirements,involving the lowest possible cost and minimal impact on the network and handset equip-ment. For example, network-based positioning systems have the advantage of working withall existing mobiles, but have the disadvantage of limited accuracy and requiring additionalinvestments in the supporting infrastructure. Similarly, mobile station based on stand-aloneGPS provides location capability with high accuracy, also in the absence of wireless cover-age or network assistance using the same infrastructure, but requires handset modifications,


receiver could take several minutes to acquire the satellite signals, and generally fails inradio shadows and indoor environments.

Another major source of error that has a strong impact on the accuracy of most ge-olocation solutions, including GPS, is due to the multipath signal propagation that maybe due to atmospheric reflection or refraction, or reflections from buildings and other ob-jects. Commonly, in most applications of radio communications and navigations [25], theline of sight (LOS) signal is succeeded by multipath components that arrive at the receiverwithin a short delay spread, that can be less than one chip interval. This causes overlap-ping fading multipath components and introduces significant errors in the LOS path time ofarrival and gain estimation. In spread spectrum CDMA systems, such as third generationmobile communications or GPS receivers, it is important to achieve accurate delay estima-tion (or code synchronization) before despreading and data detection. Also, most spreadspectrum systems use spreading codes with non-ideal correlation properties which resultin co-channel interference, or multiuser-interference, for radio communication systems,which is also called multi-transmitter interference in radio-navigation systems [25]. Gener-ally, the performance of radio communication systems is heavily affected by the multipathand multiuser-interferences.

3.3 FUNDAMENTALS OF GPS

The principle of GPS positioning is based on the concept of TOA ranging to determine theuser location. This concept entails measuring the propagation time of signals broadcast si-multaneously from satellites at known locations. Ideally, the distance between a satelliteand a user receiver is obtained by multiplying the propagation-time (or transit-time) with thespeed of light. The GPS satellites, or space vehicles (SVs), are put in medium earth orbitalplanes and they are moving in space at a speed of about 4 km/s along their orbits, repeat-ing almost the same ground track once each day. Although each SV is equipped onboardby a pair of ultra-stable atomic clocks, the satellite clocks are maintained in synchronismby the monitoring control segment that uploads the updated parameters for each SV clock,together with the navigation message broadcast by each satellite [32].

In theory, a receiver can estimate its position using three TOA measurements if the satel-lite and receiver clocks are synchronized. However, in practice the user’s receivers typicallyemploy inexpensive and low-accuracy quartz oscillators as local clocks, which are set ap-proximately to GPS time. Therefore, the receiver clock introduces certain timing offsetor clock-bias to the true GPS time which affects the observed transit-time for all satel-lites equally. The receiver can overcome this problem by measuring the distance to fourSVs in view. These measured distances are usually too short, or too long, compared to the’real’ range by a common amount and they are called pseudo-ranges. In addition to the3-dimensional coordinates of spatial position, a user needs to estimate the receiver clockoffset. The user position can then be determined by solving the four pseudo-ranges for thefour unknowns [33].

FUNDAMENTALS OF GPS 25

- 3 dB

+90˚

L2Signal

L1Signal

BPSK modulation

Modulo-2 addition

L1 Carrier

1575.42 MHz

C/A-Code

1.023 MHz

NavigationData

50 Hz

P(Y)-Code

10.23 MHz

L2 Carrier

1227.60 MHz

Fig. 3.2 GPS signal structure.

3.3.1 Satellite Signals

GPS satellite transmissions utilize direct sequence spread spectrum (DS-SS) modulation[32]. Spread spectrum techniques have been widely in military use because they can com-bat strong interference and prevent message recovery by unauthorized receivers, and theyhave also been adopted for commercial applications [16]. Spread spectrum communicationsystems involve the transmission of a signal in a radio frequency bandwidth much greaterthan the data information bandwidth to be conveyed. These systems spread the transmittedsignal spectrum over a frequency range substantially greater than the bandwidth of the mod-ulating data message. In Direct sequence (DS) systems, the spectral spreading is performedby multiplying the data signal by an auxiliary pseudo random-noise (PRN) code [34].

Each SV broadcasts two types of PRN ranging signals, as well as navigation data whichconsists of satellite ephemeris data and satellite health data, allowing users to measure theirpseudo-ranges and hence estimating their positions, velocity, and time [32]. Figure 3.2 il-lustrates the GPS signal structure. The ranging signals are pseudo-random noise codes thatmodulate the satellite carrier frequencies using binary phase shift keying (BPSK). Eachsatellite transmits continuously two microwave carrier signals called the primary carrier,L1 = 1575:42 MHz, and secondary carrier, L2 = 1227:60 MHz. The carrier frequenciesare modulated by spread spectrum codes with a unique PRN sequence associated to eachsatellite and by the navigation data [33]. The near orthogonality of the PRN code sequencespermits all the satellites to transmit on the same carrier frequencies without incurring signif-icant mutual interference [34]. PRN codes are simply deterministic binary sequences withspecific statistical random noise-like properties [34]. The family of PRN codes is mainlycharacterized by the low cross-correlation between the codes, they are nearly orthogonal,and the autocorrelation function is almost zero except at zero delay [32].


3.3.2 Signal Characteristics

The carrier frequency L1 is BPSK modulated by two PRN codes, the coarse acquisitioncode (C/A-code) and the precision code (P-code), while L2 is BPSK modulated only by theP-code. Low rate navigation data is modulated to both carriers. The P-code is available onlyfor USA Department of Defense (DoD) authorized users and, when encrypted, is called Y-code. P-code is a very long PRN code with a repetition period of one week and chippingrate of 10:23 MHz and is the basis for precise position services (PPS). The C/A-code is agold code sequence of length 1023 chips, with a repetition period of 1 ms, and a null to nullbandwidth of about 2:046 MHz, ten times smaller that of the P-code [32]. C/A codes areintended for civil standard position services (SPS) and available for all users. TheL1 carrieris modulated by C/A code sequence combined with the navigation data inphase quadraturewith the precision P-code combined with the data [36]. Thus, as shown in Fig. 3.3, themodulation of the civil C/A-code combined with the data is orthogonal to the modulationof the P-code added with the data on the L1 signal. The navigation data consists of a low-rate (50 bits/sec) bit-stream describing the satellite orbits (ephemeris and almanac), clockcorrections, and other parameters.

The received power levels on earth of the three GPS signals transmitted by the SVs areextremely weak and depend on the user elevation angle [32]. The minimum received signalpower level, correspond at an elevation angle of 5Æ from the user’s horizon are �160 dBWfor the C/A-code, �163 dBW for P(Y)-code at L1, and �166 dBW for P(Y)-code at L2.Table 3.1 summarizes the GPS signal structure and the minimum received signal power at

C/A-Code

P(Y)-Code

P(Y)-Code

f

1L

20.46 MHz

2.046 MHz

2L

Fig. 3.3 GPS signal spectrum (positive side).

an elevation of 5Æ. The change in the received power levels with respect to the user eleva-tion angle and signal structure is illustrated in 3.4. GPS is limited by the weak penetrationindoor of the satellite signal. Multipath effects are considered as one major sources of errorin precise GPS positioning services. It affects the code range, the carrier phase measure-ments, and also the signal power which is an average of the composite of signal power ofthe direct and multipath components.

MULTIPATH DELAY ESTIMATION TECHNIQUES 27

Table 3.1 GPS signal structure and minimum received signal power.

Signal P(Y) &L1 C/A &L1 P(Y) &L2

Carrier Frequency (MHz) 1575:42 1575:42 1227:60

PRN Codes (Mchips/sec) 10:23 1:023 10:23

Navigation Data (bits/sec) 50 50 50

Minimum Received Power (dBW) �163 �160 �166

-157

-160

-163

-166

4050 20 60 80 90

User elevation angle in deg

Received power in dBW

Fig. 3.4 User received minimum power levels with respect to the user elevation angle and signalstructure.

3.4 MULTIPATH DELAY ESTIMATION TECHNIQUES

3.4.1 Channel Models

In a multipath fading environment, the channel can be modeled by a linear time-variant filtercharacterized by the complex-valued lowpass equivalent impulse response [26]

h(�; t) =NXn=0

�n(t)ej�n(t)Æ(� � �n(t)) (3.1)

where N represents the number of propagation paths, �n(t), �n(t) and �n(t) are the ampli-tude, the phase, and the propagation delay of nth path, respectively. The channel is assumedto remain stationary during the considered observation time interval. In the sequel, the no-tation �n, �n and �n will be adopted for simplicity.


By assuming that h(�; t) is a zero mean, complex Gaussian random variable and wide-sense stationary, it can be described by the following fading autocorrelation function [16]

Rh(�;�t)4= E[h�(� ; t)h(� ; t +�t)] (3.2)

which gives a measure of the speed of channel variations. For �t = 0, the autocorrela-tion reduces to Rh(�; 0) � Rh(�), called the intensity profile of the channel measuring theexpected received power as a function of the delay � . The delay-spread of the channel, de-noted by Tm, is given by the region over which theRh(�) is not equal to zero and it indicatesthe degree of the time spreading in the channel. Thus, the reciprocal Bc = 1=Tm is definedas the coherence bandwidth of the channel and it is a measure of the frequency selectivity.The multipath components have different path lengths resulting in different propagation de-lays. The delay spread of a channel depends in part on the proximity of scattering objectsto the transmitter and receiver.

Due to the time variations in the medium structure, multipath delays are generally time-varying. The rate at which these-time delays vary, influences strongly the multipath channeleffects on the signal. Time-variant multipath characteristics of the channel are analyzedusing the spaced-time spaced-frequency correlation function of the channel given by [16]

RH(�f ;�t)4= E[H�(f ; t)H(f +�f; t+�t)]: (3.3)

Here H(f ; t) is the Fourier transform of h(� ; t) and the time variations in the channel aremeasured by the time spacing parameter �t.

Another way to express the rapidity of the channel variations and to reflect the Dopplereffects, is by means of the Fourier transform ofRH (�f ;�t)with respect to the time spacing�t as follows [26]

SH(�f ; �)4=

Z +1

�1

RH(�f ;�t)e�j2�� td�t: (3.4)

In particular for �f = 0, the Doppler power spectrum of the channel is defined by

S(�) � SH(0; �) =

Z +1

�1

RH(�t)e�j2��td�t: (3.5)

The width of the region over which S(�) is non-zero, is called the Doppler spread of thechannel, denoted here by Bd, and its reciprocal is called the coherence time (�t)c = 1=Bd.The Doppler spread function characterizes the rapidity of the fading and is used in model-ing Doppler shifts in the frequency domain caused by the relative motion of the transmitter,scatterers, and receiver. Slowly fading channel has a small Doppler spread Bd, whichcorresponds to a long coherence time.

Different types of fading channels do exist depending on the value of spread factor TmBd

of the channel, as well as, on the bandwidth W = 1=T of the lowpass equivalent receivedsignal [35]. For example, for a frequency nonselective flat and slowly fading channel, thespread factor is less than unity, TmBd < 1. In this case, the signal bandwidth is very smallcompared to the coherence bandwidth of the channel W � Bc, also the signalling intervalis very small compared to the coherence time T � (�t)c and the channel properties re-mains constant during a symbol interval. Similarly, if the spread factor is greater than unityTmBd > 1, it indicates that we have either a frequency selective channel W > Bc or a fastfading channel T > (�t)c.


3.4.2 Conventional ML Based Approaches

After down-conversion of the received civil signal to baseband through mixing and filter-ing, the received GPS signal from one satellite in the presence of M -path channel can bemodeled as [25], [37]

r(t) = x(t) + n(t)

=

M�1Xi=0

aiC(t� �i)ej�i + n(t); (3.6)

where C(t) is the real spread-spectrum code with a chipping rate 1=TC much higher thanthe navigation data stream rate 1=TD = 50 bits/s, and n(t) is an additive white Gaussiannoise with power spectral density N0. The time-variant ai, �i and �i represent the attenua-tion factor, time delay, and phase for the ith path, respectively. The estimation of the set ofparameters fai; �i; �ig is very essential for locating the line-of-sight path, and consequentlyfor determining the GPS position, velocity and/or time.

Assuming that the down-converted signal r(t) is completely defined in the observationtime interval TD, the conditional likelihood function based on the observation vector r fora given set of synchronization parameters � = fa; �; �g is given by

p�r�� = � exp

��

1

N0

ZTD

[r(t)� x(t)]2dt

�(3.7)

where � is just a positive constant independent of the synchronization parameters which canbe neglected, and x(t) is the local trial signal replica generated at the receiver, modeling theestimated line-of-sight and multipath signals

x(t) =

M�1Xi=0

aiC(t� �i)ej�i : (3.8)

Multipath delay estimation techniques are based on the maximum likelihood theory [25]which are trying to estimate the set of parameters fa; � ; �g by minimizing the mean squarederror of the log-likelihood function (LLF) given by

LML(a; � ; �) = Re

�ZTD

[r(t)� x(t)]2dt

�: (3.9)

Thus, the synchronization parameters are determined by differentiating the above LLF func-tion with respect to the synchronization parameters, then setting the partial derivatives equalto zero. The partial derivative with respect to the time-delay estimate is given by [25]

@

@�LML(ai; �; �i)

��=�i

= 2@

@�

�Renh

Rrc(�)�

M�1Xl=0;l6=i

alRc(� � �l)ej�l

ie�j�i

o��=�i

:

(3.10)In the above equation, Rrc(�) is the inphase/quadrature time-average cross-correlation func-tion over the observation time interval TD, given by

Rrc(�) =1

TD

ZTD

r(t)C(t� �)dt; (3.11)


and Rc(�) is the reference time-average correlation function of the spreading code, givenby

Rc(�) =1

TD

ZTD

C(t)C(t� �)dt: (3.12)

Different methods have been derived for finding the multipath delays, including coherentand non-coherent delay locked loops (DLLs) with proper early and late code spacings [34],[37] [38]. These methods are just trying to track the delay of the line-of-sight path by corre-lating the down-converted received signal with replicas of spreading codes locally generatedby the DS-SS receiver.

Normally, by spreading the transmitted signal over a bandwidth larger than the coherencebandwidth of the channel, we can decrease the effects of multipath fading. The bandwidthspread makes it possible to separate different multipath signals if their relative delay dif-ference exceeds one chip interval. Typically, distant reflectors have relatively large delayspread with more than one chip interval between successive paths, therefore, resulting multi-paths can be removed by correlator-based mitigation techniques [25]. Thus, by maximizingthe partial derivative equation of the LLF (3.9) with respect to the time-delay, the desiredith path estimate can be written as follows,

�i = argmax�

�Renh

Rrc(�)�

M�1Xl=0;l6=i

alRc(� � �l)ej�l

ie�j�i

o�: (3.13)

Similarly, by differentiating equation (3.9) with respect to the carrier phase and ampli-tude, and by solving the partial derivative equations for the M multipath components, theestimated carrier phase and amplitude for the ith path are, respectively, given by [25]

�i = arghRrc(�i)�

M�1Xl=0;l6=i

alRc(�i � �l)ej�l

i(3.14)

ai = Renh

Rrc(�i)�

M�1Xl=0;l6=i

alRc(�i � �l)ej�l

ie�j�i

o: (3.15)

Basic DLLs can estimate efficiently the multipath propagation delays if they are not over-lapping and spaced at more (1 + �)TC , where � being the loop discriminator value orearly-late code spacing. However, the effects of offset paths are ignored. Several stud-ies have been conducted to analyze the effects of multipath delays on coarse acquisitiontechniques [39] and fine acquisition techniques [40]. Delay locked loop (DLL) circuits gen-erally fail to estimate closely-spaced multipaths with less than one chip interval, besidestheir convergence can be quite slow [25], [26], [37], [38]. Fig. 3.5 shows an example ofS-curves obtained via a non-coherent DLL (NCDLL) with SNR = 20 dB for half chip andtwo chips spaced taps. We clearly notice that NCDLL fails to detect very closely spacedmultipaths [35].

Another illustrative example in Fig. 3.6 shows the effect of multipath propagation on theS-curve with different early-late spacing values. Notice that the multipath components biasthe tracking of the DLL. Smaller discriminator values may reduce this bias effect but are notable to eliminate it completely. However, as the excess path delays becomes smaller, they


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Timing Error (in chips)

S−

curv

e2 Rayleigh fading paths 1/2 chip − spaced taps 2 chips − spaced taps

Fig. 3.5 Non-coherent DLL S-curve for a multipath channel.

result in shorter delays spread. Consequently, the LOS signal overlaps with the succeedingmultipath components of short delays introducing significant error to the LOS path time andgain estimation.

Generally, multipath delay estimation techniques are based on the cross-correlation func-tion (Eq. 3.11) of the down-converted signal de-spread with a copy of the spreading codelocally generated by the DS-SS receiver. The advanced multipath estimating DLL tech-niques [38] are trying to approximate the overall cross-correlation using a set of referencecorrelation functions (Eq. 3.12) with certain delays, phases and amplitudes according toEqs. (3.13), (3.14) and (3.15).

Many other techniques for mitigating the effects of multipath delays have been devel-oped, including several subspace-based approaches. However, they are usually too complexfor practical purposes and suitable only for systems employing short codes [41], [43], andthe topic remains still an open area of research.

3.4.3 Multipath Delay Estimation Using Peak Tracking with PulseSubtraction

Assuming that the locally generated code at the receiver side is locked to the received signalcode (locally generated real code corresponds to the SV received signal code), by substitut-ing equation (3.6) into (3.11), the cross-correlation function is simply expressed by

Rrc(�) =1

TD

M�1Xi=0

aiej�i

ZTD

C(t� �i)C(t� �)dt+1

TD

ZTD

n(t)C(t� �)dt: (3.16)


−1 −0.5 0 0.5 1 1.5

−1.5

−1

−0.5

0

0.5

1

1.5

Timing Error (in Chips)

Loop

Cha

ract

eris

tic

∆=1/2∆=1/4∆=1/8∆=1/16∆=1/32

Fig. 3.6 Effect of multipath propagation on the S-curve of a non-coherent DLL using differentdiscriminator values � and two paths channel.

Generally, practical GPS receivers perform averaging over short-time observation intervalsless than the navigation data interval TD to speed up the SV signals tracking. Normally,even-though we are considering averaging over a sufficiently large observation time interval,we shall consider the following biased estimator

RT

0

D

(�) =

(1T

0

D

RT

0

D

C(t)C(t� �)dt for j� j < T0

D

0 otherwise

whose expected value or mean is given by [42]

EfRT

0

D

g = R(�)

�1�

j� j

T0

D

�= R(�)4 (�; T

0

D): (3.17)

Here, 4(�; T0

D) denotes a triangular function defined as follows

4(�; T0

D) =

�1� j� j=T

0

D j� j � T0

D

0 otherwise:(3.18)

Recalling the Gaussian nature of the random process as well as the mean of the biasedestimator (3.17), the expectation of the above cross-correlation function (3.16) becomes

EfRrc(�)g =M�1Xi=0

aiej�iRc(� � �i)

�1�

j� � �ij

T0

D

�

=

M�1Xi=0

aiej�iRc(� � �i)4 (� � �i; T

0

D): (3.19)


Notice that the expected value of the cross-correlation function is expressed simply as aweighted sum of delayed correlation functions times triangular windows with certain am-plitude and phase specific to each signal path. The correlation effectively amplifies theunderlying BPSK data signal, with an amplification factor equal to the length of the PRNGold code sequence.

The proposed peak tracking with pulse subtraction approach for estimating the multi-path delays is analogous to the advanced DLL techniques [25] in the sense that it is tryingto estimate the overall cross-correlation function using a set of reference functions, but withdifferent and simpler implementation. This approach is based on the search of the globalmaxima of the cross-correlation function (Eq. 3.16). Then, the contribution of each selectedmaximum is reduced from the overall cross-correlation using the autocorrelation functionof a rectangular pulse [54]. The proposed technique is described further by the followingalgorithm:

� find the global maximum

�1 = argmax��jRrc(�)j

�:

� compute the residual cross-correlation function

Rrc;1(�) =��Rrc(�)�Rrc(�1)4 (� � �1)

��:� find the global maximum of the residual cross-correlation function

�2 = argmax��jRrc;1(�)j

�:

� estimate successively the next Mest � 1 maxima

�k = argmax�

h��Rrc(�) �Pk�1

l=1 Rrc;l(�l)4 (� � �l; TC)��i; k = 2; 3; : : : ;Mest:

HereRrc;l(�) is the residual cross-correlation function after lth peak subtraction, Mest is anestimate for the number of multipaths of the channel (it can be taken equal to the number offingers (or correlators) as in a Rake receiver). By using the ideal code correlation pulse forsubtracting the contribution of each channel path, we are able to detect very closely-spacedmultipaths within less than one chip period (j�l � �kj < TC) [54].

Only a simplified generic structure is presented above to describe the algorithm. How-ever, instead of choosing the global maximum of the envelope of the residual correlationfunction as a true multipath component, we can take into account the merging phenomena.Hence, we try a set of multipath delays within a certain window around the global maximawith different weights, and the final estimates are those which give the best approximationof the correlation function in the mean square error sense.

3.4.4 Multipath Delay Estimation Based on Teager-Kaiser Operator

In this section, we introduce an innovative multipath delay estimation approach based on thenonlinear quadratic Teager-Kaiser operator that exploit the structure of the cross-correlationfunction (Eq. 3.11) for estimating sub-chip spaced multipath components.

The nonlinear quadratic TK operator was first introduced for measuring the real physi-cal energy of a system [44], [45]. The energy of a generating system of a simple oscillation


signal was computed as the product of the square of the amplitude and the frequency of thesignal. It was found that this nonlinear operator exhibits several attractive features such assimplicity, efficiency and ability to track instantaneously-varying spatial modulation pat-terns. Since its introduction, several applications have been derived for one-dimensionalsignal processing [S3], [46] (note that proposition 3 in [46] is erroneous as pointed in [47])and two-dimensional signal processing [48], [55].

The continuous-time TK energy operator of a complex-valued signal x(t) is defined asfollows [S3]

c[x(t)] = _x(t) _x�(t)�1

2[�x(t)x�(t) + x(t)�x�(t)]: (3.20)

Similarly, the discrete-time Teager operator of a complex valued signal is given by

d[x(n)] = x(n� 1)x�(n� 1)�1

2[x(n� 2)x�(n) + x(n)x�(n� 2)]: (3.21)

Many useful properties of this nonlinear quadratic operator have been derived, and theanalogy with the discrete-time domain has also been established [49], [56].

We notice that applying the continuous-time TK operator to the ideal reference corre-lation function of the spreading code which is characterized by the triangular shape (Eq.3.18), we obtain

c[4(t; TC)] =�(t; TC)

T 2C

+4(t; TC)Æ(t)

TC; (3.22)

where Æ(t) is the Dirac function, and �(t; TC) stands for a rectangular function with unitamplitude and duration 2TC centered at t = 0 defined as

�(t; TC)4=

�1 jtj � TC0 otherwise.

Obviously, Eq. (3.22) shows that the TK operator applied to a triangular function providesa clear time-aligned peak location of the triangular pulse in the presence of a certain ’noise’floor.

Assuming now that the cross-correlation function is the sum of M triangular pulses asfollows

R(t) =MXi=0

ai 4 (t� ti; TC)ej�i ; (3.23)

where ai, ti and �i denote the amplitude, delay and phase, respectively. By applying theTK operator to the above equation we obtain

c[R(t)] =1

T 2

C

"� MXi=0

aisign(t� ti)�(t� ti; TC) cos �i

�2

+

� MXi=0

aisign(t� ti)�(t� ti; TC) sin �i

�2#

+1

2TC

"R�(t)

� MXi=0

aiÆ(t� ti)ej�i

�+ R(t)

� MXi=0

aiÆ(t� ti)e�j�i

�#:(3.24)


Notice that the above expression becomes very large if the time variable t is equal to one ofthe true delays ti. Thus, equation (3.24) shows that the TK operator is capable of trackingvery accurately the peak locations of the triangular pulses within certain ’noise’ floor [57]independently of the delay spacing between the multipaths.

By exploiting the above properties of the quadratic operator, we found out that by ap-plying this nonlinear TK operator to the specific cross-correlation function (Eq. 3.11) ofa GPS receiver, one can easily estimate the multipath delays introduced by the channel.The multipath delays are simply estimated by selecting the time location of the highest(strongest) peaks of the Teager-Kaiser operator output function (equal to the number of fin-gers in the Rake receiver). Simulation results provided in publications [P6]-[P8] show thehigh performance of this innovative technique.

Chapter 4Summary of Publications

4.1 RECEIVER SYNCHRONIZATION STUDIES

A general approach for discrete-time modeling and simulation of continuous-time signalsand systems based on digital interpolation techniques is introduced in [P1]. We demonstratethat polynomial interpolation utilizing the generalized Farrow structure results in efficientpolynomial signal processing models with moderate computational complexity. Normally,the lowest sampling rates which are sufficient for system modeling, are not sufficient forobtaining informative and accurate models. In cases where computational efficiency is anissue, different sampling rates can be used in different blocks to reduce the computationalburden. In cases where there is a need to locate, e.g., the local maxima, minima, or zerocrossings of a signal, higher order oversampling rate is needed. It is common practice alsoto use oversampling factors which are an order of magnitude higher than what is requiredfrom the sampling theory point of view.

The generalized Farrow structure allows, with low additional computational complexity,to compute several new samples at arbitrary points between the existing samples. Thus, it isvery easy to get the signal sample values in a flexible way in what ever time instances theyare needed. Furthermore, it is possible also to implement iterative procedures for estimatingaccurately, e.g., the maximum value of a continuous-time signal as well as the location ofthe maximum. In [P1] we also develop simple modifications of the Farrow structure whichresult in efficient polynomial-based differentiators for approximating the continuous-timederivative of a signal from its discrete-time samples. The Farrow structure discussed in [P1]is the basis of the proposed ML-based synchronization algorithms. As another potentialapplication and a topic for future studies, Farrow-based derivative approximations could beutilized for efficiently calculating the continuous-time Teager function of a discrete-timesignal. This could lead to efficient implementation structures for the Teager-based delayestimation techniques.

In [P2], a polynomial-based maximum likelihood technique for jointly estimating thesymbol timing and carrier phase of digital receivers for the data-aided case is established.This technique is based on low order polynomial approximation of the log-likelihood func-

37

38 SUMMARY OF PUBLICATIONS

tion by using the Farrow structure. An extension of the proposed polynomial-based MLtechnique to the non-data-aided systems is derived in [P3]. Different simulations usingthe Lagrange and optimized interpolator are performed for both DA and NDA systems andcompared with the theoretical case. The results show the high performance of the opti-mized interpolator compared to the Lagrange interpolator especially if we want to make useof shorter block lengths to obtain fast acquisition. However, the NDA approach requiresmore computation than the DA approach for estimating the polynomial-based likelihoodfunction. Also, NDA based phase estimation is feasible only in case of PSK modulations.

In [P4], the effect of the frequency offset or Doppler shift on the performance of thedigital receiver is analyzed while adjusting the symbol timing and the carrier phase. Wedemonstrated that the proposed synchronization scheme performs quite well within a rela-tively small range of frequency offset values with respect to the symbol period. Under thesame assumption that the frequency offset is much smaller than the symbol period, sym-bol timing and carrier phase recovery can be done prior to the frequency error correction.Thus, we introduced in [P5] a frequency estimator that is directly derived from the carrierphase estimate. The proposed estimator is simple to implement and is particularly suitedfor this type of digital receiver architectures. Also, it performs fairly well when comparedwith the Fitz [50] and the Luise-Reggiannini [15] frequency estimators, in the presence ofsubstantial carrier frequency offset.

4.2 MULTIPATH DELAY ESTIMATION STUDIES

Correlation estimation properties related to DS-CDMA receivers have been studied in [P6].We showed the influence of the coherent integration and the non-coherent averaging lengthon multipath delay estimation, and the importance of the considered observation time in-terval length. In [P7], we introduced two techniques with subchip resolution capability forestimating closely-spaced overlapped multipath components suited for GSP system as wellas for other spread spectrum CDMA systems. The first innovative technique is very sim-ple and quite efficient based on Teager-Kaiser operator. It exploits the properties of thecross-correlation function between the received signal and the reference despreading codereplica generated at the receiver in order to estimate accurately overlapped multipath de-lays independently of the delay spacing between the multipaths. The second algorithm isbased on peak tracking with pulse subtraction which approximates the input correlationfunction using the superposition of a set of reference correlation pulses. Simulation resultsof a GPS receiver using different Rayleigh fading channels show the good performance ofthe proposed techniques. Also the obtained results are considered to be remarkable con-sidering the computation complexity of these approaches in comparison to the advancedmaximum likelihood based DLL techniques. Extension of the proposed algorithms to thecase of asynchronous multiuser WCDMA systems is presented in [P8]. Simulation resultsin the presence of multiple interfering users and Rayleigh fading multipath channels arepresented. It is demonstrated that the Teager-Kaiser technique is near far-resistant, and itsperformance in the presence of closely spaced multipaths is much better compared to thepeak tracking with subtraction method.

The problem of resolving closely-spaced overlapped multipath components can be con-sidered new, and has recently raised up the interest after its strong influence on the high

AUTHOR’S CONTRIBUTION TO THE PUBLICATIONS 39

precision accuracy of geolocation solutions. Few good research articles exist in the liter-ature dealing with this problem, they generally treat the case of only two paths that arepresent, and not very practical for implementation [37], [43], [51], [52].

4.3 AUTHOR’S CONTRIBUTION TO THE PUBLICATIONS

The research work of this thesis was carried out at the Institute of Communications En-gineering (formerly Telecommunications Laboratory), Tampere University of Technologyas one partner in the projects "Advanced Transceiver Architectures and Implementationsfor Wireless Communications" and "Analog and Digital Signal Processing Techniques forHighly Integrated Transceivers". The author has been a member of an active research groupinvolved in studying and developing DSP algorithms for highly integrated digital receivers,as well in building COSSAP and Matlab simulation models. The members of the researchgroup have been in close collaboration with the author and the whole project has beensupported and guided by the thesis supervisor Prof. Renfors. However, the author’s con-tribution to all of the publications has been essential in that he developed the theoreticalframework, prepared the manuscript, and performed the experiments.

This thesis includes eight publications [P1]-[P8]. They are categorized under twomain topics: estimation of the synchronization parameters for digital receivers with non-synchronized sampling clock [P1]-[P5], and estimation of closely-spaced multipath delaysin CDMA systems [P6]-[P8]. In particular, the author’s main contributions to these publi-cations are as follows. In [P1], the author formulated and simulated the general approachfor discrete-time modeling of continuous-time signals and systems based on the generalizedinterpolation technique which is efficiently implemented using minor modifications to theFarrow structure. Originally, the author introduced the concept of derivative approximationand the preliminary idea for integration using simple modifications of the Farrow structure.In [P2] and [P3], the author derived the likelihood functions for systematic synchroniza-tion schemes, made the extension for jointly estimating the symbol timing and carrier phasefor both DA and NDA systems, and carried out the simulations. In [P4] and [P5], the au-thor studied and simulated the effect of frequency offset on the synchronization scheme,and derived a new frequency estimator suitable for the proposed digital receiver type. Op-timization of the Farrow structure based filters, needed in [P1]-[P5], are performed by Dr.Vesma.

In [P6] and [P7], the author is the founder of the multipath delay estimation techniquesbased on Teager-Kaiser operator. The idea of pulse subtraction technique was originallyproposed by Prof. Renfors. The author derived and simulated the algorithms. In [P8], thesealgorithms have been applied and tested in the WCDMA multiuser environment using ex-isting COSSAP simulation model. The author conducted experimental studies and wrotethe manuscript.

Chapter 5Conclusions

Currently, the main driving force behind the software radio development is the coexis-tence of several radio communication systems as a result of the high competition betweenAmericans, Asians, and Europeans. The ultimate goal of designing a single-chip, flexiblewireless transceiver supporting multimode and multistandard compatibility introduced newchallenges in receiver architecture designs. The current trend is to push the analog/digitalinterface towards the antenna to simplify the analog parts and allow the implementationof most receiver functions digitally, increasing therefore the flexibility, configurability, andintegrability. In fact, existing communication standards are using different data rates, andconsequently they employ different master clock rates. By using a digital receiver architec-ture with a free running clock oscillator, receiver complexity can be substantially reducedby employing only a single master clock, and the other needed clock rates can be gener-ated virtually by means of sampling rate conversion. As one important functionality, thesenew receivers require flexible and efficient synchronization algorithms which depend on thesystem requirements and receiver architecture design.

The first part of this research work introduces new maximum likelihood based symboltiming and carrier phase estimators suitable for digital receivers with non-synchronized sam-pling clock. Polynomial approximations of the likelihood functions using the Farrow-basedinterpolation technique were established for both DA and NDA forms. The simulationsdemonstrate the efficiency and excellent performance of the block-based feedforward esti-mators. The proposed approach provides an efficient and practical way for approximatingvery closely the ideal maximum likelihood estimate, especially for the DA case. The Farrowstructure we are considering is particularly suitable for digital implementation.

However, the NDA case requires more computational burden because the averaging ofthe likelihood function can not be done in the polynomial coefficient domain as for the DAcase. Therefore, further studies with respect to the computational complexity of the NDAtechnique are needed for practical applications.

Another essential receiver functionality closely related to synchronization is propagationdelay estimation which is the basis for geolocation positioning solutions. The rapid evolutionof digital communication systems and the new requirements imposed by the standardiza-tion authorities brought various consumer electronics device types and a high demand on

41

42 CONCLUSIONS

new communication services, such as mobile phone positioning with high and reliable accu-racy. Even if safety was the primary motivation for mobile positioning, a lot of commercialapplications have already emerged in the market.

However, multipath effects of the mobile channel could degrade the location estimatesubstantially, specially in the case of shorter excess path delays which result in overlappedfading multipath components that introduce significant errors to the LOS path time and gainestimation. Estimation of closely-spaced multipath components is regarded as a killer issuein location estimation, considered as one of the major sources of errors in high precisiongeolocation solutions, and still it is an open topic for research work.

In the second part of this thesis work, we introduced innovative techniques with sub-chip resolution capability for multipath delay estimation based on peak tracking with pulsesubtraction and Teager-Kaiser quadratic operator both suited for spread spectrum CDMAsystems, and we focused our study to the case of GPS receivers. Simulations results con-firmed the high performance of Teager-based technique compared to the peak tracking withpulse subtraction method for all cases of closely-spaced multipaths within less than half chipperiod. The ability of solving closely-spaced paths in the presence of overlapped multipathcomponents comes from the exploitation of the properties of the cross-correlation function.This new techniques are extremely simple and very efficient for estimating closely-spacedmultipaths with much less computational complexity compared, e.g., to the conventionaldelay-locked loops and subspace-based multipath delay estimation methods [25], [41].

In general, for each shape of the reference correlation function or pulse shaping filterused, there may exist a linear or nonlinear operator capable of estimating easily the multipathdelays. It remains as a challenging topic for future work is to generalize the Teager-basedtechnique, and to derive other linear or nonlinear operators capable of tracking multipathdelays in different communication systems.

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[23] T.I Laakso, V. Valimaki, M. Karjalainen, and U.K. Laine, "Splitting the unit delay,"IEEE Signal Processing Magazine. vol. 13, pp. 30–60, January 1996.

[24] J. Vesma and T. Saramaki, "Interpolation filters with arbitrary frequency response forall-digital receivers," Proc. IEEE Int. Symp. Circuits & Syst., Atlanta, GA, May 1996,pp. 568–571.

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[26] M.K. Simon and M.S. Alouini, Digital communication over fading channels; a unifiedapproach to performance analysis. John Wiley & Sons, Inc., 2000.

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[32] E.D. Kaplan, Understanding GPS; principles and applications. Artech House Inc.,1996.

[33] P. Enge and Pratap Mistra, "Scanning the issue/technology; special issue on GPS,"Proc. of the IEEE, vol. 87, No. 1, January 1999, pp. 3–15.

[34] M. Braasch and A.J.V. Dierendonck, "GPS receiver architectures and measurements,"Proc. of the IEEE, vol. 87, No. 1, January 1999, pp. 48–64.

[35] J.S. Lee and L.E. Miller, CDMA systems engineering handbook. Artech House, 1998.

[36] http://www.colorado.edu/geography/gcraft/notes/gps/

[37] M.C. Laxton and S.L. DeVilbis, "GPS multipath mitigation during code tracking,"in Proc. of the American Control Conference, Albuquerque, New Mexico, pp. 1429–1433, June 1997.

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[48] J. Fang and L. Atlas, "Quadratic detectors for energy estimation," IEEE Transactionson Signal Processing, vol. 43, no. 11, pp. 2582–2594, November 1995.

[49] J. F. Kaiser, "Some useful properties of Teager’s energy operators," Proc. IEEEICASSP–93, vol. III, pp. 149–152, April 27–30, 1993.

[50] M.P. Fitz, "Further results in the fast estimation of a single frequency," IEEE Trans.Commun., Vol. 42, pp. 862–864, March 1994.

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[52] G. Fock, J. Baltersee, P. Schulz-Rittich and H. Meyr, "Channel tracking for Rake re-ceivers in closely spaced multipath environment," IEEE Journal on selected areas incommunications, vol. 19, no. 12, pp. 2420–2431, December 2001.

[53] R. Hamila, M. Renfors, G. Gunnarsson, M. Alanen, "Data processing for mobile phonepositioning," in Proc. VTC’99, Amsterdam, Netherlands, Sept. 1999, pp. 446–449.

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[55] F. Alaya Cheikh, R. Hamila, M. Gabbouj and J. Astola, "Impulse noise removal inhighly corrupted color images," Proc. 1996 IEEE International Conference on ImageProcessing, ICIP-96, Vol.1, pp. 997–1000, Lausanne, Switzerland, 1996.

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Publications

Publication 1

R. Hamila, J. Vesma, T. Saramaki and M. Renfors, "Discrete-Time Simulation ofContinuous-Time Systems Using Generalized Interpolation Techniques," in Proc. 1997the Summer Computer Simulation Conference, SCSC’97, Arlington, Virginia, USA, July1997, pp. 914–919.

Copyright c 1997 by SCSC’97. Reprinted, with permission, from the proceedings ofSCSC’97.

DISCRETE-TIME SIMULATION OF CONTINUOUS-TIME SYSTEMSUSING GENERALIZED INTERPOLATION TECHNIQUES

Hamila Ridha, Jussi Vesma, Markku Renfors, and Tapio SaramäkiTelecommunications Laboratory

Tampere University of TechnologyP. O. Box 553, FIN-33101 Tampere, Finland

E-mail: [email protected]

Keywords: communications, signal processing, numerical methods, discrete simulation

ABSTRACT

This paper presents a general approach for mod-elling and computer simulation of continuous-timesignals and systems. The idea is based on utilizinga polynomial approximation of the continuous-timesignal between the existing discrete-time samples.The so-called Farrow structure provides a computa-tionally efficient implementation making it possibleto get the interpolated signal values at arbitrarytime instances between the existing samples. Thisresults in efficient overall simulation models, be-cause with the aid of the proposed techniques it ispossible to obtain visually correct continuous-timewaveforms while using a modest oversampling fac-tor. This paper proposes a family of optimized poly-nomial interpolators for modelling purposes anddiscusses the filter criteria in terms of the requiredaccuracy and other simulation parameters. Inaddition, a numerical approach for estimating thederivative and integral of a discrete-time signal isestablished.

1. INTRODUCTION

Discrete-time modelling and simulation of con-tinuous-time signals and systems is a common prac-tice in many fields for research, development, andeducation purposes. From computational efficiencypoint of view, the smallest sampling rate sufficientfor accurate system modelling is determined by thesampling theory. In cases where computational effi-ciency is an issue, different sampling rates can beused in different blocks to reduce the computationalburden. Anyway, the lowest sampling rates whichare sufficient for system modelling are typically notsufficient for obtaining informative and visuallypleasing waveform plots of the signals. For this rea-son, it is a common practice to utilize oversamplingfactors which are an order of magnitude higherthan what is needed for accurate system modelling.

Also in cases where there is a need to locate,e.g., the local maxima, minima, or zero-crossingsof a signal, considerably higher sampling rates areneeded. Other difficult cases are, for example, con-version between incommensurate sampling ratesthat might be used in different model blocks to beconnected together.

Partial solutions to these problems are offered by”classical” sampling rate conversion techniques(Crochiere and Rabiner 1983) which allowsampling rate conversion by integer or fractionalfactors. Adjusting the sampling phase is difficultwith these classical techniques, but the so-calledfractional delay filters (Laakso et al. 1996; Vesmaand Saramäki 1996a; Vesma and Saramäki 1996b;Vesma and Saramäki 1997; Saramäki andRitoniemi 1996) help in this task. Another classicalapproach is the Lagrange interpolation (Erup et al.1993), which however also has its limitations.

In this paper, we consider polynomial interpola-tion with the so-called Farrow structure (Farrow1988) implementation as a general-purpose solutionto the above-mentioned tasks. The Farrow structureallows, with low additional computational complex-ity, to compute several new samples at arbitrarypoints between the existing samples. Thus, it is veryeasy to get the signal sample values in a flexibleway in what ever time instances they are needed.Furthermore, it is possible also to implement itera-tive procedures for estimating accurately, e.g., themaximum value of a continuous-time signal as wellas its location.

Also the Lagrange interpolation can be imple-mented using the Farrow structure, but we havedemonstrated that a clearly better performance canbe achieved by utilizing filter optimization tech-niques (Vesma and Saramäki 1996a; Vesma andSaramäki 1996b; Vesma and Saramäki 1997) and

also that the performance of the Lagrangeinterpolation is not sufficient for all purposes(Vesma et al. 1996).

In Section 2 of this paper, we first describe theideas of polynomial interpolation and Farrow struc-ture. In Section 3, we discuss the use of the Farrow-structured interpolators in signal and system model-ling and give some examples of interpolationfilters. In Section 4, we show how the derivativeand integral functions of a continuous-time signalcan be approximated using the Farrow-structuredinterpolators.

2. POLYNOMIAL INTERPOLATION

A continuous-time signal can be approximated atany point between the discrete-time samples using apolynomial-based interpolation. This is illustratedin Fig. 1. The approximating polynomial (solidline) is formed by adding together weighted andshifted basis functions (dashed line). Thesefunctions are weighted by the corresponding samplevalues (circles). The sample values which are usedin the approximation form a base-point set. Nowthe value of the original continuous-time signal canbe estimated by evaluating the value of the ap-proximating polynomial at the desired point.

It should be pointed out that the best approxima-tion is obtained in the middle interval of the base-point set, [0,1) in the figure. In this paper, we useonly basis functions which are piecewise polynomi-als because of the efficient implementation.

−3 −2 −1 0 1 2 3 4−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(t−mT s) / Ts

Fig. 1. Weighted basis functions (dashed line) andapproximating polynomial (solid line).

The polynomial-based interpolation can be effi-ciently implemented using a special discrete-timefilter structure (terms ‘ interpolator’ and‘ interpolation filter’ are used). This filter is the so-called Farrow structure (Fig. 2) (Farrow 1988). Inthe Farrow structure, all the filter coefficients arefixed and the only changeable parameter is thefractional interval µ ∈ [0,1), which is defined by

t mT Ts s= + µ , (1)

where Ts is the input sampling interval and

m t Ts= / . (2)

Here x stands for the integer part of x. In thiscase, [mTs, (m+1)Ts] forms the middle interval ofthe base-point set.

With different values of µ, the output of the Far-row structure gives the following approximation tothe continuous-time signal:

~( ) ( )( ) ( ) ,y t f mt mT

Tf ml

s

s

l

l

L

ll

l

L

=−

== =∑ ∑

0 0

µ (3)

where L is the degree of the interpolation and thefl(m)’s are given by

f m c m x ml l( ) ( ) * ( )= (4)

with the cl(m)’s being the coefficients of the Farrowstructure. The total number of coefficients isN(L+1), where N is the length of the branch filters.

In many applications, the output of the Farrowstructure is computed at a fixed output rate, 1/Ti,which may have an arbitrary ratio to the input sam-pling rate. In some other applications, the outputsmay be needed at arbitrary time instances, e.g., ac-cording to some iterative search procedure.

The Farrow structure has a time-varying impulseresponse. The impulse response values of this digi-tal filter are determined by the fractional interval µand they are obtained from the underlying continu-ous-time impulse response h(t) (Fig. 3). Actually,this continuous-time impulse response is the sameas the basis function. Therefore, the Farrowstructure can be analyzed and synthezised usingthis continuous-time filter h(t). In the ideal case,the continuous-time filter is an ideal low-pass filterand in the time domain it is the sinc-function. This

fact can be utilized in the filter optimization(Vesma and Saramäki 1996a).

Fig. 2. The Farrow structure for the general case. N isthe length of the filter and L is the degree of the

interpolation. Also the modifications needed for thederivative approximation are shown.

−4 −3 −2 −1 0 1 2 3 4−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

t/Ts

Fig. 3. Impulse response h(t) and impulse response ofthe Farrow structure for µ = 0.25.

3. POLYNOMIAL INTERPOLATION INSIMULATIONS

The basic problem in the computer simulationsis that we have just discrete-time samples fromsome, possibly stochastic, continuous-time system.There are also situations where the signal value isneeded between these discrete-time samples. Onesolution is to increase the sampling rate. However,this makes the simulation very slow and the signalis still discrete time.

In this paper, we introduce a new approach tothis problem. In this approach, the originalcontinuous-time signal is estimated usingpolynomials, and the value of the signal can beapproximated by evaluating the value of thepolynomial at the desired point. This polynomialinterpolation (or estimation) can be efficientlyimplemented using the Farrow structure as dis-cussed earlier.

The block diagram of the interpolation functionis shown in Fig. 4. x(mTs) is the input samplevector and y(kTi) is the output sample. The locationof this output sample t = kTi is determined by theinput parameters µ and m according Eq. (1).

Interpolatory(kTi)x(mTs)

µ m

Fig. 4. The block diagram of the interpolation functionand the input and output parameters.

What is inside the interpolation function in Fig.4 is just the implementation of the Farrow structurewith a suitable length N and order L (to bediscussed later on). The advantage of thisinterpolation function when modelling somecontinuous-time system is that there is no need forhigh over-sampling factor at the input. In additionto that, the output is really continuous time becausethe fractional interval µ is practically continuousvalued (of course this is limited by the number ofbits used to represent µ).

3.1. Interpolation Filters

The complexity of the interpolation filterdepends on the accuracy requirements of thesimulation model. The approximation error in theinterpolation (that is the error between the originalcontinuous-time signal at point kTi and theinterpolated value y(kTi)) can be decreased byincreasing the length of the branch filters or thedegree of the interpolation.

The design parameters of the interpolation filtersare related to its frequency response, as normally inthe case of linear filters. The parameters include thetolerated passband amplitude response variation,the minimum stopband attenuation, and the

passband and stopband cut-off frequencies.Basically, the task of the filter is to attenuate thealiasing/imaging components appearing due to thechange of the sampling rate. Time-domain accuracyrequirements cannot be converted easily to thecorresponding frequency-domain requirements. Inpractice, the performance of candidate interpolationfilters may be evaluated by trial simulations inorder to be able to select the most suitable one.

As mentioned earlier, interpolation filters can beanalyzed and synthezised using the continuous-timeimpulse response h(t). In the filter design criteria,the frequency response of this filter h(t) should ap-proximate unity in the signal band and zero else-where.

Table I shows four different interpolation filters.For these filters, the stop band attenuation variesfrom 53 dB to 91 dB and the number of multiplica-tions from 50 to 216. It should be noted that thecomplexity of these filters can be decreased by in-creasing the sampling rate by an integer factorwhich allows to use a wider transition band(Saramäki and Ritoniemi 1996). For example, byusing the over-sampling factor of two, the highestcomponent of the signal is below 0.25fs which helpsto reduce the filter complexity.

Table I. Four different interpolation filters. Passbandripple δp, stopband attenuation As, passband and

stopband edges (normalized frequency), and the numberof coefficients in the Farrow structure N(L+1).

δpAs[dB]

fp fs N*(L+1)

1 0.01 53 0.38 0.62 10(4+1)=50

2 0.009 80 0.38 0.62 14(6+1)=98

3 0.008 67 0.425 0.575 20(7+1)=160

4 0.012 91 0.425 0.575 24(8+1)=216

The frequency and impulse responses for the lastfilter in Table I are shown in Figs. 5 and 6. The im-pulse response in Fig. 6 is truly piecewise polyno-mial. This is difficult to see because it does not in-clude high frequency components (see Fig. 5)which makes it very smooth.

Figure 7 shows a part of a discrete-time ECGsignal and its approximated continuous-timewaveform. This waveform is obtained using the

second interpolation filter in Table I. The ECGsignal has been obtained from a patient duringanaesthesia at Helsinki University Hospital.

0 0.5 1 1.5 2 2.5 3

−100

−80

−60

−40

−20

0

Frequency / Fs

Am

plitu

de /

dB

Fig. 5. Frequency response for the interpolation filter.N = 24 and L = 8 (the last entry in Table I).

−10 −5 0 5 10−0.2

0

0.2

0.4

0.6

0.8

t/Ts

Fig. 6. Impulse response for the interpolation filter.N = 24 and L = 8 (the last entry in Table I).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

t/s

Am

plit

ude

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

t/s

Am

plit

ude

Fig. 7. Discrete-time ECG signal and its approximatedcontinuous-time waveform.

4. POLYNOMIAL APPROXIMATION OFDERIVATIVE AND INTEGRAL

In this section, we will show how the polynomialbased interpolation can be used for estimating thecontinuous-time derivative and integral from asampled signal. The design criteria for polynomial-based differentiators have been discussed in (Ridhaet al. 1997).

4.1. Differentiator

The derivative of the continuous-time approxi-mating signal at an arbitrary time instant isobtained from Eq. (3) and it is expressible by

dy t

dtlf m

t mT

Tlf ml

s

s

l

l

L

ll

l

L~( )( )( ) ( ) .=

−=−

=

−

=∑ ∑1

1

1

1

µ (5)

From this equation it is seen that the Farrowstructure used in the polynomial-based interpolationcan also be used for approximating the derivative ofthe original continuous-time signal at any desiredtime instant.

The modifications needed in the Farrow struc-ture are shown in Fig. 2, whereas the frequency andimpulse responses for the differentiator are shownin Figs. 8 and 9, respectively in the case of the lastinterpolator in Table I. Figure 10 depicts theresulting approximated continuous-time derivativefor the discrete-time ECG signal in Fig. 7.

0 0.5 1 1.5 2 2.5 3−80

−70

−60

−50

−40

−30

−20

−10

0

10

Frequency / Fs

Am

plitu

de /

dB

Fig. 8. Frequency response for the differentiator. N = 24and L = 8 (the last entry in Table I).

−10 −5 0 5 10

−1

−0.5

0

0.5

1

t/Ts

Fig. 9. Impulse response for the differentiator. N = 24and L = 8 (the last entry in Table I).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.4

−0.2

0

0.2

0.4

0.6

t/s

Am

plitu

de

Fig. 10. Approximated continuous-time derivative forthe discrete-time ECG signal in Fig. 7.

4.2. Integrator

In the following we introduce a new integrationmethod for simulating continuous-time systemsusing discrete-time computations. The developed"integrator" is also based on the polynomial gener-alized interpolation techniques.

The continuous-time integration of a signal at anarbitrary time instant is approximated from the fun-damental equation of digital interpolation, Eq. 3, asfollows

~( ) ( ) .y t dt f mll

l

l

L

∫ ∑=+

+

=

µ 1

01

(6)

The above equation represents the continuous-time integrator and is easily implemented by intro-

ducing a small modifications to the Farrowstructure used in the polynomial-based interpolationpurpose.

Assuming that the frequency response of the in-terpolation filter is H(ω), then the ideal frequencyresponse for the integrator filter is equal H(ω)/(jω)(see Fig. 11).

ω

|H(ω)/jω|

Fig. 11. Ideal frequency response for integrator.

Most of the developed digital integrators are de-signed from a polynomial perspective (Regolia1993). Further work to treat the design aspects ofthe continuous-time integrator is needed.

5. CONCLUSION

In this paper a novel approach for modelling andcomputer simulation of continuous-time signals andsystems has been introduced. We have shown howthis approach is based on the fundamental equationof digital interpolation, and that it provides a gener-alized method for approximating the continuous-time derivative and integral of a discrete-timesignal. Moreover, efficient and simpleimplementation are obtained using the Farrowstructure.

Simulation results show that accurate and goodcontinuous-time interpolation and derivative ap-proximations can be obtained by designs based onoptimized piecewise polynomial interpolators.

It remains as a topic for future work to device ef-ficient and sufficiently accurate polynomial-basedintegrators.

REFERENCES

Crochiere, R. E. and L. R. Rabiner. MultirateDigital Signal Processing. Englewood Cliffs, NJ:Prentice-Hall, 1983.

Erup, L.; F. M. Gardner; and R. A. Harris.“Interpolation in digital modems - Part II: Imple-

mentation and performance,” IEEE Trans.Commun., vol. 41, pp. 998-1008, June 1993.

Farrow, C. W. “A continuously variable digital de-lay element,” in Proc. IEEE Int. Symp. Circuits &Syst., Espoo, Finland, June 1988, pp. 2641-2645.

Gardner, F. M. “Interpolation in digital modems -Part I: Fundamentals,” IEEE Trans. Commun., vol.41, pp. 501-507, Mar. 1993.

Laakso, T. I.; V. Välimäki; M. Karjalainen; and U.K. Laine. “Splitting the unit delay,” IEEE SignalProcessing Magazine, vol. 13, pp. 30-60, Jan.1996.

Regolia, P. A. “Special filter design,” chapter 13 inHandbook for Digital Signal Processing edited byS. K. Mitra and J. F. Kaiser., John Wiley & Sons,1993.

Ridha, H.; J. Vesma; T. Saramäki; and M. Renfors.“Derivative approximations for sampled signalsbased on polynomial interpolation,” to be presentedat 13th International Conference on Digital SignalProcessing, Santorini, Greece, July 1997.

Saramäki T. and M. Ritoniemi. "An efficient ap-proach for conversion between arbitrary samplingfrequencies," in Proc. IEEE Int. Symp. Circuits &Syst., Atlanta, GA, May 1996, pp. 285-288.

Vesma, J.; M. Renfors; and J. Rinne. “Comparisonof efficient interpolation techniques for symboltiming recovery,” in Proc. IEEE Globecom 96,London, UK, Nov. 1996, pp. 953-957.

Vesma, J. and T. Saramäki. “Interpolation filterswith arbitrary frequency response for all-digital re-ceivers,” in Proc. IEEE Int. Symp. Circuits & Syst.,Atlanta, GA, May 1996, pp. 568-571.

Vesma J. and T. Saramäki. “Optimization and effi-cient implementation of fractional delay FIR fil-ters,” in Proc. Third IEEE Int. Conf. Electronics,Circuits, Syst., Rodos, Greece, Oct. 1996, pp. 546-549.

Vesma J. and T. Saramäki. “Optimization of effi-cient implementation of FIR filters with adjustablefractional delay,” to be presented at 1997 IEEEInternational Symposium on Circuits and SystemsHong Kong, June 1997.

Publication 2

R. Hamila, J. Vesma, H. Vuolle, and M. Renfors, "Joint Estimation of Carrier Phase andSymbol Timing Using Polynomial-Based Maximum Likelihood Technique," in the IEEE1998 International Conference on Universal Personal Communications, ICUPC’98, Flo-rence, Italy, October 1998, pp. 369–373.

Copyright c 1998 IEEE. Reprinted, with permission, from the proceedings of ICUPC’98.

Publication 3

R. Hamila, J. Vesma, H. Vuolle, and M. Renfors, "NDA Maximum Likelihood Approachfor Timing and Phase Adjustment by Polynomial-Based Interpolation," in Proc. 6th IEEEInternational Workshop on Intelligent Signal Processing and Communication Systems, IS-PACS’98, Melbourne, Australia, Nov. 1998, pp. 248–252.

Copyright c 1998 IEEE. Reprinted, with permission, from the proceedings of ISPACS’98.

NDA MAXIMUM LIKELIHOOD APPROACH FOR TIMING AND PHASE

ADJUSTMENT BY POLYNOMIAL-BASED INTERPOLATION

Ridha Hamila, Jussi Vesma, Hannu Vuolle, and Markku Renfors

Telecommunications Laboratory

Tampere University of Technology

P.O Box 553, FIN-33101 Tampere, Finland

e-mail: [email protected].�

ABSTRACT

In this contribution, we extend the polynomial-basedML approach derived earlier for synchronization pur-poses to the non-data-aided (NDA) case. We propose afully digitally implemented synchronization concept us-ing interpolation for jointly estimating the timing andphase. The interpolation methods used in this contextcan be implemented using the so-called Farrow struc-ture. The ML function is also expressed in terms ofthe polynomial-based interpolator �lter branches. Sim-ulation results for both NDA and DA synchronizationprinciples using two types of interpolators are providedand compared with the theoretical case. We notice thatthe optimized interpolator performs much better com-pared to the Lagrange interpolator.

1. INTRODUCTION

In conventional receivers, the sampling of the receivedsignal must be synchronized to the incoming data sym-bols. The synchronization is performed by using a feed-back or feed-forward loop. Currently, there is a trendof using digital receivers where the sampling of the re-ceived signal is not synchronized to the incoming datasymbols. With this approach, there is no need for thecomplex PLL circuits used in the conventional receivers.

In this receiver architecture, the sampling of the re-ceived signal is performed by a �xed sampling clock andthus sampling is not synchronized to the incoming sym-bols. Therefore, timing adjustment and consequentlyphase estimation must be done by digital methods af-ter sampling. One way to perform this is to calculatethe value of the signal at the desired time instants usinginterpolation. Hence, a polynomial-based interpolationtechnique using �lters for the calculation of the tim-ing and phase error estimates is applied. The so-calledFarrow structure has been used in this context becauseit o�ers an e�cient and exible realization for such aninterpolation task [1]. Simulation results show that the

This work was carried out in the project "Analog and Digital

Signal Processing Techniques for Highly Integrated Transceivers"

supported by the Academy of Finland.

optimized interpolator [2] performs much better com-pared to the Lagrange interpolator [3]-[4].

The best synchronization algorithms for digital com-munication systems are based on the maximum likeli-hood (ML) estimation theory. Estimation of joint sig-nal parameters using the ML approach provides usuallybetter estimates with respect to the variance comparedto estimates obtained from separate optimization of thelikelihood functions [5]. The feedforwad architecture weare considering provides rapid acquisition characteris-tics, which are very important especially in the mobilecommunication systems where the channel character-istics are rapidly changing, and in the case of TDMAsystem, also the transmission is bursty.

In this paper, we extend our previous DA approachfor clock and/or carrier phase synchronization of digi-tal receivers to the case of NDA systems. The proposedmethod is based on the ML function for jointly estimat-ing the symbol timing and carrier phase using a block-based feed-forward architecture. This approach is validfor M-PSK type of modulations which are commonlyused in mobile communication systems.

2. TIMING AND PHASE ADJUSTMENT

USING INTERPOLATION

The block diagram of the digital receiver with non-synchronized sampling is shown in Figure 1. The sam-pling of the received signal r(t) is performed by a �xedsampling clock, and thus, the timing adjustment andconsequently the carrier phase estimation must be doneafter the sampling using interpolation.

~

µθFIXEDCLOCKf = 1/T

m(t)r(t)

RECEIVEDSIGNAL

h(-t)INTERPOLATION

FILTERm(k)

DECISIONSYMBOLS

y(n) a(n)

TIMING & PHASEESTIMATION

ss

Figure 1: Digital Receiver with non-synchronized sam-pling.

Assuming an AWGN channel, the complex envelope

r(t) of the received noisy M-PSK modulated signal isgiven by

r(t) =Xi

cih(t� iT � � )ej� + n(t); (1)

where the couple (� ,�) are the unknown clock and car-rier phase respectively, 1=T is the symbol rate, h(t) isthe baseband pulse shape, ci is a sequence of indepen-dent equiprobable M-PSK data symbols

ci 2 fe�j�=M ; e�j3�=M ; � � �g;

and n(t) is white Gaussian noise.The interpolation �lter in Fig. 1 can be used for

both up-samplingand down-sampling, and furthermore,the output is not synchronized to the input. The func-tion of the interpolation �lter is to calculate one outputsample y(nTi) at a time using a set of adjacent inputsamplesm(kTs), within a timing error estimate (�) anda phase error estimate (�) obtained from the synchro-nization control unit. In the ideal case these outputsamples are the same sample values that would occurif the original sampling had been synchronized to thereceived symbols.

The fractional interval � determines the time instantt = nTi where the original continuous-time receivedsignal r(t) is approximated. This relation is given by

nTi = (� + k)Ts; (2)

where � 2 [0; 1) is the fractional interval, Ts is the sam-pling interval, and Ti is the output interval, whereas kis the largest integer for which kTs � nTi. In this pa-per, we assume that the original sampling rate is twicethe symbol rate T = 2Ts. Also, we use � and � to de-note the timing error with respect to the symbol andsampling interval, respectively. In addition, the errorin the carrier phase is corrected using the phase errorestimate denoted by � as shown in Fig. 1.

The polynomial-based interpolation �lters are veryeasily implemented using the Farrow structure [1]. Thisstructure is characterized by the �xed values of all the�lter coe�cients and by only one changeable parameterwhich is the fractional interval �. In the general case,the Farrow structure shown in Fig. 2 consists of L + 1FIR �lter branches and the length of each branch �lteris N , where L is the degree of the interpolation and Nis the length of the impulse response.

Timing adjustment in digital receivers using La-grange interpolation was �rst introduced by Erup et.al. [3]. Lagrange interpolation represents the conven-tional time-domain approach to the interpolation prob-lems. A new synthesis technique for polynomial-basedinterpolation �lters is proposed in [2]. In this synthe-sis technique, the design parameters are edge frequen-cies for passbands and stopbands, desired amplitudeand weight for every band, the length of the �lter, andthe degree of the interpolation. After giving these pa-rameters, the �lter coe�cients of the Farrow structure

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

Figure 2: Farrow structure for the cubic (L = 3) inter-polator.

are optimized in the frequency domain in the minimaxsense. The new designs are shown to provide clearlylower complexity or much better performance with thesame complexity as the Lagrange designs [6].

3. NDA MAXIMUM LIKELIHOOD

ESTIMATION OF PHASE AND TIMING

Under the assumption that the sequence of M-PSK datasymbols are independent and equiprobable, the likeli-hood function for NDA estimation of timing (� ) andcarrier phase (�) is given by [7];

�(~�; ~�) =NXn=1

jm(n; ~�)j2 + Re

"NXn=1

m2(n; ~�)e�j2~�

#;

(3)where m(n; ~� ), ~� 2 [��; �], and ~� 2 [0; T ) are the sam-ples from the output of the matched �lter (one sampleper symbol), carrier phase and symbol timing error tri-als, respectively. N is the number of symbols used inthe calculation and T is the symbol interval.

The above NDA likelihood approximation is onlyvalid for single amplitude type of modulations, e.g., M-PSK modulated signals.

Synchronization algorithms for digital receivers esti-mating jointly the timing and phase are based on Max-imum Likelihood estimation (MLE) theory as mentionearlier. These algorithms are trying to �nd the timingand the phase error estimates (� ; �) which maximizeEq. (3). By means of the above likelihood function,

the following algorithm for jointly estimating (� ; �) canbe established as follows [7]:

� = max~�

NXn=1

jm(n; ~�)j2 (4)

� =1

Marg[

NXn=1

(m(n; � ))M ]; (5)

where M is the number of constellation points.

An important feature here is that the ML-estimateof the timing is independent of the carrier phase. Weconsider here a block processing scheme which takes thedata samples for a block of N symbols.

In the following section, we will form a polynomialapproximation for the above likelihood function by us-ing the Farrow structure, then di�erent schemes for�nding the maximumof this polynomial can be devised.

4. POLYNOMIAL APPROXIMATION OF

THE NDA LIKELIHOOD FUNCTION FOR

TIMING AND PHASE ESTIMATION

In this section, we will extend this polynomial-basedlikelihood function for jointly estimating the timing andphase with respect to Eqs. (4) and (5). This procedureis derived for the non-data-aided (NDA) systems wherethe symbol values are not known in advance in the caseof single amplitude type of modulations. In general weshould expect some degradation of performance whenignoring the data values [4].

Here we assume that the original sampling rate istwice the symbol rate T = 2Ts and cubic interpola-tion �lter (L = 3) is used. Because the symbol in-terval is divided into two parts (two samples per onesymbol), there are two di�erent polynomial approxi-mations for the log-likelihood function as in [8]-[9]-[10].Consequently, from expression (4) the symbol timing isestimated by

� = argmax~�

��1(2~� ) 0 � ~� < 0:5�2(2~� � 1) 0:5 � ~� < 1

(6)

where

�1(2~� ) =NXn=1

jm(2n� 1; 2~�)j2;

and

�2(2~� � 1) =NXn=1

jm(2n; 2~� � 1)j2:

Also, by using the above timing estimate � , the carrierphase estimate is given by

� =1

Marg

( PNn=1[m(2n� 1; 2�)]M for 0 � ~� < 0:5PNn=1[m(2n; 2� � 1)]M for 0:5 � ~� < 1:

(7)

The value of m(n; ~� ) can be interpolated for di�erentvalues of ~� by utilizing the cubic Farrow structure andthe original samples m(k). Eqs. (6) and (7) can alsobe expressed as a function of the fractional interval ~� 2[0; 1), and the output samples of the FIR �lter branchesin the Farrow structure fi(n) in the case of a cubicinterpolation �lter, as follows:

If max[�1(~�)] > max[�2(~�)] then

� = argmax~�[�1(~�)];

else

� = argmax~�[�2(~�)]:

Where

�1(~�) =NXn=1

��3X

i=0

~�ifi(2n� 1)

��2

;

and

�2(~�) =NXn=1

��3Xi=0

~�ifi(2n)

��2

:

Finally, the above symbol timing estimate is used forcomputing the carrier phase estimate:

If max[�1(~�)] > max[�2(~�)] then

� =1

Marg

NXn=1

"3X

i=0

�ifi(2n� 1)

#M

else

� =1

Marg

NXn=1

"3X

i=0

�ifi(2n)

#M:

The overall scheme for our proposed block-based sym-bol timing and phase recovery is illustrated in Fig. 3.For the NDA feedforward (FF) phase estimator, the in- uence of the symbols is removed by the M th power[11] (see Fig. (3)).

Timing Estimator

| . |

2| . |

e

EVENODD/

-j

2

Phase Estimator

N

Σn=1

1Λ ( )µ~

Σ2( )

]M

ax[

N

i=0

n=1~

^ i

)

µ

DATA

(Λ

Λ

µ

^

M ΣN

n=1

arg(.)/M

µ

~

L

θ

Σ

µ

iµ~ if (2n)Σi=0

L

f (n)i µ

iµ~Σi=0

L

if (2n-1)

if (n)

EVEN

MEMORY

Interpolator

BRANCH

FILTERS

m(k) y(n)

( . )

ODD/

Figure 3: NDA Phase estimation and symbol timingrecovery scheme.

After the parameters � and � are estimated, tim-ing adjustment is performed by evaluating the value ofthe polynomial at the given value of �. At the sametime, sampling rate is decreased by two. Here, evensamples are discarded if max[�1(~�)] > max[�2(~�)], andodd samples otherwise. Subsequently, the carrier phase

is corrected by the factor e�j�.

5. SIMULATIONS

The proposed algorithm is simulated using M-PSK con-stellations. The transmitter and receiver pulse-shaping�lters are square root raised cosine �lters with excessbandwidth of 35%. Two interpolation �lters, cubic La-grange and third order interpolation �lter of length 8designed by the new synthesis technique [2], are com-pared. The simulations are done using the worst-casetiming o�set and some selected values of the phase er-ror.

0 10 20 30 40 50 60 70 80 90 100−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Pha

se e

rror

(ra

d)

Symbol block

Figure 4: Phase Jitter of 4-PSK signal when Eb=N0 = 9dB, phase error � = 0:6 rad, and block length N =64. The solid line shows the phase jitter of the newinterpolator, and the dashed-dot line phase jitter of theLagrange interpolator.

Figure 4 shows the phase jitter of 4-PSK signal whenEb=N0 = 9 dB, phase error equal 0:6 rad, and blocklength N = 64. Variance and mean of the phase jitterfor the interpolation �lters are also provided in Fig.5. Similarly, Figs. 6 & 7 illustrate the timing jitter,variance and mean of the timing jitter.

We noticed a dramatic di�erence in performance be-tween the conventional Lagrange and the new interpo-lator when using 4-PSK-type of modulations as shownin �g. 8. The same experiments compare the SEP asa function of the signal-to noise-ratio Eb=N0 of the DAand NDA systems with the theoretical expectations.The performance of the scheme was found to be close toideal for the DA systems even with short block lengths[10], thus having a rapid acquisition time. Similarly, byusing a longer block length, the NDA systems performfairly well.

Binary and M-PSK -type of modulations are gener-ally used in mobile telecommunications systems. Thesemodulations types do not have very demanding require-ments for the interpolator. However, linear or cubicLagrange interpolator may not be su�cient for the newsynchronization scheme. Fortunately, better interpola-tion �lters than the cubic Lagrange which are not more

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4

−3

−2

−1

0

1x 10

−3

Mea

n(ra

d)

µ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.6

0.8

1

1.2

1.4

1.6

1.8x 10

−3

Var

ianc

e(ra

d)

µ

Figure 5: Variance and mean of the phase jitter of 4-PSK signal for the Lagrange interpolator (dashed-dotline) and new interpolator (solid line).

0 10 20 30 40 50 60 70 80 90 100−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Tim

ing

erro

r

Symbol block

Figure 6: Timing jitter of 4-PSK signal when Eb=N0 =9 dB, phase error � = 0:6 rad, and block length N =64. The solid line shows the timing jitter of the newinterpolator, and the dashed-dot line the timing jitterof the Lagrange interpolator.

complicated to implement are derived in [6].

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Mea

n

µ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

−3

Var

ianc

e

µ

Figure 7: Variance and mean of the timing jitter of 4-PSK signal for the Lagrange interpolator (dashed-dotline) and new interpolator (solid line).

6. CONCLUSIONS

An extension of the polynomial-based maximum likeli-hood technique for jointly estimating the symbol tim-ing and carrier phase of NDA systems is established.Di�erent simulations using the Lagrange and the newinterpolator are also performed for both DA and NDAsystems and compared with the theoretical case. Theresults show the high performance of the new interpo-lator compared to the Lagrange interpolator. However,the NDA approach requires more computation than theDA approach for estimating the polynomial-based like-lihood function. Therefore, further studies consideringthe computation complexity of this new NDA techniquewill be carried out. Normally, by omitting decision op-erations one may reduce equipment complexity and ob-tain fast acquisition. Besides, in some circumstancesNDA systems may perform better than DD speciallyfor poor quality data decisions.

7. REFERENCES

[1] C. W. Farrow, "A continuously variable digital delayelement," in Proc. IEEE Int. Symp. Circuits & Syst.,

Espoo, Finland, June 1988, pp. 2641-2645.

[2] J. Vesma and T. Saram�aki, "Interpolation �lters witharbitrary frequency response for all-digital receivers,"in Proc. IEEE Int. Symp. Circuits & Syst., Atlanta,GA, May 1996, pp. 568-571.

[3] L. Erup, F. M. Gardner, and R. A. Harris, "Inter-polation in digital modems - Part II: Implementationand performance," IEEE Trans. Commun., vol. 41, pp.998-1008, June 1993.

0 1 2 3 4 5 6 7 8 9 1010

−6

10−5

10−4

10−3

10−2

10−1

100

4−psk

SE

P

Eb/N

0

Theoretical New interpolator, N=16, da New interpolator, N=16, nda New interpolator, N=64, nda Lagrange interpolator, N=64, nda

Figure 8: Worst case SEP with 4-PSK using cubic La-grange and the new interpolator, with di�erent blocklength N and phase error � = 0:6 rad.

[4] F. M. Gardner, "Demodulator reference recovery tech-niques suited for digital implementation," ESA Final

Report, Aug. 1988.

[5] J. G. Proakis, Digital Communication. McGraw-Hill,1989.

[6] J. Vesma, M. Renfors, and J. Rinne, "Comparison ofe�cient interpolation techniques for symbol timing re-covery," in Proc. IEEE Globecom 96, London, UK,Nov. 1996, pp. 953-957.

[7] H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital

Communication Receivers: Synchronization, Channel

Estimation and Signal Processing. John Wiley & Sons,1998.

[8] J. Vesma, V. Tuukkanen, and M. Renfors, "Maximumlikelihood feedforward symbol timing recovery basedon e�cient digital interpolation techniques," in Proc.

1996, IEEE Nordic Signal Processing Symposium, Es-poo, Finland, Sept. 1996, pp. 183-186.

[9] J. Vesma, V. Tuukkanen, R. Hamila, and M. Renfors,"Block-based feedforward maximum likelihood sym-bol timing recovery technique," in Proc. Communi-

cation Systems & Digital Signal Processing, She�eld-UK, April 1998, pp. 92-95.

[10] R. Hamila, J. Vesma, H. Vuolle, and M. Renfors, "Jointestimation of phase and timing using polynomial-basedML technique," to be presented in IEEE 1998 Inter-

national Conference on Universal Personal Communi-

cations, Florence, Italy, 5-9 October, 1998.

[11] Geert De Jonghe and Marc Moeneclaey "Asymptotic

cycle slip probability expression for the NDA feed-

forward carrier synchronization for M-PSK," in Proc.

ICC'93, Geneva, Switzerland, pp. 502-506.

Publication 4

R. Hamila, J. Vesma, H. Vuolle, and M. Renfors, "Effect of Frequency Offset on CarrierPhase and Symbol Timing Recovery in Digital Receivers," in Proc. 1998 URSI Interna-tional Symposium on Signals, Systems, and Electronics, ISSSE’98, Pisa, Italy, September1998, pp. 247–252.

Copyright c 1998 IEEE. Reprinted, with permission, from the proceedings of ISSSE’98.

Publication 5

R. Hamila, M. Renfors, "New Maximum Likelihood Based Frequency Estimator for Digi-tal Receivers," in Proc. IEEE Wireless Communications and Networking Conference, WC-NC´99, New Orleans, USA, Sept. 1999, pp. 206–210.

Copyright c 1999 IEEE. Reprinted, with permission, from the proceedings of WCNC’99.

Publication 6

R. Hamila, S. Lohan, and M. Renfors, "Effect of Correlation Estimation on MultipathDelay Estimation Techniques in DS-CDMA Systems," in Proc. of 4th International Sym-posium on Wireless Personal Multimedia Communications, WPMC’01, Aalborg, Denmark,September 2001, pp. 331–335.

Copyright c 2001 IEEE. Reprinted, with permission, from the proceedings of WPMC’01.

Effect of Correlation Estimation on Multipath Delay Estimation Techniques inDS-CDMA Systems

Ridha Hamila, Elena-Simona Lohan and Markku Renfors

Telecommunications Laboratory, Tampere University of TechnologyP.O. Box 553, FIN-33101 Tampere, Finland�

ridha,simona,mr � @cs.tut.fi, http://www.cs.tut.fi/tlt/

Abstract

Accurate detection and estimation of overlapping fading multi-path components is vital for many communication systems. Inthis paper, we analyze the effect of the coherent integration andnon-coherent averaging length over the correlation function forestimating correctly the delays introduced by a multipath chan-nel. Study of correlation estimation properties paves the wayto derive simple and efficient multipath delay estimation tech-niques with subchip resolution capability for use in direct se-quence CDMA systems.

1. IntroductionCommonly, in most applications of radio communications andnavigations [1], the line of sight (LOS) signal is succeeded bymultipath components that arrive at the receiver within shortdelays, that can be smaller than the pulse shape. This causesoverlapping fading multipath components and introduces sig-nificant errors in the LOS path time of arrival and gain estima-tion. In spread spectrum CDMA systems, such as Third gen-eration mobile communications or Global Positioning System(GPS) receivers, it is important to achieve accurate delay esti-mation (or code synchronization) before despreading and datadetection. Most spread spectrum systems use spreading codeswith non-ideal correlation properties which result in co-channelinterference, or multiuser-interference, for radio communica-tion systems, which is also called multi-transmitter interferencein radio-navigation systems [1]. Generally, the performance ofradio communication systems is heavily affected by the multi-path and multiuser-interference.

Generally, multipath delay estimation techniques in mostDS-CDMA receivers are based on the maximum likelihoodtheory [1]. Different methods have been derived, consideringmainly coherent and non-coherent delay lock loops (DLL) witharbitrary early and late code spacings [2],[3],[4]. These meth-ods are just trying to track the delay of the line-of-sight path bycorrelating the down-converted received signal with replicas ofspreading codes locally generated by the DS-SS receiver. How-ever, phase lock loop circuits generally fail to estimate closely-spaced multipaths with less than one chip interval, besides theirconvergence can be quite slow [1],[2]. Several subspace-basedapproaches have also been derived, however, they are usually

This work was carried out in the project ”AdvancedTransceiver Architectures and Implementations for WirelessCommunications” supported by the Academy of Finland. Thiswork is also supported by Tampere Graduate School in Informa-tion Science and Engineering (TISE), and Graduate School inElectronics, Telecommunications and Automation (GETA).

too complex for practical purposes, and suitable only for sys-tems employing short codes [5],[6].

The paper is organized as follows. In Section 2, we brieflyreview the correlation estimation principles over short and in-finite time observations. In Section 3, we analyze multipathdelay estimation techniques based on the maximum likelihoodtheory, and we demonstrate with simulations the impact of thecoherent and the non-coherent averaging length of the correla-tion function on multipath delay estimation. Finally, Conclud-ing remarks are drawn is Section 4.

2. Correlation Estimation PrinciplesConsider that the autocorrelation function of a stationary pro-cess �� is expressed as �� (1)

The time average of the process� �� is defined by�� !"$# %'&)( � ��*��+,�� (2)

The stationary process �� is called ergodic in the autocorre-lation function if it satisfies the following two conditions. First,the time average approaches the ensemble average in the limitas the observation interval

" #approaches infinity,-/.10&2(4365 � �� 7� �� *�� (3)

and then the variance of the time average approaches zero inthe limit as the observation interval

" #approaches infinity [Pa-

poulis, Chap. 11],-/./0&2(8365�9;:� -/./0&)(<365 !" # % : &2(>= : ��? �;@ (4) ��6@A? � ��B? �DCFE ! �HG ? G" #�I +�?� �J'�Thus, provided that the observation time interval

" #is suffi-

ciently large, the time average � �� is an unbiased estimator

and can be used to estimate �� .

However, if we are averaging over a finite short-time interval,we shall consider the following biased estimator &)( �� LK M&2(AN &2( ��B��+,� for G GPO " #J otherwise

whose expected value or mean is given by [Papoulis, Chap. 11]� � &2( �Q ��RE ! �SG G"$# I ��T��R� "$# �� (5)

and its variance is expressed as follows9 :�� ( !" # % : &2( � : ��? �$@ ��? @A�� ? � *��E ! � G G @ G ? G" # I +�?4� (6)

Here, T��P� " # � denotes a triangular function defined as followsT��P� " # � K ! � G G � " # G G � " #J otherwise � (7)

By computing the mean of the power spectrum of each of theabove estimators, we obtain� � � �� % : &2( �� 2+� �� .�� "$# �� (8)

� �� &2( �� % : &2( ��*��T�� 2+� �� .�� : " # � �� " # � : �� (9)

where � �� , & ( �� , and

�� represent the power spectracorresponding to the functions

�� , &)( �� and �� , re-spectively, defined over the finite observation interval

" #.

Notice that both estimators are biased if we are consider-ing finite time observation. However, if the observation intervaltends to infinity, both of the kernel windows � .�� ";# � � � � � � �and � .�� : � "$# � �� "$# � : �! � tend to one. Consequently,both estimators become unbiased.

3. Timing Estimation for a MultipathSignal Based on the Correlation Estimation

PrinciplesAfter down-conversion of the received signal to basebandthrough mixing and filtering, the received DS-CDMA signal inthe presence of " -path channel is modeled as [1],[4]# �� $ �� @&%8�� (10) ')(�*,+-. /�021 .43 � M- 5 /�076 5 8 . �� 5 8 . �:9 "�;�<>= �� ,?A@CB D@E%8��where �� is the spread-spectrum code, and % �� is additivewhite Gaussian noise with power spectral density F 0 �! , "�;�<G=is the symbol interval, and 1 . is the 9�HJI transmitted complexdata symbol. The time-variant 6 5 , 5 and K 5 represent the at-tenuation factor, time delay, and phase for the L�HJI path, respec-tively. The estimation of the set of synchronization parameters� 6 5 �� 5 �,K 5 � is very essential for locating the line-of-sight path,and consequently, for achieving accurate delay estimation (orcode synchronization) before despreading and data detection.

Assuming that the down-converted signal # �� is com-pletely defined in the time interval

" #, the conditional likeli-

hood function of # �� for a given set of synchronization param-eters

� 6 � R�,K,� is given byM = # ��NN � 6 � R�,K,� C PO4QGR�S KB� !F 0 %R& ( � # ��UT$ �� : +,�WV(11)

where O is just a positive constant independent of the synchro-nization parameters which can be neglected, and T$ �� is the lo-cal trial signal replica generated at the receiver, modeling theestimated line-of-sight and multipath signals. Without loss ofgenerality, we consider that the desired symbol is the zerothsymbol, the trial signal replica generated at the receiver is givenby T$ �� 3 � M- 5 /�0 T6 5 8 0 ��UT 5 8 0 �� X��Y? @JB Z � (12)

We assume that during the observation interval" #

, the chan-nel is very slowly changing, therefore the subscript 9 can beignored for convenience.

In conventional receivers, multipath delay estimation tech-niques are based on the maximum likelihood theory [1] whichare trying to estimate the set of parameters

� T6 ��T � TK�� by min-imizing the mean square error of the log-likelihood function(LLF) given by[ 37\ ��T6 �]TF� TK,� PK %R&2( � # ��UT$ �� : +,�WV � (13)

Thus, the synchronization parameters are determined by differ-entiating the above LLF function with respect to the synchro-nization parameters, then setting the partial derivatives equal tozero. The partial derivative with respect to the time-delay esti-mate is given by [1]^^ [ 3�\ ��T6 5 ��P� TK 5 � NNNN � / � @ ^^ `_ ]acb Ed�� 3 � M-e /�0 8 eCf/ 5 T6 e � �� gT e �� hY?,iCj �]�kY? @lnm � / � @ (14)

In the above equation, d�� is the in-phase/ quadrature time-

average cross-correlation function over the observation time in-terval

" #, given byEd�� !" # %'&2( # �� B��+,�� (15)

and � ��*� is the reference time-average correlation function of

the spreading code, given by � ��*� !" # % &2( �� *��+,�� (16)

Consequently, the desired time delay for the L HoI path that maxi-mizes the above expression (14) is given byT 5 qpWr,s 0 pWR� _ aEb d�� (17)

� 3 � M-e /�0 8 eCf/ 5 T6 e � ��UT e �� )Y?,i j ��hY? @ l m��Generally, multipath delay estimation techniques are based onthe cross-correlation function (Eq. 15) of the down-convertedsignal de-spread with a copy of the spreading code locallygenerated by the DS-SS receiver. The multipath estimatingDLL techniques [2] are trying to approximate the overall cross-correlation using a set of reference correlation (Eq. 16) func-tions with certain delays, phases and amplitudes.

When the locally generated code is locked to the re-ceived code, by substituting equation (11) into (15), the cross-correlation function is simply expressed byEd�� !"$# 3 � M- 5 /�0�6 5 ��? @ %P&2( 1 0 �� 5 ��;� *��+,�@ !"$# % &2( % �� B��+,�� (18)

If the data symbols,� 1 . � , are known in advance, then we can

perform coherent integration over several symbols. Otherwise,we perform the integration over one symbol and then apply anon-coherent averaging over several symbols in order to reducethe interference.

Normally, even-though we are considering averaging overa sufficiently large observation time interval, we shall con-sider the previously mentioned biased estimator. Recalling theGaussian nature of the random process as well as the mean ofthe biased estimator (5), the expectation of the above cross-correlation function becomes

� � d�� *�� 3 � M- 5 /�0 6 5 X�,? @ � ��B 5 ��T�� B 5 � "$# �� (19)

Notice that the expected value of the cross-correlation functionis expressed simply as a weighted sum of delayed correlationfunctions times triangular windows with certain amplitude andphase specific to each signal path. Thus, the correlation effec-tively amplifies the underlying data signal, with an amplifica-tion factor equal to the length of the PRN code sequence. Fig-ure 1 shows an example of a normalized autocorrelation func-tion using a Gold code sequence of length 1023 weighted by atriangular window.

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Figure 1: The upper figure shows a normalized autocorrelationfunction of one Gold code sequence within a triangular window.The lower curve illustrates their product.

Here," #

can also be referred to as a coherent integrationlength, since the integral from Eq. (18) is performed coher-ently. However, if the data modulation is not known or esti-mated in advance, the integration length

" #is limited to one

symbol interval, which is usually not sufficient to diminish the

effect of noise. Therefore, a non-coherent averaging can also beemployed in such away that the decision variable becomes

� '��-. /�0 NNNN3 � M- 5 /�0 6 5 ��,? @ �� B 5 ��T�� 5 � " # D � NNNN � (20)

where F�� denotes the number of blocks used in the non-coherent block averaging, and

";# D is the coherent integrationinterval for the 9 HJI symbol.

Also, we noticed by increasing either the coherent integra-tion length or the non-coherent averaging length, the peaks inthe correlation function can be clearly distinguished and the sidelobes decrease as illustrated in Figs 2 and 3. The coherent in-tegration provides lower side lobes in the correlation functioncompared to the non-coherent averaging, but data decisions orpilot symbols are required for performing the coherent integra-tion. It is clear that the more we increase the averaging length(coherent or non-coherent), the more the correlation functionapproaches to the ideal one (i.e., superposition of two triangularpulse shapes at the positions corresponding to the path delays).

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1 Symb. Coh. Averaging, 1 Block Non−Coh. Averaging 10 Symb. Coh. Averaging, 1 Block Non−Coh Averaging 1 Symb. Coh. Averaging, 10 Block Non−Coh. Averaging

Figure 2: Correlation function with a spreading factor of at�� F 0 � dB, using 2-path channel, with path locations at� P� ��R� �� chips and average path powers of� JP�F�� dB.

By exploiting the above study, a new technique for multi-path delay estimation using peak tracking with subtraction hasbeen derived. This approach is analogous to the DLL tech-niques in the sense that it is trying to estimate the overall cross-correlation function using a set of reference functions, but withdifferent and simpler implementation [7],[8]. Also, based onthe above analysis an innovative multipath delay estimation ap-proach based on a nonlinear quadratic Teager-Kaiser operatoris introduced [9],[10]. This new technique is extremely sim-ple and very efficient for estimating closely-spaced multipathswith much less computational complexity compared, e.g., to thesubspace-based multipath delay estimation methods [5],[6].

4. ConclusionsGenerally, the transmitted signal arrives at the receiver via mul-tiple propagation paths at different delays, which can be smallerthan the pulse shape duration, introducing overlapped multipathcomponents and causing significant errors to the prompt LOSsignal time and amplitude of arrival estimation. Correlation es-timation properties related to DS-CDMA receiver signal have

been studied. We demonstrated the influence of the coherentintegration and the non-coherent averaging length on multipathdelay estimation, and the importance of the considered observa-tion time interval length. This analysis paves the way to derivesimple and very efficient multipath delay estimation methodswith subchip resolution capability .

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1 Symb. Coh. Averaging, 1 Block Non−Coh. Averaging 50 Symb. Coh. Averaging, 1 Block Non−Coh Averaging 1 Symb. Coh. Averaging, 50 Block Non−Coh. Averaging

Figure 3: Correlation function with a spreading factor of at�� F 0 J dB, using 2-path channel, with path locations at� P� ��R� � �W� chips and average path powers of� JP�F�� dB.

5. References[1] R.Van Nee, Multipath and multi-transmitter interference

in spread-spectrum communication and navigation sys-tem, Delft University Press, 1995.

[2] R. Van Nee, J. Siereveld, P. Fenton, and B. Townsend,“The multipath estimating delay lock loop: Approachingtheoretical accuracy limits,” in Proc. of IEEE Position, Lo-cation, and Navigation Symposium, April 1994, pp. 246–251.

[3] M. Braasch and A.J.V. Dierendonck, “GPS receiver archi-tectures and measurements,” Proc. of the IEEE, vol. 87,No. 1, January 1999, pp. 48–64.

[4] M.C. Laxton and S.L. DeVilbis, “GPS multipath mitiga-tion during code tracking,” in Proc. of the American Con-trol Conference, Albuquerque, New Mexico, pp. 1429–1433, June 1997.

[5] P. Luukkanen, and J. Joutsensalo, “Comparison of MU-SIC and matched filter delay estimators in DS-CDMA,” inProc. of IEEE Personal, Indoor and Mobile Radio Com-munications, vol. 3, 1997, pp. 830–834.

[6] Z. Kostic, G. Pavlovic, “Resolving sub-chip spaced mul-tipath components in CDMA communication systems,” inProc. of IEEE VTC, vol. 1, 1993, pp. 469–472.

[7] R. Hamila, S. Lohan and M. Renfors, “Multipath delay es-timation in GPS receivers,” in Proc. of Third InternationalSymposium on Wireless Personal Multimedia Communi-cations, Bangkok, Thailand, November 2000, pp. 646–649.

[8] S. Lohan, M. Renfors, “Feedforward approach for esti-mating the multipath delays in CDMA systems,” in Proc.

of Nordic Signal Processing Symposium, Kolmarden,Sweden, pp. 125–128, June 2000.

[9] R. Hamila, S. Lohan and M. Renfors, “Nonlinear opera-tor for multipath channel estimation in GPS receivers,” inProc. of the 7th IEEE International Conference On Elec-tronics, Circuits & Systems, Jounieh, Lebanon, pp. 352–356, December 2000.

[10] R. Hamila, S. Lohan and M. Renfors, “Novel techniqueformultipath delay channel estimation in GPS receivers,” inProc. of IEEE International Conference on Third Genera-tion Wireless and Beyond, San Francisco, USA, pp. 993–998, May 2001.

Publication 7

R. Hamila, S. Lohan, and M. Renfors, "Novel Technique for Closely-Spaced MultipathDelay Estimation in DS-CDMA Systems," Submitted to Signal Processing, (EURASIP).

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Publication 8

S. Lohan, R. Hamila, and M. Renfors, "Performance Analysis of an Efficient MultipathDelay Estimation Approach in a CDMA Multiuser Environment," in Proc. of 12th IEEEInternational Symposium on Personal, Indoor and Mobile Radio Communications, PIM-RC´01, San Diego, California, USA, Sept. 2001, pp. 6–10.

Copyright c 2001 IEEE. Reprinted, with permission, from the proceedings of PIMRC’01.

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synchronization and multipath delay estimation …...[p1] r. hamila, j. vesma, t. saramaki and m....

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