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Ultra Wide Band Ad-Hoc Sensor Networks: AMulti-Layer Analysis

Dorin Panaitopol

To cite this version:Dorin Panaitopol. Ultra Wide Band Ad-Hoc Sensor Networks: A Multi-Layer Analysis. Signal andImage processing. National University of Singapore; Supélec, 2011. English. �tel-00840023�

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PhD THESISin Signal Processing and Telecommunications

Ultra Wide Band Ad-Hoc Sensor Networks:

A Multi-Layer Analysis

Dorin PANAITOPOL

OFFICIAL NUMBER: 2011-05-TH

Ecole Supérieure d’Électricité (SUPÉLEC)National University of Singapore (NUS)

PhD ThesisField: Signal Processing and Telecommunications

presented by

Dorin PANAITOPOL

for the title of

Joint-PhD SUPÉLEC-NUS

on the subject

Ultra Wide Band Ad-Hoc Sensor Networks:

A Multi-Layer Analysis

PhD SupervisorsJocelyn FIORINA

Chen-Khong THAMPeng-Yong KONG

presented on 5th of July 2011, in front of the Jury composed of:

Hikmet SARILaurent CLAVIERJocelyn FIORINA

Stephen Vaughan HANLYChen-Khong THAMPeng-Yong KONG

in the memory of my father..

i

Abstract and Thesis Overview

Abstract

The Ultra Wide Band (UWB) technology represents a promising approach al-lowing the development of new types of wireless Local Area Networks (LANs) andWireless Sensor Networks (WSNs). For twenty years, UWB has been an active fieldof research and it could be the physical layer for future multiple access wireless net-works with very high bit rates, as well as low bit rate low power networks with highaccuracy positioning abilities.

Ad-hoc networks are expected to play an important role in future communicationsystems. They are self-organizing, rapidly deployable, and require no fixed infrastruc-ture. In ad-hoc networks, the wireless nodes cooperate in order to dynamically setup communications with distributed management and administration.

Sensor networks will provide the capacity to sense the surroundings, therefore in-troducing a high degree of visibility into the real physical world. This research projectentitled ‘Ultra Wide Band ad-hoc sensor networks: a multi-layer analysis’ studies andanalyzes protocols adapted to the UWB ad-hoc sensor networks, and also the opti-mization of layers (PHY, MAC and network). The various points to be approachedin this thesis are from the physical layer toward the upper layers: antenna optimiza-tion issues, interference modeling, Impulse Radio UWB (IR-UWB) advanced receiversand transmitters with efficient wireless communication, throughput optimization, andcooperation issues.

The synthesis of these elements into a UWB-based ad-hoc sensor network willhelp in developing powerful optimized networks for which there are countless appli-cations and can lead to breakthroughs in areas such as healthcare, surveillance andenvironmental monitoring.

Thesis Overview

In this thesis we have addressed the problems of interference modeling, throughputmaximization with several applications for IR-UWB ad-hoc sensor networks, antennaoptimization, receiver and transmitter design, and relay cooperation.

After the introductory chapter, Chapter 2 presents the Impulse Radio system.In Chapter 2 we describe the Impulse Radio signal model, the classical transmissionschemes, and some generalities on UWB channel model and receiver schemes. An-other part of this chapter is concerned with multiuser interference for IR-UWB and its

iii

impact on the network organization. The network is simplified to some basic topolo-gies (e.g., two cells in uplink mode or two links Tx−Rx transmitting and interferingwith each other) that are further used in the manuscript to analyze the impact ofsome system parameters (e.g., slots, users, pulse shapes, spreading factors) on capac-ity and throughput. Thus, Section 2.3 details the IR-UWB interference model andapplies it further in some interesting applications in Chapter 4.

Our scientific contribution starts in Chapter 3 which deals with the IR-UWB non-Gaussianity of the MUI distribution. In this chapter we first recall that generalizedGaussian is a better approximation model for IR-UWB Multi User Interference. Notethat there are several cases where the Gaussian distribution could be used to approxi-mate the IR-UWB MUI distribution but these cases are not representative and cannotbe generalized. Another part of this chapter deals with a performance criterion forIR-UWB communication (see Section 3.4), that takes into account the MUI shapedistribution and several signal parameters in order to compute the capacity of thecommunication and its throughput.

Chapter 4 deals with several applications of the performance criterion developedin Chapter 3. Each of these applications uses one of the interference models pre-sented in Chapter 2 (i.e., Section 2.3). In Chapter 4 we present the effect of tuningimpulsiveness on throughput of IR-UWB Communications. In Sections 4.3, 4.4, 4.5and 4.6 we have presented different applications of several impulsiveness methods fordouble-cell scenarios, double-link communications, cognitive networks and IR-UWBantenna design respectively. While Sections 4.3 and 4.4 show that throughput can bemaximized by controlling the impulsiveness of the interference, Section 4.6 presentsthe impact of antenna design on the impulsiveness and so on the capacity of an IR-UWB system in a Multi-User Interference environment, and proposes a new antennadesign and pulse shape optimizations.

Chapter 5 is concerned with Impulse Radio UWB receivers and transmitters.Section 5.1 describes the design of a novel simple IR-UWB receiver that could beused for sensor networks and compares it with previous works presented in Section2.2.5. This receiver assumes that the probability density function describing thestatistic of the interference is a mixture-based distribution. Section 5.2 investigates anovel transmitter design based on Time Reversal (TR). We chose here to investigatethe trade-off between the number of fingers in the prefilter (i.e., at the transmitterside) of a TR-IR-UWB system versus the number of fingers in the Rake receiver. Thisallows studying the gain brought by time reversal when the complexity is switchedfrom the receiver to the transmitter i.e., when the number of fingers is increased inthe prefilter while it is reduced in the Rake receiver. Moreover, it seems that TRcombined with Rake technique changes the Probability Density Function (PDF) ofthe MUI. This chapter led to the development of a C++ platform used under Cygwin.

The work focuses further on the cooperation for wireless networks in Chapter6. While Section 6.1 proposes an incremental relaying scheme that uses multiplerelays, Section 6.2 proposes a novel and practical incremental relaying scheme thatdeals with the imperfect feedback. Section 6.3 describes a simple distributed relayselection policy without any need of information exchange between the relays.

iv

Publication List

A part of the work from the Chapters 3, 4, 5, 6 has been published as follows:

1. D. Panaitopol, J. Fiorina, A. Diet, N. Ribière-Tharaud, ‘A New Criterionto Jointly Design the Antenna and Optimize the Communication Capacity inIR-UWB’, IEEE Wireless Communications and Networking Conference, IEEEWCNC, Budapest, Hungary, 2009.

2. D. Panaitopol, J. Fiorina, A. Diet, N. Ribière-Tharaud, ‘Impact of the Designof an UWB Antenna on the Maximum Achievable Rate of the Communicationin Presence of Multi User Interferences’, Asia Pacific Microwave Conference(APMC 2008), Hong Kong, China, 2008.

3. D. Panaitopol, J. Fiorina, M. -G. Di Benedetto, ‘Trade-off Between the Numberof Fingers in the Prefilter and in the Rake Receiver in Time Reversal IR-UWB’,IEEE ICUWB’09, Vancouver, Canada, 2009.

4. D. Panaitopol, J. Fiorina, P.-Y. Kong, C.-K. Tham, ‘The Effect of Impulsive-ness in Inter-Cell Interference on Throughput of TH-IR-UWB Networks’, IEEEWCNC, Sydney, Australia, 2010.

5. D. Panaitopol, J. Fiorina, T. Erseghe, ‘A Comparison of IR-UWB ReceiversAdapted to MUI with Mixture Based Distributions’, IEEE WCNC, Sydney, Aus-tralia, 2010.

6. D. Panaitopol, P.-Y. Kong, C.-K. Tham, J. Fiorina, ‘An Efficient CooperativeTransmission Scheme Using Multiple Relays Incrementally’, IEEE PIMRC, Is-tanbul, 2010.

7. J. Fiorina, D. Panaitopol, D. Domenicali, ‘On the necessity of sensing the non-Gaussianity of the interferences: application to UWB systems’, CogART, Aal-borg, Denmark, 2009.

8. Patent Filed with NEC Technologies (UK), ‘Cooperative Transmission’ (submit-ted to the patent agent with the title ‘A Practical Multiple-Relaying IncrementalScheme for Wireless Networks’), Reference Number GB1105487.1.

9. D. Panaitopol, J. Fiorina, ‘Impact of the Generalized Gaussian Approximationon the design of IR-UWB systems’, Submitted to IEEE Transactions on Wire-less Communications Journal.

10. L. De Nardis, J. Fiorina, G. Capodanno, D. Panaitopol, and M.-G. Di Benedetto,‘Combining UWB with Time Reversal for improved communication and posi-tioning’, to appear in Telecommunication Systems Journal (Springer), SpecialIssue on Recent Advances in UWB Systems: Theory and Applications.

v

Résumé

Les systèmes Ultra Wide Band (UWB) représentent une solution prometteusepour le développement de nouveaux types de réseaux locaux sans-fils. L’UWB a étéun domaine de recherche actif depuis 20 ans et pourrait être la couche physique pourde futurs réseaux sans fils à accès multiples à très haut débit ainsi qu’à bas débit avecconsommation énergétique minimale, tout en offrant des possibilités de localisation àhaute précision.

Nous nous intéressons dans cette thèse aux réseaux ad-hoc de capteurs UWB enpartant de la couche physique pour remonter aux couches plus élevées : optimisationd’antenne, modélisation d’interférence, conception de récepteurs et transmetteurs,optimisation de débit et schémas coopératifs.

Une première contribution concerne la non Gaussianité des interférences multiutilisateurs : nous avons développé un critère de performance qui prend en compteles paramètres du système et la distribution des MUI afin de calculer la capacitédes communications et le débit utile. Nous avons montré que le débit peut êtremaximisé en contrôlant l’impulsivité des interférences et nous avons montré l’impactde la conception de l’antenne sur cette impulsivité et par conséquent sur la capacitédes systèmes UWB.

A partir de l’étude des MUI, une autre contribution concerne de nouveaux récep-teurs et émetteurs IR-UWB. Le récepteur proposé modélise la densité de probabilitédes interférences comme une distribution mélangée. Nous avons également étudié unnouvel émetteur basé sur le retournement temporel combiné à l’IR UWB et étudié legain apporté par le retournement temporel lorsque la complexité est assumée par letransmetteur à la place du récepteur, i.e. quand le nombre de doigts du pré-filtre estaugmenté et quand le nombre de doigts du récepteur à râteau est diminué. Nous avonségalement montré que le time réversal combiné à l’IR UWB change la distributiondes interférences.

Les recherches ont ensuite porté sur des schémas coopératifs. Nous avons proposédans une première étude un schéma de relayage incrémental qui utilise plusieurs relaismais avec un retour imparfait et proposé finalement une règle simple de sélectiondistribuée des relais.

Contents

Thesis Overview iii

Résumé vii

Abbreviation List xiii

1 Introduction 11.1 History of Ultra Wide Band . . . . . . . . . . . . . . . . . . . . . . . 11.2 Applications of UWB . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Contribution with Respect to the State of the Art . . . . . . . . . . . 7

2 Signal Model and Interference Scenarios 132.1 Introduction on IR-UWB Signal . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 PPM-TH-UWB . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.2 PAM-TH-UWB . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.3 DS-UWB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.4 Generalized PAM-PPM . . . . . . . . . . . . . . . . . . . . . . 152.1.5 Useful Physical Quantities for Impulse Radio . . . . . . . . . . 16

2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.1 Impulse Radio Transmitted Signal . . . . . . . . . . . . . . . . 162.2.2 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.3 Transmission Scheme and Received Signal . . . . . . . . . . . 182.2.4 Types of Interference . . . . . . . . . . . . . . . . . . . . . . . 192.2.5 Impulse Radio Receiver Generalities . . . . . . . . . . . . . . . 19

2.3 Interference Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.1 Single-Cell Scenario . . . . . . . . . . . . . . . . . . . . . . . . 242.3.2 Double-Cell Scenario . . . . . . . . . . . . . . . . . . . . . . . 272.3.3 Double-Link Scenario . . . . . . . . . . . . . . . . . . . . . . . 29

3 On the Non-Gaussianity of the IR-UWB MUI 313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3 Kurtosis Computation in Perfect Power Control and Lack of Power

Control Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.1 General Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.2 Perfect Power Control Scenario . . . . . . . . . . . . . . . . . 42

ix

CONTENTS

3.3.3 Lack of Power Control Scenario . . . . . . . . . . . . . . . . . 433.4 Channel Capacity Computation Using a Generalized Gaussian Appro-

ximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.4.1 Kurtosis Expressions . . . . . . . . . . . . . . . . . . . . . . . 453.4.2 SINR Expressions . . . . . . . . . . . . . . . . . . . . . . . . . 473.4.3 The Capacity of a Discrete Memoryless Channel with Continuous-

Valued Output . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 The Effect of Tuning Impulsiveness on Throughput of IR-UWBCommunications 514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Tuning the Impulsiveness . . . . . . . . . . . . . . . . . . . . . . . . . 524.3 Application in Double-Cell Scenario . . . . . . . . . . . . . . . . . . . 53

4.3.1 Novel Metrics Based on the Communication Capacity . . . . . 544.3.2 Tuning the Impulsiveness of the Signal through Nh,I and Nh,II 544.3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 554.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4 Application in Double-Link Communications . . . . . . . . . . . . . . 604.5 Application in Cognitive Networks . . . . . . . . . . . . . . . . . . . 64

4.5.1 Sensing the MUI . . . . . . . . . . . . . . . . . . . . . . . . . 644.5.2 Tuning w(t) in the Cognitive Network . . . . . . . . . . . . . 654.5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.6 Application in IR-UWB Antenna Testing . . . . . . . . . . . . . . . . 674.6.1 An Antenna for IR-UWB . . . . . . . . . . . . . . . . . . . . 694.6.2 A Novel IR-UWB Antenna Performance Criterion . . . . . . . 704.6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 Receiver and Transmitter Designs for IR-UWB 815.1 A Comparison of IR-UWB Receivers Adapted to MUI with non Gaus-

sian Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.1.2 Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.1.3 Modeling MUI . . . . . . . . . . . . . . . . . . . . . . . . . . 835.1.4 What is the Point of Using Receivers Adapted to Atypical MUI

Distributions? . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.1.5 The Receiver Structure for Generalized Gaussian MUI and a

Novel Block Scheme . . . . . . . . . . . . . . . . . . . . . . . 865.1.6 The Receiver Structure for Mixture MUI . . . . . . . . . . . . 875.1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.2 A Study on a Novel Type of Transmitter Using Time Reversal . . . . 965.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.2.2 TR-SRake Model . . . . . . . . . . . . . . . . . . . . . . . . . 965.2.3 Simulations and Results . . . . . . . . . . . . . . . . . . . . . 985.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

x

CONTENTS

6 Cooperation in Wireless Networks 1076.1 Cooperative Transmission Scheme Using Incremental Relaying with

Multiple Relays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.1.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.1.3 Incremental AF with Multiple Relays . . . . . . . . . . . . . . 1106.1.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 1166.1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.2 A Practical Incremental Relaying Scheme with Imperfect Feedback forWireless Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.2.2 Practical Incremental Relaying with Imperfect Feedback . . . 1226.2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 1336.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.3 On the Relay Selection and Cooperation for UWB . . . . . . . . . . . 1376.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.3.2 Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . 1396.3.3 Packet Success Probabilities . . . . . . . . . . . . . . . . . . . 1396.3.4 To Cooperate or Not to Cooperate? . . . . . . . . . . . . . . . 1416.3.5 Matlab Results and C++ Implementation . . . . . . . . . . . 1426.3.6 Further Investigation . . . . . . . . . . . . . . . . . . . . . . . 145

Conclusions and Perspectives 147

Acknowledgement 157

Remerciements 159

Bibliography 161

xi

Abbreviation List

ACK ACKnowledgementAF Amplify and Forward

AOA Angle Of ArrivalBER Bit Error Rate

BPSK Binary Phase Shift KeyingBS Base Station

CDMA Code Division Multiple AccessCF Characteristic FunctionCH Cluster Head or CHannelCM Channel Model

CNODE Cognitive NODECSI Channel State InformationDF Decode and ForwardDS Direct Sequence

Eb/N0 Energy per bit to Noise power spectral densityEM Expectation MaximizationFIR Finite Impulse Response

FRake Full RakeGG Generalized Gaussian

GGM Generalized Gaussian MixtureGM Gaussian MixtureIAF Incremental Amplify and Forward

IR Impulse RadioISI Inter-Symbol Interference

LM Laplacian MixtureLOS Line Of Sight

LPI/D Low Probability of Interference and DetectionLSE Least Square Estimator

MAC Medium Access Control (Layer)MAI Multiple Access Interference

MANET Mobile Ad-hoc NETworkMB Multi Band

xiii

MEB Mixture Estimation BlockMEM Modified Expectation Maximization

MF Matched FilterMLE Maximum Likelihood Estimator

MMSE Minimum Mean-Square ErrorMRC Maximal Ratio CombiningMUD Multi User DetectionMUI Multi User Interference

NACK Not ACKnowledgementNLOS Non Line Of SightPAM Pulse Amplitude ModulationPDF Probability Density Function

PIAF Practical IAFPHY PHYsical (Layer)PPM Pulse Position Modulation

PRake Partial RakePULSERS Pervasive UWB Low Spectral

Energy Radio SystemsRF Radio Frequency

RSS Radio Signal StrengthRx Receiver

QoS Quality of ServiceQPSK Quaternary Phase Shift Keying

SE Spectral EfficiencySGA Standard Gaussian Approximation

SINR Signal to Noise and Interference RatioSIR Signal to Interference Ratio

SNR Signal to Noise RatioSRake Selective RakeTDOA Time Difference Of Arrival

TH Time HoppingTOA Time Of Arrival

TR Time ReversalTRef Transmitted Reference or Time Reference

Tx TransmitterUWB Ultra Wide Band

WMN Wireless Mesh NetworkWPAN Wireless Personal Area Network

WSN Wireless Sensor Network

xiv

Chapter 1

Introduction

1.1 History of Ultra Wide Band

The concept of Ultra Wide Band communications originated with the first experi-ments made by Marconi and Hertz [141]. At the end of the 1890s, Marconi developedthe spark gap transmitter and he is therefore considered to be the parent of spreadspectrum systems [74, 176, 184]. The Impulse Radio based Ultra Wide Band commu-nication systems became an interesting research field twenty years ago, with the firstpublications by P. Withington and L. Fullerton [182] and by R. A. Scholtz [158, 159].Previously, the UWB systems have been classified by the US military which used thembecause of their many properties such as resistance to jamming and interference.

We can also quote as an historical invention the radio systems using Pulse Posi-tion Modulation (PPM), whose prototypes exist since 1943 [8, 15]. After the secondworld war, J. R. Pierce published the principle of IR-UWB in a Bell Labs memoran-dum [138]. We find in this article the basic ideas of the TH-IR-UWB, although thetransmitted signals are not digital but analog.

The evolutions in radar systems have also allowed the development of the UWBsystems for communications although, contrary to UWB radars [6, 106], UWB com-munication technique uses short impulses with low power. In 1987, Ross and Fontanadesigned, developed and implemented a Low Probability to Intercept and Detection(LPI/D) communications system [58], funded by the United States government.

The term UWB was first introduced by DARPA in 1990 for radar applications,and in 2002 FCC (US) established the first regulation for UWB devices. Just afterthat, European and Japanese standards followed - see Fig. 1.1.

Nowadays, not only that UWB is a physical transmission technique suitable fora great number of applications, but is also expected to be very advantageous (seeTable 1.1). UWB has emerged as a particularly appealing transmission techniquefor applications requiring either high bit rates over short ranges (on the right handside of Table 1.1) or low bit rates over medium to long ranges (on the left hand sideof Table 1.1). While the first area of interest is suitable for multimedia and highrate applications, the latter makes UWB appealing for such applications as sensornetworks.


(a)

(b) (c)

Figure 1.1 Standard - UWB Emission Limit: (a) FCC (US) emission limits for indoor andoutdoor UWB systems (b) European Commission emission limits for UWBsystems (c) Japanese emission limits for indoor UWB systems

2


Table 1.1 The position of UWB Communications← Sensors Multimedia, High Rate

Applications Applications →Technology UWB Low Zigbee Bluetooth Wi-Fi UWB High

Rate RateProtocol 802.15.4a 802.15.4 802.15.1 802.11 802.15.3a

IEEE a/b/gMemory 4Kb+ 4-32Kb 250Kb+ 1Mb+ 1Mb+

Autonomy Years months Days up to >hours4 hours

Number thousands hundreds 7 32 >32of nodesTransfer 250Kb/s- 250Kb/s 1Mb/s 11-54- >110-200Mb/sSpeed 1Mb/s 108Mb/sRange Hundred- 100m 10-100m 300m 10m-4m

tens of m

Among the various modulations schemes that have been proposed for the UWBtechnology, we can cite the most important ones: Impulse Radio UWB (IR-TH-PAMand IR-TH-PPM), Direct Sequence CDMA and Multi Band UWB (i.e., MB-OFDM).

To understand the steps taken towards the UWB technology, the future devel-opment and the applications envisaged it is imperative to discuss about the IEEEstandard 802.15 and more precisely about the Task Group 3a (i.e., 802.15.3a) andthe Task Group 4a (i.e., 802.15.4a).

The 802.15.3a standard was withdrawn in Jan 2006 due to a long lasting fightbetween the two proposals MB-UWB (sustained by WiMedia Alliance) and DS-UWB(sustained by UWB Forum). The only important task achieved was the selection oftwo candidate proposals out of 23. The final goal was to provide a High Rate UWBstandard.

It is important to say that the weakness of the MB-OFDM UWB approach is amore complex transceiver structure if compared to the original impulse radio work-ing at the baseband - a very simple radio concept. The basic idea of cheap, lowpower consumption UWB devices is lost since new functional blocks like mixers andamplifiers are needed for the up-converter and down-converter. On the other hand,the technique can offer higher data rates for medium distances than the single-bandapproach, and commercial OFDM technology is quite mature, nowadays with com-ponents available on the shelf.

On the other hand, the 802.15.4a is now complete. The first goal is to providecommunications and high precision ranging/location capability (1 meter accuracy andbetter), high aggregate throughput, and ultra low power; as well as adding scalabilityto data rates, longer range, and lower power consumption and cost. These additionalcapabilities over the existing 802.15.4 standard are expected to enable significant newapplications and market opportunities. The physical layer of this standard is basedon Low Rate UWB.

3

1.2 Applications of UWB

In particular, for ad-hoc and sensor networks, the UWB ranging capability (atthe PHY layer) is an attractive feature for location-aware applications (for MAClayer and above). Therefore, introducing positioning in low data rate networks hasrecently become the main goal of the 802.15.4a, where impulse radio ultra wide-band(IR-UWB) emerges as a most appealing principle.

Another standard with some relevance to our work is IEEE 1451.5, or the wirelesssmart sensor standard. This standard does not specifically address UWB but themain principles for wireless sensors are there.


UWB systems have potentially low complexity and low cost, are resistant to jam-ming and have very good time resolution allowing for precise location and tracking.Furthermore, as a result of their good signal properties, the UWB interference appearsas a low power spectral density noise, and therefore UWB systems should create littleinterference to other systems. Nevertheless, this interference still has to be taken intoaccount in interference management.

Among other possible applications, the inherent properties of Impulse Radio UWBtechnology make it well suited to wireless sensor network applications [32, 67, 76, 173,187] and its feasibility has been studied on real chip designs [7, 126, 150, 172].

A sensor network is a computer network composed of a large number of sensornodes. The sensors should be heterogeneous devices with low cost, low energyconsumption, small, robust with variable data rate enhancements suitable fordifferent applications. There are different sensors such as accelerometer, camera,microphone, pressure, thermal, etc. They monitor conditions such as the chemicalcomposition, direction, lightning, humidity, noise level, mechanical stress levels, move-ment, pressure, size, speed, and temperature of an habitat, highways, bridges, andother civilian infrastructure [123]. This capability leads to innovative applicationssuch as in-vehicle sensing, intrusion detection [187], weather forecast, etc. These sen-sor nodes usually contain three components: sensing, processing and communicating[4].

While sensing represents an amazing way of interacting with the physical world,with a huge number of applications, nowadays it may still be an expensive opera-tion in terms of energy consumption. For the sensing operation a sensing capabledevice usually consumes 1mJ/sample for Light (Taos [88] TSL2550 ambient lightsensor), 3mJ/sample for Humidity and Temperature (Sensirion [87] SHT1x relativehumidity sensor), while operating typically at 3V . Some other sensing features ofmore complex devices consume 3 − 5mJ for an accelerometer (Analog Devices [82]ADXL202E low-cost 2g dual-axis accelerometers), 3mJ for a barometer (Intersema[84] MS5534A barometer module) or around 10mJ for a magnetometer (Honeywell[83] HMC1001/1002 and HMC1021/1022 1-and 2-axis magnetic sensors).

Sensors networks have military applications [31] (DSN and SenIT from DARPA),environmental applications [4, 85], healthcare applications (A*STAR, Singapore), andeven home applications like "Smart Environment" [43], "Smart Kindergarten" [168] or

4


"The Intelligent Home". For most of these we need high-precision localization systems[59].

Some companies already started to commercialize UWB applications for retailmarketing, manufacturing and healthcare such as PLUS from Time Domain [89],used to tag or locate everything in real time from consumables and equipment to staffand even workflow processes. Ubisense [90] also commercializes systems using Angleof Arrival (AoA) and Time Difference of Arrival (TDoA) for precise real-time location.SandLinks [86] is developing a low-cost UWB RTLS chip that could dramaticallyreduce the cost of equipment.

In Singapore, the Agency for Science, Technology & Research (A*STAR)finances the Ultra Wide Band-enabled Sentient Computing (UWB-SC) pro-gram [81, 128] in order to extend Singapore’s R&D capabilities relevant to SentientComputing and Ultra Wide Band. Sentient computing is defined as a form of ubiq-uitous computing which emphasizes the use of sensors to perceive its environmentand react accordingly, performing intelligent actions which affect and benefit peo-ple. Research is being carried out at the Institute for Infocomm Research (I2R),Institute of Microelectronics (IME), Nanyang Technological University (NTU) andNational University of Singapore (NUS).

There are eight projects in the UWB-SC program handling research subjects fromthe physical layer to the application layer:

1. UWB Antennas and Channel Characterization,

2. Reconfigurable RF Transceiver IC Platform for Communication and Localiza-tion,

3. Adaptive Power Aware Digital Signal Processing Circuits,

4. Ranging and Low Rate Communication for an IR-UWB RFID System,

5. Localization and Identification: System architectural components and core ca-pability development for UWB-SC,

6. UWB-enabled Sentient Computing Architecture and Middleware with Coordi-nated QoS (USCAM-CQ),

7. Smart Space enabled by Sentient Computing,

8. Context Aware Middleware Services and Programming Support for SentientComputing.

The goal of the UWB-SC research program is to provide a smart space environmentfor use in indoor applications such as healthcare monitoring, customer and/or incidentmonitoring in shopping mall, etc.

Localization joint with communication is an important application for a sensornetwork and, because of their high ranging capabilities, it is particularly importantfor sensor networks using IR-UWB technology. When a technology like UWB comesalong, providing better location estimation due to its short impulses and higher band-width, having accurate information about the position of the nodes inside the networkbecomes fundamental. The benefits of knowing the position of the nodes are dividedinto two important categories:

5


– for the enhanced protocols and algorithms used in the MAC layer and above:to provide location stamps for individual sensor measurements that are beinggathered, to determine the quality of coverage, to achieve load balancing intopology control mechanism, to form clusters, to facilitate routing of informationthrough the network, to perform efficient spatial querying, or to manage thecooperation between nodes based on geographical position.

– for future applications that could be envisaged: to locate and track objects inthe environment, to secure a military area, to monitor the spatial evolution of adiffuse phenomenon over time, for search and rescue missions, for the supervisionof children or of the patients in a hospital (healthcare).

While it is not the main focus of the thesis, and therefore it is not presented inthis manuscript, we have addressed the subject of localization for Wireless SensorNetworks (WSN) during the initial phase of this study. Our contribution is about theuse of Spring Force Algorithm (SFA) for localization which has been presented in theFirst French-Singapore Workshop on Signal Processing, S2.2-B2-53/NTU, Singapore,2007 [130]. The novelty about this credibility-based algorithm is that it applies asimple principle of equivalence with a spring network: the difference between therange estimate and the real distance is seen as a compression or as an extension ofa spring. The algorithm is very simple and also it helps gathering a 3D map (pleasenote the difference between the term "ranging" [23, 75] which refers to point-to-pointdistance estimation and the term "localization" [16, 113, 129, 155] which refers to 2Dor 3D position estimation).

In this section we have shown that there are many possible IR-UWB applicationsand technological breakthroughs in the field, but IR-UWB still remains technologi-cally unavailable for WSN in the sense that there are only a few (and costly) WSNIR-UWB research platforms. For this reason, many WSN scientific studies such asthose on the MAC and network related layers (e.g., issues in network topology, whichis applicable to different PHY layers) are nowadays done using research platformsthat are not IR-UWB capable. We could cite, for example, a few motes (i.e., devicessimilar to sensors, used as research platforms for WSN scientific purposes) that donot exploit the UWB technology yet: TelosB, Mica2/MicaZ, Mica2dot (see Fig.1.2).For instance, TelosB and MicaZ [91] have IEEE 802.15.4 compliant RF transceiversoperating in a bandwidth not larger than 85MHz (in steps of 5MHz).

6

1.3 Contribution with Respect to the State of the Art

(a) (b)

(c)

Figure 1.2 (a) TelosB, (b) Mica2/MicaZ, (c) Mica2dot

1.3 Contribution with Respect to the State of the

Art

In this thesis we are concerned with Impulse Radio UWB communications andwe have addressed the problems of interference modeling, throughput optimization,receiver and transmitter design, cooperation, and antenna optimization for ad-hocsensor networks.

Interference modeling is a major research issue for any wireless network. Manyworks [11, 22, 142, 158] studied the IR-UWB interference using models provided bythe research done on CDMA networks. As we will show in the next paragraphs and inthe rest of this thesis, Impulse Radio (IR) UWB interference has different propertiesfrom any other type of interference previously studied.

The discussions around the validity of Standard Gaussian Approximation (SGA)for Impulse Radio UWB interference has recently intensified. Results suggest that thevalidity of SGA for IR schemes increases with the number of interfering users [179],but SGA may not be fully valid even with an asymptotic number of users [48, 50]. Ithas also been shown that SGA does not drive to adequate Bit Error Rate derivations[36, 56]. The analyses presented in [36, 73] show that for Time Hopping UWB systemswith and without perfect power control, when a classical correlation receiver is used,the SGA leads to more optimistic predictions of the BER floor than results obtainedby simulation. In [71] it is shown that for random uniform distributed delays and alow number of interferers, the interference will act as a "collision model". The above

7


studies are based on experimental data obtained by simulation, and they are showingthe non-validity of the standard Gaussian approximation.

The SGA hypothesis seems to cease having validity when the number of pulsesin a transmission frame is not sufficiently high. A reduced number of pulses can beassociated with scenarios with a reduced number of transmitters (e.g., low densitynetworks) or scenarios with several dominant interferers (e.g., transmitters with fixedpowers and lack of power control scenarios), or combinations of the above conditions.The transmitters may also be characterized by a low data rate with low number ofpulses per bit. All these scenarios have applications in sensor networks, which aretypically characterized by low data rates and sparse topologies. Therefore, one mayconclude that for sensor networks the Gaussian approximation is no longer valid andit has to be replaced with some other distributions that better approximate the dataobtained from experiments. Note that for high density of users (e.g., networks withmany transmitting sensors) it has also been shown that SGA may not be valid [46].

There are several works [92, 93, 152] proposing analytical approaches for IR-UWBsystem performance estimation, taking into account the non-Gaussianity of the inter-ference. In particular, [92, 93] are using the Characteristic Function (CF) techniquefollowing a consolidated procedure from CDMA systems.

Sometimes it may be very difficult to analytically compute BER under non-Gaussian approximation. One can still use different approximation methods [35]and experimental results from Monte-Carlo simulations, but the results may not beprecise. Therefore we may need to introduce novel concepts and performance criteria.

Our contribution starts with Chapter 3, which is concerned with multiuser inter-ferences for IR-UWB and its impact on the network organization. Chapter 3 proposesto further develop an IR-UWB performance criterion which we apply in some interest-ing applications in Chapter 4. This performance criterion takes into consideration notthe BER of the network but instead the achievable communication capacity of a dis-crete memoryless channel with continuous-valued output. The capacity is dependenton two important parameters: SINR and kurtosis. We give the kurtosis expressionas a function of the system parameters (it is an extension of its existing expression inthe very basic case of power control [47, 51, 133]), so our contribution allows to seethe impact of the parameters on the capacity when SGA is not valid. In Chapter 3it is also shown that kurtosis is related to physical parameters such as the distancebetween pulses and the number of users, the duty cycle of the pulse and so on. Thisis the reason for which we frequently refer to impulsiveness instead of kurtosis, as wecould see in the rest of the manuscript. In the different context of CDMA cells andusing a different approach, results were found which prove that changing the system’simpulsiveness can be used as an alternative for power control in terms of achievablebetter network capacity [64, 65]. With our performance criterion we have reachednew results which are in agreement.

In the same area of interest, Section 4.6 presents an antenna design that increasesthe capacity of an IR-UWB system in a Multi-User Interference environment. As ithas been suggested by [45], the interference in IR-UWB could be decreased by pulseshape optimizations. This part of the work proposes a statistical signal-processingtool to optimize an IR-UWB antenna, taking into account the impact of antenna’s

8


time domain response over the capacity of the discrete memoryless channel. This toolcould be also used to optimize some other basic pulse shapes.

In an IR-UWB system, both transmitters and receivers have to use pulse genera-tors. For instance, Impulse Radio receivers use Matched Filter (MF) correlation andtherefore must generate a template pulse that matches the incoming waveform. Evenif the heart of any IR-UWB system is the pulse generator, the performance of thepulse generators it is not the goal of this thesis. From our point of view, the most im-portant receiver design concerns the minimization of the bit error rate through signalprocessing tools but also of the complexity and of the power consumption. For appli-cations such as sensor networks, where low complexity and low power consumptionis a powerful demand, suboptimal receivers might be very good candidates.

Receivers for impulse IR-UWB can be broadly categorized as energy detectors[61, 149, 165] (i.e., threshold detectors), correlation detectors known also as matchedfilter (MF) receivers [94, 144, 186], and Rake receivers [119, 140, 146, 181]. Multi-user detectors (MUD) [37, 114] and hybrid Rake/MUD-UWB [115] receivers for robustnarrowband interference suppression are also becoming popular.

The Rake receiver is used in any kind of spread spectrum communication systemto accumulate the energy in the significant multipath components. The Rake receiverconsists of multiple correlators (i.e., bank of correlators). Each finger of the Rake issynchronized to a multi-path component, and the output of each finger is coherentlycombined using MRC (maximal ratio combining). Obviously, channel estimation isrequired in the combining scheme and therefore when it is applied to UWB systems,the Rake receiver suffers from a major drawback if the distortion of the pulse throughthe multipath channel changes very fast (i.e., for each transmitted pulse). In thissituation, the template in the receiver has to be recomputed faster [11]. An optimalmaximum likelihood (ML) detector is far too complex to be implemented for a realisticsystem, so in practice suboptimal coherent Rake receivers are used. Another way toreduce the complexity is to reduce the number of fingers (e.g., the number of paths),but as we will see later in Section 5.2, when the second derivative of the Gaussianpulse is used, the energy capture is relatively low for a moderate number of fingers[177].

MUD receivers usually consider minimum mean square error (MMSE) techniques.Adaptive MMSE MUD receivers are very efficient, being able to gather the multipathenergy and reject interference to a much greater extent than conventional Rake re-ceivers. The Hybrid Rake MUD UWB receivers are enhancements of Rake reception,replacing MRC combining with optimum combining based on MMSE. Unfortunately,MUD is a complex solution [185], even when suboptimal methods such as successiveinterference cancellation are used [52].

It is important to mention that the number of paths in IR-UWB is huge, makingthe estimation of the channel difficult to compute. If we cannot rely on a high numberof fingers in a Rake receiver, neither on a very frequent estimation, an alternativeapproach is the use of an autocorrelation receiver that correlates the received signalwith a previously received template [30, 80]. Under the assumption that the channelis varying slowly, this receiver is not requiring channel estimation, as both templateand useful signal are affected by the same propagation parameters. However, even

9


if the channel is varying slowly, the noise is not and it affects the demodulation.More precisely, the Time Reference (TRef) system is always suffering from the useof a noisy received template. Anyway, a potentially attractive feature of TRef UWBautocorrelation receivers is their relative robustness to synchronization problems [80].

Our contribution in the field of receiving and transmitting techniques for ImpulseRadio UWB can be found in Chapter 5, which is concerned with IR-UWB receiversand transmitters. This chapter led to the development of a C++ platform used underCygwin. As a consequence of the atypical MUI aspect in IR-UWB [34], we chose tomodel the MUI distribution in order to derive an appropriate single user receiver. Theresults in [11, 47, 50] prove that the MUI in IR-TH are not Gaussian. It has beenempirically shown [46] that after the correlation part of the receiver, the MUI (foreach impulse of the received signal) may be modeled as a generalized Gaussian. Thereceiver adapted to generalized Gaussian was developed in [46], but the parameterestimation block was not detailed. In Subsection 5.1.5 we contribute by detailing itand giving an iterative structure.

The advantage of modeling or fitting the MUI distribution with an appropriatedistribution [10, 104] is that it gives the possibility to develop a simple yet efficientreceiver [9, 45, 68, 69]. We privilege this approach in contrast with heavier multi userdetection, as the low complexity receiver is one of the targeted advantage of ImpulseRadio systems.

Section 5.1 presents the design of a novel IR-UWB receiver which assumes that theprobability density function describing the statistic of the interference is a mixture-based distribution.The receiver for Gaussian Mixture was developed in [41, 42], andin Subsection 5.1.6 we contribute by extending it to generalized Gaussian Mixtureand by giving a more simple architecture in an iterative structure where the mixtureestimation is separated from the channel estimation - which is supposed to be alreadydone and used by the Matched Filter.This simple IR-UWB receiver could be used forsensor networks and its performance is compared with previous works presented inSection 2.2.5.

In Chapter 5, we also propose a novel transmitter design based on Time Reversal(TR) Impulse Radio Ultra Wide Band which we describe in Section 5.2. Many studieshave been made on the number of fingers to be used in the Rake receiver in orderto obtain good performance [11, 24, 124, 143, 178]. Time Reversal IR-UWB systemsuse a prefilter at the transmitter side [139, 170], that has the function of convolvingthe UWB pulse with the impulse response of the channel reverted in time; when thesignal traverses the channel, the output of the channel presents the correlation ofthe channel with itself. The "fingers" from the transmitter (i.e., prefilter) could beseen similar to the fingers from the Rake. In this section, a trade-off between thenumber of fingers in the prefilter versus the number of fingers in the Rake receiverhas been done. This allows us to study the gain brought by Time Reversal when thecomplexity is switched from the receiver to the transmitter i.e., when the number offingers is increased in the prefilter while at the same time it is reduced in the Rakereceiver.

Going toward upper layers, cooperative communication techniques are used (atthe MAC layer) to improve wireless link performance. As explained in Section 1.2 on

10


the applications for UWB, one of the advantages of UWB is the positioning ability(due to the high bandwidth). The geographical informations can be used to managecooperation between adjacent nodes [161, 162, 163].

Compared to direct transmission between a source node and a destination node,cooperative transmission involves relay (neighbor) nodes that are within the radiorange of both source and destination to retransmit an overheard transmission to thedestination. Cooperative communications have some advantages in terms of through-put, connectivity, and energy consumption [153]. In literature, this type of cooper-ative communication technique is also called (cooperative) relaying. There are twocommon types of relaying, namely the decode-and-forward (DF) and amplify-and-forward (AF) [118]. In AF, a relay simply amplifies an overheard transmission with-out decoding it, before retransmitting to the destination [95]. On the other hand, inDF, the relay needs to fully decode an overheard transmission before retransmittingit to the destination, which limits DF’s performance.

In a simple AF scheme, a fixed time slot is allocated to the source and to the relay.In this fixed time slot structure, the relay retransmits the overheard transmissionto the destination node even if the original transmission from the source has beensuccessfully received. This type of simple AF scheme has been shown not to bethe best alternative in terms of spectral efficiency [108]. In [110], the authors choseto improve the performance of AF in terms of lower outage probability by usingmultiple relays that transmit concurrently in the single allocated time slot. However,this improvement does not help in achieving a higher spectral efficiency.

A few incremental relaying schemes have been proposed in order to improve spec-tral efficiency. In these schemes, a relay is dynamically allocated a time slot toretransmit an overheard transmission only if the destination has not received theoriginal transmission from the source. In order to perform incremental relaying, itis assumed that there is immediate feedback from destination to relay. When im-mediate feedback is not available, [188] proposed to dynamically allocate time slotsto the relay based on an estimated channel state information. While these existingincremental relaying schemes help in achieving a higher spectral efficiency [99], only afew exploit the existence of multiple relay nodes [97]. Nevertheless, there are severalreferences employing multiple relay strategies but they treat mainly non-incrementalschemes (e.g., AF or DF) [77, 166], while those dealing with multiple relay incremen-tal schemes [17, 19, 98, 101, 102] are referring to best single-relay selection technique(i.e., only one best relay is picked up to help) or to space time coding techniques [121](i.e., virtual antenna array).

We would like to point out that the literature for relay cooperation schemes isvast and one can find AF schemes with multiple relays, or incremental AF schemeswith best relay selection (or half-duplex), but we were not able to find existing resultsfor incremental AF schemes with multiple relays where all the relays incrementallyparticipate in cooperation.

The work in Chapter 6 focuses on the cooperation in wireless networks. In Section6.1, we propose an incremental AF relaying scheme that uses multiple relay nodes toachieve higher spectral efficiency and lower outage probability.

Even if they are the best options in terms of spectral efficiency [108], incremental

11


relaying techniques are not practical and there are difficulties to be implementedfor reasons such as non-availability of the channel state information (CSI) or theimperfect feedback [110] from the destination. There are two ways to do the dynamictime slot allocation in IAF: (a) the first way is to use the estimated channel stateinformation [17, 18, 188], and (b) the second way is to use immediate feedback fromdestination [97, 98, 99, 101].

Because of feasibility reasons related to determining the instantaneous channelstate information, in Chapter 6 we consider the use of immediate feedback fromdestination, but which can be affected by propagation impairments. In practicalwireless networks, in the presence of such imperfect feedback information, both thesource and the relay may over-react resulting in packet collisions. Therefore, in Section6.2 we propose a novel and practical incremental AF (PIAF) scheme that deals withthe imperfect feedback information. Section 6.3 describes a simple distributed relayselection policy without any need of information exchange between the relays.

This thesis covers various domains from PHY to MAC and network layers. Eachcontribution can be seen as a small piece of the whole complex puzzle which is thefield of wireless networks. The work from this thesis led to a certain number ofpublications enumerated at the beginning of this manuscript.

12

Chapter 2

Signal Model and InterferenceScenarios

Chapter Overview

This chapter presents the overall system model and the IR-UWB notation con-vention that will be used throughout the manuscript. While Section 2.1 describes theclassical Impulse Radio transmission schemes (e.g. PPM-TH, PAM-TH, DS-UWB),Section 2.2 deals with the transmitter, the channel model, and basic Impulse Radioreceivers.

We further show in Section 2.3 the primary interference scenarios that can affectour network’s performance. This section presents the basic topologies encounteredin sensor networks and theoretically develops the interference expression for all ofthe considered topologies. All topologies are further used in Chapters 4 and 5 fordifferent applications of single-cell, double-cell and double-link scenarios, or cognitivenetworks.

2.1 Introduction on IR-UWB Signal

In this introductory section we briefly describe the existing Impulse Radio UWBmodulations techniques, emphasizing on Pulse Position Modulations (PPM) andPulse Amplitude Modulations (PAM) [5, 11, 70, 147]. We are also showing that allof the IR-UWB techniques are similar: all of these techniques use pulses to transmitand therefore share similar expressions, similar interference models or similar receiv-ing methods. For instance, later in Section 3.3.1 we use this similarities between PPMand PAM to compute some important quantities necessary further in our study (e.g.,kurtosis, SINR).

13


2.1.1 PPM-TH-UWB

The signal sent in Pulse Position Modulation Time Hopping Ultra Wide Band(PPM-TH-UWB or simply called PPM-UWB) may be written as

x(t) =+∞∑

j=−∞

w(t− jTf − cjTc − bitjδP P M), (2.1)

where w(t) is the function describing the shape of the pulse, and has non-zero valuesonly inside an interval of length Tp. Tf is the frame period and usually equals NhTc

where Tc is the slot (i.e., chip) time period and Nh is the number of slots (i.e., chips)in a frame. bitj is the bit we would like to transmit, bitj ∈ {0, 1} and δP P M isthe additional time shift used for the pulse position modulation. The time hoppingcode cj is a number from a sequence Cn that defines the position of the chip intothe frame. Usually cj is chosen relative to the length of only one frame so cj ∈{

0, 1, . . . ,⌊

Tf

Tc

⌋

− 1}

.Note that different users are using different time hopping codes, which gives multi-

ple access ability and resistance to interference. One refers to orthogonal time hoppingcode when the time hopping codes of different synchronized users are such that thepulses of the different users are received in different slots: so there are no overlappingpulses and no interferences between them.

In this generation scheme we have assumed that we transmit only one bit perframe for a given user. An entire symbol could be spread into multiple frames, upona redundancy code included into [bitj]. Also, cj could be cyclic, random, or repetitiveon some portions and cyclic at the same time. On the other hand, the data bitj couldbe coded using some sophisticated methods or simply coded by repetition. w(t) isusually (but not necessary) considered as the second derivative of the Gaussian pulse,therefore

w(t) =

[

1− 4π(

t

τ

)2]

· exp

[

−2π(

t

τ

)2]

. (2.2)

Note that the pulse to be transmitted is considered to be the Gaussian pulse andthe passive components from the system of two antennas are considered to generatea second derivative operation. Therefore, w(t) already incorporates the effect of thetwo antennas (i.e., one at the transmitter side and one an the receiver side) [70]. Thispulse it is also called "Scholtz’s pulse" [180] after the name of the one who used it forthe first time in UWB communications. For this special type of pulse, the length ofthe pulse Tp = 2τ .

Usually Tp ≤ Tc for orthogonality reasons, and in many systems Tc is chosen equalto Tp (i.e., Tc = Tp).

14


2.1.2 PAM-TH-UWB

The signal sent in Pulse Amplitude Modulation Time Hopping Ultra Wide Band(PAM-TH-UWB) may be written as

x(t) =+∞∑

j=−∞

(2 · bitj − 1) · w(t− jTf − cjTc). (2.3)

PAM-TH-UWB is a hybrid modulation using both the properties of PAM andTH. The code cj gives the time hopping [38, 39] property but the data informationis contained in the amplitude of the signal. Again, different users are using differenttime hopping codes [109].

2.1.3 DS-UWB

Pulse Amplitude Modulation Direct Sequence UWB (PAM-DS-UWB) or simplycalled DS-UWB can be schematized as

x(t) =+∞∑

j=−∞

(2 · bitj − 1)Nh−1∑

i=0

biw(t− jTf − iTc), (2.4)

where we suppose we send one bit bitj per one frame Tf , which is composed of Nh

slots. In this description, the code bi ∈ {−1, +1} and it comes from a binary spreadingsequence Bn. Different users are using different spreading sequences. Note that in theexpression (2.4), w(t) is present on each chip from the frame (and not one per frame).The inconvenience for this method is that the multi-path performance is degradeddue to inter chip interference (ICI) and inter symbol interference (ISI). Since thepulse positions are fixed in frames, if one pulse was interfered with another pulse, thesubsequent pulses will be interfered too and this may cause the error bits burst. In[3], a hybrid DS-TH-UWB scheme that uses both direct sequence and time hoppingcodes has been proposed.

2.1.4 Generalized PAM-PPM

Generalized PAM-PPM [11] is a form of representation of both PAM and PPMtechniques and the generalized signal expression is

x(t) =+∞∑

j=−∞

(2 · bitP AMj − 1) · bj · w(t− jTf − cjTc − bitP P M

j δP P M), (2.5)

where– bitP AM

j = bitj and bitP P Mj = 0 for all j if the signal is PAM

or– bitP P M

j = bitj and bitP AMj = 1 for all j if the signal is PPM.

15

2.2 System Model

In equation (2.5) we have used a scrambling sequence bj with its values belongingto the set {−1, +1} [49, 184]. The symbol is transmitted over Ns frames of durationTf and each frame is decomposed into Nh slots (i.e., chips) of duration Tc. Note thatthe symbol period Ts = NsTf = NsNhTc.

If we consider that the length of the code defines one symbol or that the code iscyclic with the period Ts, the representation becomes

x(t) =+∞∑

j=−∞

(2 · bitP AMj − 1)

NS−1∑

i=0

bj · w(t− jTs − iTf − ciTc − bitP P Mj δP P M). (2.6)

In literature there are several other modulation techniques for example Pulse Or-der Time-Hopping Multiple Access ultra wide band (PO-THMA UWB) (a techniqueusing Hermite polynomial MHP pulses [96]), or On-Off Keying (OOK), but all thesetechniques used for Impulse Radio are somewhat similar because all of them use pulsesto transmit and share the same signal characteristics.

2.1.5 Useful Physical Quantities for Impulse Radio

If we transmit a single pulse w(t) in a frame Tf , then we can define the instanta-neous power pw(t), the peak power Ppeak, the average power Pavg and the energy ofthe pulse εpulse as follows:

pw(t) = |w(t)|2 (2.7)

Ppeak = max(pw(t)) = P0 (2.8)

Pavg =1Tf

∫ Tf

0pw(t)dt =

εpulse

Tf

(2.9)

εpulse = Pavg · Tf =∫ Tf

0pw(t)dt (2.10)

2.2 System Model

In Section 2.2 we present the Impulse Radio signal that we use (see Subsection2.2.1), the channel model (see Subsection 2.2.2), and the Impulse Radio receiver (seeSubsection 2.2.5). The most common types of interference are briefly described inSubsection 2.2.4. Please note that interference modeling it shall be presented in depthin Section 2.3.

2.2.1 Impulse Radio Transmitted Signal

We consider an Impulse Radio Time-Hopping Pulse Amplitude Modulation (bi-nary IR-TH-PAM) UWB system [11]; as presented before, the time is divided intoframes, each frame is divided into Nh slots with Tc being the duration of a slot (thechip time interval), and a user sends one impulse per frame in a slot position given

16

2.2 System Model

by a pseudo-random time-hopping code. This impulse is repeated over Ns frames ofduration Tf = NhTc each, therefore Ns is so-called the repetition factor.

To distinguish between users, we add the subscript k, so the time hopping codefor the user k is represented by the sequence (ck,l)l∈Z whose elements belong to{0, . . . , Nh − 1}. There is a user-specific “scrambling code” bk,l with values belongingto the set {−1, 1} [49, 184]. Note that both codes bk,l and ck,l have to be known bythe receiver when receiving signal from user k (see Fig. 2.1). The user k transmitsthe signal

xk(t) =√

εk

Ns

+∞∑

m=−∞

ak,mpk,m(t−mNsTf ) (2.11)

with

pk,m(t) =Ns∑

n=1

bk,mNs+n · w(t− (n− 1)Tf − ck,mNs+nTc), (2.12)

where ak,m is the mth symbol’s information of the kth user, bk,mNs+n is the randomizingcode and εk is the average energy per sent symbol. w(t) is the unit-energy basic pulsewaveform and is considered to be the second derivative of the Gaussian pulse, asdefined in equation (2.2).

2.2.2 Channel Model

The signal xk(t) is transmitted through the physical channel hk(t) between the userk and the receiver. While in Subsection 2.2.2.1 we present the multi-path channelmodel, in Subsection 2.2.2.2 we present a simplified channel model for free-spacepropagation.

2.2.2.1 Multi-path Channel

We consider a frequency-selective channel of the form

hk(t) =L−1∑

i=0

αk,iδ(t− iTw) (2.13)

where L is the maximum number of multi-path components in the system, and αk,i isthe attenuation of the ith path. As expected, L is a function of the imposed ChannelModel (CM), while LTw is equal to the maximum delay spread of the channel. Inour simulations we have considered the IEEE CMs proposed by Foerster and Molisch[55, 57, 116], and we have considered CM1 (LOS), CM2 and CM3 (NLOS). TheseCMs are actually a modification of the Saleh-Valenzuela channel model described in[154] that takes into account the clustering phenomenon observed in several UWBchannel measurements [107].

2.2.2.2 Single-path Channel

The single-path channel is a simplified channel model in which αk,i = 0 for everyi 6= 0. A particular case of the single-path channel is the free-space propagation

17

2.2 System Model

channel where αk,0 = 1. The single-path and the free-space propagation channels areoften used for theoretical derivations [179, 180] to show the influence of various signalparameters. In the case of multi-path channel such analytical derivations would leadto complex formulas hardly readable and handleable.

2.2.3 Transmission Scheme and Received Signal

The classical transmission-reception chain is presented in Fig. 2.1 [179, 180]. Thebits ∈ {0, 1} are transformed into polar values a, and after the repetition and multipli-cation with the scrambling code b (cf. equation (2.12)) each value is time shifted aftera Time-Hopping Code (TH Code) and finally Pulse Amplitude Modulated (PAM) andsent. We assume that the receiver is perfectly synchronized with the first user andthus △1 = 0. As a consequence the received signal becomes

r(t) = r1(t) +Nu∑

k=2

rk(t−△k) + v(t) (2.14)

where rk(t) = xk(t) ⋆ hk(t) takes into consideration the effect of the channel hk(t),with ’⋆’ denoting the convolution operation. v(t) is the additive white Gaussian noise(AWGN) and the delay △k accounts for the absence of synchronization between userk and user 1, or in other words for the timing offset between the user k and thereceiver.

Figure 2.1 Classical Transmission Scheme

18

2.2 System Model

2.2.4 Types of Interference

In this subsection we present the most common types of interference in IR-UWBsystems:

– MAI (Multi-user Access Interference) also called MUI (Multi-User Interference).For sensor networks, the coexistence of a large number of UWB transmittersin a dense environment is very important. The signals transmitted by eachsensor share the same spectrum, and simultaneous transmissions by multiplesensors are usually achieved using TH or DS codes. Ideally, it is desired to haveorthogonal codes for each sensor [44]. However, in practice the received signalfrom different sensors is not orthogonal because of multi-path, asynchronoustransmission. Also, designing perfect codes with zero auto-correlation and cross-correlation properties for all shifts is not possible [78]. In UWB, the MUI isa major problem [35, 36]. Approaches that use techniques from CDMA andMMSE [116, 147] combining of Rake fingers are recently being considered forUWB. New results show that different approaches could be envisaged (Beaulieu,Erseghe and Fiorina [9, 41, 42, 46]), in which the receiver knows the distributionof MAI and estimates its parameters.

– IFI (Inter-Frame Interference). Since a TH transmitter uses different frames totransmit consecutive pulses, IFI is the interference that arises when the mini-mum pulse-to-pulse duration is shorter than the maximum excess delay of thechannel (also called ICI if there is one pulse per frame).

– ISI (Inter-Symbol Interference) is a problem for high-data-rate systems. It refersto the interference at the level of the symbol. If we take for example the case ofTH-UWB transmissions, when we are referring to the symbols coming from asingle transmitter, the system cannot encounter ISI without IFI. When a symbolis transmitted into a single frame, on one chip, the terms ISI and IFI becomeequivalent.

– NBI (Narrowband Interference). The very low transmission power and largebandwidth enable UWB systems to co-exist with other narrowband systemsappearing as low power noise. However, the effect of the narrowband signals onthe UWB signal can be significant and may jam the UWB receiver completely.Even though the narrowband signals interfere with only a small fraction ofthe UWB signal spectrum, the performance and capacity of UWB systems canbe affected considerably. Therefore, the UWB receivers might need to employNBI suppression techniques to improve the performance, capacity and range ofcommunication.

2.2.5 Impulse Radio Receiver Generalities

Fig. 2.2 represents the classical transmission chain, where we have considered theclassical receiver proposed by Scholtz and Win [179, 180]. Please note the absence ofthe randomizing code b (in this paragraph it has been omitted on purpose). At thereceiving side, the received signal r(t) goes through the Matched Filter (MF) and itis sampled and summed up on Ns consecutive samples (of the same bit).

19

2.2 System Model

Figure 2.2 Classical Transmission Chain with the SUMF Receiver (Scholtz and Win)

We remind that in the case of UWB, the channel has unusual characteristics,for example the so-called clustering effect and the length of spreading time which ismuch larger than the length of a transmitted pulse. Obtaining precise BER curvesin MATLAB requires a lot of time, because of some of the model’s properties (e.g.,need for a high signal resolution, presence of very rich multi-path channels, need toobtain results at very Low BER rates). But still, the issue is not the time spent forsimulations, but instead the MEMORY. Unfortunately the message “out of memory”is very likely in an IR-UWB multi-user environment. A solution would be to sparsethe signal vector into sets and simulate over and over till we reach enough simulationpoints for a good resolution, but we still show a preference for a C++ based simulatorin Cygwin, since it is fast, portable and can be used on a distributed platform.

Considering the above remarks, we have developed besides the MATLAB simu-lator a basic-C++ simulator with the help of the IT++ Library (library of mathe-matical, signal processing and communication routines).

There are many studies on the performance of DS-UWB and IR-UWB techniques[54, 60, 147]. Using Cygwin we did our own simulations and experiments studyingthe performance of IR-TH UWB on several scenarios using IEEE multi-path channelmodels (see Fig. 2.3).

If we consider the entire transmission chain, we find that small changes couldbe made in order to simplify the receivers in the detriment of the transmitter (seeFig. 2.4). Nowadays there is a technique called Time Reversal [139, 170] used in otherfields as well, for example in acoustics, which uses this simple idea. We will exploitthis idea in Section 5.2.

In Section 5.2, a part of the Matched Filter (MF) [24, 124, 143, 178] includingconvolution with the estimated channel is moved in the transmitter and the receiverwill consume less, but the transmitter still needs an estimate of the channel in orderto construct the MF block.Remark: Supposing that the channel is invariant in time on small periods, thisidea of switching the complexity from the receiver to the transmitter makes sense.

20

2.2 System Model

0 1 2 3 4 5 6 7 8 9 10

10−4

10−3

10−2

10−1

Eb/N

0 [dB]

BE

R

Ns=1

Ns=2

Ns=3

(a)

0 1 2 3 4 5 6 7 810

−4

10−3

10−2

10−1

Eb/N

0 [dB]

BE

R

CM1

CM2

CM3

(b)

0 2 4 6 8 10 12 14 1610

−4

10−3

10−2

10−1

Eb/N

0 [dB]

BE

R

CM1

CM3

(c)

0 5 10 15 20 2510

−4

10−3

10−2

10−1

Eb/N

0 [dB]

BE

R

Ns=1

Ns=2

Ns=3

(d)

Figure 2.3 All-Rake IR-TH UWB, Tc = 1ns, τ = 0.5ns (a) CM1, 1 User, Just Noise, (b)Ns = 1, 1 User, Just Noise, (c) Ns = 1, 10 Users, MUI and Noise, (d) CM1,10 Users, MUI and Noise

Figure 2.4 Time Reversal Transmission Chain

21

2.2 System Model

The impact of the interference and the noise in this scenario has not been sufficientlystudied. We shall therefore consider the Section 5.2 to be a starter for further researchon this kind of approach.

The Time Reversal strategy is not the only way to implement simplified receivers.The Time Reference (or Transmitted Reference) [26, 117] receiver simplifies evenmore the original scheme because the receiver no longer estimates the channel. Inthis scheme, the receiver correlates the received pulses with the previously receivedones (i.e., previous pulses are used as reference for the correlator). The performance ofthis receiver can be found in Fig. 2.5 for LOS and NLOS channels, scenarios withoutMUI, single user case. The simulations from Fig. 2.5 have been made in C++ onUWB channels, while the theoretical curves have been obtained with the help of[26, 117].

0 2 4 6 8 10 12 14 16 18 2010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/N

0 [dB]

BE

R

LOS (CM1) Theory

NLOS (CM3) Theory

NLOS Simulations

LOS Simulations

Figure 2.5 Time Reference Performance under CM1(LOS) and CM3(NLOS)

The UWB standard is very strict regarding the emission power but we expectinterferences to play an important role: not only the interferences from differenttechnologies using the same frequency band, but also from a lot of other UWB devices.An IR-UWB sensor network is expected to have a large number of devices [145] andthe interference from all these devices is not at all negligible, taken into considerationthat synchronizing orthogonal TH codes for each possible Tx−Rx link from a randomdisposed map is not possible. It has even been suggested that less synchronizationmeans less consumed power and cheaper devices, so many articles are consideringrandom codes and asynchronous schemes [20]. Since the Multi-User Interference(MUI) is so important, we focus our interest on the study and the design of somesimple receivers that use the statistical knowledge of the interference at the receiver[45, 68].

22

2.2 System Model

Previous work on the receivers has considered Gaussian and generalized Gaussianadapted receivers. We represent the corresponding schemes in Fig. 2.6, as they havebeen defined by their authors [46, 179].

Figure 2.6 Simplified Statistical MUI-Based Receiver Schemes

In the generalized Gaussian adapted receiver two parameters have to be estimated:the signal energy and the shape of the weighting function after the MF. For the secondissue, we can assume for example that the transmitter is not transmitting for a whileso we are able to listen only to the MUI, compute the kurtosis and then extract theβ parameter that characterizes the generalized Gaussian distribution (see differentweighting shapes as a function of β in Fig. 2.7).

The weighting function (see Fig. 2.6) is derived upon the Maximum Likelihood(ML) [46], but there is also an intuition behind this approach. As a matter of fact,when the impulsiveness of MUI is high enough, the weighting function takes a non-linear shape (see Fig. 2.7) with extreme points adapted to the energy of the signal.The idea behind this intuition is that we would like to suppress the MUI that variesa lot above the energy of the useful signal, while making sure that the useful signalis not affected by this weighting. This is why it is important to find a simple way tocompute a ratio between the energy of the MUI and the energy of the signal. If thesignal’s amplitude after the Matched Filter is ε and the signal has been weighted with1/ε, one might find that the most probable value of the useful PAM signal is -1 and1 respectively (see the Ox axes of the function in the previous Fig. 2.6 and Fig. 2.7).

In Section 5.1 we propose a different scheme that computes iteratively the twoparameters to be estimated. We compare this receiver with a similar receiver whichassumes that the interference is a mixture of distributions.

23

2.3 Interference Model

−2 0 2−4

−2

0

2

4

Ou

tpu

t L

imite

r

Input Limiter−2 0 2

−4

−2

0

2

4

Input Limiter

Ou

tpu

t L

imite

r

−2 0 2−4

−2

0

2

4

Ou

tpu

t L

imite

r

Input Limiter−2 0 2

−4

−2

0

2

4

Ou

tpu

t L

imite

r

Input Limiter

β=1

β=2.374

β=0

β=0.5

Figure 2.7 Some possible weighting shapes of the generalized Gaussian


We further develop the signal model for some particular topologies: single-cell(Section 2.3.1), double-cell (Section 2.3.2), and double-link (Section 2.3.3) scenarios.These particular cases are the basic topologies of a wireless network.

As mentioned in the 802.15.4 [1] standard, it is reasonable to consider that anetwork can function in two modes:

1. in the first mode a base station BS acts as sink, and it gathers data from thesensors around (see Fig. 2.9 from Section 2.3.1);

2. in the second mode the base station BS acts as coordinator, allowing the sensorsto exchange data between each other (see Fig. 2.13 from Section 2.3.3).

Further, the network could be expanded to multi-cell scenarios. Fig. 2.10 fromSection 2.3.2 shows a particular case of double-cell scenario where two BSs are actingas sinks (i.e., uplink communication), gathering data from their connected devices.In this multi-cell scenario, the devices from one cell are interfering in uplink with thedevices from the another cell.

2.3.1 Single-Cell Scenario

In this section we consider a binary IR-TH-PAM UWB system [11] where eachuser sends one impulse per frame in a slot position given by a pseudo-random time-hopping code (see Fig. 2.8).

The single-cell scenario is depicted in Fig. 2.9. Since we are interested in theeffect of Multi-User Interferences (MUI), we consider that the receiver is synchronizedwith user 1 (transmitter 1) and that there are Nu − 1 interfering users which are notsynchronized with user 1, like the signal model presented in equation (2.14). We begin

24


(a)

(b)

Figure 2.8 IR-TH-PAM Transmission Scheme - The transmission of one bit (a) withoutand (b) with the help of a randomizing code

25


by considering the reference scenario of a free space propagation channel [179, 180].This allows us to underline the influence of the essential system parameters: Nu, Nh,the w(t) pulse shape, on the sensor network’s general performance. The conclusionsfrom free space propagation could be expanded afterwards on multi-path channelscenarios. The multi-path channel model will be considered later on, for receiver andtransmitter design (Sections 5.1 and 5.2).

Figure 2.9 Single Cell Scenario

Please note that the total received signal r is given by (2.14) with rk given by

rk(t) =√

εk

Ns

+∞∑

m=−∞

ak,mpk,m(t−mNsTf ). (2.15)

Therefore, the signal received from the kth user can be expressed similarly toequations (2.11) and (2.12), with εk the average energy per received symbol.

After being received by the antenna, the signal r(t) goes into the classical corre-lation receiver. The output of the classical correlation receiver (SUMF, Single UserMatched Filter) when the transmitted symbol is a1,0, is

rout =√

ε1

Ns

Ns∑

n=1

∫

r(t) · b1,n · w(t− (n− 1)Tf − c1,nTc)dt. (2.16)

We can simplify equation (2.16) to rout =∑Ns

n=1 rimp [n], where rimp [n] is theimpulse per impulse correlator output:

rimp [n] =√

ε1

Ns

∫

r(t) · b1,n · w(t− (n− 1)Tf − c1,nTc)dt (2.17)

and the estimated symbol for the SUMF is a1,0 = sign(rout).The impulse per impulse correlator output can be written as

rimp [n] = u [n] + y [n] + v [n] (2.18)

where u [n], y [n], v [n] are the “useful” term, the Multi-User Interference term, andthe AWGN term respectively:

26


u [n] =√

ε1

Ns

∫

r1(t) · b1,n · w(t− (n− 1)Tf − c1,nTc)dt (2.19)

y [n] =√

ε1

Ns

∫ Nu∑

k=2

rk (t−△k) · b1,n · w(t− (n− 1)Tf − c1,nTc)dt (2.20)

v [n] =√

ε1

Ns

∫

v(t) · b1,n · w(t− (n− 1)Tf − c1,nTc)dt. (2.21)

2.3.2 Double-Cell Scenario

We consider the coexistence of two unsynchronized TH-IR-UWB cells, namely cellI and cell II, as illustrated in Fig. 2.10. Here, each cell (i.e, network, where the terms“cell” and “network” are used interchangeably in this section) is composed of varioussensors (i.e., user, where the terms “users” and “sensors” are used interchangeably inthis section) and one base station (i.e., sink node, where the terms “base station” and“sink” are used interchangeably in this section).

Figure 2.10 Two Cells Scenario

We will now focus on the uplink communication from sensors to the sink. Becauseof the lack of synchronization between the two cells, cell I and cell II interfere witheach other. Both cells use IR-TH-UWB with pulse amplitude modulation. As theparameters may be different for each cell, we add the subscript i with i ∈ I, II toindicate the cell. As such, Nh,i is the frame length in terms of time slots, and theframes have a duration of Tf,i = Nh,iTc. The time slot duration for transmission ofa chip is Tc in both cells. The signal transmitted from user k inside cell i can beexpressed as

xi,k(t) =√

εi,k

+∞∑

m=−∞

ai,k,mw(t−mTf,i − ci,k,mTc), (2.22)

with ai,k,m the m-th symbol sent by user k of cell i, εk the energy per impulse andw(t) the UWB pulse. The time hopping code for the user k of cell i is represented bythe sequence (ci,k,l)l∈Z with values in {0, . . . , Nh − 1}.

27


Figure 2.11 IR-TH-PAM Transmission Scheme

As in the former section, in order to develop some analytical expressions for thecharacterization of the interference, we first consider a free space propagation channel- the usual reference scenario [11, 70, 179, 180]. This helps in clearly showing andunderstanding the influence of the various UWB signal parameters. Such result isstill representative of what would happen in multi-path channel but in this case theanalytical derivation would lead to a complex formula hardly readable and handleable.

In this specific scenario, the users of one cell are supposed to be synchronized suchas the signals they transmit do not interfere when received by their base station, asa result of orthogonal time hopping codes. However they arrive unsynchronized atthe base station of the other cell and create interferences. The signal received by thebase station I from the user 1 of the cell I, added to the noise plus the interferencefrom the Nu,II users of the cell II may be written as

rI(t) = AI,1xI,1(t) +Nu,II∑

k=1

BII,kxII,k (t−△II,k) + vI(t). (2.23)

Similarly, the signal received by the base station II from the user 1 of the cell II hasa symmetrical expression:

rII(t) = AII,1xII,1(t) +Nu,I∑

k=1

BI,kxI,k (t−△I,k) + vII(t), (2.24)

with Nu,I interfering users from cell I.The signals received from the other users have similar expressions, with vI(t) and

vII(t) the additive white Gaussian noise (AWGN) at the receiver of the 1st and 2nd

cell respectively. We note △II,k the delays that take into account the absence ofsynchronization between the k transmitters of cell II and the base station of cell I.The amplitude of the useful received signal at base station i from user k of its cell isAi,k, while the amplitude of the interfering received signal at a base station from auser k belonging to the interfering cell j is Bj,k.

At the reception, the signal rI(t) (the equivalent happens for rII(t)) goes into theclassical correlation receiver. The output of the classical correlation receiver (SUMF,

28


Figure 2.12 Cell I

Single User Matched Filter) for the symbol aI,1,0 is

rout = AI,1√

εI,1

∫

rI(t) · w(t− cI,1,0Tc)dt. (2.25)

Note that rout can be written as

rout = u + y + v, (2.26)

where u, y, v are the “useful” term, the Multi-User UWB Interference term, and theAWGN term respectively:

u = AI,1√

εI,1

∫

AI,1xI,1(t) · w(t− cI,1,0Tc)dt (2.27)

y = AI,1√

εI,1

∫Nu,II∑

k=1

BII,kxII,k (t−△II,k) · w(t− cI,1,0Tc)dt (2.28)

v = AI,1√

εI,1

∫

v(t) · w(t− cI,1,0Tc)dt. (2.29)

2.3.3 Double-Link Scenario

As mentioned in 802.15.4 [1] and 802.15.4a/D5 [2] standards, the devices froma sensor network which are in the same communication range could also exchangeinformation between each other (i.e., every device could exchange information withany other device). In this scenario, a Base Station coordinates the transmissions inthe network. Obviously, the interference model changes but we try to keep similarnotations as in previous paragraphs. As seen in Fig. 2.13, the double-link scenario(see Fig. 2.13(b)) is a simplified interference model of the scenario from Fig. 2.13(a).

The total received signal at the user 1 can be expressed as

r1(t) = A1x1,1(t) + B2x2,1 (t−△2,1) + v1(t), (2.30)

while the total received signal at the user 2 is

r2(t) = A2x2,2(t) + B1x1,2 (t−△1,2) + v2(t), (2.31)

where xi,j is the signal received by Rxi from Txj, △j,i is the synchronization delay,A is the amplitude of the useful signal and B is the amplitude of the interference.

29


(a) (b)

Figure 2.13 (a) Sensors Coordinated by a single Base Station (b) Particular case of aDouble-Link Scenario

The output of the ith user’s correlator for the mth transmitted symbol is

rout,i = Ai

√εi

∫

ri(t) · w(t− ci,mTc)dt. (2.32)

Therefore, the contributions of the useful signal ui, of the MUI yi and of the AWGNvi become:

ui = Ai

√εi

∫

Aixi,i(t) · w(t− ci,mTc)dt (2.33)

yi = Ai

√εi

∫

Bjxi,j (t−△j,i) · w(t− ci,mTc)dt (2.34)

vi = Ai

√εi

∫

vi(t) · w(t− ci,mTc)dt. (2.35)

Please also note that this model will be further used in Section 4.4, using anintermediate signaling scheme between traditional IR-UWB (PAM/PPM-TH withonly one pulse per frame) and DS-UWB (with multiple pulses per frame).

30

Chapter 3

On the Non-Gaussianity of theIR-UWB MUI

Chapter Overview

Chapter 3 deals with the non-Gaussianity of IR-UWB Multi User Interference.In Section 3.2 we present results from our Monte-Carlo simulations showing thatIR-UWB MUI is not Gaussian, which is in agreement with the known non validityof Standard Gaussian Approximation (SGA). In this section we also present severalparameters that affect the MUI distribution. Changing some of these parameters maygive the reader some basic hints on how the statistical distribution of the MUI (andfurther, the communication performance) could be tuned. Our simulations show thatin some particular scenarios MUI could be modeled with a Gaussian distribution butthis result is not representative and it cannot be generalized. On the other hand, wecheck that the generalized Gaussian fits much better the PDF of the MUI.

Based on the generalized Gaussian modeling, Sections 3.2, 3.3, and 3.4 describe aninteresting performance evaluation method that can be used to study the importanceof several parameters such as number of slots, number of users, pulse duty cycle,pulse shape, etc. This method is based on computing the communication capacityfrom the kurtosis (the statistical quantity which describes the shape of the generalizedGaussian distribution) and from the received Signal to Noise Interference Ratio, andit is further used in several applications from Chapter 4. In Chapter 3 we also providethe kurtosis expression in function of the system parameters; such an expression waspreviously known in the very basic incomplete power control scenario without additivenoise. Therefore, our contribution allows to see better the impact of the parameterson the capacity when SGA is not valid.

3.1 Introduction

In this chapter we prove by Monte Carlo simulations the non-gaussianity charac-teristic of the IR-UWB interference distribution, in agreement with [36, 46, 73]. Inorder to prove this for different interference scenarios, we have compared histograms

31

3.1 Introduction

of the interference with reference Gaussian distributions (with same variances). Oursimulations show that the peak and the tail of the histograms cannot be modeledwith a Gaussian distribution, with the exception of some particular cases (e.g., ahigh number of interferers compared with the number of available slots Nh, or highτ/Tc ratio for scenarios with Nu ≈ Nh, perfect power control and low noise). In asecond part of our simulations we show how the tail of the histogram is fitted by thegeneralized Gaussian distribution. As explained in [51], two opposite impacts of thenon-Gaussianity may appear:

– if the receiver is adapted to the classical Gaussian interference distribution, per-formances will be heavily impaired by the non-Gaussianity of the interference.Fig. 3.1 shows the difference between the performance in Gaussian interferenceand in non-Gaussian interference.

– if the receiver is adapted to the generalized Gaussian interference distribution,the performance is improved with respect to the reception in classical Gaussianinterference having the same power.

Figure 3.1 BER vs. SIRbit

It is clear that in some particular cases the generalized Gaussian approximation isnot valid, but it is recognized that this model works well in many cases and is betterthan the classical Gaussian approximation. The generalized Gaussian approximationmodel is a generally accepted model, cited in other works such as [21] (at page 16),and also validated through simulations by [10, 105].

On the basis of the approximation of the MUI with a generalized Gaussian, weuse a performance evaluation method that takes into account all the IR-UWB sys-tem’s parameters such as the shape of the w(t) pulse, the number of users Nu, thenumber of slots Nh and the power distribution of interference. This performance eval-uation method is an extension of the method appearing in [47] which was computedbefore only in a very basic scenario. This model remains valid under the following as-

32

3.2 Simulation Results

sumptions: i.i.d. transmitted bit sequence with zero mean, statistical independencebetween the interference and noise samples, and statistical independence betweenthe interference samples coming from different interferers. The model also assumesrandom uniform distributed delays of the interfering signals at the receiver side.

This performance evaluation method is described in Section 3.4.3 and providesa way to compute the communication capacity from the kurtosis of the interferenceand from the received SINR. This approach provides a useful performance metricwhich can be used in many scenarios with dense or low interference environments(e.g., hundreds of interfering users, or even a few powerful asynchronous interferers),high or low noise. It is therefore a powerful tool to study the impact of the system’sparameters over the entire communication performance.


We choose to start with a section describing experimental results, in order topresent the meaning of our further research. Our experiments done on IR-UWBnetworks show that the Probability Distribution Function (PDF) of the interferencesis not Gaussian. We thus confirm the results presented in previous papers [25, 41, 42,68]. This result has a powerful impact over the entire performance of the system. Wenote the fact that we can make use of this information for our benefit. The advantageof knowing the statistical distribution of MUI can help the devices to communicatebetter, can optimize the throughput of a network and can provide improved MACalgorithms and communication strategies.

As we have previously presented in Chapter 2, we are looking for the interferencesat the output of the correlation receiver (see equation (2.20)). In our further simu-lation scenarios we have considered uniformly distributed synchronization delays ∆.While in the perfect power scenario all the interfering users have the same receivedpowers, in the lack of power control scenario we have considered a reference scenariowhere the interferers have equal transmission powers of −30dBm, and they are ran-domly distributed after a uniform law within an area delimited by a circle with theradius of 1m and another one of 10m, both centered on the receiver. In the lack ofpower control scenario we also assume a path gain decreasing in d−2 where d is thedistance to the receiver.

In Fig. 3.2 we increase the Nh/Nu ratio, in several perfect power control scenarios.Comparing Fig. 3.2(a) with (b), (c) and (d) one can notice that the peak is increasingand the variance of the distribution is decreasing, but also that the histogram distanceitself from the PDF of the Gaussian approximation.

In Fig. 3.3 we have represented the same scenarios as in Fig. 3.2 but withoutassuring perfect power control. Comparing Fig. 3.3 (lack of power control) withFig. 3.2 (perfect power control), one can notice lower peaks but higher distributionvariances. This observation will later on help us to understand the difference betweenlack of power scenarios and perfect power control scenarios.

In Fig. 3.4 we have decreased the τ/Tc ratio in perfect power control scenarios.Comparing with Fig. 3.2, we notice that with the decrease of the τ/Tc ratio, the

33


−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

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PD

F(y

)

(a)

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0.5

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1.5

2

2.5

3

3.5

4

4.5

y

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F(y

)

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0.5

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1.5

2

2.5

3

3.5

4

4.5

y

PD

F(y

)

(c)

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3.5

4

4.5

y

PD

F(y

)

(d)

Figure 3.2 Measured MUI histograms (bars) and reference Gaussian distributions withsame variances (plain curves) - perfect power control scenarios, Tc = 1ns,τ = 0.5ns, Nu − 1 interferers, with (a) Nh = 100, Nu = 100; (b) Nh = 100,Nu = 50; (c) Nh = 250, Nu = 100; (d) Nh = 500, Nu = 100

34


−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

y

PD

F(y

)

(a)

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

y

PD

F(y

)

(b)

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

y

PD

F(y

)

(c)

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.5

1

1.5

2

2.5

y

PD

F(y

)

(d)

Figure 3.3 Measured MUI histograms (bars) and reference Gaussian distributions withsame variances (plain curves) - lack of power control scenarios, Tc = 1ns,τ = 0.5ns, Nu − 1 interferers, with (a) Nh = 100, Nu = 100; (b) Nh = 100,Nu = 50; (c) Nh = 250, Nu = 100; (d) Nh = 500, Nu = 100

35


variance increases while the interference histogram grows apart from the Gaussianapproximation.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

y

PD

F(y

)

(a)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

y

PD

F(y

)

(b)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

y

PD

F(y

)

(c)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

y

PD

F(y

)

(d)

Figure 3.4 Measured MUI histograms (bars) and reference Gaussian distributions withsame variances (plain curves) - perfect power control scenarios, Nu− 1 inter-ferers, with (a) Nh = 100, Nu = 100, Tc = 2ns, τ = 0.5ns; (b) Nh = 100,Nu = 50, Tc = 2ns, τ = 0.5ns; (c) Nh = 250, Nu = 100, Tc = 1ns,τ = 0.25ns; (d) Nh = 500, Nu = 100, Tc = 1ns, τ = 0.25ns

Fig. 3.5 and Fig. 3.6 are showing scenarios with the interference histogram close tothe Gaussian approximation. Fig. 3.5 represents two cases with a number of interferersNu−1 much higher than the number of slots Nh, while Fig. 3.6 represents a particularperfect power control scenario with higher τ/Tc and Nu = Nh. Comparing Fig. 3.5(a)with Fig. 3.5(b), we also notice that the lack of power control case needs a highernumber of users Nu in order to fit the Gaussian approximation.

In Fig. 3.7 we have considered Nu = 100 and Nh = 500, and we show thatgeneralized Gaussian fits well the tail of the histogram.

It has been shown [11, 47, 50] that the MUI plus noise in IR-TH is not Gaussian.It has also been shown [46] that the MUI signal y [n] obtained after the correlationpart of the receiver may be well fitted by a generalized Gaussian having the following

36


−5 −4 −3 −2 −1 0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

y

PD

F(y

)

(a)

−30 −20 −10 0 10 20 300

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

y

PD

F(y

)

(b)

Figure 3.5 Measured MUI histograms (bars) and reference Gaussian distributions withsame variances (plain curves) - Scenarios with interference close to a Gaussiandistribution: Tc = 1ns, τ = 0.5ns, Nh = 10 and (a) perfect power controlscenario with Nu = 100; (b) lack of power control scenario with Nu = 500

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

y

PD

F(y

)

Figure 3.6 Measured MUI histogram and reference Gaussian distribution with same vari-ance (plain curve) - Scenario with interference close to a Gaussian distribu-tion: Tc = 1ns, τ = 5ns, Nh = 100, Nu = 100 and perfect power control

0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

y

PD

F(y

)

PDF generalized Gaussian

PDF Gaussian

(a)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

y

PD

F(y

) PDF Gaussian

PDF generalized Gaussian

(b)

Figure 3.7 Measured MUI histograms (bars) and curve fitting. As shown, generalizedGaussian fits well the tail of the histogram. Nu = 100, Nh = 500, Tc = 1ns,τ = 0.5ns (a) perfect power control (b) lack of power control

37


probability density function

p(x) =c1(β)√

σ2exp

−c2(β)

∣

∣

∣

∣

∣

x√σ2

∣

∣

∣

∣

∣

2

1+β

(3.1)

where c1(β) and c2(β) are functions of β:

c1(β) =Γ

1

2

(

32(1 + β)

)

(1 + β) Γ3

2

(

12(1 + β)

) , (3.2)

and

c2(β) =

Γ(

32(1 + β)

)

Γ(

12(1 + β)

)

1

1+β

, (3.3)

β > −1 and Γ(x) is the Gamma function

Γ(x) =∫ ∞

0ux−1exp(−u)du. (3.4)

When β 6= 0 the generalized Gaussian is not a classical Gaussian. In a communi-cation system where the interference follows a generalized Gaussian distribution, thecapacity (maximum achievable rate) depends on the β parameter of the generalizedGaussian [47].

The difficulty consists in that the β parameter is hard to estimate in practicalsituations when the receiver is able to "listen" to the MUI for a certain period of time.Therefore, it is necessarily to use other parameter strongly linked to β, that is easierto estimate. It is known that the kurtosis K of a variable ζ has the expression

K Def=

E[ζ4]E[ζ2]2

− 3. (3.5)

The pth absolute moment of a generalized Gaussian variable ζ is [122]

E (|ζ|p) =Γ(

(p+1)(1+β)2

)

Γ(

1+β2

)

√

√

√

√

√

Γ(

1+β2

)

Γ(

3(1+β)2

)σζ

p

. (3.6)

Thus, from (3.5) and (3.6) it can be shown that kurtosis can be linked to β through

K(β) =Γ(

52(1 + β)

)

Γ(

12(1 + β)

)

Γ2(

32(1 + β)

) − 3. (3.7)

For small values of β we approximate (3.7) as

K(β) ≈ 3β2 − 0.3β. (3.8)

The parameter β is linked to the kurtosis K of the MUI as we can see in Fig. 3.8,where we graphically represented the dependence K(β).

38

3.3 Kurtosis Computation in Perfect Power Control and Lack of Power Control Scenarios

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4−10

0

10

20

30

40

50

β

Ku

rtosi

s

Figure 3.8 Kurtosis K = F(β)

In the next section we compute the kurtosis of the MUI plus noise (our gener-alization to already known model from [47]), and we express kurtosis as a functiondepending on the parameters of the signal such as Nh, Nu, Tc and pulse shape char-acteristics. We consider two cases: perfect power control (ppc) scenario and lack ofperfect power control (lpc) scenario.

3.3 Kurtosis Computation in Perfect Power Con-

trol and Lack of Power Control Scenarios

In the first part of this section (i.e., Subsection 3.3.1) we start by expanding theequation (3.5) in order to establish a relationship between the kurtosis K and all theIR-UWB signal parameters. Please note that the kurtosis could be computed for dif-ferent IR-UWB technologies. While in Subsection 3.3.1 we exemplify the similaritiesand differences between PPM and PAM, the next subsections are common for bothtechnologies.

In order to expand even further the expression of K, Subsection 3.3.2 and Sub-section 3.3.3 presume two complementary power control scenarios. While Subsection3.3.2 deals with perfect power control, Subsection 3.3.3 deals with lack of perfectpower control.

3.3.1 General Issues

The PAM and PPM technologies are somewhat equivalent in terms of kurtosisand we introduce similar notations to express the 2nd and 4th order moments m2(w)

39


and m4(w) respectively:

m2(w) =1Tc

∫ ∞

−∞γw(z)2dz (3.9)

andm4(w) =

1Tc

∫ ∞

−∞γw(z)4dz, (3.10)

where γw(z) represents the correlation of the received pulse w(t) with the receiver’smask.

The only difference between PAM’s kurtosis and PPM’s kurtosis resides in theexpression of the γw(z) parameter:

– PAM case:

γw(z) =∫ ∞

−∞w(t + z)w(t)dt (3.11)

– PPM case:

γw(z) =∫ ∞

−∞w(t + z) (w(t)− w(t− δP P M)) dt (3.12)

where w(t) is the shape of the pulse at the output of the receiver’s antennaand δP P M is the shift used to represent the ‘−1’ bit information in PPM. Asmentioned in the previous chapter, the pulse shape w(t) is usually (but not nec-essary) considered as the second derivative of the Gaussian pulse [5, 179]. Thisrepresentation takes into account the derivative characteristic of the antenna,which is a generally accepted antenna model. In this context, the pulse appliedat the input of the transmitter’s antenna is assumed to be the Gaussian pulse.The passive components from antennas could generate a second derivative op-eration i.e., the effect of an antenna is derivative and the effect of a system withtwo antennas is second derivative. Note that in practice, the characteristic ofthe antenna is not entirely derivative as we shall see in Section 4.6. Moreover,in Section 4.6 we are considering other shapes of the received pulse that mightimprove the system’s performance.

In the followings, we model the interference and noise in the system as the realiza-tions of a variable ζ. We recall that the kurtosis K of a variable ζ has the expression(3.5).

Single-cell or multi-cell scenario, the general expression of the interference gener-ated by the ith user is

yi [n] =√

ε1

Ns

∫

ri (t−△i) · b1,n · w(t− (n− 1)Tf − c1,nTc)dt (3.13)

We denote by yi (with i 6= 1) the sampled interference generated by the ith inter-ferer and observed at the output of the matched filter (i.e., the output of the pulse bypulse correlator). y1 is the useful signal from the user i = 1, the number of interferersis Nu − 1 and v is the noise. One could evaluate E [ζ2] as

40


Figure 3.9 Scholtz’s w(t) and its correlation for τ = 0.5ns, Tc = 2 and a duration of 2ns

E

(

Nu∑

i=2

yi + v

)2

= E

(

Nu∑

i=2

yi

)2

+ E[

v2]

+ E

[

2vNu∑

i=2

yi

]

= E

(

Nu∑

i=2

yi

)2

+ E[

v2]

+ 2E

[

Nu∑

i=2

yi

]

E [v] , (3.14)

due to statistical independence of the MUI samples yi with the noise samples v.On the other hand, one could evaluate the fourth moment E [ζ4] as

E

(

Nu∑

i=2

yi + v

)4

= E

(

Nu∑

i=2

yi

)4

+ E[

v4]

+ 4E [v] E

(

Nu∑

i=2

yi

)3

+

4E[

v3]

E

[(

Nu∑

i=2

yi

)]

+ 6E[

v2]

E

(

Nu∑

i=2

yi

)2

, (3.15)

under the same statistical independence assumption as in (3.14).Taking into account E[yi] = 0 and E[v] = 0, the equations (3.14) and (3.15)

become:

E

(

Nu∑

i=2

yi + v

)2

= E

(

Nu∑

i=2

yi

)2

+ E[

v2]

(3.16)

41


and

E

(

Nu∑

i=2

yi + v

)4

= E

(

Nu∑

i=2

yi

)4

+ E[

v4]

+ 6E[

v2]

E

(

Nu∑

i=2

yi

)2

(3.17)

Due to the statistical independence between different interferers, and taking intoaccount again that E [yi] = 0 we find that:

E

(

Nu∑

i=2

yi

)2

=Nu∑

i=2

(

E[

y2i

])

(3.18)

and

E

(

Nu∑

i=2

yi

)4

=Nu∑

i=2

(

E[

y4i

])

+ 6∑

i<j

(

E[

y2i

]

E[

y2j

])

. (3.19)

Inserting the equations (3.18) and (3.19) in the equations (3.16) and (3.17) re-spectively, we find:

E

(

Nu∑

i=2

yi + v

)2

=Nu∑

i=2

(

E[

y2i

])

+ E[

v2]

(3.20)

and

E

(

Nu∑

i=2

yi + v

)4

=Nu∑

i=2

(

E[

y4i

])

+ 6∑

i<j

(

E[

y2i

]

E[

y2j

])

+ E[

v4]

+

6E[

v2]

Nu∑

i=2

(

E[

y2i

])

. (3.21)

3.3.2 Perfect Power Control Scenario

In a perfect power control scenario, if all the users use the same pulse shapew(t) and equal symbol periods Ts, an immediate result is that all the interferencecontributions have the same variance and same fourth order moment. The equations(3.20) and (3.21) then become:

E

(

Nu∑

i=2

yi + v

)2

= (Nu − 1)(

E[

y2i

])

+ E[

v2]

(3.22)

and

E

(

Nu∑

i=2

yi + v

)4

= (Nu − 1)(

E[

y4i

])

+ 6(Nu − 1) (Nu − 2)

2

(

E[

y2i

]

E[

y2j

])

+

E[

v4]

+ 6 (Nu − 1) E[

v2] (

E[

y2i

])

. (3.23)

42


Denoting by ηp,2 = E [v2] the variance of the noise and by ηp,4 = E [v4] the fourthorder moment of the noise and presuming that we have the same shape of the pulsew(t) for every user i, one has

E[

y2i

]

= εRxm2(w)

Nh

= PRxNhTcm2(w)

Nh

= PRxTcm2(w) (3.24)

where εRx is the received energy per pulse (or PRx the received power), Ts = NhTc isthe symbol period, Nh the number of chips (i.e., slots) in the symbol and

E[

y4i

]

= ε2Rx

m4(w)Nh

= P 2RxN2

hT 2c

m4(w)Nh

= P 2RxNhT 2

c m4(w). (3.25)

Using equations (3.5), (3.22) and (3.23) we find the kurtosis in perfect powercontrol (ppc) scenario

Kppc =E[

(

∑Nui=2 yi + v

)4]

[

E[

(

∑Nui=2 yi + v

)2]]2 − 3

=(Nu − 1) P 2

RxNhT 2c m4(w) + 3 (Nu − 1) (Nu − 2) P 2

RxT 2c m2

2(w)

[(Nu − 1) PRxTcm2(w) + ηp,2]2 +

ηp,4 + 6 (Nu − 1) ηp,2PRxTcm2(w)

[(Nu − 1) PRxTcm2(w) + ηp,2]2 − 3, (3.26)

which simplified becomes

Kppc =(Nu − 1) Nhm4(w) + 3 (Nu − 1) (Nu − 2) m2

2(w)[

(Nu − 1) m2(w) + ηp,2

PRxTc

]2 +

ηp,4

P 2Rx

T 2c

+ 6 (Nu − 1) ηp,2m2(w)PRxTc

[

(Nu − 1) m2(w) + ηp,2

PRxTc

]2 − 3. (3.27)

3.3.3 Lack of Power Control Scenario

In the case of the lack of power control (lpc) scenarios, the energy received fromthe interferer i is no longer εRx, but instead εRx,i. Therefore, E [y2

i ] = PRx,iTcm2(w)(as in equation (3.24)) and:

∑

i<j

(

E[

y2i

]

E[

y2j

])

=∑

i<j

(

PRx,iPRx,jT2c m2

2(w))

=

(

∑Nui=2 PRx,i

)2 −∑Nui=2 P 2

Rx,i

2T 2

c m22(w), (3.28)

Nu∑

i=2

E[

y2i

]

= Tc

Nu∑

i=2

PRx,im2(w), (3.29)

43


andNu∑

i=2

E[

y4i

]

= NhT 2c

Nu∑

i=2

P 2Rx,im4(w). (3.30)

Further, (3.20) and (3.21) become

E

(

Nu∑

i=2

yi + v

)2

= Tc

Nu∑

i=2

PRx,im2(w) + E[

v2]

(3.31)

and

E

(

Nu∑

i=2

yi + v

)4

= NhT 2c

Nu∑

i=2

P 2Rx,im4(w) + (3.32)

6

(

∑Nui=2 PRx,i

)2 −∑Nui=2 P 2

Rx,i

2T 2

c m22(w) +

E[

v4]

+ 6E[

v2]

Tc

Nu∑

i=2

PRx,im2(w).

Denoting by Pm =∑Nu

i=2PRx,i

Nu−1the average power and Pm,2 =

∑Nu

i=2P 2

Rx,i

Nu−1the second

order moment of the random power, we get

E

(

Nu∑

i=2

yi + v

)2

= TcPm (Nu − 1) m2(w) + ηp,2 (3.33)

and

E

(

Nu∑

i=2

yi + v

)4

= NhT 2c Pm,2 (Nu − 1) m4(w) +

3(

P 2m (Nu − 1)2 − Pm,2 (Nu − 1)

)

T 2c m2

2(w) +

ηp,4 + 6ηp,2TcPm (Nu − 1) m2(w) (3.34)

and therefore the kurtosis in the lack of power control (lpc) scenario is

Klpc =E[

(

∑Nui=2 yi + v

)4]

[

E[

(

∑Nui=2 yi + v

)2]]2 − 3

=NhT 2

c Pm,2 (Nu − 1) m4(w)

[TcPm (Nu − 1) m2(w) + ηp,2]2 +

3(

P 2m (Nu − 1)2 − Pm,2 (Nu − 1)

)

T 2c m2

2(w)

[TcPm (Nu − 1) m2(w) + ηp,2]2 +

ηp,4 + 6ηp,2TcPm (Nu − 1) m2(w)

[TcPm (Nu − 1) m2(w) + ηp,2]2 − 3. (3.35)

44

3.4 Channel Capacity Computation Using a Generalized Gaussian Approximation

Note that the Kppc and Klpc kurtosis expressions are used further to characterizethe distribution of the interferences. Therefore, equations (3.27) and (3.35) show usthe dependency between the system’s parameters and the shape of the MUI distribu-tion.

3.4 Channel Capacity Computation Using a Gene-

ralized Gaussian Approximation

In order to compute the channel capacity, one needs to know both the β parameter(of the generalized Gaussian distribution which models the interference plus noise)and the Signal to Interference plus Noise Ratio (SINR). We recall that the β parameteris strongly linked to the kurtosis computed in the previous sections. Using the resultsfrom the previous section we identify two important scenarios: negligible noise andnon-negligible noise relative to MUI.

3.4.1 Kurtosis Expressions

1. Negligible Noise Relative to MUIIn the perfect power control case the kurtosis becomes (see equation (3.36))

Kppc =

[

(Nu − 1)Nhm4(w) + 6 (Nu−1)(Nu−2)2

m22(w)

]

[(Nu − 1)m2(w)]2− 3 (3.36)

with m4(w) = 1Tc

∫∞−∞ γw(z)4dz, m2(w) = 1

Tc

∫∞−∞ γw(z)2dz, and where γw(z) =

∫∞−∞ w(t+z)w(t)dt. This expression was already known in [47], we have extended

it to the cases of non negligible noise.In the lack of power control case, the kurtosis becomes (see equation (3.37))

Klpc =Pm,2Nhm4(w)

P 2m (Nu − 1) m2

2(w)− 3 · Pm,2

(Nu − 1) P 2m

(3.37)

with Pm = 1Nu−1

∑i=Nui=2 Pi = E [Pi], Pm,2 = 1

Nu−1

∑i=Nui=2 P 2

i = E [P 2i ], where Pi

is the power of the signal received from the interfering user i.

2. Non Negligible Noise Relative to MUIA priori, the existence of noise should decrease both kurtosis and SINR. Wehave computed the relationship linking the kurtosis of the MUI plus Noise tothe parameters of the signal, in the perfect power control case (see equation(3.38)) or in the lack of power control case (see equation (3.39)):

45


Kppc =

[

(Nu − 1)Nhm4(w) + 6 (Nu−1)(Nu−2)2

m22(w)

]

[

(Nu − 1)m2(w) + ηp,2(w)

PRx·Tc

]2 +

[

ηp,4(w)

P 2Rx

·T 2c

+ 6(Nu − 1)m2(w)ηp,2(w)

PRx·Tc

]

[

(Nu − 1)m2(w) + ηp,2(w)

PRx·Tc

]2 − 3 (3.38)

where PRx is the received power before the correlator in the presence of (Nu−1)UWB interfering users, ηp,2(w) and ηp,4(w) are the variance and the fourthorder moment of the contributions of the possible non UWB interferers plus theGaussian noise at the output of the correlator receiver adapted to w(t). If thereis only Gaussian noise, one can find that ηp,4(w) = 3η2

p,2(w).

Klpc =

[

Pm,2

P 2m

Nh

(Nu−1)m4(w) + 3

(

1− Pm,2

(Nu−1)P 2m

)

m22(w)

]

[

m22(w) + 2 ηp,2(w)

Tc(Nu−1)Pmm2(w) +

(

ηp,2(w)

Tc(Nu−1)Pm

)2] +

[

ηp,4(w)

T 2c (Nu−1)2P 2

m+ 6 ηp,2(w)

Tc(Nu−1)Pmm2(w)

]

[

m22(w) + 2 ηp,2

Tc(Nu−1)Pmm2(w) +

(

ηp,2(w)

Tc(Nu−1)Pm

)2] − 3. (3.39)

One can see that the kurtosis of the MUI depends not only on Nu, Nh and Tc

but also on the shape w(t) of the received impulse. Please recall that from (3.7)we already know that kurtosis K = F (β) is a monotone function of β (and is zerofor a Gaussian distribution i.e., when β = 0). Therefore, the coefficient β of thegeneralized Gaussian probability density function depends on the shape w(t) and asa consequence the capacity of the communication system [36, 40] also depends onw(t).

As it can be seen in equation (3.38), K is a function of Nh, Nu, m2(w) and m4(w).When Nu ≈ Nh and the noise is small, the kurtosis Kppc ≈ m4(w)

m22(w)

and depends onlyon the parameters τ and Tc of the pulse (see equation (3.40)) and more exactly onthe ratio τ

Tc(see also Fig. 3.10).

Kppc |Nu=Nh≈ [m4(w) + 3m2

2(w)]

[m2(w)]2− 3 =

m4(w)m2

2(w)(3.40)

Fig. 3.10 shows that for low τ/Tc values the β parameter of the generalized Gaus-sian is high (i.e., far away from the Gaussian interference model) while for high τ/Tc

values the β parameter tends to zero (i.e., closer to the Gaussian interference model).When τ/Tc is small the signal is more impulsive, so there is a relationship between

the impulsiveness of the signal and the kurtosis. This is one of the reasons for whichwe refer to ‘impulsiveness’ or ‘impulsivity’ when we are talking about maximizing thecapacity in Chapter 4. It has been also shown in [50] that it is due to the impulsiveaspect of IR signal that the SGA is not valid (contrary to the classical DS-CDMAwhich is a continuous waveform and where the SGA is valid).

46


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

τ/Tc

β (

m4(w

)/m

22(w

))

Figure 3.10 β = F−1 (K) vs. τTc

for a Scholtz Pulse

3.4.2 SINR Expressions

To compute the capacity we need β (linked to the expressions for kurtosis - equa-tions (3.38) and (3.39)) and the expressions for Signal to Interference Noise Ratio(SINR) (equations (3.41) and (3.42)). The SINR can be developed in perfect powercontrol and lack of power control scenarios as [133, 135]:

SINRppc =NhTcPRx

ηp,2(w) + Tcm2(w)(Nu − 1)PRx

, (3.41)

and

SINRlpc =NhTc

PRx

Pm

ηp,2(w)

Pm+ Tc(Nu − 1)m2(w)

. (3.42)

As it can be seen in the equation (3.41), SINR is a function of Nh, Nu and m2(w).When Nu ≈ Nh and in low noise environments, SINRppc ≈ 1

m2(w)and depends only

on the parameters τ and Tc of the pulse and more exactly on τTc

(see Fig. 3.11).Fig. 3.11 shows that m2(w) increases linearly with τ/Tc. This means that the

SINR decreases when the duty cycle of the pulse used is higher (or SINR increaseswhen τ/Tc is lower). As in the Section 3.4.1, here as well one can easily notice theimpact of the τ/Tc ratio.

Please note that the equations (3.41) and (3.42) are commonly used to describethe PAM Signal to Interference Noise Ratio. For PPM there is a light modification

47


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

τ/Tc

m2(w

)

Figure 3.11 m2(w) vs. τTc

for a Scholtz Pulse

due to the different template w (t)− w (t− δP P M) used in the correlator:

SINRppc,P P M =NhTcPRx (

∫

w (t) · (w (t)− w (t− δP P M)) dt)2

ηp,2 (w) + Tcm2(w) (Nu − 1) PRx

, (3.43)

and

SINRlpc,P P M =NhTc

PRx

Pm(∫

w (t) · (w (t)− w (t− δP P M)) dt)2

ηp,2(w)

Pm+ Tc (Nu − 1) m2 (w)

. (3.44)

However, knowing that for the unit-energy waveform w (t) one has∫

w (t)2 dt = 1,in the case of orthogonal PPM pulses (i.e., when δP P M ≥ Tp) it can be easily showedthat ∫

w (t) · w (t− δP P M) dt = 0, (3.45)

and the equations (3.43) and (3.44) have similar expressions with (3.41) and (3.42)respectively.

48


3.4.3 The Capacity of a Discrete Memoryless Channel withContinuous-Valued Output

The capacity C of a discrete memoryless channel is defined as the largest averagemutual information that can be transmitted over the channel in one use, maximizedover all input probability assignments [62]:

C = maxP (0)...P (N−1)

∑

k,j

P (k)P (j|k) log2

{

P (j|k)∑

P (i)P (j|i)

}

(3.46)

where P (k) is the probability of using integer k, with∑

P (k) = 1, and P (j|k) is theprobability of receiving integer j given that integer k is the channel input.

The capacity of a discrete memoryless channel can be extended to the case ofcontinuous-valued output [175]. The generic expression of the capacity of a discretememoryless channel with continuous-valued output is

C = maxP (0)...P (N−1)

N−1∑

k=0

P (k)∫ ∞

−∞p (z|sk) log2

{

p (z|sk)∑N−1

i=0 P (i)p (z|si)

}

dz (3.47)

where z is the channel output, s0 . . . sN−1 are values that can be taken by the trans-mitted symbols and P (k) the a priori probability associated with sk. In our case p isthe generalized Gaussian probability density function, s0 = −1, s1 = 1, N = 2 thenumber of possible values and P (0) = P (1) = 1/2. For IR-TH-PAM, the capacity(bits/channel use) of a discrete memoryless channel with continuous-valued outputbecomes

C =12

∫ +∞

−∞Q(1− log2Q)dz +

12

∫ +∞

−∞p(z| − 1)log2p(z| − 1)dz +

12

∫ +∞

−∞p(z|+ 1)log2p(z|+ 1)dz (3.48)

where Q = p(z| − 1) + p(z|+ 1), z is the channel output and ∓1 are the transmittedsymbols. Such a capacity may be evaluated by Monte Carlo averaging using a gene-ralized Gaussian random number generator with variance 1/SINR and coefficient β(see Fig. 3.12). We recall that in our scenarios SINR is computed with the equationsfrom Section 3.4.2, while β is obtained from the kurtosis computed in Section 3.4.1.

The simplified scheme from Fig. 3.12 can be used to generate performance graphsas in Fig. 3.13 or to compute the communication throughput (as will be defined andused in the next chapter).

49


Figure 3.12 Simplified Capacity Computation Scheme (the equations are valid for perfectpower control scenarios with negligible noise but the scheme is expandable toother scenarios as well)

−10

0

10

20

30

−10

12

34

50

0.2

0.4

0.6

0.8

1

SINR [dB]β

Cap

acit

y [

bit

s/ch

annel

use

]

Low Capacity Region

High Capacity Region

Figure 3.13 Capacity(SINR,β)

50

Chapter 4

The Effect of Tuning Impulsivenesson Throughput of IR-UWBCommunications

Chapter Overview

This chapter considers a generalized Gaussian MUI distribution and applies theperformance criterion developed in Chapter 3 for different applications. While inSection 4.3 the network is structured as a group of interconnected cells that are notsynchronized with each other, in Section 4.4 we deal with two unsynchronized com-munication links, in Section 4.5 we deal with double and single-cell scenarios, andin Section 4.6 we deal with unsynchronized single-cell scenarios. In the applicationspreviously described we show that network throughput can be maximized by control-ling the impulsiveness. The finding has been validated through simulations, on smallscale network model (e.g., two interfering uplink cells, two interfering connections,one uplink unsynchronized single-cell, etc.).

While Section 4.5 shows how a cognitive network which adapts its waveform tointerference can benefit from sensing its impulsiveness, in the application from Section4.6 we have presented a method to compare and to optimize IR-UWB antennas takinginto account the impact of antenna’s time domain response over the capacity of thechannel (which can be linked to throughput). For this last section we have usedagain the performance criterion developed in Chapter 3, where the pulse shape w(t)is replaced by antenna’s time domain response.

4.1 Introduction

This chapter exploits the fact that in IR-TH-UWB the MUI is not Gaussian, aspresented in Chapter 3. In the case of Gaussian MUI, the capacity may be read onFig. 3.13, taking β = 0. Therefore, if MUI would have been Gaussian, the capacitywould have been a function depending only on the SINR, whereas in non-GaussianMUI, the capacity depends on both SINR and β.

51

4.2 Tuning the Impulsiveness

In this chapter we would like to exploit the fact that MUI is not Gaussian andtherefore, for a fixed SINR, one might increase the capacity of the communicationby tuning the impulsiveness of the interference (e.g., increasing β - and implicitly thekurtosis).

4.2 Tuning the Impulsiveness

The impulsiveness of the signal is strictly related to its kurtosis. One can increasethe capacity through the increase of the kurtosis and therefore by tuning at least oneof the following parameters:

1. number of slots in a frame (Nh)

2. number of users (Nu)

3. slot length (Tc)

4. pulse spreading (i.e., using multiple smaller replicas of the w(t) pulse)

5. pulse length (Tp)

6. pulse shape (e.g., changing the derivative order of w(t), using a set of pulses, orusing a different antenna).

The 1st method, or tuning the impulsiveness through the number of slots in a frame(i.e., Nh) makes the object of a multi-cell scenario in Section 4.3. It is important tomention that tuning Nh will change both the kurtosis and the SINR. Also, increasingthe capacity per channel use in the detriment of increasing Nh is not necessary agood option, because the transmission time increases and the transmission efficiencydecreases. For all these reasons we need to evaluate the throughput as defined inSection 4.3.1, in order to determine the impact of tuning the impulsiveness. Pleasealso note that the 2nd method, based on tuning the number of users Nu, reduces toload balancing.

While the first three methods are system-related, the last three methods are basedon tuning the physical parameters. The 4th method is used on a double-link com-munication scenario in Section 4.4. What is interesting is that the capacity of thesystem can be increased even if we keep the energy of the novel pulse shape equal tothe original one as in [63, 66].

In Section 4.5 we have used the 6th method in a Cognitive Network where theaim is to optimize the consumed energy necessary for synchronization. The Cogni-tive Networks are by definition [12] networks employing conscious mechanisms in theanalysis process that is used by nodes for determining whether changes in the globalnetwork state are appropriate in order to optimize the communication. In the def-inition introduced by Benedetto [13], the radio devices operating under UWB rulesmust coexist with severely interfered environments and must control their behaviorin order to favor coexistence. In Section 4.5, we consider that a cognitive node reactsto interference by changing the waveform w(t) with the waveform from a known set(i.e., derivative orders of w(t)). On the other hand, in Section 4.6 we suppose thatw(t) incorporates the effect of an IR-UWB antenna and we present an application for

52

4.3 Application in Double-Cell Scenario

IR-UWB antenna testing. In this section we show the impact of the antenna over thecommunication capacity and we define a new criterion for antenna design.


Note that the double-cell scenario in uplink mode is a simplified case of a networkscenario, as presented in Section 2.3. On the basis of equation (3.39), we may computethe kurtosis of the MUI y plus noise v at the base station j as a function of the signalparameters:

Kj =

[

Pm,i,2

P 2m,i

Nh,i

Nu,im4(w) + 3

(

1− Pm,i,2

Nu,iP 2m,i

)

m22(w)

]

[

m22(w) + 2 ηp,2(w)

TcNu,iPm,im2(w) +

(

ηp,2(w)

TcNu,iPm,i

)2] +

[

ηp,4(w)

T 2c N2

u,iP 2

m,i

+ 6 ηp,2(w)

TcNu,iPm,im2(w)

]

[

m22(w) + 2 ηp,2(w)

TcNu,iPm,im2(w) +

(

ηp,2(w)

TcNu,iPm,i

)2] − 3. (4.1)

There are Nu,i interferences coming from the interfering cell i and

Pm,i =1

Nu,i

k=Nu,i∑

k=1

Pi,k (4.2)

and

Pm,i,2 =1

Nu,i

k=Nu,i∑

k=1

P 2i,k, (4.3)

where Pi,k is the power of the signal received from the interfering user k. The numberof slots per frame in the interfering cell is Nh,i.

Based on the equation (3.42), the Signal to Interference plus Noise Ratio (SINR)at the base station of cell j with an interfering cell i may be expressed as [135]

SINRj =Nh,jTc

Pu,j

Pm,i

ηp,2(w)

Pm,i+ TcNu,im2(w)

(4.4)

where Pu,j is the power of the useful received signal for each user of cell j (we mayconsider that, as a result of power control, all the useful signals have the same powerat reception).

The β parameter of the generalized Gaussian is linked to the kurtosis K =E[(y+v)4]E[(y+v)2]2

− 3 of the MUI plus noise as we have seen in Chapter 3. The dependenceK = F(β) is graphically represented on Fig. 3.8.

To estimate the performance of such communications we look at the capacity inbits per channel use of a single link (see equation (3.48)) and then we compute thetotal throughput (as further presented in equation (4.6)).

53


4.3.1 Novel Metrics Based on the Communication Capacity

Using expression (3.48) to compute the capacity Ci for an uplink communicationof a user in cell i (see result on Fig. 3.13), the achievable bit rate for a user in cell ican be expressed as

Ri(bits/s) =Ci

Nh,iTc

. (4.5)

The total throughput of the system considering both cells can be computed as

Rtotal(bits/s) =∑

i=I,II

Nu,iRi = Nu,ICI

Nh,ITc

+ Nu,IICII

Nh,IITc

. (4.6)

Another statistic that one might wish to compute is the fairness in terms ofthroughput. The Jain’s fairness index [103] is a common criterion used in many net-working papers such as [161]. The difference from [103, 161] is that we use the fairnessin a different context: herein the fairness is computed in terms of fair throughput percell and not per user as in [103, 161].

We thus redefine the fairness by

Fairness =(Rtotal,I + Rtotal,II)2

2(

R2total,I + R2

total,II

) (4.7)

and in our case it may be expressed as

Fairness =

(

Nu,ICI

Nh,ITc+ Nu,IICII

Nh,IITc

)2

2[

(

Nu,ICI

Nh,ITc

)2+(

Nu,IICII

Nh,IITc

)2] . (4.8)

Please note that when the fairness is 1, the balance of the throughput betweenthe two cells is fair.

It is important to mention that tuning Nh will change both the kurtosis and theSINR. Also, increasing the capacity in the detriment of increasing Nh is not necessarya good option, because the transmission time increases and the transmission efficiencydecreases. For all these reasons we need to evaluate the throughput to determine theimpact of tuning the impulsiveness.

4.3.2 Tuning the Impulsiveness of the Signal through Nh,I

and Nh,II

In order to achieve the best total throughput, we may tune the parameters Nh,I

and Nh,II . As a matter of fact, they have an impact on the system performance asseen in (4.1), (4.4) and (4.6) (changing the interference plus noise, kurtosis changesthe β parameter of their distribution and so the achievable rate, see Fig. 3.13). Theparameters Nh,I and Nh,II represent the number of slots per frame in cells I and II.Each user sends one impulse per frame, so the longer the frame is the more impulsive

54


the signal appears (for the same power there will be strong sparse impulses insteadof more continuous small amplitude impulses).

Consider two groups of users (which for instance may be sensors using UWB),one group connected to the base station BSI with Nh,I number of slots and the othergroup connected to the base station BSII with Nh,II slots. The necessary conditionsfor perfect synchronization of the users inside each cell, through time hopping codesare: Nh,I ≥ Nu,I , Nh,II ≥ Nu,II and a pulse shape not longer than the time slot Tc.We use the classical Scholtz’s pulse (see equation (2.2)) used in many UWB studies[180], which is the second derivative of a Gaussian pulse, so the condition translatesinto 2τ ≤ Tc.

We also consider the first group to be perfectly synchronized to the BSI , and thesecond group perfectly synchronized to the BSII . From the synchronization’s pointof view even a small distance between BSI and BSII may have a huge impact overthe synchronization and orthogonality of the users of one cell to the sink position ofthe other cell.

Given Nu,I and Nu,II we would like to find Nh,I and Nh,II for which the totalthroughput is maximized. This problem can be written as

maxNh,I ,Nh,II

{

Nu,I ·CI

Nh,ITc

+ Nu,II ·CII

Nh,IITc

}

(4.9)

with the constraints Nh,I ≥ Nu,I , Nh,II ≥ Nu,II . Please note that CI and CII dependalso on Nh,I and Nh,II , interference power and noise power as explained in Section3.4. As explained further in Section 4.3.3, we chose a scenario where the transmissionis at fixed SIR. Therefore, while we increase the frame length in the neighbor cell,the interference power remains the same and thus the SINR is not affected by thechange in the frame length of the interfering cells, only the interference distribution isaffected and has an impact on the communication performance of the interfered cell.

4.3.3 Numerical Results

We recall that the capacity depends on the kurtosis and on the SINR. Thekurtosis of the interference plus noise may be different for both cells. In cell I thekurtosis KI depends on Nh,II , Nu,II , and on the ratios TcPm,II

ηp,2(w)= INR1 and Pm,II,2

P 2m,II

(see

(4.1)). The ratio Pm,II,2

P 2m,II

depends on the geographical distribution of the interferers

around the base station. For instance, if all interferers did transmit with the samepower and were uniformly distributed between two circles of radii 1 meter and 10meters around the base station, the ratio would be 4.61.

In cell I the SINR depends on Nh,I , Nu,II , and on the ratios TcPm,II

ηp,2(w)= INR1 and

Pu,I

Pm,II= SIR1 (it may also be rewritten as a function depending on TcPu,I

ηp,2(w)= SNR1

and SIR1) (see (4.4)).We may notice two different cases:

High Capacity Region Case We notice that in high capacity region (for highSINR for example), the capacity changes slightly with the parameter β, and thus

55


with the kurtosis K, remaining almost constant. As Tc is considered to be fixed, themaximization problem from (4.9) becomes

maxNh,I ,Nh,II

{

Nu,I

Nh,I

+Nu,II

Nh,II

}

(4.10)

with the constraints Nh,I ≥ Nu,I , Nh,II ≥ Nu,II .The solution of this simple convex optimization problem is Nh,I = Nu,I and Nh,II =

Nu,II (see Fig. 4.3 or Fig. 4.4).

Low Capacity Region Case In low capacity region we cannot make the samepresumptions for (4.9), as the capacity C is increasing with Nh.

We have computed the total throughput as a function of Nh,I and Nh,II , usingthe equation (4.6). In (4.6), the link between Nh,I , Nh,II and CI , CII is given by(3.48), (4.1), and (4.4). We find that the solutions Nh,I and Nh,II maximizing thetotal throughput might be divided into three cases (see Fig. 4.1 or Fig. 4.2):

a) Nh,I = Nu,I and Nh,II = Nu,II

b) Nh,I →∞ and Nh,II = Nu,II ,c) Nh,I = Nu,I and Nh,II →∞.In the last two cases we may fix a maximal length for Nh (otherwise the bit rate

will tend to zero for the cell with Nh →∞).The solution between a), b) and c) depends on the ratios SIRi (and SNRi), as

seen for example in Fig. 4.1. Such results corroborate with what has been studied inthe different scenario of CDMA cells, with a different approach in [64, 65].

We chose a scenario where the transmission is at fixed power in order to exploitthe maximum of what is allowed by the regulation (another scenario would have beenfor example to keep the peak power constant and so the energy per pulse). Therefore,while we increase the frame length the interference power remains the same and onlythe interference distribution is affected and has an impact on the communicationperformance.

In our computation of (4.6), where the capacity C has been calculated as explainedin Chapter 3 by Monte-Carlo averaging, we have considered the IR-UWB parametersTc = 1ns and τs = 0.5ns. The two cells are symmetrical with Nu,I = Nu,II = 50each, SIR1 = SIR2 = SIR and SNR1 = SNR2 = SNR. In Fig. 4.1, 4.3, 4.4 isrepresented the total throughput Rtotal in different cases of SIR and SNR. Fig. 4.2is a zoom on Fig. 4.1. For low SIR (see Fig. 4.1) the total throughput is maximizedfor b) or c), while for high SIR (see Fig. 4.3 and 4.4) the maximization occurs for a).Zooming on Fig. 4.1, one can see in Fig. 4.2 that the total throughput for the pair(Nh,I = 550, Nh,II = Nu,II = 50) is the same as for the central peak region where(Nh,I = Nu,I = 50, Nh,II = Nu,II = 50).

Last but not least, the SNR impact over the total throughput is less than theimpact of the SIR, but in Fig. 4.4 one can see that the high SNR has increased thepeak in the central region with around 2 · 108 bits/sec relative to the case of low SNR- Fig. 4.3.

56


0

500

1000

1500

0500

10001500

0

1

2

3

4

5

6

x 108

Nh,IIN

h,I

Tota

l T

hro

ughput

[bit

s/se

c]

Figure 4.1 Time slotting compromise with Nu,I = 50, Nu,II = 50, Low SIR case,SIR = −25dB. The points corresponding to the pairs (Nh,I → ∞,Nh,II =Nu,II) and (Nh,I = Nu,I ,Nh,II → ∞) are higher than the central peak(Nh,I = Nu,I ,Nh,II = Nu,II)

100200

300400

500

50 100 150 200 250 300 350 400 450 500 550

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

x 108

Nh,II

Nh,I

Tota

l T

hro

ughput

[bit

s/se

c]

Figure 4.2 Zoom on Low SIR case, Nu,I = 50, Nu,II = 50, SIR = −25dB

57


0

500

1000

15000

5001000

1500

0

2

4

6

8

10

12

14

16

18

x 108

Nh,IIN

h,I

Tota

l T

hro

ughput

[bit

s/se

c]

Figure 4.3 Time slotting compromise with Nu,I = 50, Nu,II = 50, High SIR and LowSNR case, SIR = 0dB, SNR = −10dB. The peak corresponds to the pair(Nh,I = Nu,I ,Nh,II = Nu,II)

0

500

1000

15000

5001000

1500

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x 109

Nh,II

Nh,I

Tota

l T

hro

ughput

[bit

s/se

c]

Figure 4.4 Time slotting compromise with Nu,I = 50, Nu,II = 50, High SIR and HighSNR case, SIR = 0dB, SNR = 25dB. The peak corresponds to the pair(Nh,I = Nu,I ,Nh,II = Nu,II)

58


4.3.4 Conclusions

Section 4.3 considers TH-IR-UWB wireless sensor network with two co-locatedcells, each around a sink that acts as data collector. These cells are not synchronizedand thus they interfere and their inter-cell interference is not Gaussian, but generalizedGaussian. Considering uplink communications, we have shown how the throughputchanges as a function of the system parameters. The signal parameters of each cellhave an impact on the non-gaussianity (impulsiveness) of its interference over theother cell. The change in inter-cell interference distribution has an impact on thethroughput. Specifically, in the low signal to interference ratio case, it is better toincrease the impulsiveness of one of the cell by increasing the frame length Nh, whiledecreasing the impulsiveness of the signals of the other cell by taking Nh to be thesmallest value needed to keep the orthogonality inside a cell (e.g., Nh = Nu). On theother hand, at the high signal to interference ratio case, it is better to keep the lowestimpulsiveness in both cells by making Nh equal to the Nu number of users in eachcell.

However, in both cases (i.e., low or high signal to interference) maximizing through-put will impact on the fairness of the network. It can be graphically checked (seeFig. 4.5) that for the example presented in the previous subsection with Nu,I =50, Nu,II = 50, fairness is maximized for Nh,I = Nh,II (corresponding to the medianline in Fig. 4.5). Therefore, we conclude that maximizing throughput will not neces-sary maximize fairness, but fairness can still be achieved by alternatively switchingthe parameters of both cells (e.g., for a period of time we maximize throughput byincreasing the number of slots only in the second cell, and during another period oftime we maximize throughput by increasing the number of slots only in the first cell).

100200

300400

500

100

200

300

400

500

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Nh,II

Nh,I

Fai

rnes

s

50

Figure 4.5 Fairness in terms of Nh,I and Nh,II . The median line corresponds to themaximum fairness

59

4.4 Application in Double-Link Communications

Note that the capacity and therefore the throughput in a network could also beincreased by load balancing (i.e., moving the users from one cell to another). Theequation for throughput maximization by load balancing could be translated into

maxNu,I ,Nu,II

{

Nu,I ·CI

Nh,ITc

+ Nu,II ·CII

Nh,IITc

}

(4.11)

with the constraints Nh,I ≥ Nu,I , Nh,II ≥ Nu,II .


A double-link communication is a simplified case of a coordinated scenario, aspresented in Section 2.3. As mentioned, in this section we propose a system composedof 2 pairs of connections as in [63, 66]: Tx1 transmits to Rx1 and, at the same time,Tx2 transmits to Rx2. Tx1 will produce some interference to Rx2 and Tx2 will inducesome interference to Rx1. We consider a frame of Nh = 24 slots, Tc = 1ns, τ = 0.5nsand initially both pairs use each a single pulse of length 1ns. After that, we beginto vary the impulsiveness of each transmission by keeping the energy constant. Inthis section, we choose to change the impulsiveness (and therefore the kurtosis) byincreasing the number of pulse replicas from 1 to Nh on one of the links, while keepinga fixed number of pulses (e.g., 1) on the other one.

Using more pulse replicas is an intermediate signaling scheme between traditionalIR-UWB and DS-UWB (or CDMA). The impact of increasing the number of pulsesinside a frame, by keeping the energy constant, decreases the impulsiveness and there-fore the kurtosis as we have checked it by computing the capacity from Fig. 4.6. Aswe have found in Fig. 4.6, from the point of view of the capacity, decreasing theimpulsiveness leads to one of the following two situations:

– decreasing the kurtosis keeping SINR constant might increase the capacity (HighCapacity Region) - see Fig. 4.6

– decreasing the kurtosis keeping SINR constant might decrease the capacity (LowCapacity Region) - see Fig. 4.6.

Instead of using a pulse w(t) we now use a number of smaller replicas of the samepulse, but with the same energy, and we need to reevaluate the capacity. To computethe kurtosis we might use one of the two methods:

1. The Tx1−Rx1 link uses the w1(t) pulse and the Tx2−Rx2 link uses the w2(t)pulse, where w1(t) and w2(t) are spread versions of w(t). We therefore computem2(w) (defined by (3.9)) and m4(w) (defined by defined by (3.10)) where γ hasbeen replaced by γw12(z) =

∫∞−∞ w1(t + z)w2(t)dt (PAM).

2. For the particular scenarios of perfect power control, when a link uses a singlepulse of energy ε but the other uses n pulses each of energy ε/n, one can find thatthe ratio m4(w)/m2

2(w) is exactly n times smaller than in the scenario whereboth links are sharing a single pulse of energy ε. This observation simplifiesthe kurtosis computation. In perfect power control scenario, with low noise, thenth times spreading of the pulse is equivalent to n interferers each having theenergy ε/n, and the same total energy ε (see Fig. 4.7).

60


Figure 4.6 Tuning impulsiveness by spreading the transmitted pulse: High Capacity Sce-nario and Low Capacity Scenario

Figure 4.7 Equivalence model and comparison between two scenarios with same energy εbut different kurtosis

61


In Fig. 4.7 we have determined an equivalence between a system with 1 interfererusing 3 pulses of energy ε/3 and a system with 3 interferers using 1 pulse ofenergy ε/3. We find that the kurtosis 2 (i.e., K2) is different from the kurtosis 1(i.e., K1). Using the equivalence model presented above now is easy to evaluateK2 from equation (3.36).

We further use the first method to compute the kurtosis in a general context of alack of power control scenario (see equation (3.39)). For our simulations we use twosymmetrical links with SNR1 = SNR2 = SNR and INR1 = INR2 = INR, whereSNR = TcPRx,useful

ηp,2(w)and INR = TcPRx,interferer

ηp,2(w), with same Nh.

Further, we compute the total throughput of the double-link communication sce-nario using

Rtotal(bits/s) =C1 + C2

NhTc

, (4.12)

where C1 (in bits per channel use) is the capacity on the Tx1 − Rx1 link, while C2

(in bits per channel use) is the capacity on the Tx2 −Rx2 link.We find an interesting result: the total throughput may be increased (Fig. 4.8(b),

for high SNR) or decreased (Fig. 4.8(a), for low SNR) by increasing the number ofpulses (i.e., Spreading Factor SF ) on one of the Tx − Rx links. Please note thatincreasing the number of pulses inside a frame is not bandwidth spreading becausethe basic pulse has the same length as when we transmit with less pulses or a singlepulse per frame.

In Fig. 4.8(a) (lower SNR) the maximum throughput is achieved for (SF1 = 1,SF2 = 1), while in Fig. 4.8(b) (higher SNR) the maximum throughput is achievedfor (SF1 = 1, SF2 = 24) or (SF1 = 24, SF2 = 1). This result allows us to statethat in high SNR scenarios the spreading of the pulse is useful while in low SNRscenarios it is not.

62


5101520255 10 15 20 25

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

x 107

SF Link2SF Link

1

Tota

l T

hro

ughput

[bit

s/se

c]

(a)

010

2030 0 5 10 15 20 25

5

5.5

6

6.5

7

7.5

8

x 107

SF Link2

SF Link1

Tota

l T

hro

ughput

[bit

s/se

c]

(b)

Figure 4.8 Tuning the Spreading Factor of two links Tx1-Rx1 and Tx2-Rx2 with (a)INR = 10dB, SNR = −5dB and (b) INR = 10dB, SNR = 5dB

63

4.5 Application in Cognitive Networks


In [12], a cognitive UWB network has been defined. A cognitive UWB network isa network that changes the pulse shape from the transmitted signal in function of theinterference. In this cognitive radio system, sensing the interference is fundamental:it allows to estimate the performances and to change the systems parameters foran adaptation to the interference. The most classical interference estimation is themeasure of its power and of its spectral occupation. However, as we have alreadyseen in Chapter 3, the probability law of the random signal which represents theinterference is fundamental. We investigate here the impact of sensing the kurtosisof the interference on the performance of such a network.

We show here the effect of the non-Gaussianity of the IR-UWB interference onan IR-UWB system, however the same discussion can be done about the IR-UWBinterference on any other telecommunication system. The estimation of the effect ofIR-UWB interference on other telecommunication systems (for fixing the regulationon UWB deployment for instance) are usually done by computing the power of theUWB interference in the frequency band of the non-UWB system and assuming thatthis interference is Gaussian. But the allowed power limits fixed for the interferencein those cases do not corresponds to what is required when the interference is notGaussian: those power limits should be lower if the interference is not Gaussianand the receiver is adapted to Gaussian interference, or they should be higher if thereceiver is adapted to the non-Gaussianity of the interference.

4.5.1 Sensing the MUI

Sensing the kurtosis is fundamental in order to obtain good performances in IR-UWB. The interference plus noise may be sampled when the useful transmitter issilent or in the empty slots of an IR-UWB frame (except in the slots where thetransmitted pulse overlaps due to the multi-path channel, so a time guard dependingon the channel length has to be maintained after the slot used by the transmitterbefore to sense the interference plus noise).

Another way to estimate the kurtosis of the interference plus noise is to iterativelysubtract the estimated useful signal from the received signal (which contains bothinterference and useful signal). At each step the useful signal is estimated on thebase of the kurtosis using the maximum likelihood criterion, then the kurtosis of theremaining interference plus noise is updated (at first step one can take a kurtosis=0).

We may react to the interference and their non-Gaussianity by two means: byadapting the reception and by tuning the signal parameters having an impact on thoseinterferences. The next section show how w(t) may be tuned taking into account thisnon-Gaussianity of the interferences.

Part of this work has been supported by the system@tic project Urbanism and RadioCommu-nications (URC).

64


4.5.2 Tuning w(t) in the Cognitive Network

We consider a free space propagation channel, it is a reference scenario [11, 179]which allows us to underline the influence of the system parameters on performance.In the case of multi-path channel, the analytical dependency between the parame-ters and the MUI kurtosis will not have a simple formulation, however a cognitivenode may change the system parameters and learn, through sensing, the effect of theparameters changes on the kurtosis.

If we consider a first IR-UWB network using a UWB pulse shape w1(t), disturbedby another UWB network with Nu−1 users using a different UWB pulse shape w2(t)[12]. We compute the kurtosis of the interference caused by the second network onthe first one such as

K =

[

(Nu − 1)Nhk4(w) + 6 (Nu−1)(Nu−2)2

k22(w)

]

[(Nu − 1)k2(w))]2− 3 (4.13)

withk4(w) =

1Tc

∫ +∞

−∞γw12(z)4dz (4.14)

andk2(w) =

1Tc

∫ +∞

−∞γw12(z)2dz (4.15)

andγw12(z) =

∫ +∞

−∞w1(t + z)w2(t)dt. (4.16)

In this case we may find the capacity of the system in function of the two pulseshapes used in the networks.

In [72], a cognitive node (CNODE) is the receiver of uplink communications. Itsenses the noise plus interferences in order to estimate the SNR for various pulseshapes among a set. Its aim is to optimize the consumed energy necessary for syn-chronization by tuning the parameters (here the parameter is the pulse shape). Byusing a Rao Test for the detection [104], the synchronization requires the SNR to beabove a threshold level SNR0, corrected by a factor depending on the parameter β;this dependency on β shows that the synchronization performance depends not onlyon the SNR but also on the interference plus noise distribution. In perfect powercontrol, PRx being the power received from each users, in presence of (Nu − 1) UWBinterfering users, the condition may be written as

G · SNR = G · TfPRx

ηp,2(w) + Tcm2(w)(Nu − 1)PRx

≥ SNR0, (4.17)

where the correlator receiver is adapted to w(t), and

G =

[

4(1 + β)2

Γ(32(1 + β))Γ(3

2− 1

2β)

Γ2(12(1 + β))

]

(4.18)

65


is the detection gain (with the lowest gain G = 1, or G = 0dB, obtained when β = 0).The kurtosis K which serves to find β may be computed as equation (3.38) (perfectpower control).

The CNODE may find the pulse shape w(t) among a set of pulse shapes, minimiz-ing the power PRx necessary to satisfy condition 4.17. The minimum received powerto guaranty detection when using a pulse shape w(t) may be written

Pmin(w) =ηp,2(w)

Tf

(

G

SNR0

− Tc ·m2(w)(Nu − 1)Tf

)−1

. (4.19)

We have plotted the performance in term of required energy when the cognition(sensing and finding the best pulse shape w(t)) is applied taking into account or notthe non-Gaussianity. The set of pulses to be chosen was the six first derivatives of aGaussian pulse (see Fig. 4.9).

Figure 4.9 The first six derivatives of a Gaussian pulse

The Fig. 4.10 is issued from a case where Nu = 10 and non-UWB interferersrandomly appear in time, making the CNODE react by changing the waveform.

4.5.3 Conclusions

Cognitive radio systems have to deal with interference management. To this pur-pose, we have shown, in the context of cognitive IR-UWB systems, that one im-portant parameter to be sensed is the probability law of the interference (or thenon-Gaussianity). Performances of such systems are radically improved when theprobability law of the interference is estimated, also through a single and simple pa-rameter: the kurtosis, and when the network reacts and adapts itself taking intoaccount the non-Gaussianity.

66

4.6 Application in IR-UWB Antenna Testing

Figure 4.10 Performance comparison in terms of required energy when cognition is appliedunder the SGA hypothesis (interference are assumed to be Gaussian, Stan-dard Gaussian Approximation) or taking the non-Gaussianity into account(using a General Gaussian distribution to model the interference)


This section presents the impact of our criterion on the antenna design, by takinginto account the capacity of an IR-UWB system in a Multi-User Interference environ-ment. A new antenna design is investigated and it is shown using our performancecriterion that for this design, the communication capacity given by the Scholtz’s pulsemight be outperformed by using some other types of pulses. More than that, the pro-posed criterion might be used in IR-UWB not only to compare the performance ofdifferent antennas but also to compare the performance of different pulse shapes fromthe communication capacity point of view. This is due to the fact that as we haveseen before, the capacity does not depend only on the SINR but also on the kurtosisof the MUI.

From an antenna point of view, the UWB concept covers today three major typesof applications [70]:

1. Applications such as Ground Penetrating Radars (GPR, 1MHz to 10GHz),

2. Multi-narrowband applications such as signal intelligence and detection,

3. Modern UWB operating in a 3.1 to 10.6GHz frequency band.

Antennas dedicated to applications such as IR ad-hoc sensor networks are notsupposed to be multi-harmonics but they are required to be really non-dispersive andwideband, as illustrated on Fig. 4.11. Furthermore, phase transfer response must belinear with respect to frequency and ideally for any direction of emission.

Part of this work has been supported by the system@ticproject Urbanism and RadioCommuni-cations (URC).

67


(a)

(b)

Figure 4.11 Definition of wideband antenna (a) and frequency variation of main lobe (b)

68


4.6.1 An Antenna for IR-UWB

UWB antennas history begins with the work of Maxwell followed by Bose, Mar-coni, Amstrong and Carter with bi-conical 3D antennas. It goes on with horns andspherical fat structures with interesting wideband properties (Lindenbald, Schelkunoff,Friis, Kings and Kraus) [157]. In 1940, arguing that the stored reactive energy is re-duced in bulbous antenna due to smaller current concentration in thick structures,Kings claims that “fatter is better” for the design of such antennas [156]. Then, alot of work has been done on volumetric curve based or tapered antenna (Schekul-noff, Friis, Marié and Stöhr) [157]. A great contribution, concerning the frequencyindependence of some structures (based only on angles), has been added by Rumsey[151]. This was the introduction of spiral and log-periodic antennas, although thelatter are dispersive due to phase center translation in frequency. Nowadays, modernUWB antennas have to take into account several considerations: matching proper-ties (which leads to tapered profiles), minimum reactive power and hence resonance(which leads to thick or bulbous structures), low cost and small size (which leads toprinted antennas). Several technologies, already proposed for lower frequency bandapplications can be used to realize wideband antennas. Interesting states of the artabout the latter are given in [29, 120, 137]. The most important constraint is tobuild a non-resonant structure, which generally implies wide bent surfaces of conduc-tor printed on substrates. Traveling wave structures are also preferred because theyallow avoiding geometric resonances. Considering practical realization requirements(size and cost), the printed antenna is very popular for UWB. The geometry is firstdesigned for a wide radiation pattern and, then, optimized for frequency matching. Itseems that thick printed structures offer the wideband behavior necessary for UWB ina first step, bent shapes being chosen to radiate with large beam-width. In [157], theauthor suggests to take care about the free space attenuation applied in a widebandsignal transmission arguing about the complementarity between constant apertureand constant gain antenna. Also the shape of the radiation pattern (RP) has to beconserved in function of the frequency, and in function of the polarization at the sametime, to avoid an amplitude compression effect, see Fig. 4.11. The dispersion thatcan be observed on radiated pulses reveals the antenna time behavior. Integrationand production of UWB mobile transceivers imply an additional task that consistsin building a minimum-size low-cost antenna. These considerations lead to avoid vol-umetric structures (i.e., bi-conical, 3D monopoles or horns). However studying thelatter as possible UWB antennas is essential to find design rules for other technolo-gies. Micro-strip antenna shapes, for example, are often inspired by projection of 3Dexisting antennas [70, 156]. Applying the consideration developed above, the SUP-ELEC Antenna Design Laboratory focused on a modified simple monopole in orderto minimize the dispersion (see Fig. 4.13). The ground plane was opened to enable abi-directional radiation pattern in a first step although for UWB monopole it wouldbe interesting to design a ground plane around the area of the radiating part as in[27] and [28] but this has the inconvenience of narrowing the bandwidth. Herein, thereturn loss criterion was preferred and the design antenna presents the characteristicsof Fig. 4.12, where the radiation pattern is shown for 3 frequencies in the UWB band.

69


(a) (b)

(c) (d)

Figure 4.12 The antenna designed (a) and the radiation pattern (b), (c), (d) [132] inCST Microwave Studio

The main lob is slightly moving in function of the frequency. This will undesirablydistort any pulse shape transmitted by the antenna. A radio link is simulated betweentwo antennas at 1 meter, to compare with the antenna surface: 8x8 cm2.

Results of the propagation link transfer function (or S21) are shown on Fig. 4.14.The important variation of the magnitude is due to the free space path losses (FSPL)combined with the main’s lobe frequency dependence discussed above and showed inFig. 4.12. The phase of S21 in function of the frequency was not reported on Fig. 4.14but was simulated to be linear, as it is often expected for small symmetrical UWBdipole (the phase center trajectory is very small compared to the distance in far-fieldhypothesis) [120]. Now this transfer function, S21, can be used to characterize themodification of UWB pulses in our communication system.

4.6.2 A Novel IR-UWB Antenna Performance Criterion

We consider an Impulse Radio Time-Hopping Pulse Amplitude Modulation (bi-nary IR-TH-PAM) UWB system [11] and we note w(t) the received pulse after thereceiver’s antenna. We consider a single-cell scenario where the receiver is synchro-nized with user 1 (transmitter 1) and there are Nu− 1 interfering users which are notsynchronized with user 1.

70


(a) (b)

(c)

Figure 4.13 Front view (a), transversal view (b) and radiation pattern (c) of the physicalantenna

71


1 2 3 4 5 6 7 8 9 10 11 12−40

−35

−30

−25

−20

−15

−10

−5

0

Frequency GHz

dB

Return Loss

S11

(a)

1 2 3 4 5 6 7 8 9 10 11 12−60

−55

−50

−45

−40

−35

−30

−25

−20

−15

−10

Frequency GHz

dB

Transfer Function

S21

(b)

Figure 4.14 Return loss (a) and link transfer coefficient (b) at 1m

72


As seen before, the kurtosis of the MUI depends not only on Nu, Nh and Tc butalso on the shape w(t) of the received impulse. So, the coefficient β of the generalizedGaussian probability density function depends on the shape w(t) and as a consequencethe capacity of the communication system depends also on w(t).

By following this analysis we have compared the capacity of IR-TH-PAM UWBsystems using different pulse shapes at the output of the receiving antenna (seeFig. 4.15). This gives a new performance criterion for the pulses created throughthe antenna. The first pulse used is the pulse corresponding to the Dirac impulseresponse of our antennas. Such a pulse has at the output of the received antenna a−10dB bandwidth of 4GHz. The second pulse used is the classical Scholtz pulse usedin many UWB studies [180], it is the second derivative of a Gaussian pulse (see equa-tion (2.2)). We have chosen the characteristic parameters (τ = 300ps) of this Scholtzpulse to be such as having the same −10dB bandwidth as our first pulse (4GHz).The third pulse used is the response of our antennas to a rectangle impulse input(100ps square) which is easy to create with an impulse generator. At SUPELEC wecan use for example a D3186 Pulse Pattern Generator (see Fig. 4.21) from Advantest.

Figure 4.15 Capacity comparison cases

Due to the transfer function of our antennas the received impulse has a bandwidthof 3.7GHz. Thus we have considered a fourth impulse: the Scholtz’s impulse havinga −10dB bandwidth of 3.7GHz (τ = 322ps). By applying the formula (3.48) andmaking a Monte-Carlo computation of the integral as in Section 3.4.3, we obtained

73


for each pulse shape the achievable capacity (in bits per channel use, which is boundby 1 for binary PAM).

4.6.3 Results

For the simulations, we have considered two possible scenarios: perfect powercontrol and lack of power control. We assume a path gain decreasing in d−2 whered is the distance to the receiver, and that interferers with equal transmit power of−30dBm are randomly distributed with a uniform law within a ring delimited by thecircles with radius of 1m and 10m centered on the receiver (see Fig. 4.16).

dmin = 1m dmax =10m

D R

T

I

Figure 4.16 Distribution scenario

It can be proved that for the lack of power control case with Nu big enough wehave Pm,2

P 2m≈ 4.62 (see Fig. 4.17). Please note that Pm,2 and P 2

m have been defined inSection 3.3.3.

1 2 3 4 5 6 7 8 9 101

1.5

2

2.5

3

3.5

4

4.5

5

dmax

[m]

Pm

,2/P

2 m

Pm,2

/P2

m

line ([1,1],[1,4.62])

Figure 4.17 The evolution of the ratio Pm,2/P 2m with the maximum radius of deployment

74


To compute the capacity we need the expressions for Signal to Interference Ratios(SIR) and kurtosis in the cases of perfect power control and lack of power control (seeequations (3.36) and (3.37)).

As it can be seen in the equation (4.20), SIRppc (perfect power control case withnegligible noise) is a function of Nh, Nu and m2(w) while SIRlpc (lack of power controlcase with negligible noise - equation (4.21)) depends also on the ratio P1/Pm whichhas been considered to be fixed during simulations. The received power from user 1is set to be P1 = Pm

8, equivalent to a difference of 9dB.

SIRppc =Nh

(Nu − 1)m2(w)(4.20)

SIRlpc =Nh

(Nu − 1)P1

Pm

1m2(w)

(4.21)

For IR-TH-PAM, the capacity of a discrete memoryless channel with continuous-valued output is C(β, SIR) (see equation (3.48)). We recall that such a capacity maybe evaluated by Monte Carlo averaging using a generalized Gaussian random numbergenerator with variance 1/SIR and coefficient β.

Table 4.1 shows the performance of the communication with perfect power con-trol and with negligible noise (see Chapter 3), in terms of achievable capacity (inbits per channel use which is bound by 1 bit/channel use for binary PAM) for somevalues of the parameters Nu and Nh with Tc = 0.25ns. We can see that the per-formance of Scholtz’s pulse is outperformed by the pulse created by our antennas.Moreover, as expected when the −10dB bandwidth is larger, the capacity is betternot only because the spreading factor Tf/τ increases, but also because the pulse isnarrower leading to a more impulsiveness interference so a less Gaussian interference(which is the worst interference case [47]).

Table 4.1 Capacity in term of bits/channel use for various received pulse shapes - PerfectPower Control, Nu = 100

Received pulse shape Nh = 100 Nh = 250Dirac response of our antennas 0.713 b/ch.u. 0.911 b/ch.u.

Scholtz pulse 4GHz −10dB bandwidth 0.653 b/ch.u. 0.880 b/ch.u.Resp. of antennas to a 100ps sq. pulse 0.616 b/ch.u. 0.863 b/ch.u.

Scholtz pulse 3.7GHz −10dB bandwidth 0.630 b/ch.u. 0.867 b/ch.u.

By comparison with Table 4.2, we could observe that the capacity is decreasing forthe lack of power control scenario and also the capacity gap between the consideredcases modifies.

In Fig. 4.18 we have compared the dirac response of our antennas (a) and theresponse of the antennas to a square pulse (b) with the Scholtz’s pulse. As one can see,each figure contains the capacity as a function of SIR and β for each type of consideredresponse. The evaluation has been done in 5 points - denoted on the graph with 1, 2, 3,4 and 5 respectively and corresponding to the next pairs (Nh,Nu): (Nh=100,Nu=100);(Nh=250,Nu=100); (Nh=500,Nu=100); (Nh=500,Nu=50); (Nh=1000,Nu=50).

75


−10

−5

0

5

10

15

20

25

30

−1

0

1

2

3

4

50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SIR [dB]β

Cap

acit

y [

bit

s/c

han

nel u

se]

Lack of Power Control

Perfect Power Control

−1 −10

(a)

−10

−5

0

5

10

15

20

25

30

−1

0

1

2

3

4

50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SIR [dB]β

Cap

acit

y [

bit

s/c

han

nel u

se]

Perfect Power ControlLack of Power Control

5

1

2

3

4

−1 −10

(b)

Figure 4.18 Capacity(SIR,β) with (a) Dirac response of antennas ’△’ vs. Scholtz’spulse ’∇’ and (b) Response of antennas to a square pulse ’⊡’ vs. Scholtz’spulse ’∇’ in Perfect Power Control scenario and Lack of Perfect Con-trol scenario: 1, 2, 3, 4 and 5 are corresponding to the next pairs(Nh,Nu): (Nh=100,Nu=100); (Nh=250,Nu=100); (Nh=500,Nu=100);(Nh=500,Nu=50); and (Nh=1000,Nu=50) respectively

76


Table 4.2 Capacity in term of bits/channel use for various received pulse shapes - Lack ofPower Control, Nu = 100

Received pulse shape Nh = 100 Nh = 250Dirac response of our antennas 0.239 b/ch.u. 0.509 b/ch.u.

Scholtz pulse 4GHz −10dB bandwidth 0.207 b/ch.u. 0.465 b/ch.u.Resp. of antennas to a 100ps sq. pulse 0.177 b/ch.u. 0.421 b/ch.u.

Scholtz pulse 3.7GHz −10dB bandwidth 0.190 b/ch.u 0.443 b/ch.u.

For different duration of a square pulse, the pulse shape of the antenna output willbe totally different. We have noticed that the response of the antennas to a squaresignal of 70ps is the perfect equivalent in terms of capacity to the Scholtz’s pulse withthe same bandwidth at −10dB (3.84GHz, corresponding to a pulse of 306ps) - SeeFig. 4.19.

Figure 4.19 Equivalent schemes

4.6.4 Conclusions

In Section 4.6 we have seen how to apply our performance criteria for evaluatinga UWB antenna. We have seen how the shape of the waveform due to the antennahas an impact on the capacity of the communication with MUI [132]. This can beapplied to more UWB antennas and pulse shape optimizations. In Fig. 4.20 (a) and(b) we have represented the physical implemented antenna.

77


(a)

(b)

Figure 4.20 Implemented βVersion of the Monopole Antenna - (a) front view, (b) backview

78

Figure 4.21 Advantest D3186 Pattern Pulse Generator at SUPELEC

79

Chapter 5

Receiver and Transmitter Designsfor IR-UWB

Chapter Overview

The goal of this chapter is to design some simple receivers that take into accountthe MUI of the IR-UWB and can be used for sensor networks. We are looking for asimple design with low power consumption, low complexity, but good performancesas well. We also investigate in the last section a novel type of transmitter based onTime Reversal.

While in Section 2.2.5 we have presented some Impulse Radio receiver generalities,Section 5.1 describes several novel designs of non-complex IR-UWB receivers thatcould be used for sensor networks. Since we want to design a low complexity receivercapable of assuring good performances we need prior information on the statisticalproperties of the Multi User Interference (MUI). In many systems the Gaussian distri-bution is a common presumption but due to the impulsive nature of the interference,in the Impulse Radio Ultra Wide Band (IR-UWB) systems this presumption is nolonger valid when dealing with MUI. We investigate here the pertinence of using mix-tures of Gaussian, Laplacian, and generalized Gaussian random variables to modelthe MUI. While generalized Gaussian is simple and efficient (as shown in the previouschapters), it remains an approximation and it is useful to investigate the use of otherdistributions.

Section 5.2 introduces a novel transmitter design based on Time Reversal (TR).The goal of using TR at the transmitter side is for simplifying the receiver. In orderto verify this, we chose here to study the trade-off between the number of fingers inthe prefilter of a TR-IR-UWB system and the number of fingers in the Rake receiver.This allows us to analyze the gain brought by Time Reversal when the complexityis switched from the receiver to the transmitter i.e., when the number of fingers isincreased in the prefilter, while it is reduced in the Rake receiver.

81

5.1 A Comparison of IR-UWB Receivers Adapted to MUI with non Gaussian Distributions

5.1 A Comparison of IR-UWB Receivers Adapted

to MUI with non Gaussian Distributions

5.1.1 Introduction

As seen in the introductory Chapter 1, Impulse Radio Ultra Wide Band (IR-UWB)is dedicated to applications requiring either high bit rates over short ranges or low bitrates over medium to long ranges [11, 70]. Nowadays, sensor networks represent one ofthe most important applications that can benefit from this technology [184]. For suchapplications simple and low consumption receivers are required. But the receptionis facing specific multi user interferences (MUI) as such networks may have highdensity of unsynchronized users [148]. As a consequence of the atypical MUI aspectin IR-UWB, two methods have been considered in the fight against these interferences:employing multi-user detectors (MUD), or modeling the MUI distribution to derive anappropriate single user receiver. MUD is an efficient but very complex solution [185],even when suboptimal methods of minimum mean-square error MUD or successiveinterference cancellation are used [52].

The advantage of modeling or fitting the MUI distribution with an appropriatedistribution gives the possibility to develop a simple yet efficient receiver. We priv-ilege this approach in contrast with more complex multi user detection, as the lowcomplexity receiver is one of the targeted advantages of Impulse Radio systems.

In this section we present the receiver model describing our contribution (see Sub-section 5.1.2), then we propose some distributions to be compared to fit the MUI:generalized Gaussian and various mixture distributions (see Subsection 5.1.3). Wehave used the Expectation Maximization (EM) algorithm [125] to extract the param-eters of the mixture (see Subsection 5.1.6), and we have adapted this algorithm inorder to estimate the mixture cardinality n, which is a nontrivial problem. It canbe noted that the method can be expanded to generalized Gaussian mixture identifi-cation using empirical methods as the one proposed in Subsection 5.1.6.1. We thencompare our developed single user receivers adapted to different MUI distributions inSubsection 5.1.6.3.

5.1.2 Receiver

The basic structure of the receiver block from the previous subsection is presentedin Fig. 5.1. We are interested in the effect of Multi-User Interferences (MUI) after theMatched Filter (MF) and we consider that the receiver is synchronized with user 1(transmitter 1) and that there are Nu−1 interfering users which are not synchronizedwith this receiver. In the MF block the received signal r(t) is filtered by g1(−t), where

g1(t) =L−1∑

i=0

α1,iw(t− iTw) (5.1)

is used for single user detection, and α1,i are the coefficients of the channel betweenthe 1st user and the receiver (see equation (2.13)).

82


After being sampled and multiplied with the amplitude code b, the signal passesthrough a limiter function L(y) and is then summed up on Ns consecutive chips inorder to decide which bit was sent by the first user (see Fig. 5.1).

Figure 5.1 Receiver

The limiter function L(y) is related to the MUI PDF, and the receiver can computeits shape by listening to the MUI in a given time interval [46]. The reader can findmore details regarding the mechanism of the limiter function in Subsections 5.1.5 and5.1.6. In the case of generalized Gaussian MUI the limiter function is given in 5.1.5,while in the case of mixture MUI it is given in 5.1.6.2.

The main novelty introduced by Section 5.1 are the MUI PDF Parameters Es-timation Block and the comparison for different MUI distributions. Based on thisestimation, two types of iterative receivers are investigated in Subsections 5.1.5 and5.1.6.

The receiver adapted to generalized Gaussian was developed in [46], but the pa-rameter estimation block was not detailed. In Subsection 5.1.5 we contribute bydetailing it and giving an iterative structure. The receiver for Gaussian Mixture wasdeveloped in [41, 42], and in Subsection 5.1.6 we contribute by extending it to gene-ralized Gaussian Mixture and by giving a more simple architecture in an iterativestructure where the mixture estimation is separated from the channel estimation -which is supposed to be already done and used by the Matched Filter.

5.1.3 Modeling MUI

5.1.3.1 Unusual PDFs

Our observation essentially notices the fact that the distribution of the MUI is wellcharacterized by a generalized Gaussian, but it can also be modeled by a mixture ofGaussians [25, 41, 42], of Laplacians (see Fig. 5.2), of generalized Gaussians andeven by a Cauchy distribution [68] or α-Stable distributions [10, 68]. The generalizedGaussian distribution is simple because there are only two parameters describing it(i.e., β and σ) and is efficient as shown in Chapter 3, and collaborating results of[46, 47, 50]. On the other hand, the reader should keep in mind that fitting the

83


MUI distribution to the generalized Gaussian is an approximation, and is useful toinvestigate the use of other distributions.

−4 −3 −2 −1 0 1 2 3 40

0.2

0.4

0.6

0.8

1

1.2

1.4

y

PD

F(y

)

PDF Gaussian

PDF GG

PDF GMixture

PDF Cauchy

PDF Laplace

G,GM,L,GG,C

Figure 5.2 Unusual Distributions

5.1.3.2 Generalized Gaussian (GG)

As seen in Chapter 3,the PDF of a GG random variable is

p(y) =c1(β)√

σ2exp

−c2(β)

∣

∣

∣

∣

∣

y − µ√σ2

∣

∣

∣

∣

∣

2

1+β

(5.2)

with

c1(β) =Γ

1

2

(

32(1 + β)

)

(1 + β) Γ3

2

(

12(1 + β)

) , c2(β) =

Γ(

32(1 + β)

)

Γ(

12(1 + β)

)

1

1+β

where β > −1 and Γ(x) is the Gamma function.

5.1.3.3 Generalized Gaussian Mixture (GGM)

The PDF of a GGM random variable is

p(y) =n∑

i=1

pi ·c1(βi)√

σ2i

exp

−c2(βi)

∣

∣

∣

∣

∣

∣

y − µi√

σ2i

∣

∣

∣

∣

∣

∣

2

1+βi

(5.3)

with∑n

i=1 pi = 1, where pi are the weights (or mixing probabilities).When βi = 0 (∀i), we notice that the GGM becomes a Gaussian Mixture (GM)

and when βi = 1 (∀i) the GGM becomes a Laplacian Mixture (LM).

84


5.1.4 What is the Point of Using Receivers Adapted to Atyp-ical MUI Distributions?

To answer to this question, we have implemented a MUI test bank as in Fig. 5.3.Simple experimental results already show that there is an advantage of assuminghigher order interference distributions in the receiver (e.g., GG instead of Gaussian,or GM instead of Gaussian). This is basically the consequence of the fact that theGaussian distribution is a particular case of generalized Gaussian or Gaussian Mix-ture for example. If the interference is Gaussian distributed, it is expected that theperformance of the GM adapted receiver will be same as the Gaussian (G) adaptedreceiver (see Fig. 5.4(a)). On the other hand, if the interference is GM distributed,there is a performance gain if one considers a perfectly GM adapted receiver and nota G adapted receiver (Fig. 5.4(b)).

In this section we consider that the demodulation block takes the place of any MUIadapted receiver further detailed in Sections 5.1.5 and 5.1.6. Herein, the scope of theMUI test bank is to validate certain adapted receivers which are further developedand explained in those sections.

As mentioned before, we started from the presumption that MUI has a differentstructure than the Gaussian distribution. We have considered that MUI can comefrom distributions such as generalized Gaussians, Cauchy [68], Laplacians, Gaus-sians, or from mixtures of distributions. The G, GG, L or GM variables aregenerated and added to the signal obtained at the Matched Filter’s output (here theMF is represented as a pulse by pulse correlator). In this test bank, the artificialgenerated interference plays the role of the real interference.

Figure 5.3 MUI test bank

We have created a simulation environment just to test our receivers. Since afterthe Matched Filter (MF) the structure of the MUI changes, we have inserted theMUI after the MF in order to assure the previously mentioned signal properties. Weinsist that this representation is only used for testing purposes and helps to establisha relationship between different possible cases (see Fig. 5.4). In the next sections wewill study the receivers with no fake MUI but simulated ones.

Somewhat expected, under Gaussian interference (Fig. 5.4(a)) the performancesof the G, GG and GM adapted receivers are the same, while under Gaussian mixtureinterference (Fig. 5.4(b)) the best receiver is the GM adapted receiver.

Now we know that the classical receiver (i.e., Gaussian adapted) is not goodenough and that it can be outperformed by the above mentioned receivers. In real

85


0 1 2 3 4 5 6 7 8 9 1010

−3

10−2

10−1

Eb/N

0 [dB]

BE

R

G, GG and GM

(a)

0 5 10 15 20 2510

−3

10−2

10−1

Eb/N

0 [dB]

BE

R

GG receiver

G receiver

GM receiver

(b)

Figure 5.4 Gaussian, generalized Gaussian and Gaussian mixture adapted receiver per-formance under (a) Gaussian interference (b) Gaussian mixture interference

environments, IR-UWB interference (after the Matched Filter) in not Gaussian, there-fore it makes sense to search for advanced receivers for all these types of distributions.

5.1.5 The Receiver Structure for Generalized Gaussian MUIand a Novel Block Scheme

A simple GG receiver has been presented in [46], but the estimation of β was notincluded. We propose here a GG receiver with a method to estimate β. This realtime estimation is simple, practical and efficient, but is not the perfect estimation ofβ. Therefore, we call this receiver "rough GG receiver" as opposed to the GG receiverof [46].

In this subsection we propose the following 2-Steps Generalized Gaussian Blockreceiver (see Fig. 5.5). In the first step (Step 1) we consider MUI to follow theGaussian distribution and we estimate the sent bits using the classical approach (e.g.β = 0). In the second step (Step 2) we consider MUI to be generalized Gaussiandistributed and we compute the β parameter of the estimated MUI.

We work on blocks of LB bits. In the first step, the LBNs useful signal terms[u0...uLBNs−1]1 are estimated considering β = 0. The β parameter for the next step isestimated from the MUI terms [y0...yLBNs−1]1 obtained by subtracting [u0...uLBNs−1]1

from the received sampled signal.As stated in [46], the β parameter gives the shape of the non linear limiter

L(y) =(

|y + 1|2

1+β − |y − 1|2

1+β

)

. (5.4)

The β parameter is linked to the kurtosis K of the MUI as shown in Fig. 3.8,where the dependency K(β) is graphically represented. As a remark, for small valuesof β, one can use the approximation

K(β) =Γ(

52(1 + β)

)

Γ(

12(1 + β)

)

Γ2(

32(1 + β)

) − 3 ≈ 3β2 − 0.3β. (5.5)

86


Figure 5.5 Rough GG Receiver

In the Estimation Block β we compute K =E[y4]E[y2]2

−3 based on the incoming sam-ples [y0...yLBNs−1], and we extract the β parameter that characterizes the distributionusing (5.5). Then, we use the β parameter in (5.4) to design the shape of the limiterL(y).

This receiver is a practical implementation [134], but it is not the perfect receiveradapted to GG, as the estimation of β is simple and practical yet not perfect. Wewill compare this rough GG receiver with the receiver for mixture distribution in thefollowing sections.

5.1.6 The Receiver Structure for Mixture MUI

5.1.6.1 Mixture Identification using a Modified Expectation Maximiza-tion (MEM) Algorithm

The underlying assumption is that a set of m scalars y0 . . . ym−1 is drawn fromone of the n distributions. In our case, y0 . . . ym−1 are samples of the MUI passedthrough the MF block. We introduce a new variable Z denoting which one of the ndistributions the sample yj belongs to. For example, for a scalar yj with j inside theinterval [0, m− 1], verifying yj ∈ Distributioni, one has zj = i.

Our task is to estimate the unknown parameters Θ of each distribution: mean,standard deviation, weight etc. For a given number of mixtures n, we define theweight (or mixing probability) as the probability of choosing the elements y froma certain distribution i ∈ [1, n] - equivalent to saying that the weighti is equal toP(Z = i) = pi, and we define Θ1×4n as the vector of the unknown parameters

Θ1×4n = [(µ1, . . . , µn) , (σ1, . . . , σn) , (β1, . . . βn) , (p1, . . . pn)] . (5.6)

Please note that in the GM case, β1 = . . . = βn = 0.

87


The Modified EM algorithm that we propose to apply consists of three primarysteps: an estimation step over the total number of mixtures n, followed by an expec-tation step E and finally by a maximization step M. The expectation (i.e., E-Step) isobtained with respect to the unknown underlying variables Z, using the current esti-mate Θt of the parameters and conditioned by the observations Y. The maximizationstep (i.e., M-Step) then provides a new estimate Θt+1 of the parameters. These lasttwo steps are iterated until convergence is reached (by comparison of Θt+1 with Θt).The entire concept is illustrated in Fig. 5.6.

Figure 5.6 An overview of the Modified EM Algorithm

Below we describe the estimation steps for a fixed number of mixtures n in theGGM case with known βi (e.g., βi = 0 or βi = 1):

Initialization Set n; for ∀i = 1, . . . , n Set σt=0i , pt=0

i and µt=0i .

88


E-step The computation of the membership probabilities

p(zj = i | yj, Θt) =p(yj | zj = i, Θt)pt

i∑n

k=1 p(yj | zj = k, Θt)ptk

(5.7)

for j = 0, . . . , m− 1 and i = 1, . . . , n.

M-step The upgrades on the means

µi =∑m−1

j=0 p(zj = i | yj, Θt)yj∑m−1

j=0 p(zj = i | yj, Θt), (5.8)

the upgrades on the variances

σ2i =

c2(βi) · 2γi

∑m−1j=0 p(zj = i | yj, Θt) |yj − µi|

2

γi

∑m−1j=0 p(zj = i | yj, Θt)

γi

(5.9)

with γi = 1 + βi, and the upgrades on the mixing probabilities

pi =∑m−1

j=0 p(zj = i | yj, Θt)∑n

k=1

∑m−1j=0 p(zj = k | yj, Θt)

, (5.10)

for i = 1, . . . , n.For distributions with unknown βi (e.g., the distributions are known to be different

from GM or LM), we simply add the following set of equations to the M-step:– the computation of the estimated kurtosis for each distribution i from the mix-

ture

Ki =∑m−1

j=0 p(zj = i | yj, Θt)(yj − µi)4

σ4i

∑m−1j=0 p(zj = i | yj, Θt)

− 3, (5.11)

– verify that each Ki remains bounded by an upper bound value Lmax and a lowerbound value Lmin = −0.5 (to assure the convergence later on)

Lmin ≤ Ki ≤ Lmax, (5.12)

– the computation of the βi parameters

βi = F−1{

Ki

}

= F−1

Γ(

52(1 + βi)

)

Γ(

12(1 + βi)

)

Γ2(

32(1 + βi)

) − 3

, (5.13)

– the computation of c1(βi) and c2(βi) parameters.

89


Figure 5.7 Mixture Receiver (could be a GM or a LM)

5.1.6.2 The Proposed Receiver

We have followed a design approach for the mixture receiver (see Fig. 5.7) basedon the design approach of the GG block-receiver.

In [41, 42] a receiver scheme adapted to Gaussian Mixtures has been proposed,where the sampled channel is estimated inside a data decoding loop and based onthe estimation result an adapted metric is used to decode the data. We propose asimpler mixture receiver, where the metric (correction parameter) is easy to compute.The Mixture Estimation Block (MEB) can estimate Laplacians, Gaussians or othermixture distributions of the MUI.

Basically, MEB works in two steps (as the approach for the GG case): in thefirst step (Step 1) we initialize αj with 1 (∀j) and in the second step (Step 2) wecompute the number of mixtures, the membership probabilities and the variances ofeach mixture in order to create a correction parameter αj better adapted to the MUI.Recall that the initialization from Step 1 works under the Gaussian MUI assumption.During Step 2, MUI is supposed to have a GM characteristic, therefore the correctionparameter is

αj =n∑

i=1

p(zj = i | yj, Θ)σ2

i

(5.14)

and it appears as a factor rather than a parameter of a limiting function as was βfrom GG receiver (see Table 5.1).

During Step 2, the MEB works on the estimated interference samples yj whichare obtained through the use of uj computed at the Step 1 under the Gaussian MUI

90


Table 5.1 Comparison Between the Correction Parameters α (GM Receiver) and β (GGReceiver)

Parameters Step 1 Step 2

αj 1∑n

i=1p(zj=i|yj ,Θ)

σ2i

β, ∀j 0 β(KMUI)

approximation and the received signal uj + yj.

5.1.6.3 Simulation Results

The basic idea behind the correction parameter is that each sample of MUI hasa probability to belong to a certain distribution among the distributions composingthe mixture. As well, note that if the variance of the distribution is high, thereare higher chances to fall into a strong interference. Thus, we selected a correctionparameter that weights the effect of the interference, as high interference is beingpenalized. Each sample of the MUI yj is weighted with a different correction factorαj. Therefore we must use Ns ≥ 2 to observe an effect of those correction factors.

The initialization of the parameters σ2i , pi (∀i = 1, . . . , n) is very sensitive, as

we risk to fall into a non-converging case [25]. In Table 5.2, one can find the valuesrequired for the initialization, where we have considered ξ = 0.01 and γ = 25.

Table 5.2 Initialization parametersParameters i = 1 i = 2, n

Variance σ2i 1 γσ2

i−1

Mixing Probability p(zj = i), ∀j 1− ξ ξn−1

With respect to the selection of the total number of mixtures n in our simulations,we start with a large number that satisfies our requirement to cover a wide range ofenvironments. Towards this goal we start with an initial number n = 5; however,during the estimation we might fall into a smaller number of mixtures. A priori thechoice of n should depend on the environment parameters (e.g. channel model, levelof interferences, power control features, number of users).

Fig. 5.8 shows that for a correct identification we need around 100 MEB iterationsfor strong interferers (around 30dB above the useful signal), as we have consideredthat there is a high dependency between the variances plus the mixture probabilitiesand the types of the interferers. Otherwise, in the perfect power control case, anumber of maximum 20 iterations would be sufficient.

Simulations have been made over groups of packets with 1000 symbols, standardCM1 (LOS) and CM3 (NLOS) channel models, Tc = 1ns, τs = 0.5ns, Nh = 140. Wehave considered a gap of Nh slots between two consecutive transmissions in order tocancel the Inter-Chip Interference (ICI). We show a preference for the lack of powercontrol scenarios since they are describing reality with a higher accuracy but alsobecause it emphasis the effect of the non-Gaussianity of the MUI. In perfect powercontrol conditions (i.e., equal energies before the receiver), the matched filter is able

91


0 50 100 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Iterations

Mix

ing P

rob

ab

ilit

y

Mixture Probabilities Identification for GM Case

Mixture

Probabilities

Mixture

Probabilities

Identification

0 20 40 60 80 100 1200

5

10

15

20

25

30

Number of iterations

Vari

an

ce

Variance Identification for the Gaussian Mixture (GM) Case

σ2

identificationσ

2

mixture

Figure 5.8 GM identification: mixing probabilities (a) and variances (b)

92


to alleviate the effect of the UWB channels models, thus making MUI closer to theGaussian distribution (see Fig. 5.9).

1 2 3 4 5 6 7 8 9 1010

−4

10−3

10−2

10−1

100

NS

BE

R

G Receiver

GM Receiver

GG Receiver

High Noise

Low Noise

GM

G

GG

Figure 5.9 Comparison between the Gaussian Receiver, Generalized Mixture Receiver andrough Generalized Gaussian Receiver. BER as a function of Ns, two setsof graphs in Low Noise (Eb/N0 = 30dB) and High Noise (Eb/N0 = 0dB)environments, CM1 (LOS), 100 users in perfect power control with 10dBabove the useful signal

Simulations show that increasing the number of symbols in a packet helps theestimation because the receiver can "listen" to the interferences for a longer periodof time. Alternatively, increasing the repetition factor Ns has the same effect, butunder the Gaussian assumption summing on all Ns decoded symbols might lead to adisastrous result, as one of the slots might capture a high level of interference.

The rough GG receiver outperforms the Gaussian MUI adapted receiver especiallywhen there is less noise (it helps for better MUI estimation - as one can see in Fig. 5.10)and there are less number of users Nu, more slots Nh and higher power gap betweendifferent users. In our simulations we keep Nh constant, modifying only the rest ofthe parameters but the effect is still noticeable. The explanation supported by thisresult is that tuning these parameters increases the kurtosis, and when the kurtosisincreases one deviates further away from the classical Gaussian inference (see Fig. 5.9compared to Fig. 5.11).

The performance of GM receiver is incontestable and our experiments show that itperforms even better compared with the rough GG, LM or GM receivers when there isa powerful impulsive interference coming from distinct groups of users (e.g., Fig. 5.12).Comparing Fig. 5.11 (i.e., CM3 and only one strong interferer) with Fig. 5.12 (i.e.,CM1 and 2 groups of strong interferers), one can notice how the Channel Model andthe interference environment might change the BER performance gap between therough GG receiver and the GM adapted receiver for fixed Ns. The environment from

93


0 5 10 1510

−2

10−1

Eb/N

0

BE

R

Classical Receiver

GG Receiver

GM Receiver

Figure 5.10 Comparison between the (classical) Gaussian Receiver, Generalized MixtureReceiver and rough Generalized Gaussian Receiver. BER as a function ofEb/N0, Ns = 5, CM3 (NLOS), 3 users with 20dB above the useful signal, 2users with 30dB above the useful signal

Fig. 5.11 is highly impulsive but the interferences are less stronger, therefore roughGG receiver’s performance is closer to GM receiver’s performance in this scenario.However, the complexity of the mixture receiver in terms of number of iterations itis much higher.

5.1.7 Conclusions

We have proposed simple receivers based on non-Gaussian MUI distributions.These receivers outperform the classical correlation receiver adapted to Gaussian in-terferences [134]. We have compared them and showed how the performance dependson the scenario conditions (e.g., rich multipath channel, number of users and inter-ference power), because the MUI distribution also depends on the scenario.

94


1 2 3 4 5 6 7 8 9 1010

−6

10−5

10−4

10−3

10−2

10−1

NS

BE

R

G Receiver

GG Receiver

GM Receiver

Figure 5.11 Comparison between the Gaussian Receiver, Generalized Mixture Receiverand rough Generalized Gaussian Receiver. BER as a function of Ns,Eb/N0 = 30dB, CM3 (NLOS), 8 users in perfect power control with 0dBabove the useful signal, 1 user with 30dB above the useful signal

1 2 3 4 5 6 7 8 9 1010

−5

10−4

10−3

10−2

10−1

NS

BE

R

G Receiver

GG Receiver

GM Receiver

Figure 5.12 Comparison between the Gaussian Receiver, Generalized Mixture Receiverand rough Generalized Gaussian Receiver. BER as a function of Ns,Eb/N0 = 30dB, CM1 (LOS), 3 users with 20dB above the useful signal,2 users with 30dB above the useful signal

95

5.2 A Study on a Novel Type of Transmitter Using Time Reversal

5.2 A Study on a Novel Type of Transmitter Using

Time Reversal

In the previous section we have investigated receivers adapted to MUI, consideringa classical transmission. Now we look at the transmitter side and investigate a noveltype of transmitter using Time Reversal.

5.2.1 Introduction

Impulse Radio Ultra Wide Band (IR-UWB) communications are classically basedon the use of a Rake receiver. Many studies have been made on the number of fingersto be used in the Rake in order to obtain good performance [11, 24, 124, 143, 178].Time Reversal (TR) IR-UWB systems use a prefilter at the transmitter side [170], thathas the role of convolving the UWB pulse with the impulse response of the channelreverted in time; when the signal crosses the channel, the output of the channelpresents the correlation of the channel with itself. Thus, the Time Reversal prefilterhas a function somewhat analog to the Rake receiver i.e., creating the correlation ofthe channel with itself. As a matter of fact one of the main advantages which is oftenclaimed for TR is that it transfers the complexity from the receiver to the transmitter.Moreover, like the Rake receiver, the Rake of the prefilter at the transmitter sidemay be implemented ‘partially’ by selecting a number of fingers lower than the totalnumber of paths in the channel, reducing thus the complexity of implementation. Ourstudy aims at investigating how TR UWB effectively shifts the complexity from thereceiver to the transmitter. To this purpose, we analyze performance as a functionof the number of fingers in the Rake receiver and the number of fingers in the TRprefilter. As a matter of fact, while using TR, the receiver should still use a Rakeadapted to the new signal. Therefore, one can tune both parameters at once: thenumber of fingers of the prefilter versus the number of fingers of the Rake receiver.

5.2.2 TR-SRake Model

Fig. 5.13 represents a general and compact transmission chain, where the TimeReversal (TR) strategy is combined with Selective Rake (SRake). Here, the transmit-ter is composed from a TH-IR-UWB signal generator using PAM and a pre-filteringblock. The difference between the classical scheme and this one is that here thetransmitter uses a pre-filtering block and the receiver is not ‘full’ Rake. This novelapproach has the goal of reducing the complexity of the receiver by transferring someof its functions to the transmitter.

In a ‘full’ TR (Time Reversal) system the signal is convolved with h(−t) in theprefiltering block. h(t) is the channel impulse response (see Fig. 5.14):

h(t) =L−1∑

i=0

ρiδ(t− τi) (5.15)

Part of this work has been supported by the system@tic project Urbanism and RadioCommu-nications (URC).

96


Figure 5.13 Time Reversal (TR) combined with Selective Rake (SRake)

with L the total number of paths in the channel and τi the delay associated to the ith

path and ρi its amplitude.

0 5 10 15 20 25−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

t [ns]

Impuls

e R

esponse

of

the

Chan

nel

h(t

)

Figure 5.14 Example of h(t)

To reduce the complexity, we can use a pre-filter hin(t) (see Fig. 5.15) where onlya subset of paths is considered:

hin(t) =Nin−1∑

i=0

ξiδ(−(t− τ′

i )) (5.16)

with {ξi}i∈[0,Nin−1] the strongest Nin ≤ L paths of h(t), and τ′

i their associated delays.At the receiving side, the full Rake has to perform the correlation of the received

signal with w(t) ∗hin(t) ∗h(t) (see Fig. 5.16). One can also implement a partial Rake

97


−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

t [ns]

Impuls

e R

esponse

of

hin

(t)

Nin

=10

Figure 5.15 Example of hin(t)

receiver, computing the correlation of the received signal with w(t) ∗ hout(t) (see Fig.5.17), where:

hout(t) =Nout−1∑

i=0

νiδ(t− τ′′

i ) (5.17)

with {νi}i∈[0,Nout−1] the strongest Nout ≤ (2 · L) fingers of hin(t) ∗ h(t), and τ′′

i theirassociated delays. Please note that in the receiver block, instead of using the correla-tion pulse by pulse, one can also use the convolution operation. In this case, hout(t)would have been equal to

∑Nout−1i=0 νiδ(−(t− τ

′′

i )).

5.2.3 Simulations and Results

In order to obtain the performance of TR UWB in terms of BER, we have runsimulations using channel models named CM1 (line-of-sight, LOS) and CM3 (non-line-of-sight, NLOS) according to [55]. We use the classical Scholtz’s pulse w(t)defined by (2.2). System parameters are τ = 0.5ns, Tc = 1ns and Nh = 280 chips.We analyze the BER as a function of the number of fingers in the prefilter Nin andin the Rake receiver Nout. The N selected paths from the fingers correspond to theN strongest ones (Selective Rake).

Results are shown in Fig. 5.18 and 5.19; performance changes when switching thenumber of fingers from the receiver to the transmitter. For instance, when Nin = 1and Nout = 10 (i.e. without Time Reversal) performance is better than when Nin = 10and Nout = 1. It is important to mention that this effect happens for other variousvalues of Eb/N0, and it is directly related to the specific channel characteristics.

98


−10 −5 0 5 10 15 20 25−0.2

0

0.2

0.4

0.6

0.8

1

t[ns]

Impuls

e R

esponse

of

hin

(t)*

h(t

)

Figure 5.16 Example of hin(t)∗h(t)

−10 −8 −6 −4 −2 0 2 4 6 8 10−0.2

0

0.2

0.4

0.6

0.8

1

t [ns]

Impuls

e R

esponse

of

hout(t

) Nout

=20

Figure 5.17 Example of hout(t)

99


2468101214161820

10

20

30

10−5

10−4

10−3

10−2

10−1

Nout

(Receiver)Nin

(Transmitter)

BE

R

Figure 5.18 BER(Nin, Nout), Eb/N0 = 9dB, LOS

2468101214

1020

3040

10−4

10−3

10−2

10−1

Nout

(Receiver)Nin

(Transmitter)

BE

R

Figure 5.19 BER(Nin, Nout), Eb/N0 = 9dB, NLOS

100


In order to maintain the same performance when the number of fingers at theRake receiver is lowered, the number of fingers at the transmitter must be increasedin a more important way, and a small number of fingers still needs to be kept at thereceiver. For instance, a bit error rate of 0.5 · 10−2 with the initial value of Nout = 8and Nin = 1 may be kept by lowering Nout to 5 but increasing Nin to 10, as shown inFig. 5.25.

Moreover, in some situations, increasing the number of fingers at the transmitterwhile keeping the number of fingers at the receiver may reduce performance. Thiscan be explained by a major spreading of the energy on various paths: while the mainpath is strongest in Time Reversal, many sub paths may appear, thus creating a majorenergy spreading. The strengthening effect of the main path due to time reversal isvisible on Fig. 5.20 and 5.21, that show the percentage of the total received energyon the main path as a function of the number of fingers Nin in the Time Reversal pre-filter. Note that for Nout = 1, the lower energy values on the main path is obtainedfor Nin = 1, i.e. without time reversal.

Figure 5.20 LOS, Energy percentage on Nout fingers at the receiver (Nout = 1 to 10). Thecurve with Nout = 1 gives the energy percentage on the main central path

The spreading effect of the total energy on an increased number of paths in TimeReversal is shown in Fig. 5.22 and 5.23. For a fixed number of fingers Nout ≥ 10 inthe Rake receiver (LOS) or Nout ≥ 30 (NLOS), the percentage of the total energy atthe output of the Rake receiver decreases when the number of fingers in the TimeReversal pre-filter increases.

As expected, increasing the number of fingers in the Rake receiver while keepinga fixed number of fingers at the prefilter always increases performance.

101


Figure 5.21 NLOS, Energy percentage on Nout fingers at the receiver (Nout = 1 to 10).The curve with Nout = 1 gives the energy percentage on the main central path

Figure 5.22 LOS, Energy percentage of Nout fingers

102


Figure 5.23 NLOS, Energy percentage of Nout fingers [131]

In Fig. 5.24 it can be easily seen that after a certain number of Nout fingers in theRake receiver, increasing Nin in the prefilter will not change the performance of thesystem anymore.

Finally, Fig. 5.25 presents the iso-BER curves as a function of the Nin and Nout

couples that produce the same BER performance. Note that with respect to the notime reversal case (Nin = 1), a same performance can be obtained by increasing Nin

and by decreasing the number of fingers Nout in the Rake receiver only when theinitial BER is high, i.e. only when the Rake receiver without time reversal uses asmall number of fingers (less than about 10).

In summary, results show that Time Reversal helps transferring the complexityfrom receiver to transmitter, only when the receiver initially has a low number offingers, otherwise the use of Time Reversal with reduced number of fingers at theRake receiver reduces performance.

The impact of TR seems more promising however, in a scenario with multi userinterference (MUI). Fig. 5.28 shows that in the case of strong interferers, TR may leadto better performance (see (Nin = 1, Nout = 40) versus (Nin = 40, Nout = 40)). Thishappens because TR modifies the MUI distribution, while from [48, 47, 135] we knowthat the MUI distribution may have a strong impact on the IR-UWB performance.

This assumption can be simply checked by measuring the kurtosis K of the MUI(after the correlation) in three cases: (Nin = 1, Nout = N), (Nin = N, Nout = 1) and(Nin = N, Nout = N) with N ≤ L as in [127]. Please note that Nin corresponds to thenumber of fingers in the prefilter and is computed with respect to the channel of theinterferer, while Nout corresponds to the number of fingers in the receiver block andis computed with respect to the channel of the useful signal. In [127] we have found

103


0 50 100 150 200 250 300 350 400 450 5000.75

0.8

0.85

0.9

0.95

1

Nin

Fingers

Ener

gy P

erce

nta

ge

[%]

on N

out F

inger

s Nout

=1000 Nout

=500

Nout

=300

Nout

=200

Nout

=50

Nout

=100

Figure 5.24 NLOS, Energy percentage of Nout fingers

0 5 10 150

5

10

15

20

25

30

35

40

45

Nin

Fingers

Nout F

inger

s

BER=10−2

BER=0.5⋅10−2

BER=10−3

BER=0.7⋅10−3

Figure 5.25 Constant BER curves, NLOS, Eb/N0 = 9dB

104


that adding fingers in the prefilter increases the kurtosis: for (Nin = 10, Nout = 20)the kurtosis is 1.42 while for (Nin = 20, Nout = 20) it is 2.37.

0 1 2 3 4 5 6 7 8 910

−5

10−4

10−3

10−2

10−1

100

Eb/N

0 [dB]

BE

R

SRAKE with 2 fingers




Ideal RAKE

Figure 5.26 SRake without TR for binary PAM-TH, LOS

5.2.4 Conclusions

This study analyzed the impact of the number of fingers in the Rake receiver and inthe time reversal prefilter for IR-UWB. Results show that only in special conditions,i.e. for low number of fingers in the initial Rake receiver [131], transferring thecomplexity from the receiver to the transmitter by increasing the fingers in the prefilterand lowering it in the Rake receiver produces better performance. In environmentswith high interference, time-reversal combined with Rake receiver could be moreadvantageous as it appears to have an important impact over the MUI distribution.

105


0 1 2 3 4 5 6 7 8 910

−5

10−4

10−3

10−2

10−1

100

Eb/N

0 [dB]

BE

R

Ideal RAKE

Nin

=15, Nout

=1

Nin

=40, Nout

=1

Nin

=1, Nout

=40

Nin

=1, Nout

=80

Figure 5.27 TR and SRake for binary PAM-TH, NLOS

0 5 10 1510

−2

10−1

100

Eb/N

0 [dB]

BE

R

Nin

=1, Nout

=40

Nin

=40, Nout

=1

Nin

=40, Nout

=40

Figure 5.28 NLOS, strong interference from 5 users with 10dB above the useful signal

106

Chapter 6

Cooperation in Wireless Networks

Chapter Overview

Cooperative communication techniques are used at the MAC layer to improvewireless link performance. Although we do not study the geographical data providedby UWB positioning here, we investigate the cooperative schemes that can benefitfrom it. The efficiency of the information transmissions can be increased through animproved wireless link by cooperative communications. As such, cooperative commu-nications have attracted much research attention.

Recently, various cooperative relaying schemes have emerged to improve wirelesslink performance. In cooperative relaying, when a transmission from a source to adestination is not successful, a cooperative node called relay which has successfullyoverheard the original transmission, will help in relaying the packet to destination.As such, the source does not need to persistently retransmit the failed packet ifthe transmission from relay reaches the destination successfully. These cooperativerelaying schemes are useful when the wireless link between source and destination isin deep fade, and the wireless links between source and relay, and destination andrelay are good.

In reference work, cooperative transmissions can be performed in the forms ofdecode-and-forward (DF) or amplify-and-forward (AF). This work focuses on theamplify-and-forward schemes. Existing amplify-and-forward schemes exploit the useof multiple relays but have a low spectral efficiency due to their static allocationof time slots for relaying. Incremental relaying adaptively allocates time slots torelay only when the original transmission from its source fails. Hence, incrementalrelaying can achieve a better spectral efficiency compared to the simple amplify-and-forward scheme. However, existing incremental relaying schemes consider half-duplextransmissions with best-relay selection algorithms and they do not take advantage ofthe full use of multiple relays. Section 6.1 proposes an incremental relaying schemethat uses multiple relays. Analytical and numerical results show that the proposedscheme outperforms both the simple amplify-and-forward scheme and the single-relayincremental relaying scheme in terms of lower outage probability. Furthermore, wehave shown that when there is a need to pick the best relay among a group of candidatenodes, harmonic mean based selection methods are not the best solutions.

107

6.1 Cooperative Transmission Scheme Using Incremental Relaying with Multiple Relays

Cooperative relaying schemes such as amplify-and-forward and decode-and-forwardhelp in lowering the outage probability of unreliable wireless links. Compared to AFand DF, incremental relaying schemes are better options. This is because in incremen-tal relaying, a relay helps in retransmitting a packet only if the original transmissionfails, and this leads to a better spectral efficiency. However, implementing incrementalrelaying in a practical system is not a trivial task because it requires the destinationnode to provide feedback on success or failure of a transmission. In practice, the feed-back may be affected by propagation impairments and collisions. In this case, therelay and the source should coordinate among themselves in the absence of perfectfeedback information such that packet collision will not occur. In view of this chal-lenge, Section 6.2 proposes a novel and practical incremental relaying scheme thatdeals with the imperfect feedback. The key idea is to have pessimistic source andoptimistic relay. The proposed scheme is simple and easily implementable. Simula-tion results show that the proposed scheme can outperform AF and DF, as well asapproaching the performance of an idealized incremental AF with perfect feedbackinformation.

Last but not least, Section 6.3 describes a simple distributed relay selection al-gorithm without the need of any additional control message exchange between therelays.

6.1 Cooperative Transmission Scheme Using In-

cremental Relaying with Multiple Relays

6.1.1 Introduction

Compared to direct transmission between a source node and a destination node,cooperative transmission involves relay (neighbor) nodes that are within the radiorange of both source and destination to retransmit an overheard transmission to thedestination. Cooperative communications have some advantages in terms of through-put, connectivity, and energy consumption [153]. In literature, this type of cooper-ative communication technique is also called (cooperative) relaying. There are twocommon types of relaying, namely the decode-and-forward (DF) and amplify-and-forward (AF) [118]. In AF, a relay simply amplifies an overheard transmission with-out decoding it, before retransmitting to the destination [95]. On the other hand, inDF, the relay needs to fully decode an overheard transmission before retransmittingit to the destination. This limits DF’s performance. This chapter focuses on AFhereafter.

In a simple AF scheme, a fixed time slot is allocated to the source and to the relay.In this fixed time slot structure, the relay retransmits the overheard transmissionto the destination node even if the original transmission from the source has beensuccessfully received. This type of simple AF scheme has been shown not to be

This work has been done when Dorin Panaitopol was attached to the Institute for InfocommResearch under CCWnet Project (CR-2007-013). The project focuses on research and developmentof cooperative and coordinated techniques to improve performance of wireless networks.

108


the best alternative in terms of spectral efficiency [108]. In [110], the authors choseto improve the performance of AF in terms of lower outage probability by usingmultiple relays that transmit concurrently in the single allocated time slot. However,this improvement does not help in achieving a higher spectral efficiency.

A few incremental relaying schemes have been proposed in order to improve spec-tral efficiency. In these schemes, a relay is dynamically allocated a time slot toretransmit an overheard transmission only if the destination has not received theoriginal transmission from the source. In order to perform incremental relaying, itis assumed that there is immediate feedback from destination to relay. When im-mediate feedback is not available, [188] proposed to dynamically allocate time slotsto the relay based on an estimated channel state information. While these existingincremental relaying schemes help in achieving a higher spectral efficiency [99], only afew exploit the existence of multiple relay nodes [97]. Nevertheless, there are severalreferences employing multiple relay strategies but they treat mainly non-incrementalschemes (e.g., AF or DF) [77, 166], while those dealing with multiple relay incremen-tal schemes [17, 19, 98, 101, 102] are referring to best single-relay selection technique(i.e., only one best relay is picked up to help) or to space time coding techniques [121](i.e., virtual antenna array).

We would like to point out that the literature for relay cooperation schemes isvast and one can find AF schemes with multiple relays, or incremental AF schemeswith best relay selection (or half-duplex), but we were not able to find existing resultsfor incremental AF schemes with multiple relays where all the relays incrementallyparticipate in cooperation.

In this section, we propose an incremental AF relaying scheme that uses multiplerelay nodes to achieve higher spectral efficiency and lower outage probability. Therest of the Section 6.1 is organized as follows. In Subsection 6.1.2, we describe thesystem model. In Subsection 6.1.3, we define simple non-incremental AF scheme asthe baseline before presenting the proposed incremental AF scheme. Subsection 6.1.4presents and discusses the numerical results before ending Section 6.1 with concludingremarks in Subsection 6.1.5.

6.1.2 System Model

We consider a simple topology with n potential relays between a source and adestination. Let the wireless channel be divided in time domain into time slots withfixed length. Each time slot is allocated for transmission of a fixed length packetby both source and relays, with or without relaying. This model is different fromthat in [118] and [110], where the source and one relay are required to transmit at adata rate that is twice the original data rate when there is relaying so that the totaltransmission time with relaying equals the transmission time without relaying. Inour model, the total transmission time with single relay is two times that of directtransmission. This is a practical model when time slot duration cannot be changedinstantaneously.

Communications between source and destination happen in two phases. In thefirst phase, the source s broadcasts a signal. In the second phase, the relay that has

109


received (overheard) the signal will amplify and retransmit it to the destination. Insimple (non-incremental) AF, the second phase always exists. In incremental relaying,the second phase exists only if the destination has not received the transmitted signalin the first phase.

Let x be the signal transmitted by the source s in the first phase, the receivedsignal at the destination d can be written as

ys,d =√

Pshs,dx + v, (6.1)

where Ps is the transmission power from source, v is the additive white Gaussian noisewith zero-mean and hs,d is the fading channel coefficient of the source-destination link.Similarly, the received signal at relay i = 1, 2, · · · , n can be written as

ys,i =√

Pshs,ix + v, (6.2)

where hs,i is the fading coefficient of the link between source and relay i. In the secondphase, each relay i amplifies the received signal ys,i with the amplification factor

βi =

√

√

√

√

Ps/n

Ps |hs,i|2 + N0

, (6.3)

and transmits it to the destination through the channel hi,d. Notice that the trans-mission power of single relay is n times smaller than that of the source. As such, themaximum transmission power of this scheme is 2Ps. In another scenario where the

power used by the relay is Ps (i.e., the same as the source), βi =√

Ps

Ps|hs,i|2+N0

and the

maximum transmission power of the scheme equals (n + 1) Ps.The destination performs Maximal Ratio Combining (MRC) on all the orthogonal

transmissions (e.g., different transmission slots) coming from the source and from therelays that contain the useful information x. In non-incremental AF relaying, thedestination performs MRC at the end of the second phase. In incremental relaying,the destination performs MRC after each useful received transmission during thesecond phase.

In order to propose and to compare different AF schemes (i.e., cooperative strate-gies), we need to study their outage behavior. From information theory, we know thatin a transmission between two devices, an outage event happens when the spectralefficiency SE exceeds the maximum mutual information I between two devices [174].Thus, we can express the outage probability as

poutage , Prob[I < SE]. (6.4)

6.1.3 Incremental AF with Multiple Relays

In this subsection, we propose two schemes for incremental AF (IAF) with multiplerelays. The two schemes are called 1+n IAF and 1+1 IAF. Before describing the twoproposed schemes in detail, we first define a non-cooperative scheme and two non-incremental AF schemes (i.e., 1 + n AF and 1 + 1 AF) as baselines for the purpose ofcomparison.

110


6.1.3.1 Non-Cooperative Scheme

In this subsection, we define the trivial direct transmission between source anddestination as the non-cooperative scheme. One could easily notice that the totalpower consumed in a cooperative scheme (i.e., source plus n relays) is Ps +

∑

n Ps/n =2Ps. Therefore, for fairness in terms of total consumed power with a non-cooperativescheme (i.e., only the source transmits to the destination without the help of anyrelay), we let the source transmit with the power 2Ps in a non-cooperative scheme.With these assumptions, the mutual information becomes

I = log2

(

1 +2Ps

N0

|hs,d|2)

. (6.5)

We consider that the source is transmitting with spectral efficiency SE. Fromequation (6.4), an outage event occurs when I < SE. Therefore, the outage proba-bility for the non-cooperative scheme is given by

poutagenon−coop = Prob

[

|hs,d|2 <2SE − 12 · SNR

]

, (6.6)

where SNR = Ps/N0 and |hs,d|2 is an exponentially distributed random variable withthe parameter σ−2

s,d .

6.1.3.2 Non-Incremental Cooperative Schemes

In this subsection, we define two non-incremental AF schemes, called 1 + n AFand 1 + 1 AF as baselines for benchmark. In literature ([33] and [160]), these non-incremental AF schemes are also called fixed relaying schemes. As illustrated inFig. 6.1, the 1 + n AF scheme lets each relay transmit in consecutive time slots afterthe source transmits. On the other hand, in the 1 + 1 AF scheme, all the relaystransmit concurrently in the same time slot immediately after the source transmits.As such, 1 + n AF scheme consumes a total of 1 + n time slots and 1 + 1 AF schemetakes up 2 time slots, respectively.

Of course, there is always the option of single relay cooperation where the relaywill repeat n times the message, but this method is not an option for this study wherewe have considered energy limitation constraints (otherwise some devices might drainout of power too fast).

For the 1 + n AF scheme, following the method in [108], the mutual informationcan be derived as

I1nAF = log2

(

1 + SNR |hs,d|2 +n∑

i=1

(

SNR |hs,iβihi,d|2

β2i |hi,d|2 + 1

))

. (6.7)

Notice that contrary to the scenario in [108], the transmission time of the source isnot divided between relays and therefore the spectral efficiency SE of the source isnot equal to the spectral efficiency of the entire scheme when the relays cooperate.In the 1 + n AF scheme, the spectral efficiency decreases as long as we add relays

111


Figure 6.1 Two Non-Incremental AF Schemes with Maximal Ratio Combining

to cooperate. Therefore, the spectral efficiency of a consecutive transmission schemeconsists of 1 source and n relays would be SE/(n + 1). With these considerations, anoutage event occurs when I1nAF < SE/(n + 1). Hence, from (6.7), we can write theoutage probability for 1 + n AF scheme as

poutage1nAF = Prob

|hs,d|2 +n∑

i=1

(

|hs,iβihi,d|2

β2i |hi,d|2 + 1

)

<2

SEn+1 − 1SNR

, (6.8)

where |hs,d|2, |hs,i|2 and |hi,d|2 are exponentially distributed random variables withthe parameters σ−2

s,d , σ−2s,i and σ−2

i,d , respectively. The SNR term is chosen relative tothe source power and it is Ps/N0.

From [160], we compute the outage probability in high SNR regime, for Pi = Ps/nas

poutage1nAF ≈

1(n + 1)!

· 1σ2

s,d

n∏

i=1

nσ2s,i + σ2

i,d

σ2s,iσ

2i,d

2SEn+1 − 1SNR

n+1

. (6.9)

Similar to the 1 + n AF scheme, the mutual information for the 1 + 1 AF schemecan be derived as

I11AF = log2

(

1 + SNR |hs,d|2 +

(

SNR (∑n

i=1 |hs,iβihi,d|)2

∑ni=1 β2

i |hi,d|2 + 1

))

. (6.10)

Since the 1+1 AF scheme has a constant spectral efficiency of SE/2, an outage eventoccurs when

|hs,d|2 +

(

(∑n

i=1 |hs,iβihi,d|)2

∑ni=1 β2

i |hi,d|2 + 1

)

<2

SE2 − 1

SNR. (6.11)

Notice that the expressions (6.8) and (6.11) present some similarities to those in[33, 110, 111, 160] but the derivation details are different. Instead of keeping the SEconstant and dividing the source transmission time among the n relays, our derivationsneed to keep the source transmission time constant due to the system model describedearlier. This leads to a decreasing SE when relays retransmit the useful data.

112


6.1.3.3 Incremental Cooperative Scheme

In this subsection, we describe the details of the proposed 1 + n IAF scheme and1 + 1 IAF scheme. In these two schemes, there are multiple relays that will helpthe source to transmit toward the destination only if the original transmission fromsource to destination was not successful.

Figure 6.2 Two Incremental Cooperative Schemes with Maximal Ratio Combining

In the 1 + n IAF scheme, as illustrated in Fig. 6.2, duration of the second phaseis not deterministic because the phase will end immediately after the i-th relay’stransmission that leads to a successful reception at destination, where i ≤ n. Inother words, relay i will retransmit the overheard transmission only if the sourceand the relays 1, .., i− 1 transmit unsuccessfully. In this case, the total transmissionpower consumed is i · Ps/n ≤ Ps and the total allocated power for the entire scheme(i.e., source plus n relays) is i · Ps/n + Ps ≤ 2Ps. This means that for 1 + n IAF,we may actually transmit less power than the other schemes, if i < n. This issueoccurs because one cannot a priori do a perfect power allocation over relays sincethe number of relays needed to cooperate incrementally is unknown before the eventoccurs. Again, if the source is successful, the power used by the scheme will bePs < 2Ps as the source has no a priori information that it will be successful andtherefore is not consuming the full power 2Ps as in the direct transmission.

Similarly to 1 + n IAF scheme, the 1 + 1 IAF scheme may not have the secondphase if the source transmits successfully in the first phase. If the second phaseexists, all the n relays will transmit concurrently in a single time slot. In this scheme,if the source transmits in the first phase with the power Ps and if the transmission todestination is unsuccessful, all n available relays will transmit together in one singletime slot with Ps/n. Hence, the total power equals 2Ps. But, if the source is successfulin the first phase, the total transmission power of this scheme will remain Ps < 2Ps.

Practically, in both IAF schemes, the effect of the power issue is visible in theexpression of the mutual information Is,d = log2(1 + SNR|hs,d|2), instead of Is,d =log2(1 + 2SNR|hs,d|2) from the non-cooperative scheme. By using less power, theproposed IAF schemes could be suboptimal. Ideally, the IAF schemes should know

113


the exact number of relays needed and spread the allocated full power 2Ps only amongthe used relays, but this is not practical.

In 1 + n IAF scheme, the protocol operates at spectral efficiency SE when thetransmission from the source is successful, at spectral efficiency SE/2 when one relayis successful but the source is not, at spectral efficiency SE/(n + 1) when the n-threlay successfully transmits with respect to the failure of the previous transmissions,and so on (see Fig. 6.2).

For high SNR the source transmits to the destination using one slot (e.g., onlythe source transmits and the relays do not cooperate), while for low SNR the schemeuses all n + 1 slots (e.g., all the relays cooperate).

Given the average spectral efficiency SE ∈ [SE/(n + 1), SE], the outage eventoccurs in the proposed 1 + n IAF scheme when

|hs,d|2 +n∑

i=1

(

|hs,iβihi,d|2

β2i |hi,d|2 + 1

)

<2SE − 1SNR

. (6.12)

In equation (6.12), one needs to compute the average spectral efficiency SE. If thedestination performs MRC on the signal received both from the source and the relays,then the expected spectral efficiency can be expressed as

SE = SE · Prob

[

|hs,d|2 ≥2SE − 1SNR

]

+

SE

2· Prob

|hs,d|2 <2SE − 1SNR

, |hs,d|2 +|hs,1β1h1,d|2

β21 |h1,d|2 + 1

≥ 2SE

2 − 1SNR

+ . . .

SE

n· Prob


, |hs,d|2 +|hs,1β1h1,d|2

β21 |h1,d|2 + 1

<2

SE2 − 1

SNR, . . .

|hs,d|2 +n−1∑

i=1

(

|hs,iβihi,d|2

β2i |hi,d|2 + 1

)

≥ 2SEn − 1

SNR

+ (6.13)

SE

(n + 1)· Prob


, |hs,d|2 +|hs,1β1h1,d|2

β21 |h1,d|2 + 1

<2

SE2 − 1

SNR, . . .

|hs,d|2 +n−1∑

i=1

(

|hs,iβihi,d|2

β2i |hi,d|2 + 1

)

<2

SEn − 1

SNR

.

The complicated expression in equation (6.13) can be simplified by taking intoaccount the statistical independence of the events. Another way to reduce the com-plexity of this formula is to introduce an upper and a lower bound. We can com-pute the bound for SE by considering an IAF transmission scheme that worksin two steps: (a) only the source transmits and (b) n relays contribute togetherin n different slots. We can therefore assume that SE = SE × Prob[|hs,d|2 ≥

114


2SE−1SNR

] + SE/(n + 1)× Prob[|hs,d|2 < 2SE−1SNR

], which leads to

SE = SE × exp

(

− 2SE − 1SNR · σ2

s,d

)

+

SE

n + 1

(

1− exp

(

− 2SE − 1SNR · σ2

s,d

))

=SE

n + 1

(

1 + n · exp

(

− 2SE − 1SNR · σ2

s,d

))

. (6.14)

This formula provides a good approximation for SE and is easier to compute. Itcan easily be seen that the values taken by SE are inside the interval

[

SEn+1

, SE]

.Subsequently, the outage probability for 1 + n IAF can be computed using SE.

Contrary to the 1 + n IAF scheme, the outage probability for the proposed 1 + 1IAF can be determined in a simpler manner. In 1 + 1 IAF, the protocol operatesat spectral efficiency SE or SE/2 depending on whether or not there is a secondphase. If the source is successful in the first phase, the spectral efficiency is SE. Ifthe source is not successful in the first phase and n relays help simultaneously in thesecond phase, the spectral efficiency becomes SE/2 (see Fig. 6.2). Then, one cancompute the outage probability for 1 + 1 IAF as

poutage11IAF = Prob

[

IAF < SE]

, (6.15)

where an outage event occurs when

|hs,d|2 +

(

(∑n

i=1 |hs,iβihi,d|)2

∑ni=1 β2

i |hi,d|2 + 1

)

<2SE − 1SNR

. (6.16)

Since the effective spectral efficiency is no longer of constant value, one needs tocompute the average spectral efficiency SE as SE · Prob

[

|hs,d|2 ≥ 2SE−1SNR

]

+ SE/2 ·Prob

[

|hs,d|2 < 2SE−1SNR

]

. Therefore,

SE = SE · exp

(

− 2SE − 1SNR · σ2

s,d

)

+

SE

2·[

1− exp

(

− 2SE − 1SNR · σ2

s,d

)]

=SE

2·[

1 + exp

(

− 2SE − 1SNR · σ2

s,d

)]

= f(SE, SNR), (6.17)

and the average spectral efficiency is not dependent on the success of the secondphase.

At high SNR, equation (6.17) becomes SE = SE, i.e., the source is always suc-cessful. At low SNR, equation (6.17) becomes SE = SE/2, i.e., source always needsrelaying. Also, as one can see in Fig. 6.3, the condition SE ∈

[

SE2

, SE]

is alwaysvalid.

115


2

4

6

8

−100

1020

30400

1

2

3

4

5

6

7

8

SE [b/s/Hz]SNR [dB]

expec

ted S

E [

b/s

/Hz]

1

Figure 6.3 SE as a function of SE and SNR for σ2s,d = 1

One may notice that instead of describing equation (6.17) as the average value ofrandom variable SE, we can describe it as SE = SE/k, where

k =2

1 + exp(

− 2SE−1SNR·σ2

s,d

) (6.18)

is the average number of time slots used in a cycle of cooperative transmission.


Comparing the two proposed IAF schemes to the two respective AF schemes,we expect to see an improved spectral efficiency and a lower energy consumptionfor IAF in high SNR conditions, even if the power allocation is not perfect and theIAF schemes are not operating at full power 2Ps. We introduce the normalized termsSNRnorm and SEnorm, where SNRnorm is the SNR normalized by the minimum SNRrequired to achieve a spectral efficiency SE and SEnorm is the spectral efficiencynormalized by the maximum achievable spectral efficiency [108]. As the spectralefficiency and the number of used time slots are different for each of the schemes, wehave to redefine the normalization metrics for an equitable performance comparisonin terms of outage probability.

For the trivial non-cooperative scheme, the two normalized terms are computed

116


as follows:

SNRnorm =2SNR

(2SE − 1), (6.19)

and SEnorm =SE

log2

(

1 + 2SNRσ2s,d

) . (6.20)

In Table 6.1 we list the expressions of SNRnorm for different scenarios on the AFand IAF schemes. In order to compare the schemes we first have to normalize theoutage probability in terms of necessary transmission time and spectral efficiency. Forexample, the 1+1 AF scheme uses per each transmission cycle twice the transmissiontime of the source and half of the spectral efficiency.

Table 6.1 SNRnorm values for AF and IAF schemesScheme SNRnorm(SE, SNR, k) Number of slots k

1 + 1 AF SNRnorm = kSNR

2SEk −1

k = 2

1 + n AF k = n + 11 + 1 IAF SNRnorm = kSNR

2SE

k −1

k = 2

1+exp

(

− 2SE−1

SNRσ2s,d

) ∈ [1, 2]

1 + n IAF k = n+1

1+n·exp

(

− 2SE−1

SNRσ2s,d

) ∈ [1, n + 1]

The system can be parameterized by the pair (SNRnorm, SE) or by the pair(SNR, SEnorm), as the two parameterizations are duals of one another [108]. Giventhe outage events described by (6.8), (6.11), (6.12) and (6.16), and the normalizedterms defined above, one may easily recompute the expressions of the outage eventsin terms of SNRnorm.

In order to compute the probability of an outage event, we have used for simu-lations exponential distribution generators for each link s − d, s − i and i − d. Thechannels are first considered to be symmetrical with σ2

s,d = σ2s,i = σ2

i,d = 1. We havecompared the schemes in terms of their outage probabilities for different normalizedsignal to noise ratios SNRnorm.

We notice that in low SNR case (i.e., all the relays collaborate), poutage1nAF (equation

(6.8)) and poutage1nIAF (equation (6.12)) are converging towards the same result. The same

observation can be made regarding poutage11AF (equation (6.11)) and poutage

11IAF (equation(6.16)). Thus, at low SNR, 1 + n AF becomes similar to 1 + n IAF, and 1 + 1 AFbecomes similar to 1 + 1 IAF. This is confirmed by Fig. 6.4 and Fig. 6.6, respectively.

At high SNR, the gap between 1 + n AF and 1 + n IAF is almost constant. Theexplanation supported by this result is that at high SNR IAF does not use any relayso it has a spectral efficiency SE, while AF has a fixed allocated relay schedule witha spectral efficiency SE/n + 1 and it seems reasonable to conclude that the gap isproportional with the number of relays used. Also, it is reasonable to conclude thatincremental AF schemes present more robustness to fading compared to AF schemes.

117


0 5 10 15 20 25 30

10−6

10−5

10−4

10−3

10−2

10−1

100

SNRnorm

[dB]

Outa

ge

Pro

bab

ilit

y

n=1

n=2

n=10

n=1

n=10

No Cooperation

1+n AF MRC

1+n IAF MRC

Figure 6.4 Performance of 1 + n AF ’⋄’ vs. Performance of 1 + n IAF ’◦’, σ2s,d = σ2

s,i =

σ2i,d = 1, Pi = Ps/n

0 5 10 15 20 25 30

10−6

10−5

10−4

10−3

10−2

10−1

100

SNRnorm

[dB]

Outa

ge

Pro

bab

ilit

y

No Cooperation

1+n AF MRC

n=10 n=1n=1

1+n IAF MRC n=10

Figure 6.5 Performance of 1 + n AF ’⋄’ vs. Performance of 1 + n IAF ’◦’, σ2s,d = 1;

(σ2s,i = 0.1 and σ2

i,d = 10) or (σ2s,i = 10 and σ2

i,d = 0.1), Pi = Ps - sameharmonic mean and therefore equivalent Outage Probability

118


In Fig. 6.4 and Fig. 6.6, one can see at high SNR an improvement in terms ofoutage probability brought about by the proposed 1 + n IAF compared to 1 + 1 IAF.At low SNR, 1 + 1 IAF outperforms 1 + n IAF if n is large enough.

Comparing Fig. 6.5 to Fig. 6.7 one may notice that under certain conditions (e.g.,low variance of the source to relays channels, corresponding to a higher noise) 1 + nIAF outperforms 1 + 1 IAF. On the other hand, as shown by comparing Fig. 6.5 toFig. 6.8, one may notice that it makes sense to use 1 + 1 IAF when the channels fromsource to relays are good (e.g., high variance of the source to relays channels). Theexplanation supported by this result is that under 1 + 1 IAF scheme the destinationcannot perform MRC separately on each signal retransmitted by the relays sincethe relays transmit together in the same slot and therefore the diversity is not fullyexploited. Comparing 1 + 1 IAF to 1 +n IAF, we observe that 1 + 1 IAF deals with aloss in diversity but it has a gain in terms of spectral efficiency. Therefore it is alwaysa compromise between 1 + n IAF and 1 + 1 IAF, but if the channels from source torelays are good, the gain in terms of spectral efficiency seems to be higher and it isrecommended to use 1 + 1 IAF.

0 5 10 15 20 25 30

10−6

10−5

10−4

10−3

10−2

10−1

100

SNRnorm

[dB]

Outa

ge

Pro

bab

ilit

y

No Cooperation

1+1 AF MRC

1+1 IAF MRC

n=1

n=10

Figure 6.6 Performance of 1 + 1 AF ’⋄’ vs. Performance of 1 + 1 IAF ’◦’, σ2s,d = σ2

s,i =

σ2i,d = 1, Pi = Ps/n

Similarly to [17, 18], for 1 + n AF and 1 + n IAF strategies with Pi = Ps, relayselection can be done under harmonic mean policy which means that the best relay

is the relay i with the maximum|hs,i|

2|hi,d|2|hs,i|

2+|hi,d|2 and therefore switching the quality of

the channel from source-relay link to relay-destination link under the same harmonicmean is not changing the performance - see Fig. 6.5. On the other hand, for 1 + 1AF and 1 + 1 IAF this policy is no longer available. Moreover, a cooperation with

119


0 5 10 15 20 25 30

10−6

10−5

10−4

10−3

10−2

10−1

100

SNRnorm

[dB]

Outa

ge

Pro

bab

ilit

y

n=1

n=10

1+1 AF MRC

1+1 IAF MRC

No Cooperation

Figure 6.7 Performance of 1 + 1 AF ’⋄’ vs. Performance of 1 + 1 IAF ’◦’, σ2s,d = 1,

σ2s,i = 0.1, σ2

i,d = 10, Pi = Ps

0 5 10 15 20 25 30

10−6

10−5

10−4

10−3

10−2

10−1

100

SNRnorm

[dB]

Outa

ge

Pro

bab

ilit

y

n=10

n=1

1+1 AF MRC

No Cooperation1+1 IAF MRC

Figure 6.8 Performance of 1 + 1 AF ’⋄’ vs. Performance of 1 + 1 IAF ’◦’, σ2s,d = 1,

σ2s,i = 10, σ2

i,d = 0.1, Pi = Ps

120

6.2 A Practical Incremental Relaying Scheme with Imperfect Feedback for Wireless Networks

good source-relay links is better than a cooperation with good relay-destination links,under the same harmonic mean - see Fig. 6.8 compared to Fig. 6.7. For 1 + n AFand 1 + n IAF strategies with Pi = Ps/n, relay selection can be done under themodified harmonic mean policy which means that the best relay is the relay i with

the maximum|hs,i|

2|hi,d|2n|hs,i|

2+|hi,d|2 , policy that can also be noticed from equation (6.9) under

high SNR regime. As one can see, under these strategies we can no longer switch thequality of the channel on the source-relay link with the relay-destination one withoutchanging the system.

6.1.5 Conclusions

In Section 6.1 we proposed two incremental AF schemes that use multiple relays.The proposed schemes are different from the existing ones in the sense that theyare closer to what would happen in practice [136]: if the transmission from source todestination is not successful, the reasonable choice is to decrease the spectral efficiencyof the system without dynamically adjusting time slot duration to maintain a certainspectral efficiency (as in previous works, e.g., [108, 110, 111, 160]). We have derivedand compared the outage probabilities of the proposed schemes. Numerical resultsconfirm that the proposed schemes can indeed achieve a lower outage probability ora higher spectral efficiency. Lastly, it has been shown that the relay harmonic meanselection policy is valid only under certain conditions and is strictly related to thecooperation method.

6.2 A Practical Incremental Relaying Scheme with

Imperfect Feedback for Wireless Networks

6.2.1 Introduction

In literature, there are several cooperative relaying schemes derived from DF andAF, but the most efficient one is incremental AF (IAF) [108]. We recall that IAFimproves the performance of AF by dynamically allocating a time slot to a relay onlyif the destination has not successfully received the original transmission from source.There are two ways to do the dynamic time slot allocation in IAF: (a) the first wayis to use the estimated channel state information [17, 18, 188], and (b) the secondway is to use immediate feedback from destination [97, 98, 99, 101]. In a practicalwireless network, the need of channel estimation may give raise to several issues suchas computational complexity. Also, some extra energy-costly communications areprobably needed for the channel estimation, and we should always think about thepower constraints imposed by the standard and current technology. We would alsohave to consider that even if the channel state information is known, instantaneous

This work has been done when Dorin Panaitopol was attached to the Institute for InfocommResearch under CCWnet Project (CR-2007-013). The project focuses on research and developmentof cooperative and coordinated techniques to improve performance of wireless networks.

121


channel state information is very difficult to determine because the channel estimationchanges over each measurement. As a consequence, for incremental relaying, weconsider the use of immediate feedback from destination.

In a practical wireless network, the feedback from destination is affected by prop-agation impairments. In the presence of imperfect feedback information, both thesource and the relay may over-react resulting in packet collisions. In this section,we propose a practical incremental AF (PIAF) scheme that deals with the imperfectfeedback information. The rest of Section 6.2 is organized as follows. In Subsection6.2.2, we define the simple non-incremental AF scheme, and the existing idealizedIAF scheme that assumes perfect feedback information [100], as the baselines beforepresenting the proposed PIAF scheme. Subsection 6.2.3 presents and discusses thenumerical results before ending the Section 6.2 with concluding remarks in Subsection6.2.4.

6.2.2 Practical Incremental Relaying with Imperfect Feed-back

In the followings, we consider a simplified model as the one described in Section6.1.2 that involves only one relay. The performance criteria for comparing the differentAF schemes is the outage probability (see equation (6.4)).

6.2.2.1 Non-Cooperative Scheme

In this subsection we describe the classic non-cooperative scheme and its theoret-ical background. As illustrated in Fig. 6.9, in a non-cooperative scheme the source Stransmits to the destination D without the help of a relay R. In this figure, after eachtransmission from S, there is a short time period labeled as A/N (A for ACK andN for NACK) within which the destination may provide feedback on correct (ACK)or incorrect (NACK) signal reception. For simplicity, we assume that the necessarytime for the destination to transmit feedback is negligible in comparison with thetime needed to transmit a packet. Also, as it can be seen in Fig. 6.9, we considerfixed length slots with a spectral efficiency SE per slot.

Figure 6.9 Two Non-Cooperative Schemes: (a) source S transmits continuously in all theslots toward destination, (b) source S transmits with lower spectral efficiency

122


The mutual information for a non-cooperative scheme is I = log2

(

1 + SNR |hs,d|2)

where hs,d is the fading channel coefficient of the link source-destination and |hs,d|2 isan exponentially distributed random variable with the parameter σ−2

s,d .If we assume that the source is transmitting with spectral efficiency SE as in

Fig. 6.9(a), from equation (6.4), an outage event occurs when I < SE. Therefore,the outage probability for the non-cooperative scheme is given as


[


]

= 1− exp

(

− 2SE − 1SNR · σ2

s,d

)

, (6.21)

where SNR = Ps/N0 with Ps the source power, N0 is the Gaussian noise power, and|hs,d|2 is exponentially distributed.

If the source is transmitting with spectral efficiency SE/2 as in Fig. 6.9(b), theoutage probability is


[

|hs,d|2 <2SE/2 − 1

SNR

]

= 1− exp

(

− 2SE/2 − 1SNR · σ2

s,d

)

. (6.22)

6.2.2.2 AF: Non-Incremental Cooperative Schemes

The advantage of using these schemes is that the destination can perform MRC onthe same signal transmitted by the source and repeated by the relay, thus providingextra fading protection. In Fig. 6.10, we have presented two variants of AF MRCschemes. Since both use fixed time frame scheduling, the relay always cooperates evenif the source succeeds in the first place, resulting in suboptimal resource allocation.The difference between the two schemes is that the first one allows for a turn-aroundtime TR which ensures that the relay has the time to receive all the information fromthe source before transmitting it further.

Figure 6.10 Two Non-Incremental AF Schemes with Maximal Ratio Combining

Notice that the difference between this scenario and the one in [108] is that herethe transmission time of the source is not divided between the relays and thereforethe spectral efficiency SE of the source is not equal to the spectral efficiency of theentire scheme where the relays cooperate.

The mutual information for the AF scheme can be derived as

IAF = log2

(

1 + SNR |hs,d|2 +SNR |hs,rβrhr,d|2

β2r |hr,d|2 + 1

)

, (6.23)

123


where

βr =

√

√

√

√

Ps

Ps |hs,r|2 + N0

(6.24)

is the relay amplification factor, hs,r and hr,d are the fading coefficient of the linksource-relay (S − R) and relay-destination (R − D) respectively. |hs,r|2 and |hr,d|2are exponentially distributed random variables with the parameters σ−2

s,r and σ−2r,d

respectively.Since the AF scheme has a constant spectral efficiency of SE/2, an outage event

occurs when |hs,d|2 + |hs,rβrhr,d|2β2

r |hr,d|2+1< 2

SE2 −1

SNR.

6.2.2.3 IAF: Idealized Incremental Cooperative Scheme

In this subsection, we describe the proposed IAF scheme in detail. In all threemethods presented in Fig. 6.11, the relay decides to transmit the information towarddestination only if the original transmission from source to destination is not success-ful. The first two methods (i.e., Fig. 6.11(a) and (b)) are simplified in the sense thatthere is only partial feedback from destination (only ACK or only NACK). In thissubsection, all three methods are idealized in the sense that the feedback is alwayssuccessful and correct.

Figure 6.11 Three Incremental AF Schemes with Maximal Ratio Combining: (a) D sendsonly ACK, (b) D sends only NACK, (c) D sends both ACK and NACK

For the three IAF variants, with successful feedback from destination, the protocoloperates at spectral efficiency SE when the transmission from the source is successful,and at spectral efficiency SE/2 when one relay is successful but the source is not.From the theoretical point of view, if there is a successful dynamic feedback fromdestination, all the schemes share the same performance.

Since the effective spectral efficiency is no longer a constant value, one needs tocompute the average spectral efficiency SEIAF as SE ·Prob

[

|hs,d|2 ≥ 2SE−1SNR

]

+ SE 12·

Prob[

|hs,d|2 < 2SE−1SNR

]

. As equation (6.17), SEIAF = SE2·[

1 + exp(

− 2SE−1SNR·σ2

s,d

)]

=

124


f(SE, SNR). The condition SEIAF ∈[

SE2

, SE]

is always valid as one can see inFig. 6.3.

An outage event will happen when |hs,d|2 + |hs,rβrhr,d|2β2

r |hr,d|2+1< 2SEIAF −1

SNR, and the outage

probability for the IAF scheme is given by

poutageIAF = Prob

|hs,d|2 +|hs,rβrhr,d|2

β2r |hr,d|2 + 1

<2SEIAF − 1

SNR

. (6.25)

Please also note that in the case when the source S decides to help in the secondslot together with the relay, the outage probability (6.25) becomes

poutageIAF = Prob

|hs,d|2 +|hs,d|2 + |hs,rβrhr,d|2

β2r |hr,d|2 + 1

<2SEIAF − 1

SNR

. (6.26)

One may notice that instead of describing SEIAF as a mean value of random vari-

able SE, we can describe it as SEIAF = SE/k, where k = 2/(

1 + exp(

− 2SE−1SNR·σ2

s,d

))

is the average number of time slots used in a cycle of cooperative transmission.Even if the three variants look similar we will show that small differences such

as using only ACK feedback, only NACK feedback, or both ACK/NACK have a bigimpact over the entire implementation process.

We notice that these three schemes encounter no implementation issues if thesource S or the relay R receive under perfect conditions the feedback from the desti-nation D. But in a practical network, perfect feedback is a very strong assumptionas it is quite probable that the feedback link may be broken.

In the next subsection, we show what happens in any of these schemes if thefeedback link breaks. We also present a novel scheme where we avoid packet collisionsand we model the feedback from destination in terms of outage probability. This novelmodeling is very useful as it helps us to understand and to present the performanceof the incremental relaying scheme under real radio channel characteristics.

6.2.2.4 Issues Related to the Implementation of Incremental CooperativeSchemes

In this subsection, we consider that the source S has transmitted the packet andthe destination D has to feedback. One could easily notice that there are somesituations when the feedback from the destination has not been received. Thesesituations lead to implementation problems since incremental AF becomes difficultto optimally implement. Even more, if one transforms a non-incremental schemeinto an incremental scheme without a reliable feedback, collisions might easily occur.Considering the nature of the feedback (i.e., ACK or NACK) and the recipients of thecorrect feedback (i.e., source, destination, both or none), we present each situationthat might occur. Based on all these situations, we propose a suboptimal incrementalAF cooperative scheme that performs better than non-incremental AF, is easier toimplement and avoids any issues related to packet collisions or low spectral efficiencyallocation.

125


In the first strategy (see Fig. 6.12), we assume that the destination answers backwith ACK if the packet has been correctly received. If the packet has not beencorrectly decoded or the feedback link is broken, the source will not receive anything.We also assume that the source has to listen to the feedback during a guard periodin which the source verifies the state of the packet.

Figure 6.12 Strategy with ACK and without NACK transmission

Because of the proposed frame structure, if the destination sends ACK but thelink D − R is broken, the relay R will assume that an NACK has been transmittedby destination since it cannot decode the feedback. Therefore, in the second timeslot, the relay R will decide to help. On the other hand the source S has received thefeedback ACK and it will decide to send its own new packet. Therefore, both R andS will send different packets during the second time slot and a collision will occur.

For the second strategy (see Fig. 6.13), we assume that the destination sendsNACK and instead of sending ACK it simply keeps a free guard period. Now, imaginethe situation when the destination sends NACK but the link D−S is broken. In thiscase, a collision will occur as well.

From both previous schemes, we have reached the conclusion that we have to useboth ACK and NACK for the feedback (i.e., full feedback), this information beingvery useful to determine whether the link is broken or not.

126


Figure 6.13 Strategy with NACK and without ACK transmission

6.2.2.5 A Novel Practical Incremental Cooperation Scheme with Feed-back

One may notice that the full feedback condition is necessary but not sufficientas collisions might still occur. In order to avoid this in a scenario where S andR cannot exchange information (e.g., very simple devices as sensors), we propose asimple algorithm where at the same time:

– source S is pessimistic about the feedback (i.e., if the link D − S is broken, Swill consider that NACK has been send)

– relay R is optimistic about the feedback (i.e., if the link D−R is broken, S willconsider that ACK has been send).

In the next paragraphs, we will graphically show and prove (in Fig. 6.14 andFig. 6.15) that these two conditions are necessary and sufficient to avoid any collision.

One can easily notice that there are 8 possible cases resulted from 3 independentvariables that might occur: the destination sends ACK or NACK, the link D − Sbreaks or not, the link R − S breaks or not. For convenience, we divide these 8cases into 4 heterogeneous channels (see Fig. 6.14 - scenarios 1), 2), 3) and 4)) and 4homogeneous channels (see Fig. 6.15 - scenarios 5), 6), 7) and 8)).

We further consider three possible source configurations:a) source S is not retransmitting the information in the second slot,b) source S reacts by retransmitting the information in the second slot only when

NACK is explicitly received (i.e., active source),c) source S reacts by retransmitting the information in the second slot every time

NACK is received, or the feedback link is broken (i.e., proactive source).

127


Under heterogeneous channel conditions (see Fig. 6.14), we never encounter a per-fect feedback reception, but instead there is always someone who hears the feedback(e.g., either S or R). In Fig. 6.14, we have illustrated all the possible combinationsthat might appear under these assumptions.

A given combination of feedback (ACK or NACK) and broken link (D − S orD−R) is characterized by one pair of values: spectral efficiency and event probabil-ity. In Table 6.2, we describe all these pairs of values under heterogeneous channelassumptions.

Table 6.2 Useful Parameters for Heterogeneous Channel ConditionsUse Feedback Broken Spectral EventCase From D Link Efficiency Probability

1) NACK D − S SE/2 Psd(1− Pdr)Pds

2) ACK D − S SE/2 (1− Psd)(1− Pdr)Pds

3) NACK D −R SE/2 PsdPdr(1− Pds)4) ACK D −R SE (1− Psd)Pdr(1− Pds)

The parameters Psd, Pdr and Pds from Table 6.2 are the outage probabilities ofthe direct transmission link, the feedback link D − R and the feedback link D − Srespectively. These parameters are computed with the equation (6.21) as follows:

Psd = 1− exp

(

− 2SE − 1SNR · σ2

s,d

)

, (6.27)

i.e., the probability that the destination feedback is NACK;

Pdr = 1− exp

(

− 2SE − 1SNR · σ2

d,r

)

, (6.28)

i.e., the probability that the D −R link breaks; and

Pds = 1− exp

(

− 2SE − 1SNR · σ2

d,s

)

, (6.29)

i.e., the probability that the D − S link breaks.Under homogeneous channel conditions (see Fig. 6.15), both links D−S and D−R

are similar in terms of channel performance: either both fail, or both succeed. Thesehomogeneous cases could disturb the most our scheme performance, as we will shownext.

The worst scenario from this category is when the destination sends NACK butboth links D − S and D − R are broken. In this scenario, for 5a), the scheme mightfinally recover but 2 slots are implicitly lost. On the other hand, the best scenariois when the destination sends ACK and both feedback links D − S and D − R areavailable. In this scenario, we approach the ideal IAF inference, as the slots aredynamically and optimally allocated.

As in Table 6.2, we list in Table 6.3 the spectral efficiency and the probability foreach situation under homogeneous channel assumptions.

128


Figure 6.14 Practical IAF, Heterogeneous Channel Conditions

129


Figure 6.15 Practical IAF, Homogeneous Channel Conditions

130


Table 6.3 Useful Parameters for Homogeneous Channel ConditionsUse Feedback Broken Spectral EventCase From D Links Efficiency Probability

5) NACK D − S SE/2 PsdPdrPds

D −R6) ACK D − S SE/2 (1− Psd)PdrPds

D −R7) NACK none SE/2 Psd(1− Pdr)(1− Pds)8) ACK none SE (1− Psd)·

·(1− Pdr)(1− Pds)

From Fig. 6.14 and Fig. 6.15, one may notice the following facts related to thetotal of 8 possible events of the source configuration a):

– There are 4 cases where the destination D performs MRC. Among these cases,there are only two cases with a full spectral efficiency SE. In these two cases,the packet is correctly received without any other help, but we can still assumethis under MRC;

– There are 4 cases where we transmit using only direct transmission, the destina-tion D does not perform MRC and the spectral efficiency is only SE/2. Thesenon-cooperative scenarios have an event probability of Pn−c,1 = (1 − Psd)(1 −Pdr)Pds (use case 2)), Pn−c,2 = PsdPdr(1−Pds) (use case 3)), Pn−c,3 = PsdPdrPds

(use case 5)) and Pn−c,4 = (1− Psd)PdrPds (use case 6)) respectively, and all ofthem are equivalent to a direct transmission with half of the spectral efficiency.Therefore, these cases (i.e., the destination sends ACK or NACK but both feed-back links are broken, the destination sends ACK but only the feedback linkD-S is broken, and the destination sends NACK but only the feedback link D-Ris broken) where the communication is not cooperative and we do not performMRC with the help of a relay, will decrease the performance of our incrementalscheme. In the next paragraph, we describe the influence of these cases over thetotal outage probability of the scheme.

We reunite the 8 events later, in the equation (6.33). We choose to compute anaverage spectral efficiency SE

coopP IAF for the cases where we do perform MRC and a

spectral efficiency SEnon−coopP IAF = SE/2 for the case where the destination will not

perform MRC. We compute SEcoopP IAF as

SEcoopP IAF =

∑ni=1,i6=non−coop SEi · Pi∑n

i=1,i6=non−coop Pi

, (6.30)

where n = 8 is the total number of the events from the Tables 6.2 and 6.3, withthe pair (SEi, Pi) representing the spectral efficiency and probability of each event irespectively, and we find

SEcoopP IAF = SE

1− Pds + PsdPds − 12

(Psd + PsdPdr)1− Pds − PsdPdr + PdsPsd

. (6.31)

In order to analyze the bounds of SEcoopP IAF , it suffice to consider SNR→∞ (i.e.,

Psd = Pds = Pdr → 0) and SNR→ 0 (i.e., Psd = Pds = Pdr → 1).

131


In high SNR conditions SEcoopP IAF ր SE, while in low SNR conditions, although

it is less obvious, SEcoopP IAF ց SE/2. In order to prove this, we introduce a variable

z, z = Psd = Pds = Pdr → 1 (i.e., there is outage all the time).From equation (6.31) we get

limz→1

SEcoopP IAF = SE · lim

z→1

1− z + z2 − 12z − 1

2z2

1− z + z2 − z2

= SE · limz→1

−1 + 2z − 12− z

−1=

SE

2. (6.32)

Therefore, the condition SEcoopP IAF ∈

[

SE2

, SE]

remains valid.

The outage probability for the PIAF is poutageP IAF and it might be computed as

poutageP IAF =

(

1−4∑

i=1

Pn−c,i

)

· poutageIAF +

4∑

i=1

Pn−c,i · poutagenon−coop, (6.33)

where poutageIAF = f1(SE

coop

P IAF ) and poutagenon−coop = f2(SEnon−coop

P IAF ) are computed with equa-tions (6.26) and (6.22) respectively.

Similarly, from Fig. 6.14 and Fig. 6.15, one may notice the following facts relatedto the total of 8 possible events of the source configuration b):

– There are 5 cases where the destination D performs MRC;– There are 3 cases where we transmit using only direct transmission, the des-

tination D does not perform MRC and the spectral efficiency is only SE/2.These non-cooperative scenarios have an event probability of Pn−c,1 = (1 −Psd)(1 − Pdr)Pds (use case 2)), Pn−c,2 = PsdPdrPds (use case 5)) and Pn−c,3 =(1 − Psd)PdrPds (use case 6)) respectively, and all of them are equivalent to adirect transmission with half of the spectral efficiency.

From (6.30) and from the Tables 6.2 and 6.3 we can express the average spectralefficiency SE

coopP IAF for the source configuration b) as

SEcoopP IAF = SE

1− Pds + PsdPds − 12Psd (1 + PdrPds)

1− Pds + PdsPsd − PsdPdrPds

. (6.34)

It can also be found that similar to (6.31), the condition SEcoopP IAF ∈

[

SE2

, SE]

remains valid.One can easily find that, for the source configuration b), the outage probability

for the PIAF is poutageP IAF and it might be computed as

poutageP IAF =

(

1−3∑

i=1

Pn−c,i

)

· poutageIAF +

3∑

i=1



coopP IAF ) and poutage

non−coop = f2(SEnon−coopP IAF ) are computed with equa-

tions (6.26) and (6.22) respectively.

132


Finally, for the source configuration c), one may notice that:– There are 6 cases where the destination D performs MRC;– There are 2 cases which can be still approximated as non-cooperative and with-

out MRC, because the source S retransmits the information even if the packethas been already successfully received during the first transmission. These "non-cooperative" scenarios have an event probability of Pn−c,1 = (1−Psd)(1−Pdr)Pds

(use case 2)) and Pn−c,2 = (1− Psd)PdrPds (use case 6)) respectively, and bothare equivalent to a direct transmission with half of the spectral efficiency.

Further, for this source configuration c), we can express the average spectral effi-ciency SE

coopP IAF as

SEcoopP IAF = SE

1− Pds + PsdPds − 12Psd

1− Pds + PdsPsd

, (6.36)

and it can also be found that similar to (6.31) and (6.34), the condition SEcoopP IAF ∈

[

SE2

, SE]

remains valid.

For the source configuration c), the outage probability for the PIAF is poutageP IAF and

it might be computed as

poutageP IAF =

(

1−2∑

i=1

Pn−c,i

)

· poutageIAF +

2∑

i=1



coop

P IAF ) and poutagenon−coop = f2(SEnon−coop

P IAF ) are computed with equa-tions (6.26) and (6.22) respectively.


In this section we present results only for the scenario corresponding to PIAF withthe source configuration b). We have chosen this use case with respect to the sourceconfiguration a) (which is suboptimal in terms of spectral efficiency but less powerconsuming) and c) (which is better in terms of spectral efficiency but more powerconsuming). We will further generally refer to this source configuration as PIAF.We should expect that the idealized IAF which assumes perfect dynamic feedbackoutperforms the PIAF, and this can be seen even from the quantity ∆ = SEIAF −SEP IAF ≥ 0 represented in Fig. 6.16.

From the quantity ∆ one may notice that SEP IAF matches SEIAF only at thebounds (e.g. for high and low SNR), while the maximum expected performancedifference is somewhere in the middle.

Comparing the proposed PIAF scheme to the respective AF scheme, we expect tosee an improved spectral efficiency and a lower energy consumption for IAF in highSNR conditions. As the spectral efficiency and the number of used time slots are notthe same for each of the schemes, we have to define new normalization metrics forperformance comparison. Therefore, we introduce the normalized terms SNRnorm

and SEnorm.

133


2

4

6

8

−100

1020

30400

0.2

0.4

0.6

0.8

1

SE [b/s/Hz]

SNR [dB]

∆ [

b/s

/Hz]

1

Figure 6.16 ∆ = f(SNR, SE) for σ2s,d = σ2

d,r = σ2d,s = 1

For the trivial non-cooperative scheme, the two normalized terms are computedas follows:

SNRnorm =SNR

(2SE − 1), and (6.38)

SEnorm =SE

log2

(

1 + SNR · σ2s,d

) . (6.39)

For AF scheme, the two normalized terms are computed as

SNRnorm =2SNR

(

2SE

2 − 1) , and (6.40)

SEnorm =SE/2

log2

(

1 + 2SNR · σ2s,d

) . (6.41)

For IAF scheme, the two normalized terms are computed as

SNRnorm =kSNR

(

2SE

k − 1) , and (6.42)

SEnorm =SE/k

log2

(

1 + kSNR · σ2s,d

) (6.43)

where k ∈ [1, 2], with k = 2/(

1 + exp(

− 2SE−1SNRσ2

s,d

))

for IAF scheme.

134


The system can be parameterized by the pair (SNRnorm, SE) or by the pair(SNR, SEnorm), as the two parameterizations are duals of one another [108].

For simulations, we have used exponential distribution generators for each linkS−D, S−R and R−D in order to compute the probability of an outage event. Thechannels are considered with σ2

s,d = σ2s,r = σ2

r,d = 1. In Fig. 6.17 and Fig. 6.18, wehave compared the schemes in terms of their outage probabilities poutage

non−coop, poutageAF ,

poutageIAF and poutage

P IAF for different normalized signal to noise ratios SNRnorm.

−10 −5 0 5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

100

SNRnorm

[dB]

Outa

ge

Pro

bab

ilit

y

non−cooperative

AF

IAF

PIAF

Figure 6.17 Outage Probabilities for σ2s,r = σ2

s,d = σ2d,s = σ2

d,r = σ2r,d = 1, e.g., the feedback

channel has the same quality as the direct channel

The results from Fig. 6.17 and Fig. 6.18 clearly show that our proposed PIAFscheme is very close to the idealized IAF and it outperforms AF and non-cooperativeschemes.

If there is a bad feedback channel relative to the direct channel we obtain resultssimilar to Fig. 6.19 and Fig. 6.20. In this scenario, the gap between IAF and PIAFis higher and we can actually notice that PIAF performance is oscillating somewherebetween the AF performance and idealized IAF. Notice that for low SNR, PIAFis approaching the performance of both AF and IAF while for high SNR, PIAFapproaches the performance of IAF.

6.2.4 Conclusions

Section 6.2 studies the problem of a practical and implementable incrementalrelaying scheme where the feedback from destination is not perfect. In this section,we have proposed a suboptimal incremental relaying scheme called PIAF, that candeal with issues resulted from the indeterminacy of successful transmission given an

135


−5 0 5 1010

−2

10−1

100

SNRnorm

[dB]

Outa

ge

Pro

bab

ilit

y

non−cooperative

AFIAF

PIAF

Figure 6.18 Zoom on Outage Probabilities for σ2s,r = σ2

s,d = σ2d,s = σ2

d,r = σ2r,d = 1, e.g.,

the feedback channel has the same quality as the direct channel

−10 −5 0 5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

100

SNRnorm

[dB]

Outa

ge

Pro

bab

ilit

y

non−cooperative

AF

PIAF

IAF

Figure 6.19 Outage Probabilities for σ2s,r = σ2

s,d = σ2r,d = 1 and σ2

d,s = σ2d,r = 0.1,

equivalent to a bad feedback channel

136

6.3 On the Relay Selection and Cooperation for UWB

−5 0 5 10 15 2010

−4

10−3

10−2

10−1

100

SNRnorm

[dB]

Outa

ge

Pro

bab

ilit

y

non−cooperative

AF

PIAFIAF

Figure 6.20 Zoom on Outage Probabilities for σ2s,r = σ2


d,s = σ2d,r = 0.1,

equivalent to a bad feedback channel

imperfect feedback channel. In practice, such a scheme would be useful because itoutperforms the classical AF scheme under realistic conditions. Furthermore, PIAFachieves better performance by using a very simple algorithm requiring the source tobe pessimistic and the relay to be optimistic with respect to the destination’s lostfeedback. One can further improve the scheme by using 1 + 1 and 1 + n IAF orby simply using multi-sources. Another improvement could be that eventually, thesource might also help in retransmitting the information together with the relays (i.e.,proactive source).

6.3 On the Relay Selection and Cooperation for

UWB

6.3.1 Introduction

Section 6.3 deals with relay cooperation between a source S and a destination Dtaking into account UWB parameters. Questions such as ‘is it better to cooperate ornot to cooperate?’ or ‘how many sensors should cooperate?’ (too many cooperativesensors means interference, bit error, energy consumption) or ‘which is the easiestand the most efficient way to cooperate?’ have always been addressed and they stillrepresent interesting research fields [112, 183]. The truth is that there is no commonoptimal algorithm for cooperation, because there are various implementation issuesand physical layer characteristics that can not be taken into account due to highcomplexity, or simply the lack of straightforward solutions [53, 162, 167].

137


Figure 6.21 General relaying scheme between a source S and a destination D

This section proposes a simple distributed relay selection algorithm. Based on anaccess parameter Pa computed by the destination, each relay from the cooperationgroup decides locally without the need to exchange additional control information,whether to cooperate or not. Now that the decision can be made locally, the relayselection becomes distributed but it still needs a proper estimation of Pa. Therefore,our goal is to find a suitable access parameter Pa in order to maximize the packetsuccess probability Psucc at the destination.

Similarly to the previous sections, the relays are helping to retransmit the informa-tion (herein packets of bits) from source to destination. We have adopted the physicallayer of Impulse Radio UWB [147] (see Fig. 6.22) and therefore we do not considerpacket based collisions [161, 162, 163, 164, 171]. In our approach, a bad reception isnot a result of the packet collision model but instead an outcome of excessive noiseand/or interference inside the PHY layer.

Figure 6.22 General physical layer transmission scheme for Source to Destination (S-D)or Source to Relay (S-R) or Relay to Destination (R-D)

The description of UWB signal has been presented in Chapters 2 and 3. At thispoint, it is important to know only that each transmitted impulse goes through thechannel and at the other end (relay or destination) is correlated in the receiver withits know form (called mask).

138


6.3.2 Theoretical Model

The theoretical model is mainly used for the simple case of single path channel[11] with perfect power control but it can be extended to other models as well. Thismodel is used to exemplify the system’s parameters impact over the network.

Next, we would like to express the Signal to Interference and Noise Ratio SINR[11, 133, 135, 179] and we consider two possible cases, depending on the synchroniza-tion:

– Sent packets with uniform distributed delays among the relays.

SINR =Eb

σ2 + (Nu − 1) · Eb · m2(w)Nh

, (6.44)

where σ2 is the variance of the noise N0

2, Eb is the energy of one bit, m2(w) =

1Tc

∫∞−∞ γ2(z)dz is the mean of all the possible correlation interferences taking into

account a uniform distribution of the interference, and γ(z) =∫∞

−∞ w(t)w(t+z)dtis the correlation of the received pulse w(t) with its mask. Tc is the chip period,Nh is the number of slots and Nu the number of users/relays in our case. Wehave therefore considered that there is one useful relay, while Nu − 1 relays areproducing a level of interference Eb · m2(w)

Nh.

– Sent packets with perfect synchronized relays. This is equivalent to a receiveralways sampling the interference at time t = 0, so the output of the correlation isalways maximum, and it does not depend anymore on the mean m2 as previouslystated. Therefore, the SINR becomes

SINR =Eb

σ2 + (Nu − 1) · Eb · 1Nh

. (6.45)

6.3.3 Packet Success Probabilities

Under the Gaussian interference assumption

BERP AM =12

erfc

√

SINR

2

(6.46)

is the bit error probability. It is important to say that the Gaussian interference as-sumption is a simplified IR-UWB interference model that we only use in this section.Note that in Chapters 3-5 we showed that the IR-UWB interference can be bettermodeled as a generalized Gaussian or a mixture. Therefore, if the interference is ap-proximated by generalized Gaussian for example, one would have to use data profilingfrom BER curves of type BER = F(SINR, β), in order to have more precise BERvalues. These curves can be obtained by Monte-Carlo simulations (as in Fig. 3.1) andhave to be already embedded in the system. However, even if SGA is no longer valid,for uncoded packets with Ns = 1 the performance of different receivers seems to besimilar regardless of the type of interference (as we can see in Section 5.1.6.3).

139


From the bit error probability BERP AM , we can derive the packet error probabilityfor an uncoded packet as

Pe = 1− (1−BERP AM)L , (6.47)

where L is the packet length in bits. In our simulations, we send one bit per trans-mission frame and therefore the time length of one packet is Tpacket = L ·Tf . Since weconsidered uncoded packets, if one bit is not correct, the entire packet is considerederroneous [14].

6.3.3.1 Packet Success Probability (R-D)

In this subsection, we compute the packet success probability Psucc_RD for thetransmissions coming from the relays to destination, considering that all the Nu in-volved relays have perfectly decoded the information from the source S and theychoose to retransmit it using a parameter Pa. We identify two scenarios where (a)the destination can decode packets from the transmitting relays one at the time and(b) the destination can decode packets from all the transmitting relays at the sametime:

– Many relays may transmit but the destination’s receiver listens only to one (likein best relay selection):

Psucc_RD =Nu∑

j=1

CjNu

P ja (1− Pa)Nu−j Pj, (6.48)

where CjNu

= Nu!j!(Nu−j)!

, Pj = 1 − Pe (Nu = j) is the success probability when jnodes transmit together and Pa is an access parameter to be set by destination.Each relay will randomly decide either to participate or not to participate ac-cording to a uniformly distributed random number that is always compared tothe threshold Pa.

– The destination’s receiver listens to all the transmitting relays. There is a packeterror if all the packets are erroneously received (the receiver knows if the packetis correct or not). We use the random packet error distribution formula on allthe received streams from the relays:

Psucc_RD =Nu∑

j=1

CjNu

P ja (1− Pa)Nu−j

(

1− (1− Pj)j)

. (6.49)

In this case, assuming that errors on different streams are independent, andthat we can listen to everyone at the same time, we have a gain of 1−(1−Pj)j

Pj.

When only one node/relay is collaborating, this gain is 1 and is minimum.The beauty of this method resides in its simplicity. The destination does not need

to choose among the cooperative relays, therefore reducing the complexity of thescheme. The only thing that the destination does is sending the probability accessinformation Pa to the group of relays involved in cooperation and it is for them todecide. Furthermore, the relays do not have to communicate among them to decide

140


who is participating and who is not because the only information that they use issimply Pa. Therefore this algorithm becomes distributed.

6.3.3.2 Packet Success Probability (S-R-D)

In this subsection, we compute the packet success probability Psucc_SRD fromsource to relays to destination (S-R-D), considering that n relays from the Nu grouphave perfectly decoded the information and choose to transmit it with probability Pa.

We consider a few cases, depending on whether or not we can decode simultaneoustransmissions at the same time, and on whether the link S-D is available or not:

– The destination can decode the transmitting relays one by one:

Psucc_SRD =Nu∑

n=1

n∑

j=1

CnNu

CjnP j

a (1− Pa)n−j PjPn1 (1− P1)

Nu−n (6.50)

– The destination can decode the transmitting devices (including the source S)one by one:

Psucc_SRD = Psucc_SD + (1− Psucc_SD)Nu∑

n=1

n∑

j=1

CnNu

CjnP j

a (1− Pa)n−j ×

PjPn1 (1− P1)

Nu−n (6.51)

– The destination can decode all the transmitting relays at the same time:

Psucc_SRD =Nu∑

n=1

n∑

j=1

CnNu

CjnP j

a (1− Pa)n−j ×(

1− (1− Pj)j)

P n1 (1− P1)

Nu−n (6.52)

– All S-R-D and S-D links are available and the destination can listen to everyone:

Psucc_SRD = Psucc_SD + (1− Psucc_SD)Nu∑

n=1

n∑

j=1

CnNu

CjnP j

a (1− Pa)n−j ×(

1− (1− Pj)j)

P n1 (1− P1)

Nu−n . (6.53)

6.3.4 To Cooperate or Not to Cooperate?

It is somewhat obvious that if the destination has not received the useful informa-tion during the original transmission, in order to help the source transmit informationsto destination, it is better to cooperate. The question is not exactly whether or notto cooperate but more precisely which strategy should be applied:

1. Using the access parameter Pa sent by the destination to the relays

2. Each node does its best, and the destination cannot receive on multiple channels

141


3. Each node does its best, and the destination can receive on multiple channels4. Enable somehow a free channel (i.e., nodes collaborate to establish a “free chan-

nel” and only the best relay sends). The time used for collaboration means losttime for the destination in terms of received throughput (i.e., the relays are“arguing” who is transmitting instead of transmitting the information further),but also we have seen from previous sections (i.e., AF and DF schemes) that itis not the optimal strategy in high noise environments.

In Fig. 6.23 and Fig. 6.24, one can see a comparison of the first three policies.The fourth case falls into a minimum spectral efficiency policy (see the two sectionson AF and IAF schemes from the first two sections of this chapter) and it shall notbe discussed in this paragraph because comparing in terms of BER with the previouspolicies does not make sense. Comparing Fig. 6.23 (b) with Fig. 6.24, one can furthernotice that the gain of the first policy becomes more visible when L increases.

6.3.5 Matlab Results and C++ Implementation

For the experimental model we have used a C++ simulator with the followingparameters Nu = 10 relays, Nh = 10 slots, Tc = 1ns, τ = 0.5, L = 50bits for bothsingle and multi-path channels. Fig. 6.25 has been obtained under the assumption ofsingle path channel (the received pulse w(t) is considered to be the Scholtz’s pulse[179] and it has the expression from equation (2.2)). The effect of having a maximumPsucc on the graph for a certain Pa can be seen on every graph, but is underlined forsingle path channels inference. The only exception is Fig. 6.25(b), where the relaysare synchronized and send identical unscrambled packets toward destination. In thissituation, the destination could combine the signal from all the relays (i.e., MRC). Forsingle path channels it is easy to talk about the maximal ratio combining that helpscombining packets in order to increase the success probability, but our observationis that this technique can also be employed for real UWB channel if we change thetransmission technique (e.g., time reversal, technique employed in previous chapters).

Some observations that we can extract from all this figures are that the influence ofinterference becomes less important for UWB channels due to the fact that channelsare uncorrelated. In this case, the interference is lower and thus the success probabilityis higher. Fig. 6.26 has been obtained under the assumption of real UWB channel[55, 57] (the received pulse w(t) is now the result of the convolution of the channelwith the Scholtz’s pulse). As one might see, the model from Fig. 6.25(d) is very closeto the one from Fig. 6.26, a result that was not obvious.

Another remark is that Pa is not constant, but changes with the number of avail-able relays, and it has a constant average value for each channel model since it isdirectly related to the interference type.

In Fig. 6.27, one can find some comparisons between the theoretical model em-ployed and the results from simulations under Eb/N0 = 6dB. Last but not least, thetheoretical model employed is close to simulations, helping us to validate the model,as one could see in the same figure.

Again, we expect a higher gap between the max{Psucc} and the min{Psucc} whenL increases.

142


0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Eb/N

0 [dB]

Pac

ket

Succ

ess

Pro

bab

ilit

y P

s [%

]

Semi−cooperation between relays using Pa

No cooperation between relays without diversity

No cooperation between relays with diversity at destination

(a)

0 5 10 15 20 25 3010

−4

10−3

10−2

10−1

100

Eb/N

0 [dB]

Pac

ket

Err

or

Rat

e P

ER

[%

] No cooperation between relays without diversity


Semi−cooperation between relays using P

a

(b)

Figure 6.23 (a) Psucc = f(Eb/N0) and (b) PER = f(Eb/N0) for L = 50bits, Nu = 10,Nh = 10, τ = 0.5, Tc = 1ns

143


0 5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/N

0 [dB]

Pac

ket

Err

or

Rat

e P

ER

[%

] No cooperation between relays without diversity


Semi−cooperation between relays using P

a

Figure 6.24 The packet error rate PER = f(Eb/N0) for L = 20bits, Nu = 10, Nh = 10,τ = 0.5, Tc = 1ns, Perfect Power Control case

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pa [%]

PS [

%]

Eb/N

0=9dB

Eb/N

0=6dB

Eb/N

0=3dB

Eb/N

0=0dB

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pa [%]

PS [

%]

Eb/N

0=9dB

Eb/N

0=6dB

Eb/N

0=3dB

Eb/N

0=0dB

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pa [%]

PS [

%]

Eb/N

0=6dB

Eb/N

0=3dB

Eb/N

0=0dB

(c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pa [%]

PS [

%]

Eb/N

0=9dB

Eb/N

0=6dB

Eb/N

0=3dB

Eb/N

0=0dB

(d)

Figure 6.25 Psucc = f(Pa) for single-path channels with (a) synchronized users, scrambledpackets from the relays; (b) synchronized users, identical packets (with un-scrambled bits) from the relays; (c) unsynchronized users, scrambled packets;(d) unsynchronized users, identical packets (with unscrambled bits) from therelays; Nu = 10 relays, Nh = 10, Tc = 1ns, τ = 0.5, L = 50bits

144


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pa [%]

Su

cces

s P

rob

abil

ity

PS [

%]

Eb/N

0=9dB

Eb/N

0=6dB

Eb/N

0=3dB

Eb/N

0=0dB

Figure 6.26 Psucc = f(Pa) for real UWB channels, with unsynchronized users, identicalpackets (with unscrambled bits) are sent by all the relays; Nu = 10 relays,Nh = 10, Tc = 1ns, τ = 0.5, L = 50bits

6.3.6 Further Investigation

Among other interesting topics for further investigations, we could consider forexample an algorithm that adapts the length L of a packet with respect to Eb/N0,or another algorithm that maximizes the spectral efficiency through power allocationand waterfilling [79, 169].

145


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pa [%]

Pac

ket

Su

cces

Pro

bab

ilit

y [

%]

Theory

Single Path Channel, u.d. delays

Single Path Channel, u.d. delays, unscrambled Packet

Real UWB Channel CM1, u.d. delays, unscrambled Packet

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Pa [%]

Pac

ket

Su

cces

s P

rob

abil

ity

[%

]

Theory Synchr Packet

Simulation, Synchr Packet

(b)

Figure 6.27 Theoretical and simulation comparisons of Psucc = f(Pa) in case of (a) realUWB channels (b) single path channels, for Eb/N0 = 6dB; Nu = 10 relays,Nh = 10, Tc = 1ns, τ = 0.5, L = 50bits

146

Conclusions and Perspectives

In this thesis we have addressed the problems of interference modeling, throughputoptimization, receiver and transmitter design, cooperation and antenna optimizationfor Impulse Radio UWB ad-hoc sensor networks.

In Chapter 2 we have presented the IR-UWB signal model and some fundamentalinterference scenarios for sensor networks. While Section 2.1 presents the classicalImpulse Radio transmission schemes (e.g., PPM-TH, PAM-TH, DS-UWB), Section2.2 deals with the transmitter structure (in Subsection 2.2.1), the channel model (inSubsection 2.2.2), the types of multi user interference (in Subsection 2.2.4) and astate of the art on the most known IR receivers (in Subsection 2.2.5). It is importantto mention that Subsection 2.2.5 tries to establish a correspondence between thecomplexity of the receiver and the complexity of the transmitter, idea that has beenexploited further in Section 5.2. In Subsection 2.2.5 we have also presented for thefirst time the idea of adapted receivers to a certain statistical distribution, as it hasbeen presented by its developers.

In the last section from Chapter 2 (i.e., Section 2.3) we have described three funda-mental interference scenarios from a sensor network: single-cell (in Subsection 2.3.1),double-cell (in Subsection 2.3.2), and double-link (in Subsection 2.3.3) scenarios. Itcan be proved that every possible communication scenario in a sensor network it canbe reduced to one of these models. All of the interference models presented in Section2.3 are further used in Chapter 4.

Chapter 3 deals with the non-Gaussianity of IR-UWB Multi User Interference. InSection 3.2 we have presented results from Monte-Carlo simulations showing that IR-UWB MUI is not Gaussian. In Section 3.2 we have also presented several parametersthat affect the MUI distribution. Changing some of these parameters may give somebasic hints to the reader on how the statistical distribution of the MUI (and further,the communication performance) could be tuned. Our simulations show that in someparticular scenarios MUI could be modeled with a Gaussian distribution but thisresult is not representative and it cannot be generalized. On the other hand, itseems that the generalized Gaussian fits much better the PDF of the MUI. For thisreason, an interference model based on the generalized Gaussian has been presentedin Sections 3.2, 3.3, and 3.4 and further exploited in Chapters 4 and 5.

In Sections 3.2, 3.3, and 3.4 we present an interesting performance criterion thatcan be used to study the impact of several parameters as number of slots, numberof users, pulse duty cycle, pulse shape, etc. This method is based on computingthe communication capacity from the kurtosis (a statistical quantity which describes

147


the shape of the generalized Gaussian distribution) and from the received Signalto Noise Interference Ratio. In Section 3.3 we have showed the direct relationshipbetween the kurtosis and all the system’s parameters previously presented, and alsohow the kurtosis can be controlled by adjusting those parameters. While in Section3.3 we have computed the kurtosis in perfect power control and lack of power controlscenarios, Section 3.4 shows how the channel capacity may be evaluated by MonteCarlo averaging. This method is used further in Chapter 4.

In Chapter 4 we have presented the effect of tuning impulsiveness on throughputof IR-UWB Communications. Section 4.2 enumerates several methods that can beused to modify the impulsiveness and to increase the capacity of the communication.In the next sections we have presented different applications of previous describedmethods for a double-cell scenario (Section 4.3), double-link communication (Section4.4), cognitive networks (Section 4.5) and IR-UWB antenna (Section 4.6).

While in Section 4.3 the network was structured as a group of interconnected cellsthat are not synchronized with each other, in Section 4.4 we deal with two unsyn-chronized communication links, and in Section 4.5 we deal with double and single-cellscenarios. In such an unsynchronized environment, the interference is not Gaussian,but may be modeled as a generalized Gaussian. For such a non-Gaussian randominterference, its impulsiveness is determined by the kurtosis of the distribution.

In the applications previously described we show that network throughput can bemaximized by controlling the impulsiveness. The finding has been validated throughsimulations, on small scale network model (e.g., two interfering uplink cells, two inter-fering connections, etc.). Also, Section 4.5 shows how a cognitive network can benefitfrom sensing its impulsiveness. In the application from Section 4.6 we have used thesame signal processing tool to optimize an IR-UWB antenna, taking into account theimpact of antenna’s time domain response over the capacity of the channel. This toolcould be also used to optimize some other basic pulse shapes. The work from thissection (a) led to a novel performance criterion for IR-UWB antennas and (b) hasbeen applied to a physical IR-UWB antenna developed at SUPELEC.

As further contribution for the work from Chapter 4, it would be interesting tostudy load balancing algorithms (i.e., redistribution of the users to different clusterheads) and novel MAC strategies using the generalized Gaussian assumption. Onthe other hand, as further contribution for the work from Section 4.6, it would beinteresting to study an antenna design that takes into account the 3D measurements.Also, it would be interesting to study the effect of other pulse shapes (e.g., Gaussianand its derivatives, etc.) applied at the input of the designed antennas and measuringtheir performances.

In Chapter 5 we looked for a simple receiver and transmitter design with lowpower consumption, low complexity, but good performances as well. The advantageof modeling or fitting the MUI distribution with an appropriate distribution givesthe possibility to develop a simple yet efficient receiver. We have privileged thisapproach in contrast with heavier multi user detection, as the low complexity receiveris one of the targeted advantage of Impulse Radio systems. In Section 5.1 we haveproposed a novel mixture based IR-UWB receiver and numerical results showed thatit outperforms the existent MUI based receivers.

148


While the first part of this chapter deals with novel IR-UWB statistical adaptedreceivers, Section 5.2 proposed a novel type of transmitter design using Time Reversal.In Section 5.2 we have investigated the trade-off between the number of fingers in theprefilter of a Time Reversal IR-UWB system versus the number of fingers in the Rakereceiver. Also, we have shown here that TR combined with Rake receiver has animportant impact over the MUI distribution.

As further contributions for the work from Chapter 5, it would be interesting tostudy Time Reference strategies and joint Time Reversal - Time Reference strategies(Time Reversal strategies using the measurement from the Time Reference approach)and the impact on MUI.

Cooperative communication techniques are used at the MAC layer to improvewireless link performance. The proposed solution from Chapter 6 is to adopt incre-mental amplify-and-forward that uses multiple relays incrementally and capable ofdealing with imperfect feedback as well as UWB physical characteristics.

Existing incremental relaying schemes consider half-duplex transmissions or best-relay selection algorithms and they do not take advantage of the full use of multiplerelays. Section 6.1 proposed an incremental relaying scheme that uses multiple relays.Analytical and numerical results show that the proposed scheme outperforms both thesimple amplify-and-forward scheme and the single-relay incremental relaying schemein terms of lower outage probability. Furthermore, we have shown that when thereis a need to pick the best relay among a group of candidate nodes, harmonic meanbased selection methods are advised only under certain conditions.

Implementing incremental relaying in a practical system is not a trivial task be-cause it requires the destination node to provide feedback on success or failure ofa transmission reception. In practice, the feedback may be affected by propagationimpairments and collisions. In this case, the relay and the source should coordinateamong themselves in the absence of perfect feedback information such that packetcollision will not occur. In view of the challenge, Section 6.2 proposed a novel andpractical incremental relaying scheme that deals with the imperfect feedback. The keyidea was to use a pessimistic source and an optimistic relay. The proposed scheme issimple and easily implementable. Simulation results show that the proposed schemecan outperform amplify-and-forward schemes, as well as approaching the performanceof an idealized incremental amplify-and-forward with perfect feedback information.

Chapter 6 ends with Section 6.3 where we described a simple distributed relayselection algorithm without the need of any additional control message exchangebetween the relays.

As further contributions for the work from Chapter 6, it would be interesting toinvestigate incremental relaying with multiple sources (or sinks), practical incrementalrelaying strategies for multiple relays and proactive sources, optimization of packetand ACK lengths, cooperation strategies for Time Reversal and selection strategiesfor cooperation using link quality information.

This thesis covers various domains from PHY to MAC and network layers appliedto UWB sensor networks. However, each chapter presents methods and concepts thatcan be applied to other wireless technologies as well. The work from this thesis ledto a certain number of publications enumerated at the beginning of this manuscript.

149

List of Figures

1.1 Standard - UWB Emission Limit: (a) FCC (US) emission limits forindoor and outdoor UWB systems (b) European Commission emissionlimits for UWB systems (c) Japanese emission limits for indoor UWBsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 (a) TelosB, (b) Mica2/MicaZ, (c) Mica2dot . . . . . . . . . . . . . . . 7

2.1 Classical Transmission Scheme . . . . . . . . . . . . . . . . . . . . . . 182.2 Classical Transmission Chain with the SUMF Receiver (Scholtz and

Win) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 All-Rake IR-TH UWB, Tc = 1ns, τ = 0.5ns (a) CM1, 1 User, Just

Noise, (b) Ns = 1, 1 User, Just Noise, (c) Ns = 1, 10 Users, MUI andNoise, (d) CM1, 10 Users, MUI and Noise . . . . . . . . . . . . . . . 21

2.4 Time Reversal Transmission Chain . . . . . . . . . . . . . . . . . . . 212.5 Time Reference Performance under CM1(LOS) and CM3(NLOS) . . 222.6 Simplified Statistical MUI-Based Receiver Schemes . . . . . . . . . . 232.7 Some possible weighting shapes of the generalized Gaussian . . . . . . 242.8 IR-TH-PAM Transmission Scheme - The transmission of one bit (a)

without and (b) with the help of a randomizing code . . . . . . . . . 252.9 Single Cell Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.10 Two Cells Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.11 IR-TH-PAM Transmission Scheme . . . . . . . . . . . . . . . . . . . 282.12 Cell I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.13 (a) Sensors Coordinated by a single Base Station (b) Particular case

of a Double-Link Scenario . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1 BER vs. SIRbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Measured MUI histograms (bars) and reference Gaussian distributions

with same variances (plain curves) - perfect power control scenarios,Tc = 1ns, τ = 0.5ns, Nu − 1 interferers, with (a) Nh = 100, Nu = 100;(b) Nh = 100, Nu = 50; (c) Nh = 250, Nu = 100; (d) Nh = 500, Nu = 100 34

3.3 Measured MUI histograms (bars) and reference Gaussian distributionswith same variances (plain curves) - lack of power control scenarios,Tc = 1ns, τ = 0.5ns, Nu − 1 interferers, with (a) Nh = 100, Nu = 100;(b) Nh = 100, Nu = 50; (c) Nh = 250, Nu = 100; (d) Nh = 500, Nu = 100 35

151

LIST OF FIGURES

3.4 Measured MUI histograms (bars) and reference Gaussian distributionswith same variances (plain curves) - perfect power control scenarios,Nu − 1 interferers, with (a) Nh = 100, Nu = 100, Tc = 2ns, τ = 0.5ns;(b) Nh = 100, Nu = 50, Tc = 2ns, τ = 0.5ns; (c) Nh = 250, Nu = 100,Tc = 1ns, τ = 0.25ns; (d) Nh = 500, Nu = 100, Tc = 1ns, τ = 0.25ns 36

3.5 Measured MUI histograms (bars) and reference Gaussian distributionswith same variances (plain curves) - Scenarios with interference closeto a Gaussian distribution: Tc = 1ns, τ = 0.5ns, Nh = 10 and (a)perfect power control scenario with Nu = 100; (b) lack of power controlscenario with Nu = 500 . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.6 Measured MUI histogram and reference Gaussian distribution withsame variance (plain curve) - Scenario with interference close to a Gaus-sian distribution: Tc = 1ns, τ = 5ns, Nh = 100, Nu = 100 and perfectpower control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.7 Measured MUI histograms (bars) and curve fitting. As shown, genera-lized Gaussian fits well the tail of the histogram. Nu = 100, Nh = 500,Tc = 1ns, τ = 0.5ns (a) perfect power control (b) lack of power control 37

3.8 Kurtosis K = F(β) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.9 Scholtz’s w(t) and its correlation for τ = 0.5ns, Tc = 2 and a duration

of 2ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.10 β = F−1 (K) vs. τ

Tcfor a Scholtz Pulse . . . . . . . . . . . . . . . . . 47

3.11 m2(w) vs. τTc

for a Scholtz Pulse . . . . . . . . . . . . . . . . . . . . . 483.12 Simplified Capacity Computation Scheme (the equations are valid for

perfect power control scenarios with negligible noise but the scheme isexpandable to other scenarios as well) . . . . . . . . . . . . . . . . . . 50

3.13 Capacity(SINR,β) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1 Time slotting compromise with Nu,I = 50, Nu,II = 50, Low SIRcase, SIR = −25dB. The points corresponding to the pairs (Nh,I →∞,Nh,II = Nu,II) and (Nh,I = Nu,I ,Nh,II → ∞) are higher than thecentral peak (Nh,I = Nu,I ,Nh,II = Nu,II) . . . . . . . . . . . . . . . . 57

4.2 Zoom on Low SIR case, Nu,I = 50, Nu,II = 50, SIR = −25dB . . . . 574.3 Time slotting compromise with Nu,I = 50, Nu,II = 50, High SIR and

Low SNR case, SIR = 0dB, SNR = −10dB. The peak correspondsto the pair (Nh,I = Nu,I ,Nh,II = Nu,II) . . . . . . . . . . . . . . . . . 58

4.4 Time slotting compromise with Nu,I = 50, Nu,II = 50, High SIR andHigh SNR case, SIR = 0dB, SNR = 25dB. The peak corresponds tothe pair (Nh,I = Nu,I ,Nh,II = Nu,II) . . . . . . . . . . . . . . . . . . . 58

4.5 Fairness in terms of Nh,I and Nh,II . The median line corresponds tothe maximum fairness . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.6 Tuning impulsiveness by spreading the transmitted pulse: High Ca-pacity Scenario and Low Capacity Scenario . . . . . . . . . . . . . . . 61

4.7 Equivalence model and comparison between two scenarios with sameenergy ε but different kurtosis . . . . . . . . . . . . . . . . . . . . . . 61

152

LIST OF FIGURES

4.8 Tuning the Spreading Factor of two links Tx1-Rx1 and Tx2-Rx2 with(a) INR = 10dB, SNR = −5dB and (b) INR = 10dB, SNR = 5dB 63

4.9 The first six derivatives of a Gaussian pulse . . . . . . . . . . . . . . 664.10 Performance comparison in terms of required energy when cognition

is applied under the SGA hypothesis (interference are assumed tobe Gaussian, Standard Gaussian Approximation) or taking the non-Gaussianity into account (using a General Gaussian distribution tomodel the interference) . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.11 Definition of wideband antenna (a) and frequency variation of mainlobe (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.12 The antenna designed (a) and the radiation pattern (b), (c), (d) [132]in CST Microwave Studio . . . . . . . . . . . . . . . . . . . . . . . . 70

4.13 Front view (a), transversal view (b) and radiation pattern (c) of thephysical antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.14 Return loss (a) and link transfer coefficient (b) at 1m . . . . . . . . . 724.15 Capacity comparison cases . . . . . . . . . . . . . . . . . . . . . . . . 734.16 Distribution scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.17 The evolution of the ratio Pm,2/P 2

m with the maximum radius of de-ployment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.18 Capacity(SIR,β) with (a) Dirac response of antennas ’△’ vs. Scholtz’spulse ’∇’ and (b) Response of antennas to a square pulse ’⊡’ vs.Scholtz’s pulse ’∇’ in Perfect Power Control scenario and Lack of Per-fect Control scenario: 1, 2, 3, 4 and 5 are corresponding to the nextpairs (Nh,Nu): (Nh=100,Nu=100); (Nh=250,Nu=100); (Nh=500,Nu=100);(Nh=500,Nu=50); and (Nh=1000,Nu=50) respectively . . . . . . . . . 76

4.19 Equivalent schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.20 Implemented βVersion of the Monopole Antenna - (a) front view, (b)

back view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.21 Advantest D3186 Pattern Pulse Generator at SUPELEC . . . . . . . 79

5.1 Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.2 Unusual Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 845.3 MUI test bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.4 Gaussian, generalized Gaussian and Gaussian mixture adapted receiver

performance under (a) Gaussian interference (b) Gaussian mixture in-terference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.5 Rough GG Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.6 An overview of the Modified EM Algorithm . . . . . . . . . . . . . . 885.7 Mixture Receiver (could be a GM or a LM) . . . . . . . . . . . . . . 905.8 GM identification: mixing probabilities (a) and variances (b) . . . . . 925.9 Comparison between the Gaussian Receiver, Generalized Mixture Re-

ceiver and rough Generalized Gaussian Receiver. BER as a functionof Ns, two sets of graphs in Low Noise (Eb/N0 = 30dB) and High Noise(Eb/N0 = 0dB) environments, CM1 (LOS), 100 users in perfect powercontrol with 10dB above the useful signal . . . . . . . . . . . . . . . . 93

153

LIST OF FIGURES

5.10 Comparison between the (classical) Gaussian Receiver, GeneralizedMixture Receiver and rough Generalized Gaussian Receiver. BER asa function of Eb/N0, Ns = 5, CM3 (NLOS), 3 users with 20dB abovethe useful signal, 2 users with 30dB above the useful signal . . . . . . 94

5.11 Comparison between the Gaussian Receiver, Generalized Mixture Re-ceiver and rough Generalized Gaussian Receiver. BER as a functionof Ns, Eb/N0 = 30dB, CM3 (NLOS), 8 users in perfect power controlwith 0dB above the useful signal, 1 user with 30dB above the usefulsignal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.12 Comparison between the Gaussian Receiver, Generalized Mixture Re-ceiver and rough Generalized Gaussian Receiver. BER as a function ofNs, Eb/N0 = 30dB, CM1 (LOS), 3 users with 20dB above the usefulsignal, 2 users with 30dB above the useful signal . . . . . . . . . . . . 95

5.13 Time Reversal (TR) combined with Selective Rake (SRake) . . . . . . 975.14 Example of h(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.15 Example of hin(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.16 Example of hin(t)∗h(t) . . . . . . . . . . . . . . . . . . . . . . . . . . 995.17 Example of hout(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.18 BER(Nin, Nout), Eb/N0 = 9dB, LOS . . . . . . . . . . . . . . . . . . 1005.19 BER(Nin, Nout), Eb/N0 = 9dB, NLOS . . . . . . . . . . . . . . . . . 1005.20 LOS, Energy percentage on Nout fingers at the receiver (Nout = 1 to

10). The curve with Nout = 1 gives the energy percentage on the maincentral path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.21 NLOS, Energy percentage on Nout fingers at the receiver (Nout = 1 to10). The curve with Nout = 1 gives the energy percentage on the maincentral path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.22 LOS, Energy percentage of Nout fingers . . . . . . . . . . . . . . . . . 1025.23 NLOS, Energy percentage of Nout fingers [131] . . . . . . . . . . . . . 1035.24 NLOS, Energy percentage of Nout fingers . . . . . . . . . . . . . . . . 1045.25 Constant BER curves, NLOS, Eb/N0 = 9dB . . . . . . . . . . . . . . 1045.26 SRake without TR for binary PAM-TH, LOS . . . . . . . . . . . . . 1055.27 TR and SRake for binary PAM-TH, NLOS . . . . . . . . . . . . . . . 1065.28 NLOS, strong interference from 5 users with 10dB above the useful signal106

6.1 Two Non-Incremental AF Schemes with Maximal Ratio Combining . 1126.2 Two Incremental Cooperative Schemes with Maximal Ratio Combining 1136.3 SE as a function of SE and SNR for σ2

s,d = 1 . . . . . . . . . . . . . 1166.4 Performance of 1 + n AF ’⋄’ vs. Performance of 1 + n IAF ’◦’, σ2

s,d =σ2

s,i = σ2i,d = 1, Pi = Ps/n . . . . . . . . . . . . . . . . . . . . . . . . 118

6.5 Performance of 1+n AF ’⋄’ vs. Performance of 1+n IAF ’◦’, σ2s,d = 1;

(σ2s,i = 0.1 and σ2

i,d = 10) or (σ2s,i = 10 and σ2

i,d = 0.1), Pi = Ps - sameharmonic mean and therefore equivalent Outage Probability . . . . . 118

6.6 Performance of 1 + 1 AF ’⋄’ vs. Performance of 1 + 1 IAF ’◦’, σ2s,d =

σ2s,i = σ2

i,d = 1, Pi = Ps/n . . . . . . . . . . . . . . . . . . . . . . . . 119

154

LIST OF FIGURES

6.7 Performance of 1 + 1 AF ’⋄’ vs. Performance of 1 + 1 IAF ’◦’, σ2s,d = 1,

σ2s,i = 0.1, σ2

i,d = 10, Pi = Ps . . . . . . . . . . . . . . . . . . . . . . . 1206.8 Performance of 1 + 1 AF ’⋄’ vs. Performance of 1 + 1 IAF ’◦’, σ2

s,d = 1,σ2

s,i = 10, σ2i,d = 0.1, Pi = Ps . . . . . . . . . . . . . . . . . . . . . . . 120

6.9 Two Non-Cooperative Schemes: (a) source S transmits continuouslyin all the slots toward destination, (b) source S transmits with lowerspectral efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.10 Two Non-Incremental AF Schemes with Maximal Ratio Combining . 1236.11 Three Incremental AF Schemes with Maximal Ratio Combining: (a)

D sends only ACK, (b) D sends only NACK, (c) D sends both ACKand NACK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.12 Strategy with ACK and without NACK transmission . . . . . . . . . 1266.13 Strategy with NACK and without ACK transmission . . . . . . . . . 1276.14 Practical IAF, Heterogeneous Channel Conditions . . . . . . . . . . . 1296.15 Practical IAF, Homogeneous Channel Conditions . . . . . . . . . . . 1306.16 ∆ = f(SNR, SE) for σ2

s,d = σ2d,r = σ2

d,s = 1 . . . . . . . . . . . . . . . 1346.17 Outage Probabilities for σ2

s,r = σ2s,d = σ2

d,s = σ2d,r = σ2

r,d = 1, e.g., thefeedback channel has the same quality as the direct channel . . . . . 135

6.18 Zoom on Outage Probabilities for σ2s,r = σ2

s,d = σ2d,s = σ2

d,r = σ2r,d = 1,

e.g., the feedback channel has the same quality as the direct channel . 1366.19 Outage Probabilities for σ2

s,r = σ2s,d = σ2

r,d = 1 and σ2d,s = σ2

d,r = 0.1,equivalent to a bad feedback channel . . . . . . . . . . . . . . . . . . 136

6.20 Zoom on Outage Probabilities for σ2s,r = σ2


d,s =σ2

d,r = 0.1, equivalent to a bad feedback channel . . . . . . . . . . . . 1376.21 General relaying scheme between a source S and a destination D . . . 1386.22 General physical layer transmission scheme for Source to Destination

(S-D) or Source to Relay (S-R) or Relay to Destination (R-D) . . . . 1386.23 (a) Psucc = f(Eb/N0) and (b) PER = f(Eb/N0) for L = 50bits, Nu =

10, Nh = 10, τ = 0.5, Tc = 1ns . . . . . . . . . . . . . . . . . . . . . . 1436.24 The packet error rate PER = f(Eb/N0) for L = 20bits, Nu = 10,

Nh = 10, τ = 0.5, Tc = 1ns, Perfect Power Control case . . . . . . . . 1446.25 Psucc = f(Pa) for single-path channels with (a) synchronized users,

scrambled packets from the relays; (b) synchronized users, identicalpackets (with unscrambled bits) from the relays; (c) unsynchronizedusers, scrambled packets; (d) unsynchronized users, identical packets(with unscrambled bits) from the relays; Nu = 10 relays, Nh = 10,Tc = 1ns, τ = 0.5, L = 50bits . . . . . . . . . . . . . . . . . . . . . . 144

6.26 Psucc = f(Pa) for real UWB channels, with unsynchronized users, iden-tical packets (with unscrambled bits) are sent by all the relays; Nu = 10relays, Nh = 10, Tc = 1ns, τ = 0.5, L = 50bits . . . . . . . . . . . . . 145

6.27 Theoretical and simulation comparisons of Psucc = f(Pa) in case of(a) real UWB channels (b) single path channels, for Eb/N0 = 6dB;Nu = 10 relays, Nh = 10, Tc = 1ns, τ = 0.5, L = 50bits . . . . . . . . 146

155

List of Tables

1.1 The position of UWB Communications . . . . . . . . . . . . . . . . . 3

4.1 Capacity in term of bits/channel use for various received pulse shapes- Perfect Power Control, Nu = 100 . . . . . . . . . . . . . . . . . . . . 75

4.2 Capacity in term of bits/channel use for various received pulse shapes- Lack of Power Control, Nu = 100 . . . . . . . . . . . . . . . . . . . 77

5.1 Comparison Between the Correction Parameters α (GM Receiver) andβ (GG Receiver) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2 Initialization parameters . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.1 SNRnorm values for AF and IAF schemes . . . . . . . . . . . . . . . . 1176.2 Useful Parameters for Heterogeneous Channel Conditions . . . . . . . 1286.3 Useful Parameters for Homogeneous Channel Conditions . . . . . . . 131

156

Acknowledgement

I am deeply grateful to the academic staff of Ecole Supérieure d’Électricité - ServiceRadio, led by Dr. Hikmet SARI, and to the academic staff of National Universityof Singapore, Department of Electrical & Computer Engineering (ECE) - ComputerNetworks and Distributed Systems (CNDS) Lab, led by Dr. Chen-Khong THAM, forgiving me the chance to undertake this interesting research.

I would like to express my gratitude to my supervisors and collaborators: Dr.Jocelyn FIORINA, Dr. Chen-Khong THAM and Dr. Peng-Yong KONG for theirguidance and mentorship over the last 3 years. They have taught me a lot and theirknowledge and insight into research were invaluable for me. Thank you for yourunderstanding, patience and for believing in me!

With all my sincere thoughts, I am deeply grateful to my PhD Examiners Dr.Stephen V. HANLY and Dr. Laurent CLAVIER, and I would like to assure themthat their experience and their remarks were highly appreciated, as I could improvemy manuscript and my overall presentation.

All my thanks to Mrs. Catherine MAGNET, Mrs. Anne BATALIE, Mr. JoseFONSECA and Mr. Huu-Hung VUONG (Supélec) for their kindness and availability.

I would also like to thank all my friends who were near me during all this period:my special thanks to Nassar for his support during the difficult beginning, to Stefanfor his LATEX template, to Ileana for her moral support, to Andrei and Ioana forlending me their computers in Singapore.

Last but not least, I thank my parents and my sister Corina for all they have donefor me.

Remerciements

Je suis profondément reconnaissant à mes professeurs de Supélec, le Service Ra-dioélectricité et Électronique, dirigé par Dr. Hikmet SARI, de même qu’à mes pro-fesseurs de l’Université Nationale de Singapour, le Département Electrical & Com-puter Engineering (ECE) - Computer Networks and Distributed Systems (CNDS)Lab, dirigé par Dr. Chen-Khong THAM, pour m’avoir offert l’opportunité de menercette recherche intéressante.

Je voudrais exprimer ma gratitude à mes directeurs de recherche et collaborateurs:Dr. Jocelyn FIORINA, Dr. Chen-Khong THAM et Dr. Peng-Yong KONG pour leuraide et leur soutien au cours de ces dernières trois années. Ils m’ont beaucoup enseignéet leurs connaissances, de même que leur vision dans la recherche, ont été inestimablespour moi. Merci beaucoup pour votre compréhension et patience et pour avoir euconfiance en moi!

Je suis profondément reconnaissant à mes examinateurs: Dr. Stephen V. HANLYet Dr. Laurent CLAVIER dont l’expérience et remarques pertinentes ont été forte-ment appréciées. C’est grâce à eux que j’ai pu améliorer mon manuscrit et ma présen-tation globale.

Je remercie également à madame Catherine MAGNET, à madame Anne BATALIE,à monsieur Jose FONSECA et à monsieur Huu-Hung VUONG (Supélec) pour leuramabilité et disponibilité.

Un merci chaleureux s’adresse à tous mes amis qui ont été à côté de moi pendanttoute cette période. Je voudrais remercier particulièrement Nassar pour son aidependant le début difficile, Stefan pour son modèle de thèse en LATEX, Ileana pour sonsoutien moral, Andrei et Ioana pour m’avoir emprunté leurs ordinateurs à Singapour.

Finalement, je suis reconnaissant à mes parents et à ma soeur Corina pour toutce qu’ils ont fait pour moi.

Issy-les-Moulineaux, France Dorin Panaitopol25 Juillet, 2011

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