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SPSC Dissertation Series

Jac Romme

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UWB

Doctoral Thesis

UWB Channel Fading Statistics and

Transmitted-Reference

Communication

ir. Jac Romme

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Signal Processing and Speech Communication LaboratoryGraz University of Technology, Austria

Supervisor: Univ.-Prof. Dipl.-Ing. Dr.techn. Gernot KubinExternal Evaluator: Prof. Sergio BenedettoCo-supervisor: Dipl.-Ing. Dr. Klaus Witrisal

Graz, March 2008

“The scientist is not a person who gives the right answers,he is one who asks the right questions.”

[Claude Levi-Strauss, 1964]

Zusammenfassung

Die Robustheit der Ultra WideBand (UWB) Ubertragung gegenuber Small-Scale-Fading(SSF) in Mehrwegekanalen als Folge der großen Bandbreite ist hinlanglich bekannt. Den-noch gibt es bislang kein Modell, das die Variation der Empfangssignalstarke in Ab-hangigkeit von der Signalbandbreite und genereller Kanaleigenschaften wie dem Leistungs-verzogerungs-Profil beschreibt. Ein solches Modell wurde dem Kommunikationsingenieurerlauben, die Nachteile einer großen Bandbreite wie z.B. steigende Systemkomplexitatgegenuber dem Vorteil einer erhohten Systemrobustheit abwagen zu konnen. In dieserDissertation wird ein Modell vorgestellt, das diese Analyse erstmals ermoglicht, indem esdie statistischen Eigenschaften von SSF als Funktion der Signalbandbreite und des Lei-stungsverzogerungs-Profils beschreibt. Zudem wird eine Berechnung der resultierendenBitfehlerrate bei Verwendung von BPSK Modulation vorgestellt.

Die hohe Bandbreite der UWB-Systeme ist zwar vorteilhaft bei der Bekampfung vonSSF, fuhrt aber im Empfangerdesign zu Problemen. Koharente Empfangerkonzepte sindsehr komplex, so dass bereits in 2002 von den Autoren Tomlinson und Hoctor ein alterna-tives Konzept vorgeschlagen wurde, das das Transmitted Reference (TR) Verfahren miteinem Autokorrelationsempfanger kombiniert und eine Kanalschatzung vermeiden kann.Aufgrund der nichtlinearen Struktur des Empfangers war es bislang schwierig, sein ex-aktes Verhalten vorherzusagen. Diese Dissertation gibt nun Einblicke in das prinzipielleVerhalten des TR-Autokorrelationsempfangers und zeigt zusatzlich Verbesserungen auf,die es ermoglichen, einige Nachteile des Konzepts abzuschwachen. Weiterhin werden ver-schiedene Interpretationen des TR UWB-Prinzips prasentiert, die z.B. den Einfluss vonIntersymbolinterferenz auf das System erklaren.

Basierend auf dem theoretischen Verstandnis von TR-UWB wird im Anschluss einhochratiges Ubertragungssystem mit Datenraten im Bereich von einigen 100 Mbps beieiner Bandbreite von 1 GHz entwickelt. Es verwendet eine Kombination von trellis-basierter Entzerrung, Turbo Entzerrung, Turbo-Dekodierung und Verarbeitung in meh-reren Bandern, die es erlaubt, die gewunschte Datenrate mit moderater digitaler Signal-verarbeitung zu erzielen. Bereits ein Eb/N0 von 12 dB ist fur eine Bitfehlerrate kleinerals 10−6 ausreichend.

i

Abstract

It is well known that Ultra WideBand (UWB) transmission is inherently robust againstsmall-scale-fading (SSF) that arises in multipath scattering environments, due to its largesignal bandwidth. However, no model with a physical interpretation exists that relatesthe variations of received signal strength to the signal bandwidth and general channelparameters, like e.g. the average channel power delay profile. Such a model would be ofrelevance for e.g. system designers, who have to make tradeoffs between system aspects,like complexity and energy efficiency on one hand, and robustness against small-scale-fading on the other hand. In this thesis, a model is presented that allows for such a tradeoffanalysis, relating the average power delay profile parameters and signal bandwidth to thestatistical properties of the SSF. Additionally, it is shown how the uncoded and codedBER of BPSK modulation can be computed in a closed-form for a given average powerdelay profile and signal bandwidth.

As stated before, UWB communication is inherently resilient against SSF. Unfortu-nately, coherent receivers become rather complex in the UWB case. In 2002, Tomlinsonand Hoctor proposed to combine Transmitted Reference (TR) signaling with an auto-correlation receiver (AcR) for UWB communications, to dispose of the need for channelestimation. Due to the non-linear structure of the AcR, little was known with respect toits behaviour in various situations. This thesis aims to provide better insight in the be-haviour of such systems. Not only is the principle of TR UWB communication explained,also several extensions to the TR principle are proposed, which relieve some of its draw-backs. Additionally, novel interpretations for TR UWB systems are presented, whichexplain the behaviour of TR systems e.g. in the presence of inter-symbol-interference.

After understanding the behaviour of TR UWB systems, the design of a high-rateTR UWB system is presented that supports data-rates up to 100 Mb/s, while occupying1 GHz of bandwidth. Using a combination of trellis-based equalization, multiband pro-cessing, turbo equalization and turbo coding, a system is obtained which is moderatelycomplex with respect to digital signal processing and requires an Eb/N0 of only 12 dB toobtain a BER better than 10−6.

iii

Contents

Zusammenfassung i

Abstract iii

Acronyms ix

1 General Introduction 1

1.1 Wireless Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Ultra-WideBand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Framework and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Thesis Outline and Contributions . . . . . . . . . . . . . . . . . . . . . . . 6

2 Theory of Fading UWB Channels 13

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.1 The Radio Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Radio Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.3 Channel Characterizing Parameters . . . . . . . . . . . . . . . . . . 152.1.4 Impact of the Channel on Radio Signals . . . . . . . . . . . . . . . 16

2.2 Frequency Domain Properties of UWB Channels . . . . . . . . . . . . . . . 182.2.1 Frequency Domain Correlation . . . . . . . . . . . . . . . . . . . . 192.2.2 Eigenvalues and Their Physical Interpretation . . . . . . . . . . . . 202.2.3 Asymptotic Behaviour of the Eigenvalues . . . . . . . . . . . . . . . 22

2.3 Diversity of UWB Channels . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.1 The Mean Power Gain . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.2 Statistical Characterization of the NLOS Mean Power Gain . . . . . 262.3.3 Generalization of the Statistics to LOS Scenarios . . . . . . . . . . 282.3.4 Diversity Level of UWB Channels . . . . . . . . . . . . . . . . . . . 29

2.4 BER on UWB Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4.1 BER of BPSK on Fading Channels . . . . . . . . . . . . . . . . . . 312.4.2 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Fading of Measured UWB Channels 37

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Description of Radio Channel Measurements . . . . . . . . . . . . . . . . . 373.3 Overview of Measurement Results . . . . . . . . . . . . . . . . . . . . . . . 38

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vi CONTENTS

3.3.1 Delay Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.2 Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Principal Components of Measured UWB Channels . . . . . . . . . . . . . 433.4.1 Estimation of the Eigenvalues and Principal Components . . . . . . 433.4.2 Verification of the NLOS Eigenvalues and Principal Components . . 443.4.3 Verification of the LOS Eigenvalues and Principal Components . . . 46

3.5 Analysis of the Mean Power Gain . . . . . . . . . . . . . . . . . . . . . . . 463.5.1 Estimation of the Diversity Level . . . . . . . . . . . . . . . . . . . 473.5.2 Verification of the Diversity Level . . . . . . . . . . . . . . . . . . . 483.5.3 Verification of the Mean Power Gain . . . . . . . . . . . . . . . . . 49

3.6 BER Comparison on Measured and Theoretical UWB Channels . . . . . . 513.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Theory of TR UWB Communications 57

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2 Principle of Transmitted Reference Communication . . . . . . . . . . . . . 58

4.2.1 Transmitted-Reference Signaling . . . . . . . . . . . . . . . . . . . . 584.2.2 Autocorrelation Receiver . . . . . . . . . . . . . . . . . . . . . . . . 584.2.3 The Drawbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2.4 Implementation Considerations . . . . . . . . . . . . . . . . . . . . 61

4.3 Extensions of the TR Principle . . . . . . . . . . . . . . . . . . . . . . . . 624.3.1 Weighted Autocorrelation and Fractional Sampling AcR . . . . . . 624.3.2 Complex-Valued Autocorrelation Receiver . . . . . . . . . . . . . . 654.3.3 TR M-ary Phase Shift Keying . . . . . . . . . . . . . . . . . . . . . 67

4.4 Generic TR System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.4.2 Continuous-Time System Model . . . . . . . . . . . . . . . . . . . . 684.4.3 Discrete-Time Equivalent System Model . . . . . . . . . . . . . . . 69

4.5 Interpretation of the TR System Model . . . . . . . . . . . . . . . . . . . . 724.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.5.2 Vector Notation for Volterra Kernels . . . . . . . . . . . . . . . . . 754.5.3 Extension of the Vector Notation . . . . . . . . . . . . . . . . . . . 774.5.4 Linear MIMO Model . . . . . . . . . . . . . . . . . . . . . . . . . . 784.5.5 Data Model as Finite State Machine . . . . . . . . . . . . . . . . . 794.5.6 Reduced Memory Data Model . . . . . . . . . . . . . . . . . . . . . 82

4.6 Statistical Properties of the TR System Model . . . . . . . . . . . . . . . . 844.6.1 Statistics of the Signal Term . . . . . . . . . . . . . . . . . . . . . . 854.6.2 Statistics of the Gaussian Noise Term . . . . . . . . . . . . . . . . . 854.6.3 Statistics of the Non-Gaussian Noise Term . . . . . . . . . . . . . . 874.6.4 Analysis of the Noise Term . . . . . . . . . . . . . . . . . . . . . . . 88

4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5 Analysis of TR UWB Communication 91

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.2 Description of the Linear Weighting . . . . . . . . . . . . . . . . . . . . . . 915.3 System Performance in the Absence of ISI . . . . . . . . . . . . . . . . . . 92

CONTENTS vii

5.3.1 Influence of the Weighting Criteria and Fractional Sampling Rate . 935.3.2 Influence of Delay and Fractional Sampling Rate . . . . . . . . . . . 945.3.3 Influence of Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . 955.3.4 Influence of Modulation . . . . . . . . . . . . . . . . . . . . . . . . 95

5.4 System Performance in the Presence of ISI . . . . . . . . . . . . . . . . . . 975.4.1 Influence of the Weighting Criteria and Fractional Sampling Rate . 975.4.2 Influence of Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . 985.4.3 Influence of Modulation . . . . . . . . . . . . . . . . . . . . . . . . 98

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6 Design of a High-Rate TR UWB System 103

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.2 Design Considerations for a High-Rate TR UWB System . . . . . . . . . . 103

6.2.1 Trellis-Based Equalization . . . . . . . . . . . . . . . . . . . . . . . 1036.2.2 Power Spectral Density of TR Signals . . . . . . . . . . . . . . . . . 1046.2.3 Volterra System Identification . . . . . . . . . . . . . . . . . . . . . 1056.2.4 Multiband Transmitted Reference . . . . . . . . . . . . . . . . . . . 1066.2.5 The Role of Forward Error Control . . . . . . . . . . . . . . . . . . 1066.2.6 Principle of Turbo Equalization . . . . . . . . . . . . . . . . . . . . 107

6.3 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.3.1 Description of the TX Architecture and RX RF Front-End . . . . . 1086.3.2 Forward Error Control . . . . . . . . . . . . . . . . . . . . . . . . . 1096.3.3 Turbo Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.3.4 SISO Decoder Structure . . . . . . . . . . . . . . . . . . . . . . . . 1126.3.5 Stop Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.3.6 Measure of Complexity . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.4 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.4.1 Impact of Equalizer Complexity Without Turbo Equalization . . . . 1156.4.2 Benefit of Turbo Equalization . . . . . . . . . . . . . . . . . . . . . 117

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7 Conclusions and Recommendations 123

A Estimation of the Nakagami-m Parameter 127

B Complex-Valued AcR 135

C PSD of Scrambled QPSK-TR UWB Signals 137

D Derivation of the Log-MAP Algorithm 139

Bibliography 143

Acknowledgments 153

Curriculum Vitae 155

viii CONTENTS

Acronyms

1G First Generation

2G Second Generation

3G Third Generation

4G Fourth Generation

ADC Analogue to Digital Converter

AcR Autocorrelation Receiver

AMPS American Advanced Mobile Phone System

APDP Average Power Delay Profile

AWGN Additive White Gaussian Noise

BCJR Bahl, Cocke, Jelinek and Raviv

BER Bit Error Rate

BPF Band Pass Filter

BPSK Binary-Phase-Shift-Keying

CC Convolutional Code

CDF Cumulative Distribution Function

CEPT European Conference of Postal and TelecommunicationsAdministrations

CEV Circulant Eigenvalue

CFR Channel Frequency Response

CIR Channel Impulse Response

CV Complex-Valued

CW Carrier Wave

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x CONTENTS

DAA Detect and Avoid

DARPA Defence Advanced Research Projects Agency

DFT Discrete Fourier Transform

DLL Data Link Layer

DS Direct Sequence

DSL Digital Subscriber Loop

DSP Digital Signal Processing

DS-UWB Direct Sequence - UWB

ECC Electronic Communications Committee

ECMA European Computer Manufacturers Association

EM Electro-Magnetic

FCC Federal Communications Commission

FDM Full Data Model

FEC Forward Error Control

FER Frame Error Rate

FH Frequency Hopping

FIR Finite Impulse Response

HMM Hidden Markov Model

FSFC Frequency Selective Fading Channel

FSM Finite State Machine

FSR Fractional Sampling Rate

I&D Integrate and Dump

IEEE Institute of Electrical and Electronics Engineers

ISI Inter Symbol Interference

ISP Internet Service Provider

ITU-R International Telecommunication Union Radiocommunication Sector

LLV Log-Likelihood Value

CONTENTS xi

LMS Least Mean Square

LOS Line-of-Sight

LPF Low Pass Filter

LS Least Squares

MAC Multiple Access Layer of the OSI model

MAP Maximum A-Posteriori

MB-OFDM Multi-Band Orthogonal Frequency Division Multiplexing

MIC Ministry of Internal Affairs and Communications

MIMO Multiple-Input, Multiple-Output

MLSD Maximum-Likelihood Sequence Detection

MMSE Minimum Mean Square Error

MPG Mean Power Gain

MRC Maximum Ratio Combining

NLOS Non-Line-of-Sight

NMT Scandinavian Nordic Mobile Telephone

OFDM Orthogonal Frequency Division Multiplexing

OSI Open Systems Interconnection

PC Principal Component

PCA Principal Component Analysis

PDF Probability Density Function

PDP Power Delay Profile

PHY Physical Layer of the OSI model

PIAM Pulse Interval and Amplitude Modulation

PN Pseudo Noise

PPM Pulse Position Modulation

PSD Power Spectral Density

QAM Quadrature Amplitude Modulation

xii CONTENTS

QPSK Quadrature-Phase-shift-Keying

R&O Report and Order

RF Radio Frequency

RMDM Reduced Memory Data Model

RMS Root Mean Square

RSCC Recursive Systematic Convolutional Code

RV Random Value

RX Receiver

SIMO Single-Input, Multiple-Output

SISO Soft-Input, Soft-Output

SNIR Signal-to-Noise-and-Interference Ratio

SNR Signal-to-Noise Ratio

SOVA Soft-Output Viterbi Algorithm

SSF Small-Scale Fading

SVD Singular Value Decomposition

TACS British Total Access Communication System

TG3a Task Group 3a

TR Transmitted Reference

TX Transmitter

UB Upper Bound

US Uncorrelated Scattering

USB Universal Serial Bus

UWB Ultra-Wideband

VOIP Voice over IP

WPAN Wireless Personal Area Network

WLAN Wireless Local Area Network

WMAN Wireless Metropolitan Area Network

WWAN Wireless Wide Area Network

WRAN Wireless Regional Area Network

Chapter 1

General Introduction

1.1 Wireless Communications

In the 1860s, James Clerk Maxwell, a Scottish physicist, proposed a set of differentialequations, which together describe the behaviour of electric and magnetic fields, as wellas their interactions with each other and matter. Based on these equations, he predictedthe existence of self-sustaining, oscillating waves composed out of an electric and magneticfield that travel through space. Nowadays, these waves are referred to as electro-magneticwaves. Also he was the first to propose that light is a type of electromagnetic wave.

Through experimentation using a spark-gap transmitter and a spark-gap loop antennaas detector, Heinrich Rudolph Hertz proved in 1886 that a spark at the transmitter caninduce a spark at the receiver, showing that electromagnetic waves can travel throughfree space over some distance, as Maxwell predicted.

Fascinated by these results, Lodge, Marconi and Popov began almost simultaneouslytransforming radio into a way of wireless communication. In 1896, Marconi and Popovboth sent radio messages over short distances. In 1899, Marconi signalled the first wire-less signal across the English Channel and two years later, he already telegraphed fromEngland to Newfoundland. These experiments made the world considerably smaller, sincenow the transport of information was only limited by the speed of light.

Nowadays, the use of wireless communications for the transport of voice and datahas been integrated into everyday’s life. The penetration of mobile telephony in thewestern world is often above 80% and the deployment of wireless local area networks hasbecome normal. It is therefore hard to imagine that only as recently as the early nineties,commercial wireless communications was a rare commodity for many.

After the introduction of First Generation (1G) mobile radio telephony, the successof wireless communications started. The 1G mobile radio telephony emerged in theearly eighties, although in its early years the term portable communications was moreappropriate. Due to limitations in technology, these phones deployed analogue modulationand were still rather big. These systems were developed for voice communication only andevery country used their own frequency bands, disabling the possibility of internationalroaming. Nevertheless, these systems allowed for the opening of a mass-market for wirelessvoice communication. Some of the most successful 1G systems were American AdvancedMobile Phone System (AMPS), British Total Access Communication System (TACS) and

1

2 CHAPTER 1. GENERAL INTRODUCTION

Scandinavian Nordic Mobile Telephone (NMT).

With the introduction of Second Generation (2G) systems, a true revolution tookplace in the use of mobile radio telephony. The reasons for their success are manifold.Not only used 2G digital modulation, but more importantly 2G was standardized. Asa result, devices became significantly smaller, cheaper, allowed for international roamingand had longer operating time on single charge of battery. All these aspects contributedto the success of 2G.

The commercial success of 2G together with the Internet boom, caused an explosion ofdevelopments to penetrate mobile communications further into everyday’s life. Nowadays,many wireless communication systems are in use aside each other, each developed for itsown application scenarios.

Roughly speaking, five network types can be distinguished, namely WPAN, WLAN,WMAN, WWAN and WRAN. No strict definitions exist to distinguish between them andoften the terms are used more for market-technical reasons than technical ones. Standardsbelonging to different types may therefore compete with each other for the grace of thecustomers. Nevertheless, we will make an effort to characterize them coarsely:

A Wireless Personal Area Network (WPAN) is a network technology to interconnectdevices around a workspace or person using wireless radio technology. The typical rangeis around 10 meters and often mobility is not supported, meaning that the connectionbreaks down when leaving the coverage area.

A Wireless Local Area Network (WLAN) is used to connect computers and otherWLAN enabled devices to the wired network directly via base-stations. The range coveredby a base-station is on the order of 100 m depending on the environment. Furthermore,a network of base-stations can be installed to support mobility within the area coveredby the network of base-stations.

Wireless Metropolitan Area Network (WMAN) is basically an extension of the WLANconcept to support ranges in the order of 1 km, such that with less base-stations a largerarea can be covered reducing the cost of the infra-structure. An Internet Service Provider(ISP) managing the WMAN network will provide Internet access to its subscribers as analternative for cable and Digital Subscriber Loop (DSL). Most likely, access the Internetis limited to the coverage area of the ISP and the underlying technology supports onlymoderate velocities. Nevertheless, WMAN could go in competition with the cellularnetwork operators, especially now Voice over IP (VOIP) is taking off as well.

The term Wireless Wide Area Network (WWAN) is typically used for standards de-veloped by/for the cellular network operators. The typical range is 10 km and differs fromWLAN and WMAN, because of the use of cellular network technology. These cellulartechnologies provide for nationwide and international access using roaming. In this senseWWAN provides higher mobility and higher velocities.

A recent development is the Wireless Regional Area Network (WRAN) introduced byInstitute of Electrical and Electronics Engineers (IEEE) standard 802.22, which has asmandate to develop a standard for a cognitive radio-based PHY/MAC/air interface foruse by license-exempt devices in spectrum allocated to TV Broadcasting. The target ap-plication of WRANs is wireless broadband Internet access in areas with sparse costumers,such as rural areas and developing countries. As a result, the typical coverage range of asingle base station is aimed to be up to 100 km to reduce the cost of the infra-structure.

1.2. ULTRA-WIDEBAND 3

11n

10 kb/s 100 kb/s 1 Mb/s 10 Mb/s 100 Mb/s 1 Gb/s 10 Gb/s

∼ 10 m

∼ 100 m

∼ 1 km

∼ 10 km

∼ 100 km

ZigBee 802.15.4a WiMedia 802.15.3c

GSM

802.22

TerraHz

UMTS HSDPA

802.11 11b 11a/g

WiMax802.20

WWAN

WMAN

WLAN

WPAN

WRAN

100 Gb/s

EDGEGPRS 3G-LTE

802.16d/e

Bleutooth(W-USB)

data rate

max

range

Standard

Under development

UWB Standard

Figure 1.1: Overview of communication standards

Another way to distinguish standards is with respect to supported data rates. Typi-cally, recent standards provide higher data rates than the older ones. When violating thisgeneral rule, the new standard will have some distinct benefits with respect to existingones, e.g., with respect to cost or added functionality like ranging or localization. Anoverview of currently successful standards and promising future standards can be foundin Fig. 1.1, separated with respect to coverage area and data rate. The overview containstwo standards related to the topic of this thesis, namely 802.15.4a and WiMedia. Thesewill be discussed in more detail in Sec. 1.2. More details on the other radio communicationstandards can be found in [1, 2, 3].

1.2 Ultra-WideBand

Although Ultra-Wideband (UWB) is often considered a new radio technology, UWBtechnology has been around for many years. In fact, the first wireless transmission ex-periments conducted by Hertz and Marconi could be considered a pulse based UWB.The use of a spark gap to generate radio signals inherently results in the radiation ofa pulse that is UWB. Radio communications took another course with the inventionof the Alexanderson radio alternator radio-frequency source, which allowed for CarrierWave (CW) communications. Not only because CW allowed for simpler transmitters, butalso because the low bandwidth of CW signals allowed selective Band Pass Filters (BPFs)to be used in the receiver to block out most of the noise and interference. Therefore, radioregulatory bodies started to assign frequency bands to specific systems, such that theycould co-exist without interfering with each other.

The success of CW systems resulted in UWB to be forgotten for more than 60 years.The interest in UWB came back with the invention of sub-nanosecond pulse generators in


the sixties. Shortly after, the potential of UWB for radio communications was identified,eventually resulting in the first US patent on pulse-based UWB radio communicationsin 1973 [4]. In those days, the main applications were radar and positioning, becauseof the inherent ability of UWB to resolve objects with a high spatial resolution, andmilitary communication systems, because of the inherent covertness of UWB signals.Most developments were therefore conducted in the military or funded by governmentsunder classified programs. Interestingly, UWB in those days was called either baseband,carrier-free or impulse technology. The term UWB itself was first used in a radar studyby the Defence Advanced Research Projects Agency (DARPA) in 1990. Despite theseearly developments, CW remained to govern commercial wireless radio communications.

The interest in UWB for commercial wireless radio communications revived with aseries of papers by Scholtz and Win [5, 6, 7] and the UWB activities of U.S. based com-panies like XtremeSpectrum, Multispectral Solutions and Time Domain. The lobbyingactivities of these companies resulted in a Notice of Inquiry by the Federal Communica-tions Commission (FCC) in September 1998 on the allowance of UWB on an unlicensedbasis under Part 15 of its rules [8]. This eventually led to a Report and Order (R&O)of the FCC in February 2002, to allow UWB under part 15 of its regulation [9]. Here,UWB emitters are allowed to operate in a frequency band from 3.1 to 10.6 GHz with aPower Spectral Density (PSD) of -41.3 dBm/MHz, the same as allowed by part 15 forunintentional radiators. The main intent of the R&O is to provide re-use of scarce radiospectrum while enabling high data rate WPAN as well as radar, imaging and localizationsystems.

At first, UWB was thought to be a pulse-based system, but the FCC defined UWB interms of a transmission from an antenna for which the emitted signal bandwidth exceedsthe lesser of 500 MHz or 20% of the center frequency. This allows Orthogonal FrequencyDivision Multiplexing (OFDM) and Direct Sequence (DS) systems to be operated underthe UWB regulation. The opening of several GHz of bandwidth for commercial applica-tions resulted in an avalanche of academic research and industrial efforts, which eventuallylead to the standardization of UWB for WPAN [10, 11].

The road to standardization has been rather rocky. In December 2002, the IEEEgranted the project authorization request as Task Group 3a (TG3a) part of the 802.15standards family for WPAN. The aim of TG3a was to specify a standard PHY forhigh-data-rate, short-range, low-power, and low-cost wireless networking technology usingUWB. In total 23 UWB PHY specifications were submitted, which quickly merged intotwo proposals. The WiMedia Alliance proposed a Multi-Band Orthogonal FrequencyDivision Multiplexing (MB-OFDM) PHY, which is a combination of Frequency Hopping(FH) and OFDM, while the UWB Forum proposed a Direct Sequence - UWB (DS-UWB)PHY. Over two and a half years, both consortia debated to come to a single PHY-proposal.Eventually, both agreed to not agree, resulting in a withdrawal of TG3a.

The withdrawal of TG3a did not mean the end of UWB for high-data-rate WPAN.Both parties continued their effort on their own. In December 2005, the European Com-puter Manufacturers Association (ECMA) released two ISO-based standards for UWBbased on the WiMedia UWB proposal [10, 11]. It supports data rates up to 480 Mb/s,but future extensions are expected to support data rates above 1 Gb/s. Furthermore, theWiMedia PHY has been selected for wireless Universal Serial Bus (USB) under the name

1.3. FRAMEWORK AND OBJECTIVES 5

Certified Wireless USB [12]. After initial activities of the UWB Forum, it became ratherquiet after the Freescale’s departure from the UWB Forum. Therefore, it seems that theWiMedia Alliance is winning the race.

Besides UWB being considered for high data rate WPAN, also joint low data rateand localization is considered for WPAN. In March 2004, the IEEE launched task group802.15.4a for a mandate to develop an alternative PHY as optional extension to the802.15.4 PHY, which provides low data rate communications and high precision rang-ing/location capability, while being low power and low cost. In March 2007, P802.15.4awas approved as a new amendment to 802.15.4 by the IEEE. Besides the mandatoryDSSS PHY of 802.15.4, one of the two alternative PHYs in 802.15.4a provides UWB inthree frequency bands, allowing for data rates between 110 kb/s up to 27.24 Mb/s andlocalization [13].

Following the FCC, the International Telecommunication Union Radiocommunica-tion Sector (ITU-R) has published a Report and Recommendation on UWB in Novemberof 2005. National bodies are expected to adopt their regulation to allow UWB. InSeptember 2005, a draft decision was released by the European Conference of Postal andTelecommunications Administrations (CEPT). In March 2006, the Electronic Commu-nications Committee (ECC) decision was issued, allowing UWB for frequencies between6 and 8.5 GHz. The frequencies between 3.1 and 4.8 GHz are expected to follow soon.In Japan, the Ministry of Internal Affairs and Communications (MIC) launched a regu-latory proposal. The foreseen allocated bandwidths are the frequencies between 3.4 until4.8 GHz and 7.25 until 10.25 GHz, with the same PSD limits as allowed by the FCC. Incontrast to the FCC, the European and Japanese regulation bodies may demand UWBsystems to use so-called Detect and Avoid (DAA) to avoid interference with current andfuture wireless services [14, 15].

1.3 Framework and Objectives

The work presented in this thesis is the partial outcome of an objective defined at theIMST GmbH to develop understanding on UWB technology. Starting in 2000, the ob-jective was to acquire know-how on the theory and implementation of low-cost UWBsystems for communication and localization. The objective resulted in the participationin several projects both on a European level as well as on a regional level. The projectsfunded by the 5-th and 6-th framework of the IST program of the European Union in achronological order are Whyless.com, Europcom and Pulsers 2. The projects funded inthe scope of the Nordrhein-Westfalen Zukunftswettbewerb are Bison and PulsOn

While having many benefits, the implementation of UWB systems is significantly morecomplex than those of narrowband systems, since many of the hardware components mustbe well-behaving over a larger frequency range. Crudely spoken, more bandwidth moreproblems, at least with respect to implementation and cost. On the other hand, onewould like to take advantage of the fundamental benefits of UWB. Hence, during sys-tem design a trade-off is required between both aspects. One of the benefits of UWBis inherent resilience against small-scale-fading, which allows the Physical Layer of theOSI model (PHY) to operate with higher energy efficiency. The first aim of this thesisis to understand and mathematically model the Small-Scale Fading (SSF) behaviour of


General Introduction

Chapter 1:

Chapter 2:

Theory of fading UWB channels

Chapter 3:

Fading of measured UWB channels

Chapter 6:

Design of a high-rate TR UWB system

Chapter 4:

Theory of TR UWB communication

Chapter 5:

Analysis of TR UWB communication

Figure 1.2: Organization of the thesis

the UWB radio channel to ultimately allow for an educated trade-off between systemperformance and complexity. Having low-cost and low-complexity in mind, the secondaim of the work is to model and understand the fundamental behaviour of UWB wire-less communications using Transmitted Reference (TR) signaling and AutocorrelationReceivers (AcRs). Based on the developed understanding on UWB, SSF and UWB TRcommunications, the final aim is to design a low-cost UWB PHY for WPAN operatingat a data rate of 100 Mb/s to unveil the potential of UWB TR communications.

1.4 Thesis Outline and Contributions

In this section, the outline and the scientific contributions of the thesis are presented.After the general introduction to the topic presented in this chapter, the thesis outline

follows two parallel branches, which can be read and understood independently. The firstbranch consists of the subsequent Chapters 2 and 3, which deal with the theory andpractice of SSF on UWB channels, respectively. The second branch deals with the theoryand practice of TR UWB systems in Chapter 4 and 5, respectively. The insight gainedin both branches is used for the design of a high-rate TR-UWB system in Chapter 6. Agraphical impression of the thesis outline can be found in Fig. 1.2.

In the following, a short summary of each chapter is presented, including the author’scontributions.

Chapter 2

Chapter 2 relates the statistics of SSF on UWB channels and its dependence on bandwidthin closed-form. By assuming Uncorrelated Scattering (US), first a statistical model is

1.4. THESIS OUTLINE AND CONTRIBUTIONS 7

presented for radio channels in the frequency domain. Based on US, the eigenvaluesare derived in closed-form for UWB channels. Using the eigenvalues, the expectation,variance and diversity level is derived in closed form both for Line-of-Sight (LOS) andNon-Line-of-Sight (NLOS) UWB channels. The diversity level is shown to scale linearlywith respect to the Root Mean Square (RMS)-delay-spread-by-bandwidth product, bothfor LOS and NLOS channels.

Finally, upper bounds for the uncoded and coded Bit Error Rate (BER) for ideal UWBsystems will be presented using the eigenvalues of the channel. These bounds allow for atrade-off analysis between bandwidth and BER performance of UWB systems on NLOSUWB channels. Assuming a typical RMS delay spread for indoor environments, theupper bound for the performance of Multiband OFDM systems using frequency hoppingis found to be only 1 dB less energy efficient than an infinite bandwidth system.

The main contributions are:

• Introduction of a single measure to quantify the diversity level of (UWB) radiochannels [16].

• Derivation of a lower bound for the diversity level of UWB channels, which convergesto the actual diversity level with increasing bandwidth. The lower bound shows alinear relationship between the diversity level, bandwidth and RMS-delay-spread,both for LOS and NLOS channels. This relationship is well-known, but, to ourknowledge, has never derived before in closed form. [to be published ].

Chapter 3

In Chapter 3, the theoretical model presented in Chapter 2 is verified using measurementdata of UWB radio channels both emphasizing its strengths and short-comings. Firstly,the channel measurement campaign is described briefly. The statistical properties of themodel are validated using the measurement data in both the time and frequency domain.The statistical properties of the Principal Components (PCs) of the measured UWB radiochannel have been analyzed. The diversity level as function of bandwidth of measuredradio channels is compared with the theoretical results. Finally, the BER predicted bytheory is compared with the BER on measured channels.


• On NLOS channels, the theoretical model was found to be reasonably accurate, butnot exact because the independence assumption of the PC is not valid for the usedmeasurement data. It is expected that a better prediction is obtained for richermultipath environments. [to be published ].

• For LOS channels, the predicted diversity level of the theoretical model is consider-ably lower than for measured LOS channels. In practice, the LOS eigenvalue doesnot share a PC-dimension with the largest NLOS eigenvalue, but one which is con-siderably smaller. The result is considerably less fading. The mechanism(s) behindhave not been unveiled. [to be published ].


Chapter 4

Firstly, a brief introduction of TR signaling is presented including its strengths and short-comings with respect to performance and implementation. To overcome some of theseshortcomings, several extensions of the TR principle are proposed. First, a fractionalsampling AcR structure is proposed to relax synchronization and allow for weightedautocorrelation, while simplifying the implementation. Second, a complex-valued AcRis proposed to make the system less sensitive against delay mismatches. Additionally,complex-valued modulation for TR signaling is proposed. To understand the system’sbehaviour, a general-purpose discrete-time equivalent system model is derived and pre-sented, where general-purpose means that all extensions are taken into account for. Sev-eral interpretations for the system model are presented, which allow for more insight inthe behaviour of TR systems in various situations. Finally, the statistical properties ofTR UWB system are presented.


• Proposal of a fractional sampling autocorrelation receiver to relax synchronizationand allow for weighted autocorrelation demodulation [17].

• Proposal of a complex-valued autocorrelation receiver to relax delay implementationand allow for complex-valued TR signaling [18].

• Development of a general-purpose model for TR UWB systems, which illustratesthat TR systems in the presence of ISI can be modelled using a second-order FIRVolterra model [17].

• Development of a linear Multiple-Input, Multiple-Output (MIMO) model for thesecond-order FIR Volterra model for TR systems, modulated with finite-alphabetsymbols [19]. The model shows that more ISI in a TR system can be suppressedwith increasing fractional sampling rate [17]. The model explains how the amountof ISI that can be suppressed is influenced by the TR modulation [19].

• Finite state machine description for the finite-alphabet, second-order FIR Volterramodels, taking reference-pulse scrambling into account. The model shows thatreference-pulse scrambling may lead to a time-variant finite state machine, butdoes not complicate a trellis-based equalizer significantly [to be published ].

• Derivation of a reduced memory Finite State Machine (FSM) description for finitealphabet, second-order FIR Volterra models, optimal in the sense of the MMSEcriterion. The model allows for tradeoff analyses between equalizer complexity andsystem performance [20].

Chapter 5

In Chapter 5, the impact of different parameters on the system performance is analyzed.The evaluated system parameters are Fractional Sampling Rate (FSR), bandwidth, delay,weighting criterion and modulation, both in the absence and presence of ISI.



• Closed-form derivation of the weighting coefficients, optimal in the sense of theMRC and MMCE criteria [18].

• In the absence of ISI, an FSR of 2 is sufficient to obtain close to optimal performance.

• The non-Gaussian noise term has a significant impact on the system performance,such that smaller bandwidth TR systems perform better, in the absence of fad-ing [17].

• In the presence of ISI, more ISI can be suppressed using linear weighting if the FSRis increased [17].

• In the presence of ISI, the modulation has a significant impact on the amount ofISI that can be suppressed using linear weighting [19].

Chapter 6

In Chapter 6, the design of a high-rate TR UWB system is presented. The design aim is aTR-UWB PHY supporting a data rate of 100 Mb/s, while occupying a 1 GHz bandwidth.In the design, the insight gained in the previous chapters has been taken into account. Theuse of trellis-based equalization is considered, to support high data rate. To reduce theequalizer complexity, the multiband concept, originally proposed for energy detectors,is applied to TR signaling. The system performance is analyzed taking into accountForward Error Control (FEC) and using turbo equalization.


• Proposal of scrambled QPSK-TR signaling, which avoids spectral spikes, while pre-serving the time-invariant character of the FSM describing the Volterra model [tobe published ].

• Proposal of multiband TR signaling to reduce the equalizer complexity, while al-lowing for higher data rates. Application of the multiband concept allows for animprovement of 3 dB, while reducing the equalizer complexity by a factor 16 [20].

• Application of turbo equalization to (multiband) TR UWB systems. A performanceimprovement of 1.5-3 dB is observed with respect to the Frame Error Rate (FER) [tobe published ].

List of Publications

In this section, an overview is provided of the author’s academic publications.

Journal Papers

[17] J. Romme and K. Witrisal, ”Transmitted-Reference UWB Systems using WeightedAutocorrelation Receivers,” IEEE Transactions on Microwave Theory and Techniques,Apr. 2006, vol.54, pp.1754-1761, Special Issue on Ultra-Wideband Systems


Conference Papers

[21] G. Durisi, J. Romme and S. Benedetto, ”A general method for SER computation ofM-PAM and M-PPM UWB systems for indoor multiuser communications,” IEEE GlobalTelecommunications Conference (GLOBECOM), Dec. 2003, vol.2, pp.734-738

[22] D. Manteuffel, T.A. Ould-Mohamed and J.Romme, ”Impact of Integration in Con-sumer Electronics on the performance of MB-OFDM UWB,” International Conference onElectromagnetics in Advanced Applications, 2007. ICEAA 2007, Sept. 2007, pp.911-914,Torino, Italy

[23] L. Piazzo and J.Romme, ”Spectrum control by means of the TH code in UWBsystems,” IEEE Semiannual Vehicular Technology Conference (VTC-Spring), Apr. 2003,vol.3, pp.1649-1653 Seoul, Korea

[24] J. Romme and G. Durisi, ”Transmit Reference Impulse Radio Systems Using WeightedCorrelation,” Internal Workshop on UWB Systems Joint with Conference on UWB Sys-tems and Technologies, May 2004, pp.141-145, Kyoto, Japan,

[16] J. Romme and B. Kull, ”On the relation between bandwidth and robustness of indoorUWB communication,” IEEE Conference on Ultra Wideband Systems and Technologies,Nov. 2003, pp.255-259, Reston, VA

[25] J. Romme and L. Piazzo, ”On the power spectral density of time-hopping impulseradio,” IEEE Conference on Ultra Wideband Systems and Technologies, 2002, pp.241-244,Baltimore, MA

[20] J. Romme and K. Witrisal, ”Reduced Memory Modeling and Equalization of Sec-ond Order FIR Volterra Channels in Non-Coherent UWB Systems,” European SignalProcessing Conference (EUSIPCO), Sep. 2006, Florence, Italy, invited paper

[19] J. Romme and K. Witrisal, ”Impact of UWB Transmitted-Reference Modulation onLinear Equalization of Non-Linear ISI Channels,” IEEE Vehicular Technology Conference(VTC), May 2006, pp.1436-1439, Melbourne, Australia

[18] J. Romme and K. Witrisal, ”Analysis of QPSK Transmitted-Reference Systems,”IEEE Internal Conference on Ultra-Wideband (ICU), Sep. 2005, pp.502-507, Zurich, CH

[26] J. Romme and K. Witrisal, ”Oversampled Weighted Autocorrelation Receivers forTransmitted-Reference UWB Systems,” IEEE Vehicular Technology Conference (VTC),May 2005, pp.1375-1380, Stockholm, Sweden

[27] J. Romme and K. Witrisal, ”On Transmitted-Reference UWB Systems using Discrete-Time Weighted Autocorrelation,” COST273, COST 273 TD(04)153, Sep. 2004, Duisburg,Germany

[28] W. Xu and J. Romme, ”A Class of Multirate Convolutional Codes by Dummy Bit In-sertion,” IEEE Global Telecommunications Conference (GLOBECOM), Nov. 2000, vol.2,pp.830-834, San Francisco, CA


Miscellaneous

K. Witrisal, J. Romme, M. Pausini and C. Krall ”Signal Processing for Transmitted-Reference UWB Systems,” IEEE International Conference on Ultra-Wideband (ICUWB),Waltham, MA, Sep. 2006, Half-Day Tutorial

J. Romme and B. Kull ”A low-datarate and localization system,” UWB4SN: Workshopon UWB for Sensor Networks, Nov. 2005, Lausanne, CH

Unpublished

J. Romme and K. Witrisal, ”Estimation of Nakagami m Parameter for Frequency SelectiveRayleigh Fading Channels,” IEEE Communications Letters, In Preparation

Chapter 2

Theory of Fading UWB Channels

2.1 Introduction

Understanding the mechanisms behind radio propagation is mandatory for any engineerevaluating and optimizing the performance of wireless radio communication systems.This chapter is on the theory of SSF of UWB channels, having in mind indoor datacommunication. The goal is to relate the statistical properties of the SSF to generalchannel parameter like bandwidth and channel delay spread.1

As an introduction, the remainder of this section is on the basics of the radio channel.In Sec. 2.2, the statistical properties of frequency selective fading channels are derivedand an insightful channel model is derived using the eigenvalues of the radio channel.Additionally, the eigenvalues of UWB channels are derived in closed-form. In Sec. 2.3, thefrequency diversity of radio channels in general and UWB channel specifically is quantifiedusing the eigenvalues of the channel. In Sec. 2.4, the uncoded and coded BER for idealUWB systems are presented based on the eigenvalues of the channel, which is useful fortrade-off analyses between bandwidth and BER performance. Finally, conclusions aredrawn in Sec. 2.5.

2.1.1 The Radio Channel

Consider a radio communication system consisting of a transmitter and receiver operatingin an indoor environment. To allow for radio communication, both deploy antennas toconvert electrical signals into radio signals.

In its most elementary form, an antenna consists of two conductive objects, whichare electrically isolated from each other. By applying a time-variant Radio Frequency(RF) signal to the antenna connectors, electrical and magnetic fields form around theantenna. The combined fields generate self-sustaining Electro-Magnetic (EM) waves,allowing energy to ”release” itself from the antenna and to propagate into the surroundingenvironment.

In the environment, the EM waves will interact with the objects they encounter. Atypical indoor environment contains many objects, e.g. walls cabinets and chairs. Three

1Strictly speaking, the radio channel itself has no bandwidth. It is the bandwidth of the transmitsignal that determines how the radio channel is experienced.

13

14 CHAPTER 2. THEORY OF FADING UWB CHANNELS

types of interactions that are relevant for radio communication can be distinguished,namely reflection, scattering and diffraction.

Reflection occurs when a radio wave encounters an object with large dimensions andsmooth surface compared to the wavelength. Examples of such objects are a wall orcabinet. In this case, the well-known optical ray model holds, i.e. reflections occur.

Scattering is similar to reflection with the difference that the dimensions of the en-countered object are in the order of the wavelength or less and causes the radio signalto re-radiate in many directions. Examples of scattering objects are pens, scissors, cups,wall with a rough surface etc.

Diffraction occurs when an object is positioned such that its edge is near the ray-path of the radio signal, where near is with respect to the wavelength. In this case,the ray-model does no longer apply. However, the more sophisticated Huygens-principlecan model the behaviour of radio wave propagation in such scenarios [29, 30]. Since theobject blocks part of the Huygens sources, the radio signal bends around the object. Thisphenomenon is also referred to as shadowing , because EM energy can reach the receiver,although it is in the ”shadow” of the object.

Due to these interactions with the environment, numerous EM waves will reach thereceiver, each with its own delay, direction, distortion and intensity. Each EM wave willgenerate a signal in the antenna such that the overall signal at the antenna connectors isthe superposition of all individual contributions.

2.1.2 Radio Channel Model

To obtain insight in the influence of the indoor radio channel on a radio signal, the mul-tipath radio channel model is introduced. In this model, the radio signal is assumed topropagate from the transmitter to the receiver along distinct paths, where each path in-troduces its own attenuation and delay, see Fig. 2.1. This phenomenon is called multipathpropagation and the channel over which the radio signal propagates is referred to as themultipath channel . Most often, the propagation environment will vary in time such thatpath delays and path attenuations will be a function of time. For instance, the transmit-ter and/or the receiver can move. Even if both are static, the environment itself may besubject to change.

Based on the described mechanisms of indoor radio propagation, a model for the radiochannel can be obtained. Each time-variant path is characterized by a delay τn(t) andamplitude gain βn(t), where n identifies the path. Based on this assumption, the receivedsignal appears as a train of identically shaped transmit pulses, which possibly overlap intime. The time-variant Channel Impulse Response (CIR) h(τ, t) can thus be formulatedas

h(τ, t) =

Np(t)∑

n=1

βn(t)δ(τ − τn(t)), (2.1)

where Np(t) denotes the number of observed multipath components at time t.2

2The mathematical representation is both valid for passband and baseband representations of pass-band channels. In the baseband case, βn(t) is complex-valued and its phase is related to the path delayτn(t) according to arg(βn(t)) = 2πfcτn(t)[rad], where fc denotes the center frequency

2.1. INTRODUCTION 15

!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!

Scatterer

TX Antenna RX Antenna

Scatterer

Scatterer

Figure 2.1: The multipath radio channel

Following the discussion in the previous section, it is evident that the multipath chan-nel model is an oversimplification of reality. For instance, the ray-model of (2.1) does notinclude diffraction. Nevertheless, the assumption is widely accepted, because the resultingmodel is intuitive, practical and, more importantly, the results closely resembles realityfor narrowband channels. Although yet to be proven for UWB channels, the multipathmodel will be used throughout this thesis to obtain simple, traceable results.

2.1.3 Channel Characterizing Parameters

It is useful to introduce some parameters that capture the nature of radio channels. ThePower Delay Profile (PDP) is defined as the power of the CIR as a function of τ . TheCIR h(τ, t) has a PDP given by

P (τ, t) = |h(τ, t)|2 =∑

n

|βn(t)|2δ (τ − τn(t)) , (2.2)

The mean excess delay is the first moment of the PDP and is given by

τ(t) ,

∞∫

−∞

P (τ, t)τdτ

∞∫

−∞

P (τ, t)dτ

(2.3)

and can be seen as the weighted average delay of the radio channel [31].

The RMS delay spread is defined as the squared root of the second central moment


of the PDP, i.e.

τd(t) ,

√√√√√√√√

∞∫

−∞

(τ − τ(t))2 P (τ, t)dτ

∞∫

−∞

P (τ, t)dτ

(2.4)

The RMS delay spread represents the RMS of the path delays around the mean excessdelay using the normalized path energies as a weighting function.

The RMS delay spread is often averaged over space. In this manner, it does nolonger characterize a single CIR, but a certain propagation environment. The averageRMS delay spread is an important measure to characterize radio channels and used tomodel the Average Power Delay Profile (APDP). An exponential decay model is a widelyaccepted model for the APDP in NLOS environments for UWB and radio channels ingeneral [32, 33, 34]. This model is described by the equation,

E[|h(τ)|2

]=

A2

σexp

(− τ

σ

)∀ τ ≥ 0,

0 ∀ τ < 0.(2.5)

where E [.] denotes a mathematical expectation and the parameters σ and A2 allow themodel to mimic specific NLOS radio environments and should be chosen such that σ = τd

and A2 =∑Np

n=1 |βn|2.The model can be generalized to include LOS scenarios, by adding an additional

component to the APDP,

E[|h(τ)|2

]=

A2K

(K+1)δ(τ) + A2

σ(K+1)exp

(− τ

σ

)for all τ ≥ 0,

0 for all τ < 0.(2.6)

where K denotes the ratio of LOS gain with respect to cumulative gain of all radio paths.This ratio is referred to as the Ricean K factor. Due to the generalization, σ is re-definedto

σ = τdK + 1√2K + 1

. (2.7)

These parameters will be used throughout this thesis report as characterization of theradio channel.

2.1.4 Impact of the Channel on Radio Signals

The effect of a multipath radio channel on a narrowband radio signal is well-known notonly to radio communication engineers. Anyone who listens to their car radio is likelyto have observed the following phenomenon. While stopping at a traffic light, first thereception is very poor, but by moving the car only slightly the audio signal qualityimproves drastically. This phenomenon is referred to as fading .

In case of a narrowband signal y(t) with a center frequency fc, the impact of the radiochannel can be well approximated by a scalar multiplication, such that the received signalwill be

r(t) ≈ H(fc, t)y(t). (2.8)

2.1. INTRODUCTION 17

In this case, the channel is referred to as flat fading , since all frequency components ofy(t) are scaled equally [31].

The scalar multiplication factor H(f, t) is the Channel Frequency Response (CFR) attime t, which is equal to the Fourier transform of h(τ, t) with respect to τ , i.e.

H(f, t) =

Np(t)∑

n=1

βn(t) exp (j2πfτn(t)) (2.9)

The equation shows that each radio path has its own distinct phase. Since H(f, t) is thesummation of all paths, the paths can interfere destructively with each other. By movingslightly, the number of paths and the path amplitude gains will not change. Howeverthe phase of each path can change significantly. Hence, the interference between pathsis possibly/likely no longer destructive, such that the reception can improve drastically.This phenomenon is referred to as SSF.

Although the phase of each path is a deterministic function of the environment, thevariation of H(f, t) as function of time is often modelled as a complex-valued3 Gaussiandistributed RV, see [35]. This model is accurate if the environment is rich of scatters,which is typically valid for indoor NLOS environments, such that none of the βn(t) is trulydominant. For this case, Rice has proven that |H(f, t)| has a Rayleigh distribution [31].For these scenarios, the Rayleigh distribution has proven itself to successfully predict thestatistics of measured channel gain with good accuracy.

If one of the rays is dominant, which is often the case in LOS environments, a gener-alization of the Rayleigh distribution, called the Rice distribution, accurately models thestatistics of measured channel gain [31]. More on the Rice distribution will follow in theremainder of this chapter.

To illustrate the effect of fading, the Rayleigh distribution is depicted in Fig. 2.2.The figure shows that the received radio signal on a Rayleigh fading channel can varyextensively. For 1 percent of time, the received signal power will be 20 dB lower than itsaverage. To complicate matters, the received power can vary rapidly and unpredictably,making it difficult for the transmitter to compensate for the variations using power con-trol.4 Therefore, radio communication systems often use large fading margins, whichinevitably reduces the system’s energy efficiency.

Fortunately, one can reduce the probability of such deep fades and waste less TX poweron fading margins. If the information is communicated over two or more independentlyfaded channels, evidently the probability that all channels are in a deep fade simultane-ously becomes smaller. This probability decreases with every additional channel used.The principle described here is referred to as diversity and the amount of independentlyfading channels is called the diversity level . Diversity can be found in three directions ofthe radio channel, namely space, time and frequency.5

The availability of independent fading channels is not sufficient. To exploit the diver-sity, it should be ensured that the radiated energy related to a single unit of information

3Assuming a baseband notation.4Assuming a return channel to inform the transmitter on the channel state.5In literature also the terms polarization diversity and path diversity are used. However, polarization

diversity can be seen as a type of spatial diversity. Path diversity is actually another perspective onfrequency diversity.


−30 −25 −20 −15 −10 −5 0 5 100

0.5

1

p(|

H(ω

0)|

= r

)

20log10(r)

−30 −25 −20 −15 −10 −5 0 5 1010

−3

10−2

10−1

100

E[|H

(ω0)|

≤ r

]

20log10(r)

Figure 2.2: The Rayleigh distribution

is spreads over multiple and at best all available fading channels. The drawback is thatparts of the TX signal are communicated over independent fading channels and thus af-fected differently. Inherently, the receiver has to conduct signal processing on the receivedsignal in order to exploit the diversity. This type of signal processing is referred to asdiversity combining.

Several signal processing techniques for diversity combining exist, each with its ownperformance and complexity. Assuming Gaussian noise and the absence of Inter SymbolInterference (ISI), Maximum Ratio Combining (MRC) is the optimal one with respect toboth the Signal-to-Noise Ratio (SNR) and BER. Other techniques are Minimum MeanSquare Error (MMSE) combining , switched combining , selective combining and equal-gain combining . More information on diversity and diversity combining can be found inliterature [36, 31].

Due to their large bandwidth, UWB systems inherently allow for a large amount of fre-quency diversity, explaining the large interest of both industry and academic society. Thefocus of this part of the thesis is on frequency diversity in UWB systems. In this chapter,a theoretical framework is developed to understand the underlying mechanisms. In thesecond chapter, the frequency diversity is analyzed using radio channel measurements tovalidate the insight obtained in this chapter.

2.2 Frequency Domain Properties of UWB Channels

In this section, the statistical properties of UWB channels are investigated in the fre-quency domain. Using principal component analysis, the CFR will be decomposed into

2.2. FREQUENCY DOMAIN PROPERTIES OF UWB CHANNELS 19

the smallest possible set of uncorrelated Random Values (RVs) driving the CFR. Theseresults are not only of statistical relevance, but also explain the mechanism of frequencyselective fading channels. Furthermore, the eigenvalues of UWB US channels are derivedin closed-form, which allow for further insight into the properties of UWB channels.

2.2.1 Frequency Domain Correlation

In Sec. 2.1.4, the CFR H(f, t) at a given frequency f can be modelled using a complex-valued, zero-mean Gaussian function. Evidently, the CFR at two distinct frequenciesf1 and f2 at the same time instant t will be correlated if the two frequencies are closetogether. To capture the statistical properties of CFR, we introduce the correlationfunction of the frequency response

φ(f1, f2) , E [H(f1)H∗(f2)] . (2.10)

For the US case, the result is well-known [37], namely

φ(f1, f2) =

∞∫

−∞

E[|h(τ)|2

]e−j2π∆f τdτ (2.11)

where ∆f is defined equal to f1−f2. Since its value depends only on the frequency differ-ence, φ(f1, f2) is inherently Hermitian and Toeplitz. Furthermore, E [|h(τ)|2] is definitionthe APDP as defined in Sec. 2.1.3.

Substitution of the NLOS APDP model of 2.5 into (2.11) leads to the following ex-pression for the frequency correlation,

φ(f1, f1 − ∆f ) = ρ(τd∆f ) (2.12)

where

ρ(x) =A2

1 + j2πx(2.13)

This result is easily generalized to include LOS scenarios, by adding a constant. Forillustrative purposes, the magnitude of ρ(x) has been depicted in Fig. 2.3.

The frequency correlation function is closely related to the coherence bandwidth. Nogenerally accepted definition exists for the coherence bandwidth, but in most cases, it isdefined as the frequency separation ∆f for which ρ(τd∆f ) equals 1/2. This definition willalso be used in this thesis. The coherence bandwidth in case of the APDP model is

Bcoh =√

3/(2πτd) ≈0.28

τd

. (2.14)

Hence, the analytical model states a reciprocal relation between Bcoh and τd. The recip-rocal relationship between the RMS delay spread and coherence bandwidth is confirmedby measurements (see [37]).


0 0.2 0.4 0.6 0.8 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

|ρ(∆

f)|

τd∆f

Figure 2.3: The normalized frequency domain correlation function ρ(x)

2.2.2 Eigenvalues and Their Physical Interpretation

A considerable amount of information on the statistical properties of the radio channelas ”experienced” by a UWB signal or in fact by any radio signal can be obtained from itseigenvalues. Therefore, let us assume a radio signal with a power spectral density function|Y (f)|2. In this case, the PSD of the received signal R(f) will be equal to |H(f)|2|Y (f)|2.For simplicity, the transmit power is assumed uniformly distributed over a bandwidth Baround a center frequency fc, such that

|Y (f)|2 =

Pt

Bfor |f − fc| ≤ 1

2B,

0 otherwise.(2.15)

The advantage of this definition is that the channel properties can be investigated withoutany influence of the TX signal spectrum, except for the influence of bandwidth and centerfrequency. To simplify the derivations, unit transmit power is assumed, i.e. Pt = 1. With-out having impact on R(f), H(f) may be assumed to be zero outside the spectral maskof Y (f) as well. Consequently, the two-dimensional autocorrelation function φ(f1, f2) isdefined for a finite square area from fc − B/2 to fc + B/2 in both dimensions f1 and f2,and zero otherwise.

Using Principal Component Analysis (PCA), the bandwidth-limited function φ(f1, f2)can be decomposed into the most efficient set of eigenfunctions and eigenvalues, givingus information on the uncorrelated random processes driving the CFR. In [38] PCA isdescribed as follows:

“The central idea of principal component analysis is to reduce the dimension-


ality of a data set in which there are a large number of interrelated variables,while retaining as much as possible of the variation present in the data set.This reduction is achieved by transforming to a new set of variables, the prin-cipal components, which are uncorrelated, and which are ordered so that thefirst few retain most of the variation present in all of the original variables.”

In our context, the data consist of many realizations of the CFR for the frequency rangeunder consideration. More information on PCA can be found in [39, 38].

Using PCA and the fact that φ(f1, f2) is Hermitian, φ(f1, f2) can be decomposed intothe following form,

1

Bφ(f1, f2) =

∞∑

k=1

λ[k]Gk(f1)Gk(f2), (2.16)

where λ[k] and Gk(f) denotes the k-th eigenvalue and its eigenfunction, respectively. Thedivision by B in (2.16) ensures that the eigenvalues and eigenfunctions are dimensionlessand simplifies derivations later on. Since φ(f1, f2) is Hermitian, the eigenfunctions areorthogonal with respect to each other.

Although the summation index k theoretically goes to infinity, it can be truncatedto N without losing much accuracy by choosing N sufficiently large. The low-pass-characteristic of ρ(x) ensures that only a finite number of significant eigenvalues exist,i.e. eigenvalues will vanish with increasing index. The application of PCA ensures thatthe truncated summation represents the best possible approximation using only N com-ponents.

The principal components can rarely be found in closed-form, except for some asymp-totic cases, see Sec. 2.2.3. Fortunately, numerical tools exist to obtain them, like SingularValue Decomposition (SVD). In Fig. 2.4, the eigenvalues are depicted obtained usingSVD for different RMS-delay-spread-by-bandwidth products. It shows that the numberof significant eigenvalues increases with an increasing RMS-delay-spread-by-bandwidthproduct. This result is confirmed by the analysis of UWB measurement data in [40, 41]and Chapter 3.

PCA is not only of mathematical relevance, but it also allows for a physical interpreta-tion of radio channels. Any band-limited radio channel can be thought to be decomposedby the eigenfunctions into N sub-channels, such that

H(f) =N∑

k=1

u[k]Gk(f). (2.17)

where u[k] is by definition equal to the inner-product < H(f), Gk(f) >. Assuming H(f)to be a complex-valued Gaussian distributed random function, u[k] will be a complex-valued Gaussian distributed RV with a variance λ[k], which will be referred to as the k-thPC of the channel. Using the orthogonality of the eigenfunctions, it can be shown thatu[k] is independent of u[l] if k is unequal to l. Hence, the radio channel can be seen asthe sum of N parallel independent fading channels.

Furthermore, the radio channel can be thought to decompose the transmit signal y(t)


5 10 15 20 250

10

20

30

40

50

60

70

80

index k

λ[k

]τdB = 0.5τdB = 2τdB = 5

Figure 2.4: Dependence of the eigenvalue distribution on the τdB product

into N sub-signals using a filter bank, since

R(f) = H(f)Y (f) =N∑

k=1

u[k]

k-th sub-signal︷︸︸︷

Gk(f)︸︷︷︸

k-th filter

Y (f) . (2.18)

where the k-th sub-signal is multiplied with the RV u[k], i.e. all sub-signals experience flat-fading. A graphical representation of this interpretation for the time-domain is presentedin Fig. 2.5.

2.2.3 Asymptotic Behaviour of the Eigenvalues

In Sec. 2.2.2, the eigenvalues of the channel were investigated. However, the eigenvaluescould not be obtained in closed form. Analytical expressions however often lead to moreinsight in the behaviour of the system with respect to its parameters. In this section, aclosed form approximate relationship will be presented between the channel eigenvalueson one side and parameters like bandwidth and RMS delay spread on the other side,which is exact for B going to infinity.

Already in Sec. 2.2.1, φ(f1, f2) was shown to have a Toeplitz structure. Furthermore,φ(f1, f2) is a banded function in the UWB case, i.e. the significant values are aroundthe main diagonal of φ(f1, f2) and virtually zero otherwise. An illustration of a UWBφ(f1, f2) can be found in the left-hand sub-plot in Fig. 2.6.

As stated before, no generally valid, closed-form expression for the eigenvalues ofbanded Toeplitz functions exists. However, for a special case of Toeplitz functions the


...y(t)

+

(y ∗ gN )(t)

(y ∗ g2)(t)

(y ∗ g1)(t)

u[N ]

r(t)

u[2]

u[1]

Figure 2.5: Physical interpretation of the eigenfunctions and eigenvalues of radio channels

φ(f1, f2)

−5

0

5

−5

0

5

0

0.2

0.4

0.6

0.8

1

τd(f1−fc)

τd(f2−fc)

|φ(f1,f2)| [dB]

φc(f1, f2)

−5

0

5

−5

0

5

0

0.2

0.4

0.6

0.8

1

τd(f1−fc)

τd(f2−fc)

|φ(f1,f2)| [dB]

Figure 2.6: Comparison between φ(f1, f2) and φc(f1, f2) of a UWB channel


eigenvalues can be derived in closed-form, namely for circulant functions. Furthermore,Gray has proven that every banded Toeplitz matrix has a circulant counterpart thatis asymptotically identical, such that their eigenvalues are asymptotically identical aswell [42].

Hence, the strategy is to derive a circulant function φc(f1, f2), which is asymptoticallyidentical to φ(f1, f2). Like all Toeplitz functions, the value of φ(f1, f2) and φc(f1, f2)depends only on the difference between both arguments. In case of φ(f1, f2), its value isdetermined by ρ(f1−f2). To obtain a Toeplitz function φc(f1, f2), which is asymptoticallyidentical to φ(f1, f2), we defined

φc(f1, f1 − ∆f ) = ρc(∆f )

= ρ(∆f ) + ρ∗(B − ∆f ) (2.19)

To illustrate their relation, a comparison of φc(f1, f2) with φ(f1, f2) is presented in Fig. 2.6.

In [42], circulant matrices are shown to have the following properties:

1. The eigenvalues of a circulant matrix are equal to the Discrete Fourier Transform(DFT) of the first row.

2. Using linearity of the DFT, the k-th eigenvalue λA[k] of a circulant matrix A mustbe equal to the sum of λB[k] and λD[k], if B and D are also circulant matrices andλB[k] and λD[k] their k-th eigenvalue, respectively.

Applying property 1 to the circulant function φc(f1, f2), the k-th Circulant Eigenvalue(CEV) will be equal to

λc[k] =

1∫

0

ρc(xB) exp (−j2πkx)dx (2.20)

This equation shows that the eigenvalues λc[k] can be seen as the weights of the Fourierseries of the frequency domain autocorrelation function ρc(xB), where the eigenfunctionsexp (−j2πkx) are the Fourier modes [43]. Substitution of (2.19) into (2.20) and using thefact that ρ(x) is an Hermitian function, this result can be further simplified to

λc[k] =

1∫

0

(ρ(xB) + ρ∗(B − xB)) exp (−j2πkx)dx (2.21)

=

1∫

−1

ρ(xB) exp (−j2πkx)dx

In the UWB case, B is so large that ρ(Bx) is zero at the integration interval edges,

2.3. DIVERSITY OF UWB CHANNELS 25

so that the upper limit may be replaced by ∞ without altering the result.

λc[k] =

∞∫

−∞

ρ(xB) exp (−j2πkx)dx

= Fρ(xB)

= F A2

1 + j2πτdxB

=A2

τdBexp(− k

τdB) (2.22)

Hence, the eigenvalues drop exponentially with increasing k in the asymptotic UWB caseat a pace inverse proportional to both B and τd.

It has already been mentioned that by adding a constant to (2.13), also LOS scenarioscan be modelled. Since a constant function is also circulant and using property 2 ofcirculant matrices on page 24, it can be understood that the eigenvalues for LOS scenariosare equal to

λc[k] = λc,L[k] + λc,N[k] (2.23)

with

λc,L[k] =A2K

(K + 1)δ[k] (2.24)

λc,N[k] =A2

σB(K + 1)exp(− k

σB), (2.25)

where σ has been defined in (2.7).

This result shows that the LOS component shares a dimension with the PC with thelargest eigenvalue/variance of the NLOS part of the APDP. Since the eigenvalue of thisPC decreases with increasing bandwidth, the LOS component asymptotically has its owndimension.

2.3 Diversity of UWB Channels

The gain of the radio channel is a valuable measure of the signal quality. Detailed knowl-edge on its statistical properties is relevant for any system engineer, not only to predictthe average BER performance, but also how the BER will vary in time/space. Previouslyin this thesis, the radio channel was modelled as a random process. Since the receivedpower depends on the radio channel, it will be modelled as random process as well. Inthis section, the statistical properties of the power gain of the channel will be derived inclosed form as experienced by a UWB signal with bandwidth B.


2.3.1 The Mean Power Gain

Given a Transmitter (TX) signal with a PSD |Y (f)|2, the received signal power will beequal to

Pr =

∞∫

−∞

|H(f)|2|Y (f)|2df (2.26)

Using the definition of the TX PSD |Y (f)|2 of (2.15), the received power will be

Pr =Pt

B

fc+B/2∫

fc−B/2

|H(f)|2df. (2.27)

To isolate the transmit power from channel properties, let us define the Mean PowerGain (MPG) of the channel as follows,

gc =1

B

fc+B/2∫

fc−B/2

|H(f)|2df (2.28)

such thatPr = gcPt (2.29)

This definition of the MPG has similarities with the signal quality as defined in [44].Using the physical interpretation of the radio channel, H(f) can be substituted by (2.17),such that

gc =1

B

fc+B/2∫

fc−B/2

∣∣∣∣∣

N∑

k=0

u[k]Gk(f)

∣∣∣∣∣

2

df =N∑

k=0

N∑

l=0

u[k]u∗[l]1

B

fc+B/2∫

fc−B/2

Gk(f)G∗l (f)df (2.30)

Due to the orthonormality of the eigenfunctions, this simplifies to

gc =N∑

k=0

|u[k]|2 (2.31)

which shows that the MPG of the channel is related one-on-one to the value of the RVsdriving the CFR.

2.3.2 Statistical Characterization of the NLOS Mean Power Gain

The statistical properties of the MPG are of great significance for system designers, sincethey give information on the behaviour of the radio channel. Here, the relationshipbetween the MPG and the RVs driving the CFR simplifies the derivations greatly and istherefore used as starting point. Hence,

E [gc] = E[

N∑

k=0

|u[k]|2]

, (2.32)


since the RVs driving the radio channel u[k] are uncorrelated. This simplifies to

E [gc] =N∑

k=0

E[|u[k]|2

]=

∞∑

k=0

λ[k] (2.33)

where the fact is used that E [|u[k]|2] is by definition equal to λ[k].Only in the UWB case, λ[k] can be accurately approximated by λc[k]. Otherwise,

the approximation for the eigenvalues will be inaccurate. However, the sum over alleigenvalues λc[k] will be identical to the sum over all λ[k] independent of the bandwidth.The sum over all eigenvalues λ[k] and λc[k] is namely equal to trace of the functionφ(f1, f2) and φc(f1, f2), respectively. Since the functions φ(f1, f2) and φc(f1, f2) haveidentical main diagonals, their trace and thus their sum over all eigenvalues are inevitablyequal to each other. Hence, λ[k] can be substituted by λc[k] without affecting the result.The expected MPG will thus be

E [gc] =∞∑

k=0

λc[k] (2.34)

without the need to make assumptions regarding the channel bandwidth nor the environ-ment.

In the UWB case, the function for the eigenvalues λc[k] changes slowly for consecutivek’s. Hence, the summation over λc[k] can be approximated by an integration over λc(ϑ)if λc(ϑ) = λc[ϑ]. Because k is incremented with unit steps, no step-size factor is required,such that

E [gc] =

∞∫

0

λc(ϑ)dϑ =

∞∫

0

A2

τdBexp(− ϑ

τdB)dϑ =

[

−A2 exp(− ϑ

τdB)

]∞

0

= A2 (2.35)

Hence, the expected MPG is equal to the accumulated path powers. This is not surprising,since E [|H(f)|2] =

∑Np

n=1 β2n for all f and the MPG is the frequency domain average of

|H(f)|2. Therefore, this result is generally true, including flat-fading channels and LOSenvironments.

Let us continue with the derivation of the variance of the MPG, i.e.

Var[gc] = Var

[N∑

k=0

|u[k]|2]

(2.36)

Without making any additional assumptions, no further simplifications are possible. Al-though the RVs u[k] are due to the PCA ensured to be uncorrelated, the variance of theMPG involves fourth-order moments of u[k]. The RVs however need to be independent,to allow for further simplification. In this case, the expression simplifies to

Var[gc] =N∑

k=0

Var[|u[k]|2

](2.37)


In a NLOS scenario, u[k] is assumed to be an independent Gaussian distributed RV, i.e.|u[k]| has a Rayleigh distribution. In this case, it is well-known that Var[|u[k]|2] = λ2[k],such that

Var[gc] =∞∑

k=0

λ2[k]. (2.38)

Hence, the variability of the MPG depends on the distribution of the eigenvalues, whichon their turn depend on the RMS delay spread and bandwidth. For any bandwidth, itcan be proven that

∞∑

k=0

λ2[k] ≤∞∑

k=0

λ2c [k] (2.39)

To obtain φc(f1, f2), additional correlation terms were introduced in φ(f1, f2). Theseadditional correlation terms unavoidably lead to an increase of the sum over the squaredeigenvalues. Nevertheless, they are asymptotically identical for an RMS-delay-spread-by-bandwidth product going to infinity. Hence, the following upper-bound can be derivedfor the variance of the MPG,

Var[gc] ≤∞∑

k=0

λ2c [k]. (2.40)

As stated before, λc[k] changes slowly for consecutive k’s in the UWB case, such that thesummation can be replaced by an integration without altering the result, i.e.

∞∑

k=0

λ2c [k] =

∞∑

k=0

(A2

τdBexp(− k

τdB)

)2

=∞∑

k=0

A4

τ 2d B2

exp(−2k

τdB)

≈∞∫

0

A4

τ 2d B2

exp(−2ϑ

τdB)dϑ ≈

[

− A4

2τdBexp(−2

ϑ

τdB)

]∞

0

≈ A4

2τdB. (2.41)

In other words, the variance of the MPG is smaller or equal to A4

2τdB, which shows a

reciprocal relation between the MPG variance and the RMS-delay-spread-by-bandwidthproduct.

2.3.3 Generalization of the Statistics to LOS Scenarios

As stated before, the expectation for the MPG of NLOS channel given by (2.35) alsoapplies to LOS channels. However, the variance of the MPG for both channel types willbe different. In this section, its variance will be computed for LOS channels.

In Sec. 2.2.3, the LOS component was found to share the dimension with index k = 0with the largest NLOS RV. The power of the LOS PC and the power of the NLOS PChave been found to be λc,L and λc,N, respectively. As in the NLOS case, the NLOS PCis assumed to be a circular zero mean complex Gaussian distributed RV and the LOS


component is modelled as a circular complex RV with a random phase and constant mag-nitude. This corresponds to the traditional model for LOS flat-fading channels . In otherword, the resulting PC, obtained from the superposition of the constant magnitude RVand the Gaussian distributed RV, will have a magnitude that is Ricean distributed. TheRice distribution is characterized by κ and Ω, which are the shape and scale parameter,respectively. The shape parameter is the ratio of the power received via the LOS PC tothe power contribution of the non-LOS PC, i.e. κ = λc,L/λc,N, which after substitution of(2.24) and (2.24) gives that κ = σBK.6

As in the NLOS case, the PCs with an index k larger than zero will be Rayleighdistributed. As a result, the MPG will be the superposition of a squared Rice distributedRV and N − 1 squared Rayleigh distributed RVs of which the variance will be smallerthan

Var[gc] ≤A4

(K + 1)2

(

2K

σB+

∞∑

k=0

1

σ2B2exp

(

−2k

σB

)

)

(2.42)

In the asymptotic UWB case, the summation can again be replaced by an integration.Using the results of the previous subsection, the variance of the MPG will be smaller than

Var[gc] ≤A4

2σB

4K + 1

(K + 1)2. (2.43)

Similar to the NLOS case, the variance is found to be reciprocal with respect tobandwidth. Looking at the impact of the Ricean K-factor, a remarkable insight can beobtained. First, let us consider the case that K = 0, which actually relates to a NLOSscenario. Realizing that σ will be equal to τd, this result is indeed identical to (2.41).Now let us start transferring energy from the NLOS part to the LOS component, i.e.increase K starting from zero while keeping σ constant. At first the variance of the MPGwill increase and a maximum is obtained at K = 1/2 at which the variance will be 4/3times the variance at K = 0. Only from there on, the variance starts to decrease andultimately goes to zero if K approaches infinity. This result is rather counterintuitive,since the presence of a LOS component is often thought to decrease the variation of theMPG. This result is only observed in the UWB case.

2.3.4 Diversity Level of UWB Channels

In the previous section, the MPG variance was found to depend on the distribution ofthe eigenvalues, which in turn depends on RMS-delay-spread-by-bandwidth product andthe accumulated path powers A2. The dependency on A2 makes it less useful as measurefor the frequency diversity of the radio channel. To obtain such an objective measure, weconsider,

m =E [gc]

2

Var[gc](2.44)

6 Here the Ricean κ factor defines the ratio of the LOS component gain with respect to overallchannel gain in the first dimension only. The Ricean K-factor as defined in (2.6) is the ratio of the LOScomponent gain with respect to gain of the complete APDP, i.e. with respect to the channel gain over

all dimensions, see (2.35).


which will be referred to as diversity level or diversity order . An intuitive explanation forthe diversity level can be given as well. A channel with a diversity level m has the samediversity level as a channel composed out of m independent, identically distributed (i.i.d.)Rayleigh fading sub-channels. Consequently, a higher diversity level indicates that thesignal experiences less fading.

Obtaining a closed form expression for the diversity level is difficult if not impossible.Using the results of the previous section, two lower bounds for the diversity level can becomputed using the circulant eigenvalues. For the derivation of the first lower-bound,the results of the LOS channel will be used, which also incorporates NLOS channels asspecial case. Later on the results will be simplified to the NLOS scenarios. The diversitylevel computed using the CEVs will be denoted by mc, where

mc =(K + 1)2

2KσB

+∞∑

k=0

1σ2B2 exp

(

−2 kσB

)

(2.45)

The second lower-bound for the diversity level is obtained by approximating the sum-mation by an integration using the UWB assumption. Using the closed-form expressionfor the asymptotic UWB case of the eigenvalues, the m value can be approximated. Thediversity level computed using the UWB assumption will be denoted by mUWB, where

muwb = 2σB(K + 1)2

4K + 1(2.46)

which shows that in the UWB case both in LOS and NLOS scenarios the diversity levelis proportional to the bandwidth. For NLOS scenarios, the result further simplifies tomuwb = 2τdB, which is a rather intuitive result already. Both mc and muwb can be usedas lower-bound for the actual diversity level m. Since the variance of the MPG is less orequal to the sum of squared eigenvalues λc[k], it is evident that mc is a lower-bound form.

In Fig. 2.7, both lower-bounds for the diversity level of NLOS scenarios are comparedwith the diversity level obtained using SVD for a NLOS scenario, i.e. K = 0. As reference,the coherence bandwidth has been depicted as well.

If the RMS-delay-spread-by-bandwidth product is small, the diversity level is con-stantly equal to 1, which means that the signal experiences a flat-fading channel. For abandwidth in the order of the coherence bandwidth, the diversity level starts to increase.Finally, the diversity level becomes a linear function of the normalized bandwidth with aslope equal to two. Furthermore, the diversity level comes close to the lower-bound if thebandwidth is approx. 5 − 10 times the coherence bandwidth. This linear increase of thediversity level with the bandwidth is confirmed by analyses of UWB channel measurementdata, see [41] and Chapter 3.

In the narrowband case, the diversity level increases with increasing Ricean K-factor.However in the UWB case, the diversity level decreases if the Ricean K-factor is onlymarginally increased starting from zero, which is rather counter-intuitive. When increas-ing the Ricean K-factor further, the diversity first starts to increase for all bandwidths,which is more inline with intuition. The exact diversity level has not been depicted,because the eigenvalues could not be obtained numerically.

2.4. BER ON UWB CHANNELS 31

10−1

100

101

100

101

102

τdB [ ]

m[]

mmcmuwb

τdBcoh

K=0.5K=0

K=10

Figure 2.7: Relation between diversity level, bandwidth and RMS delay spread

2.4 BER on UWB Channels

2.4.1 BER of BPSK on Fading Channels

In this section, the average bit error probability of a Binary-Phase-Shift-Keying (BPSK)modulation scheme is analyzed, incorporating the fading induced by the radio channel.Hereby, the receiver is assumed to have perfect knowledge on the channel and to performoptimal detection [37]. The system does not suffer from ISI. Taking the fading intoaccount, the average uncoded BER of BPSK over a frequency selective Rayleigh fadingchannel, denoted by Qf (.), is given by

Qf

(

Eb,TX

N0

)

=

∞∫

−∞

Q

(

√

2Eb,TXgc

N0

)

p (gc) dgc, (2.47)

where Eb,TX denotes the transmitted energy per bit and N0 the noise power spectraldensity. For performance analysis it is common to express the BER as function of theaverage received energy per bit Eb over the noise spectral density. The variable Eb isrelated to the MPG according to

Eb = Eb,TXE [gc] (2.48)


so that the average uncoded BER of BPSK over a frequency selective Rayleigh fadingchannel is equal to

Qf

(

Eb

N0

)

=

∞∫

−∞

Q

(√

2Eb

N0

gc

E [gc]

)

p (gc) dgc, (2.49)

The PDF of the MPG is dictated by the eigenvalues of the radio channel. In [45], a closedform expression has been derived for the BER of BPSK as function of Eb/N0 on diversitychannels that requires only the eigenvalues of the diversity channel. Hence, the functionQf (.) is defined as follows,

Qf

(

Eb

N0

)

=1

2

L−1∑

k=0

%kI(λn[k], m[k]) (2.50)

where m[k] and %k denote the number of occurrence and k-th residue in the partial-fraction expansion of the k-th normalized eigenvalue λn[k], respectively. The normalizedeigenvalue λn[k] is equal to λ[k] normalized as follows

λn[k] =Eb

N0

∑Lk=0 λ[k]

λ[k], (2.51)

where the fact has been used that the expected MPG is equal to the sum of eigenvalues.Furthermore, the k-th residue in the partial-fraction expansion is defined as

%k =L−1∏

l=0,l 6=k

λn[k]

λn[k] − λn[l](2.52)

and

I(c, m) =

(

1

2− 1

2

√

c

1 + c

)m m−1∑

k=0

(

m − 1 + k

k

)(

1

2+

1

2

√

c

1 + c

)k

(2.53)

which concludes the derivation of the closed-form expression of the average bit error rateusing the eigenvalues. This expression will be used to quantify the impact of the diversitylevel on the BER.

Using the union bound, it is straightforward to obtain an upper bound (UB) for theaverage coded BER on FSFCs from the uncoded one. Equivalent to the coded BERon AWGN channels (see [31]), the coded BER on Frequency Selective Fading Channels(FSFCs) is

Pb ≤∞∑

d=df

adQf

(

dEb

N0

)

(2.54)

where df denotes the free distance of the deployed code and ad denotes the numberof corrupted information bits accumulated over all erroneous paths with an Euclideandistance of d.

2.4. BER ON UWB CHANNELS 33

An approximation (AP) for the average coded BER of FSFC, which is accurate athigh Eb/N0-values can also be derived, namely

Pb ≈ adfQf

(

dfEb

N0

)

. (2.55)

If the approximation is close to the upper bound, it is known that both are close to theactual BER.

2.4.2 Performance Analysis

In this subsection, the BER performance is presented for BPSK modulation on a fre-quency selective Rayleigh fading channel as function of the RMS delay-spread-by-band-width product, using the eigenvalues of the radio channel. The eigenvalues are obtained intwo manners. Firstly, by applying SVD on the discrete equivalent of the autocorrelationfunction φ(f1, f2), which represents the actual eigenvalues of the channel. Secondly, theCEVs are used, which are obtained in closed form in Sec. 2.2.3. These eigenvalues con-verge to the actual eigenvalues with increasing RMS-delay-spread-by-bandwidth product.Additionally, the BER on an Additive White Gaussian Noise (AWGN) channel has beendepicted. In [31], the BER performance of an infinite bandwidth signal on a frequencyselective fading channel is proven to be identical to the BER performance on an AWGNchannel. Hence, the AWGN performance can be used as lower-bound for the averageBER on any FSFC.

In Fig. 2.8, the uncoded BER performance is depicted. As expected, an enlargement ofthe RMS-delay-spread-by-bandwidth product leads to an improvement of the performancein terms of the BER as function of the Eb/N0. Furthermore, the BER curve computedusing the analytical eigenvalues converges indeed to the actual BER. If the RMS-delay-spread-by-bandwidth product is larger or equal to 5, the CEVs can be used for BERanalysis without introducing any significant error.

In Fig. 2.9 and Fig. 2.10, the Upper Bound (UB) and the approximation (AP) arepresented for the coded BER of (UWB) radio systems using convolutional coding onfrequency selective Rayleigh fading channels. The convolutional codes (CCs) of rate 1/2and 1/3 used in WiMedia standard are assumed. The rate 1/3 CC has the generatorpolynomials g0 = 1338, g1 = 1658, g2 = 1718. The rate 1/2 CC is obtained by puncturingthe second output g1. Due to the absence of ISI, both the diversity gain and coding gainare assumed to be fully exploited. Hence, the lower-bounds apply to all systems deployingthe same bandwidth and convolutional code, including OFDM systems.

In Fig. 2.9 and Fig. 2.10, the BER bounds are presented for the system deploying aCC of rate 1/2 and 1/3, respectively. For both rates, the following conclusions apply.

The upper bound for coded BER computed using the CEVs is in any case higher thanthe upper-bound using the actual eigenvalues. Hence, it is a useful bound for the BERperformance analysis on FSFC, although the bound is rather loose if the RMS-delay-spread-by-bandwidth product is small. In case of an RMS-delay-spread-by-bandwidthproduct of 2, the CEV upper-bound is approx. 1 dB more conservative than the upper-bound computed from the actual CEVs for both code rates. If the product is equal to5, the CEV upper-bound is at most 0.2 dB more conservative than the upper-boundcomputed using the actual CEVs.


−5 0 5 10 15 20 25

10−4

10−3

10−2

10−1

100

Eb/N0 [dB]

BE

RSVD,τdB = 0.5SVD,τdB = 1SVD,τdB = 2SVD,τdB = 5AWGNCEV,τdB = 0.5CEV,τdB = 1CEV,τdB = 2CEV,τdB = 5

Figure 2.8: The BER of BPSK modulation on frequency selective Rayleigh fading channelswith different RMS-delay-spread-by-bandwidth products

−5 0 5 10 15

10−4

10−3

10−2

10−1

100

Eb/N0 [dB]

BE

R

UB,SVD,τdB = 2UB,SVD,τdB = 5UB,AWGNUB,CEV,τdB = 2UB,CEV,τdB = 5UB,CEV,τdB = 15AP,SVD,τdB = 2AP,SVD,τdB = 5AP,AWGNAP,CEV,τdB = 2AP,CEV,τdB = 5AP,CEV,τdB = 15

Figure 2.9: Bounds for the rate 1/2 CCd BER of BPSK modulation on frequency selectiveRayleigh fading channels with different RMS-delay-spread-by-bandwidth products

2.5. CONCLUSIONS 35

−5 0 5 10 15

10−4

10−3

10−2

10−1

100

Eb/N0 [dB]

BE

R

UB,SVD,τdB = 2UB,SVD,τdB = 5UB,AWGNUB,CEV,τdB = 2UB,CEV,τdB = 5UB,CEV,τdB = 15AP,SVD,τdB = 2AP,SVD,τdB = 5AP,AWGNAP,CEV,τdB = 2AP,CEV,τdB = 5AP,CEV,τdB = 15

Figure 2.10: Bounds for the rate 1/3 CCd BER of BPSK modulation on frequency selec-tive Rayleigh fading channels with different RMS-delay-spread-by-bandwidth products

The lower-bound for the coded BER computed using the CEVs is higher than theactual lower-bound. By nature, this makes little sense and therefore useless as lower-bound for BER performance analysis. However, if the RMS-delay-spread-by-bandwidthproduct is sufficiently large (≥ 5), it is rather tight to the actual lower-bound.

The BER performance for an RMS-delay-spread-by-bandwidth product equal to fiveapply to systems with a bandwidth of approx. 500 MHz on channels with an RMS delay-spread of 10 ns, e.g. multiband OFDM systems without frequency hopping. Using theBER bounds for an AWGN channel as reference, and using the fact that the UB are closeto the actual performance at low BER, an energy efficiency gain of 3.1 dB at a BER of10−4 is possible.

The BER performance for an RMS-delay-spread-by-bandwidth product equal to fif-teen, apply to systems with a bandwidth of approx. 1.5 GHz on channels with an RMSdelay-spread of 10 ns, e.g. multiband OFDM systems with frequency hopping. Due tonumerical stability problems with the eigenvalues obtained using SVD, only the BERbounds are presented using the CEV. In case of frequency hopping, the energy efficiencycan be improved only by 1.1 dB at a BER of 10−4. Here, losses in terms of energyefficiency due to e.g. a cyclic prefix or code termination are not taken into account.

2.5 Conclusions

After an introduction of the basics of radio channels, a mathematical model has been pre-sented for the fading statistics of UWB radio channels both for LOS and NLOS channels.


The model describes in closed form the relationship between the eigenvalue distributionof UWB radio channels, the signal bandwidth and the RMS-delay-spread. The NLOSeigenvalues are found to follow an exponential curve of which the decay-factor dependssolely on the RMS-delay-spread-by-bandwidth product. Additionally, the LOS compo-nent was found to be contained in the largest eigenvalues together with a componentresulting from the NLOS part of the PDP.

A single, insightful measure was proposed for the diversity level of fading channels andclosed-form under-bounds were derived for UWB fading channels both for the LOS andNLOS case. In both cases, the diversity level was found to scale linearly with the RMS-delay-spread-by-bandwidth product. Based on UWB radio channel measurements, thesame linear relationship has already been observed in [46, 41], but also sub-linear scalinghas been reported [40]. Furthermore, the theoretical model predicts that the presence ofan LOS component will increase the fading, if the Ricean K-factor has a value less thantwo.

Additionally, upper bounds for the uncoded and coded BER for ideal UWB systemswere presented using the eigenvalues of the channel. These bounds are shown to beaccurate and useful for trade-off analyses between bandwidth and BER performance ofUWB systems on NLOS frequency selective Rayleigh fading channel. Assuming a typicalRMS delay spread for an indoor environment, the upper bound for the performance ofMultiband OFDM systems using frequency hopping was only 1 dB less energy efficient,compared to an infinite bandwidth system.

In line with the goal of the chapter, a theoretical fading model has been derived,which gives an elegant insight in the UWB channel and the role of bandwidth, while onlyneeding to describe the APDP of the channel. Using the model, the performance of UWBsystems can be evaluated in closed-form up to the coded BER. In chapter 3, the fadingmodel will be verified using measurement data of UWB radio channels both emphasizingits strengths and shortcomings.

Chapter 3

Fading of Measured UWB Channels

3.1 Introduction

In Chapter 2, a theoretical statistical model for the fading properties of UWB channels wasderived. By definition, a model is a representation of a system that allows for investigationof the (statistical) properties of the system. To achieve this goal, a model makes a seriesof simplifying assumptions from which it deduces how the system will behave. It is adeliberate simplification of reality. For a proper use of the model, the strengths andweaknesses of the model have to be known by its user.

To accommodate these needs, the fading model of Chapter 2 will be verified in thischapter using measurement data of UWB radio channels both emphasizing its strengthsand short-comings. The outline of the chapter is as follows. Firstly, the channel measure-ment campaign is introduced in brief in Sec. 3.2, followed by a discussion of the indoorUWB radio channel in the time and frequency domain in Sec. 3.3. The statistical prop-erties of the PCs of the measured UWB radio channel are analyzed in Sec. 3.3.1 and usedfor a statistical analysis of the MPG in Sec. 3.5.

3.2 Description of Radio Channel Measurements

The measurement data used in this thesis have been obtained during a measurementcampaign conducted at the premises of IMST GmbH in Kamp-Lintfort [47]. Using avector network analyzer, the complex CFR was measured for the frequency range fromf1 = 1 GHz to f2= 11 GHz. Two identical bi-conical horn antennas were used with again of approx. 1 dBi at both the transmitter and receiver, that is approx. constant overthe whole frequency range. Both antennas were positioned at a height of 1.5 m abovefloor level.

The RX antenna was mounted on a tripod and positioned at various positions withinthe environment. The TX antenna was mounted on a rail and moved along the rail insteps of 1 cm over a distance of 150 cm. During the measurement, the rail was movedto obtain parallel tracks spaced 1 cm apart. As a result, the CFRs were obtained fora 150 cm by 30 cm rectangular grid, where Hi(f) denotes the i-th CFR measured atfrequency f and position x[i]. For notational convenience, all frequencies measured at

37

38 CHAPTER 3. FADING OF MEASURED UWB CHANNELS

Figure 3.1: Ground plan of the office measurement environmentc© J.Kunisch, IMST GmbH

the same position are gathered in a vector h[i] of length F . The frequency step size ofthe measurements is 6.25 MHz, such that F = 1601.

The 1 cm spatial grid ensures that the spatial resolution is better than half a wave-length over the complete measurement frequency band, to allow for the analysis of thechannel response as a function of space. During the channel measurements, the environ-ment was ensured to remain static, allowing for a time-invariant characterization of theradio channel.

The deployed measurements were performed in an office of approx. 5 m by 5 m with aheight of 2.6 m. Within the office, positions were selected to obtain two different visibilityconditions, namely line-of-sight (LOS) and non-line-of-sight (NLOS). The positions of TXgrid and the RX during the LOS measurement were TxC and RxB, respectively. Duringthe NLOS measurements, the TX grid and RX are positioned respectively at TxA andRxA. The LOS path has been blocked using a metal cabinet of size 1.78 m x 0.42 m x1.96 m. A plan of the office environment can be found in Fig. 3.1.

3.3 Overview of Measurement Results

3.3.1 Delay Domain

Although the analysis of the UWB channel concentrates on the frequency domain, itsbehaviour in the delay domain is of relevance since both domains are related by theFourier transform.

All CIRs presented in the thesis are in the baseband to provide for a better viewon the radio paths, due to the absence of a carrier. Additionally, all CIRs have beencompensated for the propagation delay of the LOS component, i.e. the LOS component

3.3. OVERVIEW OF MEASUREMENT RESULTS 39

−10 0 10 20 30 40 50 60 70 80−115

−110

−105

−100

−95

−90

−85

−80

−75

−70

−65

|h(τ

)|2 [dB

(10G

Hz)]

excess delay [ns]

Single CIRPath Enh. CIRs

LOS

NLOS

Dense Multipath

Figure 3.2: Example of local PDP in the LOS office environment

arrives at τ = 0. For illustrative reasons, the PDP1 of a single CIR is depicted in Fig. 3.2.The LOS component can be easily identified, but no NLOS paths can be identified visually.To emphasize these paths, averaging has been conducted on the PDPs of the CIRs froma small geometric area. Due to the limited size of the geometric area, distinct pathspresent in each CIR arrive more or less with the same delay. Nevertheless, only the LOScomponent truly adds up coherently. To ensure that the magnitude of the NLOS pathshas the proper relation to the LOS magnitude, an additional Gaussian filter has beenapplied over the delay domain with a width of approx. 0.5 ns. The resulting APDP isalso depicted in Fig. 3.2 and reveals the presence of distinct NLOS paths.

In [47], the distinct NLOS radio paths are shown to originate from reflections onthe walls. In this respect the UWB indoor radio channel differs from narrowband in-door radio channels. In a typical indoor environment, the rays of different radio pathsarrive shortly after each other. Hence, only (ultra) wideband signals allow for the sepa-ration/identification of these distinct radio paths.

Besides the presence of distinct radio paths, dense multipath can be identified in thePDP. After the arrival of the LOS, the power contained in the dense multipath firstrapidly increases, achieves its maximum value after approx. 8 ns and, then follows anexponential decay. The dense multipath is caused by the interaction of a radio signal withobjects like walls, which is more complex than a mere reflection. Although a significantpart is reflected, part of the ray energy is at first absorbed by an object to be releasedat a later time instant. As a result, each reflection is followed by a tail with decaying

1Every PDP is normalized by the measurement bandwidth to ensure independence of the measurementbandwidth


magnitude.For the LOS as well as the NLOS measurements, the Ricean APDP model parameters

K and τd have been extracted by [48]. Their results are listed in Tab. 3.1. These param-

Table 3.1: APDP Model parameters [48]Scenario K τd [ns] σ [ns]

LOS 1.26 8.75 10.54NLOS 0 11.2 11.2

eters will be used when comparing the analytical results of Chapter 2 with measurementdata.

3.3.2 Frequency Domain

In Sec. 2.2, the CFR has been statistically characterized by the function φ(f1, f2). As inany measurement campaign, the CFR was measured at distinct frequencies only. There-fore a discrete equivalent matrix Φ of φ(f1, f2) is introduced. In this case h[i] can beseen as the i-th realization of a random vector h, which is statistically characterized byΦ. Due to the finite number of measured realizations, only an estimate for Φ can beobtained, which will be denoted as W. The estimate W is as follows,

W =1

MHHH (3.1)

where M denotes the number of used CFRs and

H =[h[1] h[2] . . . h[M ]

]. (3.2)

In Sec. 2.1, the expected gain of the CFR was assumed to be frequency independent.In practice, this assumption does not apply due to the frequency dependent gain ofthe antennas. To obtain insight in the average channel gain as function of frequency,the spatial average of the frequency domain power gain function is computed, whichis equivalent to the main diagonal of the auto-covariance matrix W. Using the wholemeasurement grids, W is computed and its main diagonal is presented for both LOS andNLOS scenario in Fig. 3.3.

For the measurements, bi-conical antennas have been used, which have approx. aconstant gain. For constant gain antennas, the Friis transmission equation predicts a6 dB gain loss with each doubling of the frequency [29]. Therefore, the measured channelpower gain decreases with increasing frequency. Due to the logarithmic scaling of bothaxis, the frequency gain follows an approx. linear curve with a slope of −7 dB with eachdoubling of the frequency. Additionally, spectral spikes can be observed above 8 Ghzin both the LOS and NLOS scenario, most likely caused by interferers present duringmeasurement that are mixed to these frequencies. As a consequence, all measurementdata above 8 GHz will be considered less trust-worthy.

Additionally, the function φ(f1, f2) was assumed to be banded. Let us verify whetherthis also applies to the matrix W. Due to the frequency dependent power gain of the

3.3. OVERVIEW OF MEASUREMENT RESULTS 41

Figure 3.3: Spatial RMS average of the CFRs as function of frequency in a NLOS envi-ronment (lower dashed curve) and a LOS environment (upper solid curve).

average CFR, its correlation-coefficient matrix C will be presented instead. The elementat row k and column l of the matrix C is defined as

C[k, l] =W[k, l]

√

W[k, k]W[l, l](3.3)

The correlation-coefficient matrix has been computed for the LOS and NLOS scenarioboth using the complete measurement grid. Both results are presented in Fig. 3.4.

As expected for the NLOS environment, the measured correlation-coefficient matrixis indeed a band-limited matrix. Somewhat larger out of band cross-correlations areobserved at frequencies lower than 3.5 GHz. The finite grid size in combination with theslower spatial de-correlation of the CFRs at lower frequencies reduces the effective numberof uncorrelated observations, which is possibly causing the larger correlation coefficientsat lower frequencies.

The same analysis has been conducted for the LOS scenario. Based on the RiceanAPDP model, one expects the correlation to approach a certain floor with increasingfrequency separation whose amplitude depends on the Ricean K-factor. The measurementresults confirm the validness of the model.

To obtain another view on the banded character of W, the correlation coefficients havebeen averaged over a frequency range from 1 until 3 GHz as function of the frequencydifference. This procedure is repeated several times while increasing the center frequencyin 2 GHz steps. The results have been depicted in Fig. 3.5.


(a) (b)

Figure 3.4: Estimated Frequency domain correlation function of the CFR in a NLOS andLOS scenario in subplot (a) and (b), respectively.

100

101

102

103

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

∆f [MHz]

|ρn(∆

f)|

1−3 [GHz]

3−5 [GHz]

5−7 [GHz]

7−9 [GHz]

9−11 [GHz]

Theory

NLOS

LOS

Figure 3.5: Correlation of CFR as function of the frequency separation

3.4. PRINCIPAL COMPONENTS OF MEASURED UWB CHANNELS 43

In the NLOS case, the curves have essentially the same behaviour for every frequencyrange, indicating that the correlation indeed depends mostly on the frequency differenceand less on the center frequency. Starting from a frequency difference equal to zero, first afast decrease of the correlation is observed with increasing frequency separation ∆f . If ∆f

is approx. 20 MHz, the correlation is about 0.5, i.e. the empirically determined coherencebandwidth equals approx. 20 MHz. A complete de-correlation cannot be expected dueto the finite-sized measurement pool. Taking this into account, a rather good match isobserved with respect to the theoretical model, even though the parameters for the modelpresented in Tab. 3.1 were derived from the full bandwidth APDP parameters [48].

In the LOS case, a similar behaviour can be observed, evidently with the differencethat a correlation floor exists. Again a good match is observed between the measurementdata and the theoretical model. However, the floor is slightly higher for the frequencyrange from 1-3 GHz, indicating that the optimal APDP parameters are weakly frequencydependent.

3.4 Principal Components of Measured UWB Chan-

nels

Following the same structure as in Chapter 2, the PC of the radio channel measure-ments are analyzed and compared with the theoretical results. Firstly, the algorithmsare described to extract the desired information from the measurements, followed by acomparison between theory and practice.

3.4.1 Estimation of the Eigenvalues and Principal Components

The PCA is applied on a sub-matrix of the measurement matrix H, containing onlythose elements corresponding to the frequency range under evaluation. For notationalconvenience, the sub-matrix will be denoted by H with H ∈ C

F ,M , where F denotes thenumber of frequency point in the frequency range under evaluation.

An estimate for the eigenvalues of the PCs is obtained by applying SVD on F−1W =

VΛVH

, where V is a unitary matrix with eigenvectors and Λ is a diagonal matrixcontaining the eigenvalues. Although the eigenvalues are exact with respect to F−1W,they are only estimates of the actual eigenvalues of the channel. Therefore, the k-thestimate of the channel eigenvalues will be denoted by λ[k]. Without loss of generality,the eigenvalues are assumed to have a descending order. The division by F can be seenas the finite element equivalent of the division by B in (2.16).

Using the results of the SVD, the measured realization for the PCs are obtained asfollows,

U =1

FVHH, (3.4)

where the element at row k and column l denotes the l-th realization of the PC u[k]. Fornotational convenience, all realizations of u[k] are gathered in the vector u[k].


In the previous chapter, the PCs are assumed to be complex-valued Gaussian dis-tributed RVs, except for the PC containing the LOS component. To analyze whether thisassumption applies to the measurement data, the kurtosis of each PC is computed. Thekurtosis of a zero-mean, complex-valued RV x is defined as2

k(x) =µ4(x)

µ22(x)

− 2, (3.5)

where x contains the realizations of x and µm(x) is an estimate of the m-th order momentof x. For a zero-mean RV, an estimate for the m-th order moment is given by

µm(x) ,1

N

N∑

n=1

|x[n]|m (3.6)

where N denotes the length of the vector x. The closer the magnitude of the kurtosis isto zero, the better the validity of the Gaussian assumption.

In Sec. 2.2.3, the largest eigenvalue λc[0] in LOS scenarios was shown to be the super-position of the eigenvalues λc,L[0] and λc,N [0]. To accommodate its validation, a procedure

will be presented for the division of the estimate of the largest eigenvalue λ[0] in its twocomponents.

The procedure consists of two steps. Firstly, the Ricean κ factor is estimated using amethod of moments [49], where the Ricean κ estimate for a RV x is shown to be obtainedby

κ =−2µ2

2(x) + µ4(x) − µ2(x)√

2µ22(x) − µ4(x)

µ22(x) − µ4(x)

, (3.7)

where x denotes the vector containing the observations of x. Using the estimate κ, bothcomponents are obtained as follows

λc,L[0] =κ

κ + 1λ[0] (3.8)

λc,N [0] =1

κ + 1λ[0], (3.9)

which concludes the description of the procedure to separate the largest eigenvalue in itstwo components.

3.4.2 Verification of the NLOS Eigenvalues and Principal Com-

ponents

The eigenvalues and kurtosis of all PCs are depicted in Fig. 3.6 for the NLOS scenariofor UWB signals with a bandwidth of 1 GHz and a center frequency of 4.5 GHz. Asreference, the theoretical eigenvalues and kurtosis have been depicted as well using theAPDP parameters presented in Tab. 3.1.

2A value of two has been subtracted, to ensure that a complex-valued Gaussian RV has a kurtosisequal to zero. In the real-valued case, a value equal to three is subtracted.

3.4. PRINCIPAL COMPONENTS OF MEASURED UWB CHANNELS 45

0 10 20 30 40 50 60−100

−95

−90

−85

−80

−75

−70

−65

−60

Index k

Gain

[dB

]

0 10 20 30 40 50 60

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

kurt

osi

s

λest

[k]

λc[k]

k(u[k])

k(uc[k])

Figure 3.6: The eigenvalues and kurtosis of the PCs of a NLOS UWB channel with abandwidth of 1 GHz

0 10 20 30 40 50 60−100

−95

−90

−85

−80

−75

−70

−65

−60

Index k

Gain

[dB

]

0 10 20 30 40 50 60

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

kurt

osi

s

λest

[k]

λc[k]

λest

(NLOS)[0]

λc

(NLOS)[0]

k(u[k])

k(uc[k])

Figure 3.7: The eigenvalues and kurtosis of the PCs of a LOS UWB channel with abandwidth of 1 GHz.


It is seen that the eigenvalues of the PCs do not follow exactly an exponential decayas expected based on the theoretical eigenvalues, possibly caused by the use of frequencydependent gain antennas. Nevertheless, the measured eigenvalues match well with thetheoretical ones, especially the significant ones with small index.

Additionally, Fig. 3.6 show that all significant eigenvalues have a kurtosis near zero,indicating that the CV Gaussian assumption is indeed valid for the PCs, i.e. |u[k]| isapprox. Rayleigh distributed for all k in the NLOS case. Overall, a reasonably goodmatch can be observed between the theory and practice.

3.4.3 Verification of the LOS Eigenvalues and Principal Com-

ponents

The same analysis has been repeated for the LOS measurement, again using the theoret-ical eigenvalues and kurtosis as reference, see Fig. 3.7. Remember that the largest PC isassumed to be a Ricean distributed RV, while all other PCs are assumed to be Rayleighdistributed. Again a rather good match is found between theory and practice, although inthe LOS case the eigenvalues are shifted in weight towards the eigenvalues with a smallerindex.

Nevertheless, all is not as it seems. Fig. 3.7 reveals a 4 dB difference between theexpected and measured NLOS eigenvalues denoted by λ

(NLOS)c [k] and λ

(NLOS)est [k], respec-

tively. The theoretical model predicts the LOS component to share its dimension with thePC containing the largest NLOS eigenvalue, where in practice it is considerably smaller.In fact, when re-ordering the NLOS eigenvalues, it would be around the 10-th position.This also explains the minor difference between the measured and expected kurtosis. Thekurtosis is however not very sensitive in the vicinity of −1 for Ricean distributed RVsand explains why the difference is so small. This discrepancy has a significant impact onthe statistical properties of the MPG, as will be shown in the following section.

3.5 Analysis of the Mean Power Gain

Analogous to the definition of the MPG in Sec. 2.3.1, the MPG of the i-th measured CFRfor a signal with center frequency fc and bandwidth B is defined as,

gc[i] =1

F

∥∥∥h[i]

∥∥∥

2

(3.10)

where the h[i] contains only those elements within the corresponding frequency range[fc − 1

2B, fc + 1

2B]. Since frequency-domain oversampling is applied to Hi(f), (3.10) can

be considered to be the discrete equivalent representation of (2.28).For illustrative purposes, the MPG is depicted for a 30-by-30 cm grid for the NLOS

scenario for a signal with B = 10 MHz and B = 1 GHz in Fig. 3.8 in subplots (a) and(b), respectively.

Subplot (a) shows that the MPG varies extensively for a signal with a relatively smallbandwidth of 10 MHz. Furthermore, the spatial separation between local maxima andminima is in the order of half a wavelength in both directions indicating that the angle

3.5. ANALYSIS OF THE MEAN POWER GAIN 47

(a)

05

1015

2025

30

0

5

10

15

20

25

30

−90

−85

−80

−75

−70

−65

−60

ij

Gi,j

(b)

05

1015

2025

30

0

5

10

15

20

25

30

−90

−85

−80

−75

−70

−65

−60

ij

Gi,j

Figure 3.8: The MPG for a signal with B = 10 MHz,fc = 4.6 GHz in (a) and B = 1 GHz,fc = 4.6 GHz in (b) as function of measurement grid position with a grid spacing of 1 cm

of arrival of each multipath component is widely spread. Due to the inherent frequencydiversity for 1 GHz bandwidth signals, the MPG depicted in subplot (b) varies much less.

3.5.1 Estimation of the Diversity Level

Three different estimates are presented for the diversity level of the measured UWBchannel data. The first estimate is applied to both LOS and NLOS channels, while thesecond estimate and the third estimate are exclusively used for NLOS and LOS channels,respectively.

The first estimate is obtained by applying a method of moments to the pool of MPGsmeasured in a local area, i.e.

mm =µ2

1(gc)

µ2(gc) − µ21(gc)

. (3.11)

Hence, this estimate makes no assumptions regarding the statistical properties of thePCs, but uses the moments of the MPG instead.

The second estimate is based on the NLOS fading model of Sec. 2.3.1 that all PCs areindependent Rayleigh distributed RVs. In this case, the second moment of the PCs, i.e.the eigenvalues of the channel, fully describe the diversity level of the MPG, such that

mR =

(∑N

k=0 λ[k])2

∑Nk=0 λ2[k]

. (3.12)

The third estimate is based on the LOS fading model of Sec. 2.3.1, which is referredto as the Rice-Rayleigh fading model. Here, the largest PC is assumed to be a Riceandistributed RV, while all others are assumed to be Rayleigh distributed. In contrast to theRayleigh distribution, the Rice distribution has an additional shape parameter κ, i.e. itsProbability Density Function (PDF) is not fully described by the estimated eigenvalues.


The κ-parameter has been estimated using the method described in Sec. 3.4.1. Based onthe Rice-Rayleigh (RR) model, the estimate for the diversity level becomes,

mRR =

(∑N

k=0 λ[k])2

∑Nk=0

(

1 − δ[k] κ2

(1+κ)2

)

λ2[k], (3.13)

where the estimate κ is obtained using the method of moments described by (3.7).The estimate mm is widely used for the estimation of the m parameter of Nakagami

distributed RVs. As a result, the estimation behaviour is well described in literature.In [50], it is reported that typically large sets are required to obtain accurate estimates,depending on the actual m-value. If the set is chosen too small, not only the variance ofthe estimates will be high, but also a bias will be present, which is proportional to theactual diversity level.

Due to the experimental nature of the PCs based estimates mR and mRR, little isknown regarding their behaviour for finite measurement sets. In Appendix A, an an-alytical evaluation is presented for the estimate mR. It was found to have a superiorperformance with respect to the estimation variance compared to the moment basedestimate mm, in case the PCs are indeed independent Rayleigh distributed RVs. Further-more, the mR is found to be asymptotically unbiased. For finite set-sized, it is found toproduce downwards biased. One can compensate for this bias if the number of indepen-dent observations is known. Due to spatial correlation, this is not the case. Therefore, noeffort has been made to compensate for any bias. Due to their similar nature, it is likelythat the estimate mRR is downwards biased as well.

3.5.2 Verification of the Diversity Level

As stated in the previous subsection, relatively large data sets are needed to obtainaccurate estimates for the diversity level. Unfortunately, the luxury of large data setsinherently does not apply to SSF analyses. The local area over which the diversity levelis estimated may not be to large. If chosen too large, the probability that distinct radio-paths will appear and/or vanish becomes too high and by definition one can no longerspeak of SSF. Additionally, the data set of a single local-area will not contain uncorrelatedobservations/measurements, due to spatial correlation. As a result, the effective area-sizewill be reduced. Taking both aspects into consideration, the local-areas are limited toa 30-by-50 cm rectangular area. Hence, the 150-by-30 cm measurement grids could bedivided into 3 adjacent local areas.

Furthermore, the diversity level is independent of the center frequency, at least froma theoretical point of view. The validity of this assumption is partially covered by theresults of Sec. 3.3.2, where it is shown that the frequency domain correlation dependsmainly on the frequency difference and only little on the frequency range. Therefore,the frequency range from 3 until 7 GHz has been divided into 4 adjacent bands. In thismanner, in total 12 subsets are obtained and used to extract the diversity level of themeasured channels.

The estimates for the diversity level are presented for both the NLOS and LOS channelin Fig. 3.9 and Fig. 3.10, respectively. For both scenarios the average estimated diversity

3.5. ANALYSIS OF THE MEAN POWER GAIN 49

level is depicted including markers identifying the standard deviation from the average.In the NLOS case, the increase of all estimates is approx. proportional to the band-

width, as expected based on (2.45). Nevertheless, a difference can be observed betweenthe two estimates. At small bandwidths, the difference is still rather small, because theunderlying assumption of the NLOS fading model that the PCs are approx. independentis valid. The diversity level mc is too small, because the circulant approximation forφ(f1, f2) is not accurate for such small RMS-delay-spread-by-bandwidth products.

With increasing bandwidth, the estimate mR under-estimates mm. This is caused bythe presence of distinct NLOS paths reported in Sec. 3.3.1, which are more and moreresolved with increasing bandwidth. As a result, the PCs remain uncorrelated but areno longer independent, explaining the discrepancy between both estimates. Finally, mR

converges to mc. It is expected that in environments with richer multipath, like e.g.industrial environments, the difference between theory and practice is smaller. In anycase, mc was found to lower-bound mm, possibly making it a useful conservative estimatefor the actual diversity level of NLOS channels. Whether this observation is universallyvalid has not been determined.

In the LOS case, both estimates agree again rather well for small bandwidths. Withincreasing bandwidth, a similar behaviour is observed as in the NLOS case; the estimatemRR under-estimates mm. Hence, the same reasoning can be applied as in the NLOScase. However, the theoretical model mc constantly under-estimates the diversity levelby far. Even at higher bandwidth, where the model is expected to be accurate. Thediscrepancy can be explained as follows. The theoretical model namely predicts the LOScomponent to share a dimension with the largest NLOS eigenvalue, which leads to thesmallest diversity level. In combination with any other eigenvalue, the diversity level willbe higher, i.e. it represents the worst-case and can therefore be used as lower-bound forthe diversity level. In Fig. 3.7, the measured NLOS eigenvalue is shown to be considerablysmaller than the expected NLOS eigenvalue

The difference is responsible for the difference between the theoretical and measureddiversity level.3 A possible cause is the alteration of the pulse-shape due to the frequencydependency of the antennas, causing the LOS component to occupy another dimensionthen the one expected assuming a frequency independent antenna gain. To incorporatethis phenomenon, an extension of the theoretical model is mandatory. When correctingfor this phenomena using the measurement data, one obtains the estimate mRR since ituses κ. This explains why mRR performs considerably better.

3.5.3 Verification of the Mean Power Gain

The differences between theory and practice with respect to mm should not be over-valued. At higher diversity levels the Cumulative Distribution Function (CDF) of theMPG becomes significantly less sensitive to under-valued estimates. To illustrate this,the measured CDF has been depicted in Fig. 3.11 together with the CDFs obtained usingthe different estimates for the diversity level, assuming the square-root of the MPG tobe a Nakagami distributed RV. This distribution is not only selected because its CDF

3In contrast to the legendary words of W.C. Jakes:”Nature is seldom kind.”, the UWB fading appearsto be one of those rare exceptions [35].


0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

30

35

40

B [MHz]

mm

m

mc

mR

Figure 3.9: Comparison of the different diversity level estimates as function of bandwidthin the NLOS scenario

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

B [MHz]

m

mm

mc

mRR

Figure 3.10: Comparison of the different diversity level estimates as function of bandwidthin the LOS scenario

3.6. BER COMPARISON ON MEASURED AND THEORETICAL UWB CHANNELS51

provides a good fit with the measured CDFs, but also because closed-form expressions areavailable for the BER on Nakagami m fading channels. Furthermore, a Nakagami fadingmodel is intuitive; a channel with a diversity level m is composed out of m independent,identically distributed (i.i.d.) Rayleigh fading sub-channels.

A comparison of the MPGs for both the NLOS and LOS scenario is conducted forthree different bandwidths of 200 MHz, 500 MHz and 1 GHz depicted in sub-figure (a), (b)and (c), respectively.

In general, the CDFs obtained using the estimates for the diversity level fit ratherwell to the measured CDF, indicating the validity of the Nakagami m distribution. Thoseestimates that produce a bad-fit are mc in the case of a 200 MHz bandwidth (N)LOSchannel and a 500 MHz bandwidth LOS channel, and mRR for all LOS channels. Thereasons for these bad fits have already been explained when discussing the diversity levelin Sec. 3.5.2.

3.6 BER Comparison on Measured and Theoretical

UWB Channels

In this section, the BER on the measured UWB channels is compared with the BER onUWB MPG models, using the estimates for the diversity levels of the previous section.The aim is to evaluate the usefulness of the theoretical models developed in the previoussections for system performance analysis with respect to the BER.

As reference, the average BER of a local-area is used, which is obtained in two steps.Firstly, the BER is computed for each measured MPG using the so-called Gaussian Q-function. The average local area BER is obtained by averaging over the BER of allMPGs within that area. Both the average uncoded and the UB for the coded BERs arepresented for both the NLOS and LOS scenario. The convolutional code of rate 1/3 isused as presented in Sec. 2.4. Two bandwidths have been considered, namely 200 MHzand 500 MHz.

In Fig. 3.12, the obtained BERs are presented for the NLOS case. In Fig. 3.12(a) ona 200 MHz bandwidth channel, the theoretical model is approx, 2 dB more conservativethan the BER based on the measured MPG at a BER of 1e−4 for reasons already presentedin Sec. 3.5.2. In the coded cases, depicted in Fig. 3.12(b), the differences becomes 5.5 dBdue to nature of coding. On good channels, the coding ensures practically error-freecommunication, but when the signal comes below a certain SNR-threshold, the BERrapidly becomes poor.

When increasing the bandwidth to 500 MHz, the differences become significantlysmaller. In the uncoded case, depicted in Fig. 3.12(a), the difference between the theo-retical MPG model and the measured MPGs is merely 0.7 dB at a BER of 1e−4. Thedifference will decrease further with increasing bandwidth. When comparing both inthe scenario with FEC, the difference will increase to 1.8 dB, which is still significantlysmaller than in the 200 MHz case. Both in the uncoded and coded case of Fig. 3.12, thetheoretical model performs approx. equally well as the model based on the estimateddiversity level mRR, indicating the validity of the model.

In Fig. 3.13, the obtained BERs are presented for the LOS case. In all cases, the


(a)

−70 −68 −66 −64 −62 −60 −58 −560

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Abscissa [dB]

CD

F[P

(MP

G¡A

bsc

issa

)]

Meas

mm

mc

mR

mRR

NLOS LOSNLOS LOS

(b)

−70 −68 −66 −64 −62 −60 −58 −560

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Abscissa [dB]

CD

F[P

(MP

G¡A

bsc

issa

)]

Meas

mm

mc

mR

mRR

NLOS LOSNLOS LOS

(c)

−70 −68 −66 −64 −62 −60 −58 −560

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Abscissa [dB]

CD

F[P

(MP

G¡A

bsc

issa

)]

Meas

mm

mc

mR

mRR

NLOS LOSNLOS LOS

Figure 3.11: Comparison of the CDF of the measured MPG with the CDFs using the esti-mated diversity levels for both the NLOS and LOS scenario for a bandwidth of 200 MHz,500 MHz and 1 GHz in sub-figure (a), (b) and (c), respectively

3.7. CONCLUSIONS 53

theoretical model fails to deliver exact results, because of reasons explained in Sec. 3.5.2.However, since the BER results are so much tighter using the model with the estimateddiversity level mRR, the principle validity of the Rice-Rayleigh fading channel model isconfirmed.

3.7 Conclusions

In this chapter, the fading model derived in the previous chapter was verified usingmeasured channels to obtain insight in the strength and weaknesses of the model. Aftera short description of the radio channel measurement set-up, the typical behaviour hasbeen presented of the UWB channel in the delay domain and frequency domain.

To validate the model, the statistical properties of the PCs of measured UWB chan-nels have been compared with the expectations based on the theoretical model. Severalalgorithms have been described to obtain estimates for the eigenvalues, the PCs andthe kurtosis from the measurement data. The resulting estimates were compared withexpectation derived from the theoretical model both for a LOS and a NLOS scenario.

For the NLOS scenario, a good match was found between the theoretical modeland practice. Also for the LOS scenario, the estimates for the eigenvalues and kurtosismatched reasonably well with theory. However, a 4 dB difference was observed betweenthe expected and measured NLOS part of the largest PC. When validating the diversitylevel, this discrepancy was found to have a significant impact in the LOS scenario. Inboth scenarios, the theoretical model was found to accurately describe the change in theeigenvalue distribution of channel with increasing bandwidth.

For UWB NLOS scenarios, the diversity of the MPG predicted with the theoreticalmodel fitted rather well to the measured diversity. Both reveal a linear increase withbandwidth. However, the theoretical model consistently under-estimated the diversitylevel slightly. With increasing bandwidth, more and more distinct radio paths are re-solved, such that the PCs are no longer independent. It is expected that in environmentswith richer multipath, like e.g. industrial environments, the difference between theoryand practice becomes smaller. In any case, the theoretical model was found to be aconservative estimate for the actual diversity of UWB NLOS channels.

For UWB LOS scenarios, the theoretical model consistently under-estimated the ac-tual diversity level by far. In this case, the theoretical model predicts the LOS componentto share a dimension with the largest NLOS eigenvalue, which leads to the smallest diver-sity level. The 4 dB smaller measured NLOS eigenvalue with respect to the theoreticalone, leads to a significant larger diversity level in practise. When compensating for thisdiscrepancy, the theoretical model is performing considerably better, indicating the basicvalidity of the Rice-Rayleigh fading model. This gives hope that the theoretical fadingmodel for LOS channels can be refined.

Also the CDF of the MPG has been presented for signal with different bandwidthsin both a LOS and NLOS case. To convert the previously described estimates for thediversity level, the MPG was assumed to be Nakagami distributed RV, where the diversitylevel was used as shape parameter. The Nakagami distribution was shown to accuratelydescribe the CDF of the actual MPG.

Finally, the diversity level was used to obtains estimates for both the uncoded and


(a)

0 2 4 6 8 10 12 14 16

10−4

10−3

10−2

10−1

MPG,200MHz

mm

,200MHz

mcir

,200MHz

mR

,200MHz

MPG,500MHz

mm

,500MHz

mcir

,500MHz

mR

,500MHz

(b)

0 2 4 6 8 10 12 14 16

10−4

10−3

10−2

10−1

MPG,200MHz

mm

,200MHz

mcir

,200MHz

mR

,200MHz

MPG,500MHz

mm

,500MHz

mcir

,500MHz

mR

,500MHz

Figure 3.12: Comparison of average BER on measured and modelled UWB NLOS channelwith different bandwidths both coded and uncoded, in sub-figure (a) and (b), respectively.

3.7. CONCLUSIONS 55

(a)

0 2 4 6 8 10 12 14 16

10−4

10−3

10−2

10−1

MPG,200MHz

mm

,200MHz

mcir

,200MHz

mRR

,200MHz

MPG,500MHz

mm

,500MHz

mcir

,500MHz

mRR

,500MHz

(b)

0 2 4 6 8 10 12 14 16

10−4

10−3

10−2

10−1

MPG,200MHz

mm

,200MHz

mcir

,200MHz

mRR

,200MHz

MPG,500MHz

mm

,500MHz

mcir

,500MHz

mRR

,500MHz

Figure 3.13: Comparison of average BER on measured and modelled UWB LOS channelwith different bandwidths both coded and uncoded, in sub-figure (a) and (b), respectively.


coded BER, assuming BPSK modulation. It was shown that for UWB NLOS channels,the theoretical model is useful to obtain conservative but rather tight estimates for theactual BER performance, although the theoretical model in the NLOS case is fully definedby the RMS delay spread only. For UWB LOS channels, the theoretical model was overlyconservative, making it less useful for BER analysis. Therefore, it is recommended torefine/revise the theoretical model for LOS scenarios to obtain more accurate predictions.Especially since the underlying Rice-Rayleigh model was found to be accurate, only itsparameters are incorrect.

In general, it is concluded that the theoretical model accurately describes the statisti-cal behaviour of NLOS UWB channels. For a LOS UWB channels, the theoretical modeldoes not match well to reality and a refinement of the model is needed. The analysisshowed that the Rice-Rayleigh distribution is able to accurately describe the statisti-cal nature of measured LOS channels. The distribution parameters, obtained using thetheoretical model of Chapter 2, are however inaccurate.

Chapter 4

Theory of TR UWB

Communications

4.1 Introduction

As shown in part one, UWB communication is inherently resilient against SSF. Unfor-tunately, this advantage does not come without a price. A coherent receiver as used incurrent spread-spectrum systems becomes rather complex in the UWB case. For example,a rake receiver collecting the signal energy of distinct radio paths will need many rakefingers, due the richness of the UWB channel, i.e the large number of resolvable radiopaths [7, 51]. Additionally, each finger has to be synchronized to a distinct radio pathwith high accuracy, due to the large signal bandwidth and the channel gain of each pathhas to be estimated. To complicate matters further, each path distorts a UWB signaldifferently [52], such that the template waveform at each rake-finger has to be adaptablein order to be optimal.

Tomlinson and Hoctor proposed to combine TR signaling with an AcR for UWBcommunications, to dispose of the need for channel estimation, while still capturing thecomplete pulse energy [53]. Furthermore, its simple structure may sustain the promise ofUWB technology to bring low-cost wireless communication. As result, the UWB societyshowed a great interest in this concept, resulting in many scientific studies, one of thembeing presented in this thesis.

The aim of this chapter is to provide better insight in the behaviour of TR UWBsystems in various situations. Firstly, the principle of TR UWB communication willbe introduced, including a discussion of its pro’s and con’s with respect to performanceand implementation. Several extensions of the TR principle will be proposed. Firstly, afractional sampling AcR structure will be proposed to relax synchronization and allowfor weighted autocorrelation, while simplifying the implementation. Secondly, a complex-valued AcR will be proposed to make the system less sensitive against delay mismatches.Additionally, the complex-valued AcR allows for the extension of the TR signaling schemeto complex-valued modulation.

To understand the system’s behaviour, a general-purpose discrete-time equivalent sys-tem model will be derived, where general-purpose means that all extensions are taken intoaccount. Several interpretations for the system model will be presented, which allow for

57

58 CHAPTER 4. THEORY OF TR UWB COMMUNICATIONS

more insight in the behaviour of TR systems in various situations. Finally, the statisticalproperties of TR UWB system will be presented.

4.2 Principle of Transmitted Reference Communica-

tion

4.2.1 Transmitted-Reference Signaling

A TR symbol in its essence consists of two identically-shaped pulses p(t) transmittedwith a predefined time separation in between. The first pulse is left unmodulated, whilethe second pulse is data modulated with b[n]. The time-separation in seconds D is shortcompared to the coherence time of the channel, such that both pulses are distorted equallyby the channel. The TR UWB TX signal y(t) in a mathematical notation is as follows

y(t) =∞∑

n=−∞

p(t − nTs) + b[n]p(t − nTs − D), (4.1)

where Ts denotes the symbol duration in seconds, such that T−1s is equal to the TR symbol

rate.In Fig. 4.1, a TR signal is depicted before and after the channel, which are denoted

by TX signal in Fig. 4.1(a) and Receiver (RX) signal in Fig. 4.1(b), respectively. The RXsignal is not only distorted by the multipath channel, but it is also corrupted by noise.For simplicity, BPSK modulation is assumed and the time-interval between both pulsesis the same for all symbols. In this example, the time-interval D is 10 ns and the symbolduration Ts is 100 ns.

4.2.2 Autocorrelation Receiver

Assuming the RX is aware of the time-separation D, it can use the first pulse as areference for the demodulation of the second pulse, by computing essentially the short-term autocorrelation of the received signal at delay lag D. Similar to a matched filter, thefirst pulse is used as reference for the demodulation of the modulated second pulse. Sinceboth pulses are corrupted equally by the channel, there is no need for channel estimation.Furthermore, the autocorrelation can be performed using analog components. In the mostsimple case, a single sample is generated for each TR symbol, which is further processedusing digital circuitry. The rate at which the digital circuitry operates is thus no longerdictated by the bandwidth of the TR signal, such that the digital sampling and clockrates can be significantly lower than the Nyquist rate. This allows for a reduction in costand power consumption for the digital circuitry.

In an AcR, the demodulation is performed in several stages. In the first stage, band-pass filtering is applied to the received signal to mitigate out-of-band noise and inter-ference. The signal after the RX BPF r(t) will consist out of the desired signal andnoise

r(t) =∞∑

n=−∞

q(t − nTs) + b[n]q(t − nTs − D) + n(t). (4.2)

4.2. PRINCIPLE OF TRANSMITTED REFERENCE COMMUNICATION 59

Figure 4.1: Signals at different stages in TR system


BPF

Delay

∫ b

a.dt DSP

Figure 4.2: Block diagram of an elementary AcR

Here, q(t) denotes the convolution of the TX pulse p(t) with the radio channel h(t)and the RX BPF frx(t), i.e. q(t) = (p∗h∗frx)(t). Assuming white Gaussian noise onthe channel with a double-sided spectral density N0/2, the noise after the BPF will becoloured Gaussian noise with an autocorrelation function

rnn(τ) =N0

2

∫ ∞

−∞

frx(t + τ)frx(t)dt (4.3)

In stage two, r(t) is divided over two parallel branches. The first branch leaves thesignal unaltered, while the second delays the signal by D seconds. The RX signal and itsdelayed version are depicted in Fig. 4.1(b). Please notice that the modulated pulse andreference pulse on the parallel branches now overlap in time.

In stage three, the output of both branches are multiplied with each other. Themultiplier output is depicted as a solid line in Fig. 4.1(c). By integrating the multiplieroutput over the proper interval in stage four, the computation of the autocorrelation iscompleted and the integrator output can be sampled. The block diagram of the describedAcR is depicted in Fig. 4.2.

An illustration of the signals in stage 3 can be found in Fig. 4.1(c). Here, the dottedline represents the output of the integrator and the start and duration of integrationinterval are denoted by means of a box. After ending the integration, the integratoroutput is stable, i.e. it can be sampled for further processing by the digital circuitry. Thevalue of the received signal will be

u[n] =

∫ nTs+Tend

nTs+Tstart

r(t)r(t − D)dt (4.4)

Afterwards, the integrator will be reset to zero and ready for the next TR-symbol. In theabsence of pulse-overlapping, noise and assuming appropriate integration intervals, thevalue of the n-th sample u[n] will be equal to

u[n] =

∫ nTs+Tend

nTs+Tstart

r(t)r(t − D)dt (4.5)

=

∫ nTs+Tend

nTs+Tstart

b[n]q2(t − nTs − D)dt (4.6)

= b[n]Eq (4.7)

4.2. PRINCIPLE OF TRANSMITTED REFERENCE COMMUNICATION 61

where Eq denotes the energy of the pulse q(t). The sign of u[n] will be equal to the BPSKmodulation b[n] applied, allowing for a simple threshold detection scheme in the digitalcircuitry.

4.2.3 The Drawbacks

Nothing comes without a price and TR UWB communication is no exception. Theuse of TR UWB leads to a loss of at least 6 dB compared to an ideal matched filterreceiver; 3 dB due to noise contained within the reference and 3 dB due to usage of twopulses per bit, instead of one. The loss can be even higher. If the integration intervalduration is set too long (which is not the case in Fig. 4.1), additional noise is accumulatedduring the integration. These noise terms can be identified in the multiplier output inFig. 4.1(c) in the intervals 〈10, 20〉, 〈50, 60〉, 〈110, 120〉 and 〈150, 160〉, where all valuesare in nanoseconds. Furthermore, an additional noise signal exists not present in linearRXs, resulting from the multiplication of the noise signal with a delayed version of itself.This causes the multiplier output to vary from zero, even if no signal is received. Thiseffect can be identified in the multiplier output in Fig. 4.1(c) in the interval from 60 nsuntil 100 ns. A performance loss of at least 6 dB is rather high, but compared to morerealistic, sub-optimal rake receivers, equipped with only a limited amount of fingers andimperfect channel state information, the difference diminishes [51].

4.2.4 Implementation Considerations

Although the TR principle itself is rather straight-forward, its implementation has severalopen issues. For instance, the implementation of UWB analog delays is not straight-forward [54, 55]. Although this thesis is not on the design of analog circuitry, like delay-lines, one should consider the RF front-end complexity during system design. In [26],the complexity of the delay is shown to be approximately proportional to the product ofbandwidth and delay, which should be kept small to allow for a low cost implementation.Having this in mind, the delay hopping signaling scheme as proposed by Hoctor andTomlinson has not been considered, since it requires long delays [53]. Therefore, thefocus is on the most elementary TR signaling scheme as described in this section, usingonly two pulses per symbol and a single delay. Multi-user access functionality should beprovided by one of the other OSI-layers, for instance by the Data Link Layer (DLL) usingan Aloha-like access scheme.

Besides having to select an appropriate value for the delay, the delay unavoidably willvary from one device to another and from time to time due to variations in the productionprocess, temperature, etc. Any difference between the transmitter’s and receiver’s delay,will increase the system’s sensitivity to noise. Evidently, these variations can be keptsmall using sophisticated delays, but will increase the cost of the devices. Taking thedelay variation into account during system design is therefore a must to obtain a low-costsystem. In Sec. 4.3.2, a complex-valued AcR is proposed, which not only decreases thesystem’s sensitivity to delay mismatches, but also allows for an increase in data rate, seeSec. 4.3.3.

Another topic of this thesis is the proper setting of the start and duration of the inte-


BPF

Delay

∫.dt SIGN

w(t − nTs)

Reset

α[n] b[n]r(t)

DSP

c

Figure 4.3: Block diagram of a weighted AcR

gration interval. In a practical implementation, most likely both variables will be underthe control of the digital circuitry. This requires both additional DSP and additionalinterfaces between the RF front-end and the digital circuitry, increasing both complexityand cost. In Sec. 4.3.1, a fractional sampling AcR is proposed, which allows for synchro-nization to be obtained in the digital domain, reducing the power consumption and costof the devices.

4.3 Extensions of the TR Principle

4.3.1 Weighted Autocorrelation and Fractional Sampling AcR

In the original TR system model proposed by Tomlinson and Hoctor [53], the receiversignal is multiplied with a delayed version of itself, followed by an integration of themultiplier output. Hence, no weighting is applied to multiplier output signal, although itsSignal-to-Noise-and-Interference Ratio (SNIR) can vary over the duration of the symbol.Therefore, the usage of a weighted correlation stage at the demodulator is proposed, toimprove the performance of the AcR receiver. The weighting function is also used tosynchronize the RX to the received signal. In addition, it is proposed to add a constant cto the AcR output to compensate for any DC-offset. The resulting decision statistic α[n]in a mathematical description is given by

α[n] =

∫ ∞

−∞

w(t − nTs)r(t)r(t − D)dt + c. (4.8)

Assuming BPSK modulation, the sign of α[n] can be used as decision for the transmittedsymbol. If neither delay-hopping nor time-hopping is used, the TR signal will be cyclo-stationary, such that the weighting function can be the same for every symbol. As theweighting is applied in the analog domain of the receiver, the proposed AcR is called ananalog weighted AcR. The proposed structure is depicted in Fig. 4.3. For simplicity, areal-valued AcR is assumed, but the principle can be applied to complex-valued AcRs aswell, see Sec. 4.3.2.

In order to be optimal, the weighting function must be adapted to the conditions on thechannel, like channel impulse response, SNR and delay. A likely implementation would beto control the weighting function from the digital domain of the receiver. Unfortunately,the implementation of an adaptable wideband weighting function is not low complexity.Assuming a single AcR front-end, the weighting applied to a TR symbol must be finalized,

4.3. EXTENSIONS OF THE TR PRINCIPLE 63

before the weighting for the following symbol can start. This hardware limitation willlead to sub-optimal results if the TR symbols overlap in time.

To overcome these shortcomings, it is proposed to restrict the degrees-of-freedom ofthe weighting function w(t) at the cost of some performance. Concretely, the weightingfunction w(t) is restricted to the following general expression with ML degrees of freedom

w(t) =M∑

k=0

L−1∑

α=0

w[k, α]hclk(t − (α/L + k)Ts), (4.9)

where hclk was defined in Sec. 4.4.2 as a rectangular function, which equals one for 0 ≤t < Ts/L and zero otherwise. The ML samples w[k, α] fully describe the shape of w(t).By substituting (4.9) into (4.8), we obtain

α[n] =

∫ ∞

−∞

M∑

k=0

L−1∑

α=0

w[k, α]hclk(t − (α/L + k + n)Ts)r(t)r(t − D)dt + c

=M∑

k=0

L−1∑

α=0

w[k, α]

∫ ∞

−∞

hclk(t − (α/L + k + n)Ts)r(t)r(t − D)dt

︸︷︷︸

= u[n + k, α]

+c. (4.10)

After changing the order of the summations and the integration, the samples generatedby a so-called fractional sampling AcR u[n, α] can be identified. This allows us to writethe value of the decision statistic at time n as a weighted sum of fractional samples of anAcR. In other words,

α[n] =M∑

k=0

L−1∑

α=0

w[k, α]u[n + k, α] + c (4.11)

with

u[n, α] =

((α+1)/L+n)Ts∫

(α/L+n)Ts

r(t)r(t − D)dt. (4.12)

This illustrates that the analog weighted AcR with limited degree of freedom definedby (4.9) is mathematically equivalent to applying weighting to the fractional samples ofan AcR. From an implementation point-of-view, applying weighting in the digital domainis simpler and allows for overlapping weighting functions for consecutive symbols.

In essence, a fractionally sampled AcR divides the symbol period into several integra-tion intervals, where one sample is generated per interval. For simplicity, all intervals areof equal duration Tclk, which is an integer fraction of the symbol duration, i.e. Tclk = Ts/Lwith L ∈ N, such that L samples are generated per TR symbol. To simplify the imple-mentation, these intervals are by no means synchronized to the received signal.

Regarding the implementation of fractional sampling AcRs, two mathematically equiv-alent schemes are possible. For example, an integrator with reset can be used. After


Figure 4.4: Signals in a fractional sampling AcR

BPF

Delay

I&D DSP

Figure 4.5: Block diagram of a fractional sampling AcR

receiving a reset signal, the integrator output is sampled, forced to zero and starts inte-grating again until the next reset is received. This operation is often referred to as anIntegrate and Dump (I&D). The block diagram of a fractional sampling AcR using anI&D is depicted in Fig. 4.5. Alternatively, the integrator can be replaced by a Low PassFilter (LPF) with a rectangular impulse response of duration Tclk. The output of the LPFis sampled at a rate of T−1

clk , i.e. again L samples are taken per symbol. The mathematicalequivalence of both AcR implementations is depicted in Fig. 4.4. The dashed line rep-resents the integration value in a I&D sampler, where the dot-dashed line represents theLPF output. In both cases, the markers identify the sample moment and value. Pleasenote that the sample values are the same for both implementations.

Both implementations have their own pro’s and con’s. A drawback of the I&D inte-grator is that after receiving the reset signal, the integrator will be shortly insusceptibleto the input signal. The LPF based implementation is at all time susceptible to theinput signal, but the implementation of a LPF with a rectangular impulse response isimpossible. The appropriate choice depends on the application scenario.

Applying adaptive weighting has several distinct advantages. Firstly, the synchroniza-tion process can fully take place in the digital domain. Secondly, it implicitly controls theeffective integration duration, such that noise is suppressed more effectively [24]. Thirdly,fractional sampling with weighting also allows for the suppression of more non-linear ISI,allowing the system to operate at higher rates, see Sec. 4.5.4 and Sec. 5.4.1.

Fractional sampling, weighted AcR have been proposed almost simultaneously by


several authors, including the author of this thesis [24, 56, 57]. The main novelty ofthis work is that this thesis also takes inter-symbol-interference (ISI) into account. Both[56, 57], assume neither inter-pulse interference nor ISI. In [58] only inter-pulse interferenceis considered. However, the introduction of ISI leads to effects, which are fundamentallydifferent to ISI in a linear receiver, see Sec. 4.5.

4.3.2 Complex-Valued Autocorrelation Receiver

For a transmitted-reference (TR) system to operate efficiently, the RX delay Drx must bewell-matched to the TX delay Dtx. Any delay mismatch δ means that Rq(τ) is sampledat lag δ instead of lag zero, where Rq(x) denotes the autocorrelation of the RX pulse q(t)defined as

Rq(τ) =

∫ ∞

−∞

q(t + τ)q(t)dt (4.13)

Assuming BPSK modulation, the Euclidean distance between both symbols 2|Rq(δ)| willdecrease with any delay mismatch, making the system more susceptible to noise. TheEuclidian distance has been depicted as a solid line in Fig. 4.8 for Rq(0) = 1. In case ofa normal AcR, the figure shows that a delay-mismatch of 1/(8fc) ≈ 31 ps already resultsin a 3 dB loss in the system’s energy efficiency and a delay-mismatch of 1/(4fc) = 62.5 pswill make communication completely impossible. Note that the multipath channel hasno impact on the sensitivity of TR systems to delay mismatches [59].

To increase the robustness of the system against delay mismatches, we propose to usea Complex-Valued (CV) AcR. In addition to the autocorrelation branch used in a normalAcR, the CV AcR has a second autocorrelation branch, which samples the autocorrelationfunction at lag Drx +1/(4fc). Hence, the autocorrelation function is sampled at two lags,Rp(δ) and Rp(δ + 1/(4fc)). In Appendix B, it is shown that the proposed AcR computesthe short-term complex-valued autocorrelation of the received signal at delay lag D andis therefore referred to as such. Its operation in a baseband notation is as follows,

u[n, α] = exp(jωcD)

∫ ((α+1)/L+n)Ts

(α/L+n)Ts

r(t)r∗(t − D)dt. (4.14)

A block-diagram of the proposed receiver architecture can be found in Fig. 4.6. Since adelay of 1/(4fc) seconds represents a 90 phase shift, it is depicted as such.

To illustrate the benefit of using a CV AcR, the value of both autocorrelation functionshas been set out against each other as function of the delay-mismatch δ. The mismatchvalue is given in picoseconds within the figure. The resulting trajectory resembles adamped spiral, meaning that both BPSK constellation points are rotated around theorigin and the Euclidean distance between both points slowly decreases with an increasingmismatch. This rotation must be compensated for, before a decision on the symbolvalue is made, which is a well-known problem in traditional Quadrature-Phase-shift-Keying (QPSK) systems, see e.g. [60].

In Fig. 4.8, the Euclidian distance for a CV AcR has been depicted as a dashed line asfunction of the delay mismatch. In case of a CV AcR, the decrease of Euclidean distancedepends on the pulse envelope and thus the bandwidth. Fig. 4.8 shows that the Euclidian


clkTD

I&D

90˚

0˚DSP

I&D

Figure 4.6: QPSK-TR receiver architecture, where 90 denotes the delay 1/(4fc)

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Rq(δ)

Rq(δ

+1/(4f c

))

−480

−450−420

−390

−360

−330 −300

−270

−240

−210

−180

−150

−120

−90

−60

−30

0

30

60

90

120

150

180

210

240

270

300330

360

390

420 450

480

Figure 4.7: IQ-diagram shift as function of the delay mismatch in picoseconds for a 1 nspulse with a rectangular envelope and a 4.0 GHz carrier frequency


Figure 4.8: Euclidian distance as function of the delay mismatch in picoseconds for a 1 nspulse with a rectangular envelope and a 4.0 GHz carrier frequency demodulated usingeither a RV AcR or a CV AcR.

distance decreases smaller than 1 dB for any delay mismatch less than 200 ps. Hence,the sensitivity of the system to delay mismatches depends now on the bandwidth insteadof the carrier frequency, decreasing its sensitivity by an order of magnitude.

The use of multiple AcR branches to overcome delay mismatches has previously beendescribed in [61, 62]. Our proposal is fundamentally different, since it exploits the band-pass characteristics of UWB signals.

4.3.3 TR M-ary Phase Shift Keying

In [63], it is proposed to modulate the time-interval between both pulses, which theauthors referred to as TR Pulse Interval and Amplitude Modulation (PIAM). A CVAcR is able to demodulate certain types of TR PIAM signaling, without the need toextend the RF front-end of the AcR. Assuming the pulse time-interval can assume twodistinct values, and if these are equal to D and D + 1/(4fc). The so-called pulse intervalmodulation factor TD equals 1/(4fc) and allows for the translation of the time-shift of themodulated pulse into a phase-shift of its carrier by 90 using a first order approximation.Furthermore, the BPSK modulation applied on the modulated pulse is equivalent to 180

phase-shift of its carrier. Hence, the carrier-phase of the modulated pulse can assumefour distinct values, 0, 90, 180 and 270, i.e. the modulated pulse is QPSK modulated.Therefore, this specific type of TR PIAM signaling is called QPSK TR signaling. It isstraight-forward to extend this concept to higher-order PSK modulation or QuadratureAmplitude Modulation (QAM). Only BPSK and QPSK modulation are considered inthis thesis.

Independently, the combination of higher order Pulse Position Modulation (PPM) andadditional AcR branches to overcome delay mismatches has been proposed simultaneouslyin [62].


4.4 Generic TR System Model

4.4.1 Introduction

In this section, a baseband time-discrete equivalent system model for TR signaling de-modulated using AcRs will be derived. The model has been kept general, to include allextensions proposed in section 4.3. The discrete-time equivalent notation is used, sinceit is better suited for incorporation of the characteristics of imperfect RF components,like BPFs. The baseband signals allow for a reduction of the number of discrete timeobservations to characterize the signal, leading to shorter computation and simulationtimes. The term observation is used instead of sample to emphasize that the signal is notactually sampled by the RX.

4.4.2 Continuous-Time System Model

Assuming TR signaling in which a single TR symbol is transmitted each Ts seconds. Inthis case, the time-line is divided in frames/symbol periods of Ts seconds, where eachframe contains a single TR symbol. On its turn, a frame is again divided into Nh chipsof duration Tc. In each frame, two identically shaped pulses are transmitted; a referencepulse followed by a modulated one. Let us focus on the waveform transmitted within then-th frame. To obtain a general-purpose system model, both pulses are allowed to bemodulated. This may be beneficial for several reason, e.g. to avoid spectral peaks [25].

The amplitude of the first pulse is modulated with the scrambling factor b[n], whilethe delayed pulse in the n-th frame is modulated by both the scrambling factor and theinformation bearing symbol b[n]b[n]. The term scrambling factor is used to emphasizethat b[n] is not used for the signaling of information. The scrambling code b[n] may begenerated using a PN generator, but could depend on the information to be transmittedas well.

The reference pulse will be transmitted in the c-th chip and the modulated pulseis transmitted d chips later. To allow for a compact mathematical representation, theposition of the reference pulse within the frame is represented by the column vectors = [si], where si = δ[i− c] for i = 1, 2, ...Nh and δ[i] denotes the Kronecker delta.Similarly, the position of the modulated pulse can be denoted by the column vectors = [si] where si = δ[i−c−d] for i = 1, 2, ...Nh. The received signal after the RX bandpassfilter (BPF) in a baseband notation can be written as

r(t) =∑

n

q (t, nTs)TSb[n] + n(t) (4.15)

with S = [s, s] and b[n] = [b[n], b[n]b[n]]T . Furthermore, the received pulse shape q(t) isthe convolution of the TX pulse, the radio channel including antennas and the RX-BPF,such that

q (t, τ) = [q(t, τ), q(t, τ +Tc), . . . , q(t, τ +(Nh−1)Tc)]T (4.16)

with q(t, τ) = qm(t − τ) exp(−jωcτ), where ωc = 2πfc and qm(t) represents the envelopesignal of the bandlimited UWB signal q(t). Due to the RX BPF, the complex-valuedGaussian noise signal n(t) is coloured with an autocorrelation function rnn(τ).

4.4. GENERIC TR SYSTEM MODEL 69

In the original notation, the matrix S was labelled with symbol-index index n toobtain a generic model, which includes time-hopping and delay hopping. It has alsobeen implemented in the simulation environment, but finally not used. Therefore, thesymbol-index n has been omitted.

As stated in Sec. 4.1, the usage of two autocorrelation branches is proposed, wherethe first one is matched to a lag D and the second to a lag D + 1/(4fc). Without lossof generality, D is chosen equal to dTc with d ∈ N. Each branch is sampled L timesper symbol, such that the sampling period Tclk = Ts/L and L denotes the fractionalsampling ratio (FSR) with L ∈ N. Hence, the AcR generates two parallel real-valuedsample streams, which can be seen as a single complex-valued sample stream. Appendix4.3.2 shows the input-output relation of the proposed AcR in a complex-valued basebandnotation is

u[n, α]= exp(jωcD)

((α+1)/L+n)Ts∫

(α/L+n)Ts

r(t)r∗(t−D)dt, (4.17)

where α ∈ 1, 2, . . . , L.

4.4.3 Discrete-Time Equivalent System Model

Due to the finite bandwidth of r(t), a discrete-time equivalent model of the system can bedeveloped by taking an observation of r(t) every Tr seconds, where Tr will be chosen tofulfill the Nyquist criterion. The analog received signal is modelled using its discrete-timeequivalent.

Since neither delay-hopping nor time-hopping is used, the received signal is cyclo-stationary with period Ts. In this case, a finite interval [nTs, (n+1)Ts〉 is sufficient to fullycharacterize the received signal, i.e. only Nob observations with Nob = Ts/Tr are needed.The vector containing these Nob observations will be denoted by r[n]. Without loss ofgenerality, the n-th symbol b[n] is assumed to be under demodulation. Evidently, b[n]will also influence r[n+1]. Due to the cyclo-stationarity, this relationship is the identicalto the relationship between b[n − 1] and r[n]. Because of the finite duration of q(t), afinite number of symbols M + 1 can influence the observation interval, independently ofwhether the ISI is caused by a reference pulse, a modulated pulse or both. Based oncausality, only symbols with an index equal or smaller than n can influence the interval.Based on this reasoning, the observation vector r[n] can be described using the expression

r[n] = QSd[n] + W1n[n], (4.18)

where the column-vector d[n] contains the modulation applied to both pulses of the TRsymbol with time index n and the M previous TR symbols. Hence, d[n] will have 2M +2elements constructed according to

d[n] =[

b[n − M ]T , b[n − M + 1]T , . . . , b[n]T]T

. (4.19)

The matrix S contains the positioning of the pulses within each symbol period andrelated to S as follows,

S = IM+1 ⊗ S, (4.20)


where ⊗ denotes the Kronecker product, such that S ∈ 0, 1Nh(M+1),2M+2.The matrix Q is the channel matrix Q = [qkl] with qkl = q(kTr, NlTs − lTc), for

k +1 = 1, 2, . . . , Nob samples and l = 0, 1, . . . , (M +1)Nh − 1 chips, such that the channelmatrix Q ∈ C

Nob,(M+1)Nh .The noise vector n[n] contains complex-valued, identically distributed, zero-mean,

Gaussian random variables (RVs), characterized by the discrete autocorrelation function

rnn[k − l] =1

2N0Rnn((k − l)Tr). (4.21)

The noise vector has Nob+Nd elements, where Nd denotes the delay in samples Nd = D/Tr.Hence, it contains more elements than the vector r[n]. The purpose of these Nd additionalelements becomes apparent when introducing the delayed version of the received signal.The matrix W1 is constructed such that W1n[n] equals the last Nob values of n[n] in theproper order. Consequently,

W1 =[

0Nob,NdINob

]. (4.22)

Using the same methodology, the discrete equivalent signal of the delayed version of thereceived signal r(t − D), denoted by rd[n], can be written as

rd[n] = QDSd[n] + W2n[n]. (4.23)

The signal components have been delayed by introducing a delay matrix D, such that

D = exp(−jωcD)

[01,(M+1)Nh−1 0I(M+1)Nh−1 0(M+1)Nh−1,1

]d

. (4.24)

The matrix W2 is constructed such that W2n[n] is a column-vector containing the firstNob elements of n[n]. In this fashion, the system model takes into account that the noisecontained in both r[n] and rd[n] originates from the same noise process. The matrix W2

is constructed as

W2 =[

INob0Nob,Nd

]. (4.25)

In the third stage of an AcR, the received signal is multiplied with its delayed versionto form the multiplier output, see Fig. 4.2. In a vector notation, this multiplication willbe modelled using an element-wise multiplication of r[n], rd[n]. Therefore, the diagonaloperator Λ(a) is introduced. Assuming that the vector a contains N elements, Λ(a)denotes an N by N matrix with the elements of a on its main diagonal and zeros otherwise.Consequently, the multiplier output during the interval can be written as Λ(r[n]) rd[n].

In stage four, the multiplier output is fed into an integrate and dump (I&D) operator,generating L samples during each symbol interval. The α-th sample generated during then-th symbol interval u[n, α] is equal to,

u[n, α] = h[α]TΛ(r[n]) r∗d[n], (4.26)

where the k-th element of h[α] equals h(kTr − αTclk).

4.4. GENERIC TR SYSTEM MODEL 71

Substituting (4.18) and (4.23) into (4.26), followed by some re-ordering and re-definition,leads to the expression

u[n, α] =s[n, α] + η[n, α] (4.27)

where s[n, α] is the signal term containing the desired information as well as intra- andinter-symbol-interference terms and η[n, α] is a noise term. A more detailed derivation ofthe system model can be found in [18].

The signal term can be represented by the following structure,

s[n, α] = d[n]HKαd[n], (4.28)

which is known in literature as an FIR, second-order Volterra system [64]. The matrixKα is defined as

Kα =

hTαΛ

(Q∗D∗Se1

)QS

hTαΛ

(Q∗D∗Se2

)QS

...hT

αΛ(Q∗D∗SeL

)QS

(4.29)

A detailed analysis of the signal term can be found in Sec. 4.5.The noise term η[n, α] is the superposition of two noise terms, each having different

statistical nature,

η[n, α] = ηg[n, α] + ηz[n, α], (4.30)

The term denoted by ηg is called the Gaussian noise term, while the term denoted byηz will be referred to as the non-Gaussian noise term. The terminology has been chosenfor the following reasons. The Gaussian noise term is a superposition of two Gaussiansub-terms. They not only have a similar structure but also the same statistical nature.Both are the cross-product of the noise signal and received signal,

ηg1[n, α] = d[n]HLα,1n[n], (4.31)

ηg2[n, α] = d[n]TLα,2n[n]∗. (4.32)

Hence, both ηg1[n, α] and ηg2

[n, α] are the superposition of the Gaussian distributed RVscontained in n[n]. Hence, both terms and ηg are all Gaussian distributed. Although notindependent, both Gaussian noise sub-terms are uncorrelated, since the cross-correlationbetween the noise vector and its conjugate is zero. The matrices Lα,1 and Lα,2 arestructured as follows

Lα,1 =

hTαΛ

(Q∗D∗Se1

)W1

hTαΛ

(Q∗D∗Se2

)W1

...hT

αΛ(Q∗D∗SeL

)W1

(4.33)

Lα,2 =

hTαΛ

(QSe1

)W2

hTαΛ

(QSe2

)W2

...hT

αΛ(QSeL

)W2

(4.34)


The noise term ηz is independent of the TX signal y(t) and is given by the followingequation,

ηz[n, α] = hTαz[n]. (4.35)

where the non-Gaussian noise vector z[n] is related to the Gaussian noise vector accordingto

z[n] = Λ(W1n[n])W2n[n]∗. (4.36)

Hence, the elements in z[n] are the product of two Gaussian RVs, which are possibly cor-related, and thus themselves not Gaussian distributed. Therefore, ηz[n, α] is not Gaussiandistributed either and referred to as such. The actual distribution depends on system pa-rameters like bandwidth, and integration duration. Based on the central limit theorem,it can be understood that the distribution of ηz[n, α] converges towards a Gaussian onewith an increasing product of bandwidth and integration duration, where a product largerthan 20 leads to an almost complete convergence [65, 66].

4.5 Interpretation of the TR System Model

4.5.1 Introduction

In the previous section, it has been shown that the relationship between the transmittedsymbols and the I&D output is described by an Finite Impulse Response (FIR) second-order Volterra system. Volterra systems are widely used for the modeling of non-linearsystems. However, Volterra systems of TR-UWB systems differ to some extent from thoseused e.g. for the modeling of analog components. The difference is not so much causedby the Volterra systems themselves, but in the way they are excited. When used forthe modeling of analog components, Volterra systems are typically excited by continuousvalued signals. In our case, the Volterra system is excited using a digitally modulatedsignal which is by nature finite alphabet. The difference in excitation allow for alternativeinterpretations of the Volterra system. In this section, some of those interpretations willbe presented to provide insight in the behaviour of TR systems. More information onnon-linear system modeling can be found in [64]. An extensive bibliography on non-linearsystem modeling and other aspects can be found in [67].

In (4.28), the Volterra system describing the relationship between the fractional sam-ples and the modulation was shown to be

s[n, α] = d[n]HKαd[n],

where the vector d[n] contains both the modulation applied to the reference pulses as wellto the information bearing pulses. Furthermore, the elements in d[n] are shifted by twopositions with each increment of the time-index n. In this respect, the Volterra systemas defined here differs from the typical definition for Volterra systems [64].

However, an alternative equivalent interpretation of the system model, presented inSec. 4.4.3, can be obtained. The introduction of the decimator known from multi-ratesystems allows us to use the default definition of Volterra system as described in [64]. In

4.5. INTERPRETATION OF THE TR SYSTEM MODEL 73

b[n]

b[n]

P

Sd[n]

K1 2

u[n, 1]

KL 2

u[n, L]

Mod.

PN sequence

Constant

Options

a[n]

a[n]

Figure 4.9: Block diagram of the SIMO FIR Volterra model

Fig. 4.9, its block diagram is depicted describing the complete system model includingthe modulation. Please be aware that the decimator undoes the rate increase introducedby the parallel-to-serial conversion, such that the overall system generates L I&D samplesfor each symbol. Hence, a fractional sampled AcR can be seen as a single-input, multiple-output FIR Volterra system. The scrambling applied to the reference pulse b[n] is notinterpreted as input, since it is either fully described by the TX symbol b[n] or a PseudoNoise (PN) sequence or left unmodulated. More details can be found in Sec. 4.5.5.

The size and composition of the Volterra kernel(s) is influenced by system parameterslike the delay, channel, FSR, BPFs, symbol rate etc. The memory in the Volterra systemis determined primarily by the radio channel and symbol-rate, i.e. increasing either thechannel delay spread or symbol rate will also increase the memory of the Volterra system.

The Volterra system models the ISI, which depends in a non-linear fashion on thetransmitted symbols. In this respect, TR communication differs from ”ordinary” com-munication systems, where the ISI is modelled using a FIR structure. To illustrate thenon-linear ISI and its dependency on the data rate, several constellation diagrams havebeen depicted in Fig. 4.10 of TR systems using fractional sampled CV-AcR with the FSRequal to twice the symbol rate, deployed in an indoor environment. QPSK-TR signalingis assumed and the data rate is either 10, 20 or 40 Mb/s.

At a bit-rate of 10 Mb/s, the ISI is negligible, such that the Volterra has no memory.Additionally, only one output contains information regarding the transmitted symbol. Asa result, only one of the constellation diagrams contains the four constellation points ofthe QPSK modulation.

Moderate ISI can be observed when the data rate is increased to 20 Mb/s. In theleft-hand side constellation-diagram, the four QPSK constellation points are still visible.However, ISI and an offset is observed in both constellation diagrams. In contrast to the10 Mb/s case, both outputs contain information regarding the transmitted symbol. Thenature of the observed ISI cannot be modelled using an FIR structure for two reasons.Firstly, an FIR structure can not account for any offset in the constellation diagram. Sec-ondly, an FIR structure inherently results in a constellation diagram, which is rotational


(a)

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Real

Imag

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Real

Imag

(b)

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Real

Imag

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Real

Imag

(c)

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Real

Imag

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Real

Imag

Figure 4.10: IQ diagrams at both outputs of a TR-QPSK system, operating in a multipathenvironment at 10, 20 and 40 Mb/s, in sub-figure (a),(b) and (c) respectively


symmetrical by n times 90 degrees. This is clearly not the case in Fig. 4.10(b).When increasing the data rate further to 40 Mb/s, both outputs are distorted severely,

such that the four QPSK constellation points can not be identified visually. Additionally,the constellation diagram contains highly non-linear components, resulting in a densecloud of points.

4.5.2 Vector Notation for Volterra Kernels

For the statistical derivations and to obtain more insight in the behaviour of the system,it is convenient to write the Volterra system in a vector notation of the following form

s[n, α] =d[n]Tkα. (4.37)

In [64] the vectors d[n] and kα are defined to be equal to vec(d[n]d[n]H

)and vec (Kα),

respectively. The operator vec (K) creates a column vector by stacking the columnsof K. However, the elements in vec

(d[n]d[n]H

)are likely correlated, due to the finite

alphabet/digital modulation. For example, let us assume a constant modulus modulation.In this case, all elements on the main diagonal of the matrix d[n]d[n]H will be the samefor all realization of the random vector d[n].

To allow for a decomposition of vec(d[n]d[n]H

)into its uncorrelated components, its

non-central auto-covariance matrix is introduced

A , E[

vec(d[n]d[n]H

)vec

(d[n]d[n]H

)H]

. (4.38)

If A is not full-rank, i.e., the rank Nk = rank(A) is less than (2M + 2)2, it meansthat it indeed contains correlated elements. This allows for the following interpretation.The vector vec

(d[n]d[n]H

)can be thought to be driven by Nk uncorrelated variables.

Assuming these variables to be gathered in d[n], a linear transformation matrix T exists,which fulfils the following two criteria:

Td[n] = vec(d[n]d[n]H

)(4.39)

E[

d[n]d[n]H]

= INk,Nk. (4.40)

The fact that all components in d[n] are uncorrelated, unit power RVs makes the vectornotation powerful for statistical analysis.

Assuming T to be available, d[n] and kα can be obtained by

d[n] = T†vec(d[n]d[n]H

), (4.41)

kα = THvec (Kα) . (4.42)

For all modulation types considered in this thesis, the composition of A was ratherstraight-forward. The elements of vec

(d[n]d[n]H

)are either fully correlated or uncorre-

lated. In other words, the matrix A contains only zero- and one-valued elements, makingthe identification of identical rows and columns relatively easy as well as the constructionof transformation matrix T.


Table 4.1: Dependence of Nk on the modulation and channel memory M .Ref. Pulse Mod. Pulse M = 0 M = 1 M = 2 M = 3 M = 4 M = 5

BPSK QPSK 3 12 28 51 81 118BPSK BPSK 2 7 16 29 46 67’none’ QPSK 3 7 13 21 31 43’none’ BPSK 2 4 7 11 16 22

Alternatively, we notice that the transformation matrix T can also be obtained byapplying SVD on A = EΛEH , where E is the unitary matrix of eigenvectors and Λ is adiagonal matrix of eigenvalues. The transformation matrix T is then obtained from

T = ENZ

√

ΛNZ (4.43)

where the zero-eigenvalues and the corresponding eigenvectors are skipped, as denotedby the subscript NZ . It is unclear, whether this generally yields an appropriate mapping,or only in our context. Furthermore, the SVD may become unstable for large A, leadingto incorrect results.

Without loss of generality, some assumptions are made with respect to the compositionof d[n]. Firstly, the first element of d[n] is assumed to be a constant equal to 1. Secondly,the modulation b[n] and its M predecessors are assumed to be present on the M +1 subsequent position. The remaining components are assumed to be present on theremaining positions. How many additional elements d[n] has depends on the statisticalproperties of d[n], i.e. the applied modulation. To summarize, d[n] is assumed to havethe following structure

d[n] = [ 1 b[n]T︸︷︷︸

Linear Info-Terms

b[n]b[n − 1] . . .︸︷︷︸

Non-linear Terms

]T, (4.44)

where b[n] ,[b[n] b[n − 1] . . . b[n − M ]

]T. Since the components are uncorrelated,

unit power and only the first element is a constant, d[n] has the following statisticalproperties,

E[

d[n]]

= e1 (4.45)

E[

d[n]d[n]H]

= INk,Nk(4.46)

Due to the composition of d[n] described by (4.44), its is correlated to b[n] according to

E[

d[n + m]b[n]∗]

=

e2+m ∀m ∈ 0, 1, . . . ,M,0 otherwise.

(4.47)

As stated before, Nk depends on the applied modulation. In Tab. 4.1, the numberof uncorrelated elements is presented as function of the modulation scheme and channelmemory M . In the case of a memory-less channel, i.e. in the absence of ISI, Nk is equal totwo. The first one is the desired term, while the second is a constant to model DC offsets


in the constellation diagram. Only in the case of QPSK modulation, an additional termexists, because intra-symbol interference is still possible. In the presence of ISI, Nk issuper linear with respect to the channel memory. Assuming a constant channel memory,Nk is reduced if the modulation can assume less values, i.e. if the degree of freedom ofthe modulation is reduced.

4.5.3 Extension of the Vector Notation

If the SIMO FIR Volterra model has memory, a better performance might be obtained ifthe symbol decision is delayed. Similar to how I&D samples at time index n are influencedby M preceding symbols, the symbol b[n] will influence samples with a time index betweenn and n+M . In other words, these samples may contain information on the value of b[n]and involving them in the symbol decision process may improve the system performance.Therefore, it makes sense to delay the decision by at least M , assuming a channel withmemory M and taking only information theoretical consideration into account and noimplementation aspects.

Taking only the signal part into account, these samples are assumed to be gatheredin s[n], which has the following composition

s[n]=[s[n, 1], . . . , s[n, L], s[n+1,1], . . . , s[n+M, L]

]T. (4.48)

The data samples can be related to the TX symbols in the following manner,

s[n] = d[n]T K. (4.49)

The most straight-forward way to define d[n] and K is as follows,

d[n] =[

d[n]T d[n + 1]T . . . d[n + M ]T]T

, (4.50)

K = K⊗ IM+1,M+1 (4.51)

where K =[k1 k2 . . . kL

]. In this case the elements in d[n] are surely correlated if M

is larger than zero, where Ni denotes the number of uncorrelated elements. By applyingthe same mathematical trick as in Sec. 4.5.2, an alternative definition is obtained suchthat d[n] has the same statistical properties as d[n], i.e. (4.45)-(4.47) also applies to d[n].The matrix K is defined as follows

K = TH (K⊗ IM+1,M+1) , (4.52)

where the matrix T is obtained in the same manner as T using the autocorrelation matrixof d[n] as defined in (4.50).

Similar to Nk, also the relationship between Ni, M and modulation has been com-puted. The results are gathered in Tab. 4.2. Similar to Nk, the number of uncorrelatedelements Ni decreases if the degree of freedom of the modulation is reduced. In case theVolterra model has no memory, the value for Nk and Ni are equal, because samples witha time-index larger than n do not contain information on b[n] and are thus not included.In other words, d[n] and d[n] are identical. When M is larger than zero, all elements in


Table 4.2: Dependence of Ni on the modulation and channel memory M .Ref. Pulse Mod. Pulse M = 0 M = 1 M = 2 M = 3 M = 4

BPSK QPSK 3 21 60 120 201BPSK BPSK 2 12 34 68 114’none’ QPSK 3 11 25 45 71’none’ BPSK 2 6 13 23 36

d[n] are also contained in d[n], i.e. the elements in d[n] are a subset of the elements ind[n]. Assuming the same memory and modulation, Ni is thus bounded to be equal orlarger than Nk. This result of reasoning is confirmed by Tab. 4.2. Furthermore, it showsthat Ni grows considerably faster than Nk with increasing memory.

4.5.4 Linear MIMO Model

In [68], a linear MIMO interpretation was introduced for SIMO FIR Volterra systems,using the linearity of a Volterra model with respect to its kernel elements [64]. In thissection, the MIMO interpretation is presented as described in [68] with two differences.Firstly, the model is presented in the notation deployed in this thesis. Secondly, theMIMO model regards only uncorrelated, modulated inputs as different inputs, where theMIMO model as presented in [68] regards every element of vec

(d[n]d[n]H

)as input. The

MIMO model presented here allows for understanding the role of modulation on the BERperformance in the presence of ISI, see Sec. 5.4.3.

Eq.(4.49) shows that s[n] is a superposition of Ni vectors, which are gathered in K,that are modulated by Ni uncorrelated RVs gathered in d[n]. In other words, the systemcan be interpreted as a MIMO system with Ni uncorrelated inputs. The number ofoutputs is ML or 2ML for a RV AcR and a CV AcR, respectively. However, the firstelement in d[n] is by definition a constant to account for any DC-offset in the outputs,which can be compensated for using DSP. For simplicity, it will be assumed that s[n] iszero mean, i.e. that k1 is an all zero vector, where kn denotes the n-th column of K. TheMIMO model has now Ni − 1 uncorrelated inputs, which modulate Ni − 1 vectors in anmulti-dimensional linear vector space. A simplified representation of the linear vector isgiven in Fig. 4.11.

The RX can apply linear weighting on s[n] to form a decision statistic based on whicha decision is made on the value of b[n]. Assuming MMSE weighting in the absence ofnoise, the MMSE solution will suppress all ISI if k2, which describes the linear relationshipbetween b[n] and s[n], has a component that is perpendicular to the space spanned bythe remaining interfering terms. In other words, if

(I − PISI) k2

?

6= 0

true: full ISI suppression possible,

otherwise: no full ISI suppression possible.(4.53)

where PISI denotes the space spanned by the ISI terms, e.g. obtained using GramSchmidtorthonormalization. Furthermore, if k2 is orthogonal with respect to PISI, the ISI can besuppressed without increased sensitivity to noise. On the other hand, if the ISI projectionmatrix PISI is full-rank, no linear weighting vector exists, which fully suppress the ISI.


k1

k2

k3

dim 1

dim 3

dim 2

k4

Figure 4.11: Vector space spanned by the MIMO kernel K

The actual rank of PISI is upper-bounded by the number of interfering MIMO inputsNi − 2 and secondly depends on on the composition of K. Since, K is a random matrixdue to the radio channel, it’s impact on the system performance has a random natureas well. Nevertheless, it is likely that the ISI can be fully suppressed if the number ofMIMO inputs Ni is smaller than the number of output ML, such that PISI can never befull-rank. With all other parameters being the same, more ISI can be suppressed withdecreasing Ni and/or increasing number of MIMO outputs.

The number of MIMO outputs can be changed by increasing the FSR or by usingadditional autocorrelation branches. For instance, extending a real-valued AcR to acomplex-valued AcR means that the effective number of MIMO outputs is increased bya factor 2.

The number of MIMO inputs changes when modifying the modulation scheme. InFig. 4.12, the dependence of the constellation diagram on the modulation type beforeand after MMSE weighting in the absence of noise. Fig. 4.12 shows that the constellationdiagram the MMSE weighting is able to suppress more ISI if the reference pulse is leftunscrambled, which increases the robustness of the system against noise. In this example,the scrambled QPSK-TR results in detection error even in the absence of noise, sincesome of the constellation points belonging to the TR-QPSK symbol in quadrant 1 fall inquadrant 2.

4.5.5 Data Model as Finite State Machine

The FIR Volterra system is in fact a Hidden Markov Model (HMM), resulting from theradio channel. After the RX identifies the HMM, the HMM reduces to a Markov modelor FSM with a non-linear relationship between state-transitions and outputs. In this


(a)

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Real

Imag

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Real

Imag

Scrambled QPSK−TR

Unscrambled QPSK−TR

Scrambled QPSK−TR


(b)

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

Real

Imag

Scrambled QPSK−TR


Figure 4.12: Dependence of the constellation diagram(s) of the modulation before (a) andafter MMSE weighting (b).


a[n]

b[n], Sp[n]

s[n,α]

PN seqn

constant

Table fα

∈ 2Nb

Sz=2(Np+Nb)(M+1)

S0

S1

S0

S1

b[n], St[n]

Synchrone?

Memory Memory-less

NSt=2NbM

NSp=2NpM

bits

∈ 2Np

Random Entries

+

η[n,α]

u[n,α]

Figure 4.13: FSM description for a FIR Volterra model

section, it will be shown that on the non-linear FSM trellis-based equalization can beapplied with the same complexity as needed for equalization of linear channels, assumingthe same channel memory for both. Also the role of scrambling of reference pulses on theidentification and equalization of FIR Volterra systems will be discussed, showing thatscrambling does not significantly increases the equalizer complexity.

To obtain a generic system model, the modulation applied to both pulses has beenkept general so far. For the description of the system as FSM, it is required to introducea more formal description of the modulation. In practice, it may be assumed that aninteger amount of (channel) bits Nb are mapped on a single TR-symbol. For notationalconvenience, the k-th (channel) bit mapped on the n-th TR symbol will be denoted byc[n, k] and the vector c[n] gathers all bits with time-index n. Assuming these bits tobe i.i.d. RVs in B = 0, 1, the n-th TR-symbol is identified by i.i.d. RV a[n], wherea[n] ∈ 0, 1, . . . , 2Nb − 1 with equal probability, where the modulation b[n] depends onlyon a[n] and thus the bits to be transmitted. In the same fashion, an identifier a[n] isdefined, which drives the modulation applied to the reference pulse b[n].

In the presence of ISI, the Volterra model becomes a FSM of which the memorysize depends on the symbol rate and CIR. Evidently, the radio channel does not distin-guish between scrambled or modulated pulses, meaning that the memory applies to both.Without loss of generality, the FSM will be divided into two parallel FSMs with the samememory depth, one driven by the symbol identifiers a[n] and the other one driven by thescrambling identifiers a[n]. The state of both FSMs at time n will be denoted by St[n]and Sp[n], respectively. The principle structure of both FSMs is known by the RX, i.e.it knows the possible state-transitions and their probabilities a-priori. In this respect,the HMM differs from those used e.g. for speech processing, where the state-transitionprobabilities are unknown a-priori. Based on the states and inputs of both FSMs, amemory-less relationship exists for each output fα(a[n], St[n], b[n], Sp[n]). In Fig. 4.13,the structure of the system has been depicted.


a[n]s[n,α]

Table fα

Sz=2(Nb)(M+1)

S0

S1 b[n], St[n]

Memory Memory-less

NSt=2NbM

bits

Random Entries

+

η[n,α]

u[n,α]

Figure 4.14: Simplified FSM description for a FIR Volterra model in the absence of PNscrambling

If the scrambling code is driven by a PN sequence, the RX can reconstruct the stateSp[n] at all time, assuming the RX is aware of the initial state, phase and structure of thePN generator. In this case, the input-output relationship of the system can be describedby a time-variant function,

s[n, α] = fα(a[n], St[n], n) (4.54)

The number of state transitions, an important measure for the complexity of a trellis-based equalizer [69], is not affected by the use of a PN sequence.

Applying no scrambling to the reference pulse can be seen as time-invariant PN se-quence. In this case, the table function becomes time-invariant fα(a[n], St[n]). However,the PSD of the TX signal will contain spectral spikes, if no scrambling is applied.

Alternatively, the scrambling may be driven by a[n], i.e. a[n] = a[n] for all values ofn. In this case, both FSMs will be running synchronously, i.e. Sp[n] is fully describingSt[n]. The system model of Fig. 4.13 is simplified to a single FSM. Additionally, the tablefunction becomes time-invariant fα(a[n], St[n]) containing at most 2Nb(M+1) entries as inthe unmodulated case, but since the reference pulse is scrambled, the PSD will be smoothif appropriate modulation is applied. The simplified block diagram has been depicted inFig. 4.14.

4.5.6 Reduced Memory Data Model

The usage of trellis-based algorithms for the equalization of FIR Volterra channels isa promising technique, since the information contained in non-linear ISI terms is takeninto account. Unfortunately, the complexity of a trellis is proportional to the numberof channel-states, i.e. it is exponentially proportional to the channel memory. As aresult, trellis-based equalization becomes quickly too complex for practical application,if the full channel memory is taken into account. In this subsection, a Reduced MemoryData Model (RMDM) is introduced, which mimics the behaviour of the Full Data Model(FDM), while using less memory. Using the RMDM, trellis-based algorithms can equalizethe channel with less complexity, at the cost of an increased sensitivity to noise.

The structure of the RMDM is in essence the same as that of the FDM, except for thefact that the incoming symbols are delayed by m and the memory N of the RMDM isless or equal to the FDM’s memory M . In a vectorial notation, the output of the RMDM


is defined as

u[n, α] =d[n − m]H kα + η[n, α]. (4.55)

where d[n] is of length Nr and contains a subset of the elements in d[n].It is the challenge to find the optimal combination of kernel kα and delay m, such

that

[m, kα] =argminx∈N,k∈CNr,1

[L∑

α=1

E[∣∣∣d[n]Hkα − d[n − x]Hk

∣∣∣

2]]

(4.56)

where N and C denote the sets of non-negative integers and complex numbers, respec-tively. To our knowledge, (4.56) cannot be solved in closed form. Therefore, a divide-and-conquer approach is applied to the problem. Firstly, the MMSE solution for kα isderived for a single given output and delay x, denoted by k

(x)α , such that

k(x)α = argmin

k∈CNr,1

[

E[∣∣∣d[n]Hkα − d[n − x]Hk

∣∣∣

2]]

(4.57)

Since both d[n] and d[n− x] contain per definition only uncorrelated variables, it is easyto prove that the optimal kernel in the sense of the MMSE criterion is given by

k(x)α = Ckα (4.58)

with

C, E[

d[n − x]d[n]H]

(4.59)

where C ∈ 0, 1Nr,Nk . The under-modeling error, i.e. the average squared differencebetween the RMDM and the FDM, for the delay under evaluation and output σ2

u,α(x)equals

σ2u,α(x) , E

[(

d[n]Hkα − d[n − x]H k(x)α

)2]

= kHα

(I − CHC

)kα (4.60)

Using the previously obtained result, the MMSE delay m is selected using

[m] = argminx∈0,...,M

[L∑

α=1

σ2u,α(x)

]

(4.61)

where the fact is used that an m greater than M can never be optimal.As stated before, the RMDM does not completely describe the FDM, such that an

equalizer deploying the RMDM will not exploit fully the information present in the RXsignal. On the contrary, the unused part will have a noise-like effect from the equalizer’spoint of view, deteriorating the system performance. Hence, the noise variance at theα-th output of the RMDM can be written as,

σ2η,α = σ2

η,α + σ2u,α(m). (4.62)


Figure 4.15: Constellation diagram at the outputs of an FDM and its RMDM. The left-hand plot refers to α = 1 and the right-hand plot to α = 2

where σ2η,α denotes the variance of the noise term η[n, α]. Here, it is assumed that the noise

term is white with respect to both n and α. The validity of this assumption will be shownin Sec. 4.6. However, the under-modeling-error σ2

u,α(m) is most likely not white. Hence, aclosed-form expression for the performance degradation due to the under-modeling is noteasily obtained. An indication for the performance is obtained by computing the ”overallSNR” seen from the equalizer’s perspective as,

SNRRMDM =L∑

α=1

∥∥kα

∥∥

2

σ2η,α + σ2

u,α(m). (4.63)

The ability of an RMDM to mimic its FDM can be visualized by comparing theirconstellation diagrams. In Fig. 4.15, the constellation diagram is depicted of a memory-four FDM describing a QPSK-TR system, together with the constellation diagram ofits memory-one RMDM. The RMDM’s constellation diagram resembles the constellationdiagram of the FDM reasonably well, in the sense that the general structure of the FDM’sconstellation is preserved. Nevertheless, the number of states is reduced by a factor of64, simplifying the complexity of a trellis-based equalizer by the same amount.

4.6 Statistical Properties of the TR System Model

In this section, the statistical properties of the I&D samples are derived. In Sec. 4.4.3,each I&D samples was shown to be the superposition of three types of terms, a signal terms[n, α], a Gaussian noise term ηg[n, α] and a non-Gaussian noise term ηz[n, α]. Althoughthey are statistically dependent, it is straight-forward to prove that they are uncorrelatedand as only the first and second order moments of the I&D samples are derived, they canbe solved separately. In case of all three types of terms, the expectation, co-variance and

4.6. STATISTICAL PROPERTIES OF THE TR SYSTEM MODEL 85

cross-correlation with the symbol under demodulation will be derived in the followingthree sub-sections. In the final section (Sec. 4.6.4), some claims regarding the noise termare validated.

4.6.1 Statistics of the Signal Term

Although ordinary rather complicated, deriving the statistical properties of the signalterm has become rather straight-forward, due to the extended linear model presented inSec. 4.5.3. Using this linear model, the expectation for s[n, α] is

E [s[n, α]] = E[

dT [n]]

K (4.64)

Using the statistical properties of d[n] presented in (4.45), the expectation becomes

E [s[n]] = KTe1. (4.65)

The correlation between s[n] and b[n] is by definition as follows

E [s[n]b∗[n]] = KTE[

d[n + m]b∗[n]]

. (4.66)

Using (4.46), it is evident that,

E [s[n]b∗[n]] =

KTe2+m ∀m ∈ 0, 1, . . . ,M,0 otherwise.

(4.67)

The covariance of the signal vector s[n] using the linear model notation is as follows

C[s[n], sH [n]]

]= C

[

KT d[n], dH [n]K∗]

(4.68)

= KTC[

d[n], dH [n]]

K∗ (4.69)

Using (4.45) and (4.46), it is straight-forward to show that the covariance of s[n] is equalto

C[s[n], s[n]H ]

]= KT

(I − e1e

T1

)K∗, (4.70)

which concludes the derivation of the statistical properties of s[n, α].

4.6.2 Statistics of the Gaussian Noise Term

In this subsection, the mean and covariance of the Gaussian noise term ηg will be com-puted. As stated before, the term ηg is in fact the superposition of two uncorrelatedGaussian noise terms, ηg1

and ηg2. Since both terms are uncorrelated and since the inter-

est is only in first and second order moments, both terms will be treated independently.The computation of the mean of both terms is rather simple. The noise vector n[n]

contains elements resulting from a zero mean random process. Since both ηg1[n, α] and


ηg2[n, α] depend linearly on n[n], it is evident both Gaussian noise terms ηg1

[n, α] andηg2

[n, α] are zero mean as well.The cross-correlation between the first Gaussian noise and b[n] in a mathematical

notation is as follows

C[ηg1[n, α], b[n]] = E

[b∗[n]d[n]H

]Lα,1E [n[n]] . (4.71)

Since the expectation of the noise vector is equal to zero, the first Gaussian noise is notcorrelated to b[n]. Using a similar derivation, it is straight-forward to show that the samealso applies to the second Gaussian noise term ηg2

.Computation of the covariance of both terms is not so straight-forward. Both are the

cross-product depend on two random processes, the noise vector n[n] and the transmittedsymbols d[n]. Fortunately, both processes are independent, which allows us to computethe covariance in two consecutive steps. Firstly the covariance of both terms will becomputed conditioned on the transmitted symbols, after which the statistical propertiesof the transmitted symbols are taken into account. The covariance between ηg1

[n, α] andηg1

[m, β] conditioned on d[n] and d[m] is given by

C[ηg1[n, α], ηg1

[m, β]|d[n],d[m]] = d[n]HLα,1C[n[n],n[m]H

]LH

β,1d[m] (4.72)

= d[n]HLα,1NnmLHβ,1d[m] (4.73)

where

Nnm , E[n[n]n[m]T

], (4.74)

Nnm[k, l] = rnn((n − m)Ts + (k − l)Tr) (4.75)

In the same manner, the conditional covariance of the second Gaussian noise termηg2

[n, α] is found to be

C[ηg2[n, α], ηg2

[m, β]|d[n],d[m]] = d[n]TLα,2N∗nmLH

β,2d[m]∗ (4.76)

In practice, the integration duration is long compared to the correlation time of thenoise. As a result, the noise matrix Nnm is approx. an all zeros-matrix if n is unequalto m. To simplify both derivation and the noise model, it will be assumed that they arefully uncorrelated for n 6= m. The validity of this assumption will be shown in Sec. 4.6.4.

Considering only the case n = m, both Gaussian noise terms can be combined to asingle Gaussian ηg noise term of which the variance can be described using a quadraticVolterra model. The Volterra description for the Gaussian noise terms has been firstreported for traditional AcR receivers in [70, 71]. In our case, the model has the followingform,

C[ηg[n, α], ηg[m, β]|d[n]] =

d[n]HHα,βd[n] if n = m,

0 otherwise.(4.77)

with

Hα,β = Lα,1NnnLHβ,1 + (Lα,2N

∗nnL

Hβ,2)

T . (4.78)


To simplify the statistical derivation, the linear model is also applied to the Volterranoise model, i.e.

C[ηg[n, α], ηg[m, β]|d[n]] = d[n]hα,β (4.79)

where

hα,β = THvec (Hα,β) . (4.80)

This notation greatly simplifies the derivation of unconditional covariance. In the uncon-ditional case, the covariance becomes,

C[ηg[n, α], ηg[m, β]] = δ[n − m]E[

d[n]]

hα,β (4.81)

Using statistical property 1 of d[n] given by (4.45), the unconditional covariance of thejoint Gaussian noise terms is found to be

C[ηg[n, α], ηg[m, β]] = δ[n − m]eT1 hα,β, (4.82)

which concludes the derivation of the first and second order moments of the Gaussiannoise terms.

4.6.3 Statistics of the Non-Gaussian Noise Term

Before deriving its statistical properties, the non-Gaussian noise term will be re-writtento simplify interpretation. Previously, the non-Gaussian noise term was defined as

ηz[n, α] = hTαz[n] (4.83)

with z[n] equal to Λ(W1n[n])W2n[n]∗. The matrices W1 and W2 are however block-selection matrices, containing only binary entries with only one-valued elements on itsmain diagonal. Using this structure, the k-th element of the vector z[n], which will bedenoted as z[n, k], can be written as.

z[n, k] = n[n + k]n∗[n + k − Nd]. (4.84)

This insight greatly simplifies the statistical derivations.Using this definition and the stationarity of the signal n[n] makes it straight-forward

to prove that the expectation for the non-Gaussian noise term is equal to,

E [z[n, k]] = E [n[n + k]n∗[n + k − Nd]] , rnn(D), (4.85)

which means that

E [η[n, α]] = hTα1rnn(D), (4.86)

which in turn means that the expectation of the non-Gaussian noise term depends on theimpulse response of the RX BPF, the delay and the integration interval duration only. Inpractise, the delay duration will be larger than the duration of the impulse response of


the BPF, i.e. rnn(D) = 0, so that it is reasonable to assume that the non-Gaussian noiseterm is zero mean.

Since it does not depend on b[n], it is evident that the non-Gaussian noise term is notcorrelated to b[n].

Let us continue with the derivation of the covariance of the non-Gaussian noise term.This term is by definition given by

C[ηz[n, α], ηz[m, β]] = C[hT

αzn, zHmhβ

]= hT

αZn,mhβ, (4.87)

where Zn,m = C[zn, z

Hm

]. In [72], the fourth-order moment of a complex Gaussian random

signal is presented. Using the notation deployed in this thesis, it is stated that in thecomplex-valued case,

Zn,m[k, l] = |rnn((n − m)Ts − (k − l)Tr)|2. (4.88)

where Zn,m[k, l] denotes the element at position k, l in the matrix Zn,m. Using the samereasoning as applied to Nn,m, in practise Zn,m will be virtually an all zero matrix. Thevalidity of this assumption will be verified in Sec. 4.6.4. This also concludes the derivationof of the statistical properties of the non-Gaussian noise term.

4.6.4 Analysis of the Noise Term

The noise present at each output originates from the same noise process, so that thesamples nn,α and nm,β are potentially correlated, when the difference between n and m issmall. In the derivation of the statistical properties of the noise in the previous subsections4.6.2 and 4.6.3, they were assumed to be uncorrelated. In this subsection, the validity ofthis assumption will be shown.

To compute the covariance between samples related to different outputs, the L outputsare multiplexed into a single sample stream. The covariance matrix of the resultingcyclo-stationary sample stream with period L is investigated. The assumed scenario is aresidential NLOS environment in which the TR system is operated at 10 and 80 Mb/s andfour times oversampling. This represents two extreme scenarios, one without ISI and withsevere ISI, respectively. Both the Gaussian and the non-Gaussian term can dominate theoverall noise term, depending on the value of Eb/N0. Therefore, the covariance of bothnoise terms will be analyzed separately.

In Fig. 4.16, the covariance matrix of the Gaussian noise term is depicted at both datarates. As can be seen on the main-diagonal, the variance varies from output to output.Furthermore, the cross-covariance is very low. At 80 Mb/s the cross-correlation betweentwo consecutive samples is well below 0.25. At lower data rates, this correlation is evenless. Strictly speaking, the Gaussian noise term is not truly white, but the approximationerror will be small when assuming it to be white.

The same procedure is repeated for the non-Gaussian noise term. In Fig. 4.17, thecovariance of this noise term is presented. As expected, every element on the maindiagonal has the same value, due to the stationary nature of this noise term. Comparingboth data rates, the variance is 8 times higher at 10 Mb/s compared to the 80 Mb/scase, because the integration duration is 8 time longer as well. In either case, the de-correlation is rapid and zero if one or more samples are in between the samples under


(a)

[n,0] [n,1] [n,2] [n,3] [n+1,0] [n+1,1] [n+1,2] [n+1,3][n,1]

y [n,3]

[n+1,1]

[n+1,3]0

1

2

3

4

C[n

g[x

],n

g[y

]]

x

(b)

0.2

0.4

0.6

0.8

[n,0] [n,1] [n,2] [n,3] [n+1,0] [n+1,1] [n+1,2] [n+1,3][n,1]

y [n,3]

[n+1,1]

[n+1,3]0

1

C[n

g[x

],n

g[y

]]x

Figure 4.16: The covariance matrix of Gaussian noise in the sample-stream of a fourtimes fractionally sampled AcR operating at 10 and 80 Mb/s in sub-figure (a) and (b),respectively

(a)

6

8

10

[n,0] [n,1] [n,2] [n,3] [n+1,0] [n+1,1] [n+1,2] [n+1,3][n,1]

y [n,3]

[n+1,1]

[n+1,3]0

2

4

C[n

z[x

],n

z[y

]]

x

(b)

0.2

0.4

0.6

0.8

[n,0] [n,1] [n,2] [n,3] [n+1,0] [n+1,1] [n+1,2] [n+1,3][n,1]

y [n,3]

[n+1,1]

[n+1,3]0

1

C[n

z[x

],n

z[y

]]

x

Figure 4.17: The covariance matrix of Non-Gaussian noise in the sample-stream of a fourtimes fractionally sampled AcR operating at 10 and 80 Mb/s in sub-figure (a) and (b),respectively


consideration. The correlation between two consecutive samples is higher at 80 Mb/s,because the impulse response of the BPF at the receiver front-end is longer comparedto the integration duration at this data rate. Nevertheless, the correlation between twodifferent samples is always lower than 0.15, making this process quasi-white. This resultis inline with the conclusions drawn in [58].

4.7 Conclusions

In this chapter, the principle of UWB Transmitted-Reference communication was intro-duced, including a discussion of its pro’s and con’s with respect to performance andimplementation. Furthermore, several extensions of the TR principle were proposed.Firstly, a fractional sampling AcR structure was proposed to allow for synchronization andweighted autocorrelation, which also simplifies the implementation. Secondly, a complex-valued AcR was proposed to make the system less sensitive against delay mismatches.Additionally, the complex-valued AcR allows for the extension of the TR signaling schemeto complex-valued modulation.

A general-purpose discrete-time equivalent system model was presented for the analy-sis of TR systems, where general-purpose means that all extensions are accounted for. Itwas shown that the I&D samples generated by a fractional sampling AcR in a TR systemconsist of two contributions with a different nature, a signal term and a noise term. Thesignal term could be modelled using a SIMO FIR Volterra model. The noise terms wasshown to consist of two types of noise, a Gaussian noise sub-term and a non-Gaussiannoise sub-term.

Several interpretations for the SIMO FIR Volterra model have been presented, whichallow for more insight in the behaviour of TR systems. Firstly, the Volterra modelhas been written in a normal vector notation and extended vector notation, to allowfor simplified statistical analysis. The extended vector notation also allowed for theinterpretation of the SIMO FIR Volterra model as a linear MIMO Model. Furthermore,the SIMO FIR Volterra model was shown to be a finite state machine, meaning thattrellis-based algorithms can be used for the equalization of TR systems. In this line ofreasoning, a reduced-memory system model was introduced, which mimics the behaviourof TR systems, but with a significant memory reduction.

Finally, the statistical properties were derived of the signal term as well as bothnoise terms and the noise was shown to be quasi-white, with an output dependent noisevariance.

Chapter 5

Analysis of TR UWB

Communication

5.1 Introduction

In chapter 4, the theory of TR UWB systems was presented. However, the role of manysystem parameters on the system performance was left undefined, but required to obtain asolid system design based on the TR UWB communication. In this chapter, the impact ofdifferent parameters on the system performance will be analyzed. The evaluated systemparameters are FSR, bandwidth, delay, weighting criteria and modulation both in theabsence and presence of ISI. One of the main contributions presented in this chapter isthat linear weighting can also suppress non-linear ISI, if the AcR is fractionally sampled.

5.2 Description of the Linear Weighting

In Sec. 4.3.1, the concept of fractionally sampled, weighted AcR was presented. In thissection, the weighting applied to the I&D samples and the decision process for the detectedbits is presented. For notational convenience, the samples on which linear weighting isapplied are assumed to be gathered in a vector denoted by u[n]. Its composition is asfollows,

u[n]=[u[n, 1], . . . , u[n, L], u[n+1,1], . . . , u[n+M, L]

]T. (5.1)

Linear weighted combining is applied on u[n] to generate a single decision statistic α[n],such that

α[n] = wTu[n] + c, (5.2)

where w is a vector containing the weighting coefficients. Due to the random nature ofthe channel, both w and c should be adaptable. Therefore, adaptive weighting algorithmswill be deployed that thrive to find the weighting coefficients according to some criteria.In this thesis, two weighted combining criteria are considered to shape w and c, namelyMRC and MMSE combining [73]. In either case, the first and second order moments of

91

92 CHAPTER 5. ANALYSIS OF TR UWB COMMUNICATION

u[n] are required to derive the optimal coefficients in closed form. In [73], the weightingvector is given by,

w =

R−1uup MMSE criterion

p MRC criterion(5.3)

where

Ruu = C[u[n],u[n]] , (5.4)

p = C[u[n], bn] .

In either case, the offset equalization factor will be equal to

c = −wE [u[n]] . (5.5)

The expressions of Ruu, p and E [u[n]] have been presented in Sec. 4.6, such that all themathematical tools are available to compute the weighting vector in a closed form.

The value of the detected b[n], denoted by b[n], is based on the sign of α[n], such that

b[n] = sign (α[n]). (5.6)

The probability of an erroneous decision for b[n] can be computed in closed-form. Adetailed description can be found in [17].

5.3 System Performance in the Absence of ISI

In this section, the impact is investigated of system parameters like FSR, bandwidth andmodulation on the system performance in the absence of ISI. To allow for comparison,the general system set-up and communication environment will be kept the same, exceptfor the system parameter(s) under evaluation.

The general system set-up is as follows; to allow for reference with work by others, aTR signaling scheme using BPSK signaling is assumed, demodulated using a traditionalreal-valued AcR. The symbol rate is chosen equal to 10 MHz, which in case of BSPKmodulation results in a channel bit rate of 10 Mb/s. The delay is chosen equal to 40 ns.The bit rate and delay are chosen such that hardly any pulse-overlapping will occur, i.e.there is neither ISI nor intra-symbol-interference. The TX and RX delays are assumedto match perfectly. The bandwidth of the TX signal is approx. 500 MHz with a centerfrequency of 4.5 GHz. The RX BPF is matched to the TX signal. The FSR L is chosenequal to two. The default weighting principle is MMSE principle, except stated otherwise.The RX is assumed to have perfect side-information on the statistical properties of theI&D samples, so that the weighting vector is optimal. However, no time-synchronizationis assumed between TX and RX, i.e. due to the cyclo-stationarity of the TX signal withperiod Ts, the time-offset has been modelled as a RV with an uniform distribution overthe interval [0, Ts〉. The propagation environment is the NLOS environment as describedin Sec. 3.2. To obtain better insight on the role of the system parameters, SSF has notbeen taken into account. The role of bandwidth with respect to SSF has already beenthoroughly investigated in chapters 2 and 3.

5.3. SYSTEM PERFORMANCE IN THE ABSENCE OF ISI 93

5 10 15 20 25

10−4

10−3

10−2

10−1

100

Eb/N0[dB]

P(e

)

FSR=1,MMSE

FSR=1,MRC

FSR=2,MMSE

FSR=2,MRC

FSR=4,MMSE

FSR=4,MRC

FSR=8,MMSE

FSR=8,MRC

Figure 5.1: Influence of the weighting criteria and FSR on the system performance in theabsence of ISI

5.3.1 Influence of the Weighting Criteria and Fractional Sam-

pling Rate

In Sec. 5.2, two weighting principles have been proposed, MMSE combining and MRC. Inthis subsection, the difference in performance between both principles and the role of theFSR will be investigated. Four different FSRs have been considered, namely 1, 2, 4 and8. Additionally, the performance results also show the ability of the RX to synchronizeto the RX signal. The BER performance is depicted in Fig. 5.1.

As one might expect, if no fractional sampling is used, the RX has two problems.Firstly, the RX cannot synchronize to the RX signal with sufficient accuracy, since itcannot influence the time-offset of the integration interval. As a result, the I&D samplesmay contain information not only regarding the symbol under demodulation, but alsoof other symbol. In other words, the I&D samples suffer from ISI, not caused by pulse-overlapping, but because the integration interval is gathering information of multiplesymbols. Secondly, the AcR is accumulating more noise due to the long integrationduration. Comparing both weighting principles, MMSE combining is coping better withISI than MRC weighting.

If the FSR is increased, both weighting principles obtain more information from thechannel, so that the problems occurring in a system without fractional sampling can beresolved by the RX. As a result, MRC has approx. the same performance as its MMSEcombining counterpart, over the complete BER range depicted. In case of a low data rate,a single pulse will fall almost completely into a single integration window, i.e. a singlesample contains the information on b[n]. In this case, the MRC and MMSE weighting


5 10 15 20 25

10−4

10−3

10−2

10−1

100

Eb/N0[dB]

P(e

)FSR=2,D=40ns

FSR=2,D=10ns

FSR=4,D=40ns

FSR=4,D=10ns

FSR=8,D=40ns

FSR=8,D=10ns

Figure 5.2: Influence of the delay and FSR on the system performance in the absence ofISI

vector will be virtually the same, explaining the similar performance. At (very) low Eb/N0

values, the stationary, non-Gaussian noise term is dominant and the SNR of the samplesin u[n] becomes proportional to p, so that the MMSE combining vector converges to theMRC vector.

As to be expected, increasing the FSR will improve the system performance, but theimprovement cannot justify the additional complexity. The avoidable part of Gaussiannoise namely falls largely in the samples that precede and follow the sample containingthe desired information term, even when the FSR is equal to two. This is caused by thedelay, which is approx. half the value of the symbol period. As will be shown in Sec. 5.3.2,if the delay is smaller, a further increase of the FSR can improve the performance of thesystem.

5.3.2 Influence of Delay and Fractional Sampling Rate

As stated in Sec. 4.1, BPSK-TR signaling demodulated using an AcR performs approx.6 dB worse compared to a perfect matched-filter receiver. However, if the reference pulseand modulated pulse arrive overlapped, the variance of the Gaussian noise terms willincrease and an additional performance loss of 3 dB can be anticipated. To illustrate this,the 6 systems of Sec. 5.3.1 are compared with two differences. Only MMSE combiningis considered and the delay is decreased to 10 ns. The resulting performances have beendepicted in Fig. 5.2.

As expected, the performance of all systems degrades with decreasing the delay value.In case of the 10 ns delay, the RX is able to suppress more noise if the FSR is increased,

5.3. SYSTEM PERFORMANCE IN THE ABSENCE OF ISI 95

5 10 15 20 25

10−4

10−3

10−2

10−1

100

Eb/N0[dB]

P(e

)

BW=250MHz

BW=500MHz

BW=1GHz

Figure 5.3: Influence of bandwidth on the system performance in the absence of ISI

but is unable to fully compensate for the additional noise.

5.3.3 Influence of Bandwidth

In this subsection, the impact of the bandwidth on the system performance will be ana-lyzed. In linear systems, the BER performance on AWGN channels does not depend onthe bandwidth but only on the Eb/N0 ratio [31]. This rule however does not apply toAcRs. Their non-linear structure leads to the presence of the non-Gaussian noise termwith a variance proportional to the RX BPF bandwidth.

To obtain insight in the impact of this additional noise term on the overall perfor-mance, the performance of TR systems is compared for three different bandwidths inFig. 5.3, namely 250 MHz, 500 MHz and 1 GHz.

The BER curves in Fig. 5.3 show that the non-Gaussian term has a significant impacton the overall system performance. As to be expected, the system with the smallestbandwidth outperforms the others. At a BER of 10−2, the performance decreases ap-proximately 1 dB with each doubling of the bandwidth. The distance between the BERcurves has the tendency to decrease with increasing Eb/N0-ratio. However, the curves arestill far from convergence at a BER of 10−5.

5.3.4 Influence of Modulation

In Sec. 4.5.4, the MIMO model for Volterra models excited using finite-alphabet modula-tion was introduced. The modulation has been shown to influence the number of MIMO


Table 5.1: Properties of the considered TR systems to analyze the role of modulationSystem mod(b[n]) mod(b[n]) Receiver L

1 1 BPSK Real-Valued AcR 42 1 BPSK Complex-Valued AcR 43 1 QPSK Complex-Valued AcR 44 BPSK QPSK Complex-Valued AcR 4

5 10 15 20 25

10−4

10−3

10−2

10−1

100

Eb/N0[dB]

P(e

)

BPSK,RV−AcR

BPSK,CV−AcR

unscrambled QPSK,CV−AcR

scrambled QPSK,CV−AcR

Figure 5.4: Influence of modulation on the system performance in the absence of ISI

inputs whenever the Volterra model has memory. In the absence of intra-and-ISI, thenumber of MIMO inputs will be unaffected by the modulation. Therefore, it is expectedthat the modulation has little impact on the performance of the TR system in the absenceof ISI. This expectation will be validated in this section.

The performance of 4 TR systems will be compared. The first TR system used BPSKmodulation and a real-valued AcR. Two other systems employ QPSK-TR, one appliesno scrambling on the TR waveform, while other one does. Since in case of complex-valued modulation, a complex-valued AcR is required. To allow for a fair comparison,the performance of an additional BPSK-TR system is presented, demodulated using acomplex-valued AcR. Note that all systems operate at the same symbol rate of 10 MHz,meaning that the general structure of the signaling scheme is equal for all, allowing for afair analysis of the role of modulation. An overview of the properties of the 4 TR systemscan be found in Tab. 5.1. The performance of the four systems in the absence of ISI hasbeen depicted in Fig. 5.4.

In the absence of ISI, all modulation schemes have the same Euclidian distance be-tween the symbols, such that approximately the same BER performance is expected for

5.4. SYSTEM PERFORMANCE IN THE PRESENCE OF ISI 97

5 10 15 20 25

10−4

10−3

10−2

10−1

100

Eb/N0[dB]

P(e

)

FSR=1,MMSE

FSR=1,MRC

FSR=2,MMSE

FSR=2,MRC

FSR=4,MMSE

FSR=4,MRC

Figure 5.5: Influence of the weighting criteria and FSR on the system performance in thepresence of ISI

all four systems. As expected, Fig. 5.4 shows that all four systems have approximatelythe same performance, indicating that the modulation scheme has little to no influenceon the performance in the absence of ISI.

5.4 System Performance in the Presence of ISI

In this section, the analysis of Sec. 5.4 is repeated for scenarios with ISI. To allow forcomparison, all system parameters are the same as in the previous section, except thatthe symbol rate is increased to 40 Mb/s and the delay has been decreased to 10 ns.

5.4.1 Influence of the Weighting Criteria and Fractional Sam-

pling Rate

In Sec. 5.3.1, it was concluded that the weighting criteria and FSR had hardly any influ-ence on the system performance in the absence of ISI, provided the FSR is at least equalto 2. In this section, it will be investigated whether this conclusion holds in the presenceof ISI. The system performance for the ISI scenario has been depicted in Fig. 5.5.

As to be expected, the presence of ISI has a negative effect on the energy efficiencyof the system. Furthermore, MRC performs considerable worse than MMSE combining,since it incorrectly presumes the noise and ISI to be stationary. This in contrast toISI-free conditions, where both weighting principles performed almost equally well. Toillustrate the effect of the FSR, it is varied from 1, 2 to 4. In any case, the performance


improved when increasing the FSR. In case of MRC weighting, the BER floor decreasesonly slightly with increasing FSR, while MMSE combining is able to use the additionaldegree of freedom for weighting more effectively, leading to a significant performanceimprovement.

This does not explain why increasing the FSR improved the performance in the pres-ence of ISI. For traditional linear narrowband systems, the Nyquist criteria states that asampling rate larger than twice the symbol-rate will not provide the RX more informa-tion on the channel and thus not on the transmitted symbol. Hence, increasing the FSRabove 2 will not lead to a performance improvement in this case. The explanation for theperformance improvement in the case of FSR TR systems can be given using the MIMOinterpretation presented in Sec. 4.5.4.

In the MIMO model, both linear and non-linear ISI terms were modelled as additionalinputs, where all inputs are fed using symbols with the same first- and second-orderstatistical properties as the symbol under demodulation b[n]. Without changing themodulation or the symbol rate, the number of MIMO inputs cannot be altered. However,by increasing the FSR, the number of outputs of the MIMO model will be increased.Hence, it is reasonable to assume that more non-linear ISI can be suppressed with anincreasing ratio of outputs with respect to the inputs, i.e. more ISI will be suppressedwith an increasing FSR L. 1

5.4.2 Influence of Bandwidth

Previously, its was shown that with increasing bandwidth, the amount of non-Gaussiannoise in the detector input increases as well, leading to a decreased system performance.In this subsection, the role of bandwidth is analyzed in the presence of ISI. The resultshave been shown in Fig. 5.6.

Firstly, in the Eb/N0 region in which the noise is dominant, i.e. at lower Eb/N0-values, the system with the smallest bandwidth still outperforms the other. However,with increasing Eb/N0 the noise becomes less significant and the system becomes ISIlimited. In this case, Fig. 5.6 shows that large bandwidth TR systems suffer less from ISIthan their narrowband counterparts. The system’s sensitivity to ISI is namely related tothe amplitude of the autocorrelation side-lobes of the received pulse. Generally speaking,a larger TX bandwidth gives a smaller variance for the autocorrelation side-lobes [74],explaining the difference in performance. Consequently, the larger bandwidth systemseventually outperform their smaller bandwidth counterparts with an increasing Eb/N0.

5.4.3 Influence of Modulation

In Sec. 5.4.1, it was shown that with an increasing FSR more ISI can be suppressed. Notso obvious is however the role of modulation in the presence of ISI. In this section, theBER performance of four TR systems is compared under ISI conditions, to show that

1These insights are expected to hold for delay-hopped differential signaling and for systems deployingon/off keying in combination with an energy detector. All these systems can namely be modelled usinga Single-Input, Multiple-Output (SIMO) FIR Volterra system, assuming the related detectors to befractionally sampled.

5.4. SYSTEM PERFORMANCE IN THE PRESENCE OF ISI 99

5 10 15 20 25

10−4

10−3

10−2

10−1

100

Eb/N0[dB]

P(e

)

MMSE,BW=250MHz

MRC,BW=250MHz

MMSE,BW=500MHz

MRC,BW=500MHz

MMSE,BW=1GHz

MRC,BW=1GHz

Figure 5.6: Influence of bandwidth on the system performance in the presence of ISI

modulation indeed has a profound impact on the performance. The same four systemsas in Sec. 5.3.4 are evaluated. An overview of their properties can be found in Tab. 5.1.

In Fig. 5.7, the average BER performance of each system as a function of Eb/N0 isdepicted. Comparing the performance of both BPSK TR systems, one can see that theuse of a complex-valued AcR can improve the system performance significantly. In thecomplex-valued case, in fact, the sampling rate is increased by a factor 2, meaning thatthe MMSE weighting vector has twice as much degrees of freedom to suppress ISI. Usingthe linear MIMO system interpretation, one can say that the amount of MIMO inputs isunaltered, but the number of outputs is increased by a factor 2, such that more ISI canbe suppressed.

Comparing the three TR systems using a complex-valued AcR, the performance de-creases starting from BPSK via unscrambled QPSK to scrambled QPSK. In other words,the performance decreases whenever the modulation obtains more degree of freedom.More freedom degrees for the modulation namely results in more MIMO inputs, whichon its turn means that the ISI is spread over more dimensions in the space spanned bythe vector u[n]. As a result, the probability becomes larger if more ISI interferes with thedesired term, such that linear weighting can suppress less ISI. On the same channel, itcan be expected that more ISI can be suppressed when the modulation is more restrictivewithout needing to reduce the symbol rate.

This result also means that scrambling of the reference pulses, i.e. to avoid spectralpeaks in the PSD of the TX signal, may decrease the system performance depending onthe channel conditions. Furthermore, with increasing Nt, the number of MIMO inputsis super linear with respect to M , while the number of outputs is linearly related to


5 10 15 20 25

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10−3

10−2

10−1

100

Eb/N0[dB]

P(e

)

BPSK,RV−AcR

BPSK,CV−AcR

unscrambled QPSK,CV−AcR

scrambled QPSK,CV−AcR

Figure 5.7: Influence of modulation on the system performance in the presence of ISI

M . Hence, it can be expected that the system performance will quickly deterioratewith increasing channel memory, making linear weighting only suitable for channels withmoderate ISI.

5.5 Conclusions

An alternative AcR has been proposed for basic TR signaling, that allows for synchro-nization using DSP and is able to suppress more ISI if MMSE combining is deployed. Astatistical characterization of the system has been presented, which allows for the com-putation of the weighting vector. To analyze the performance, a method to computethe BER has been described, which is exact with respect to ISI, but assumes Gaussiandistributed noise at the demodulator output. Simulation results for several TR systemshave been compared.

In the absence of ISI, a TR system with a smaller bandwidth will outperform onewith a larger bandwidth, not taking SSF into account. MRC results in approximatelythe same performance as MMSE combining.

In the presence of ISI, large bandwidth TR systems are inherently less sensitive toISI. However, the performance loss can be partly compensated by increasing the FSR,illustrating that with proper linear filtering and fractional sampling, non-linear ISI canbe suppressed. Specifically, the 40 Mb/s, 250 MHz system with a FSR of 4 and MMSEcombining performs reasonably well, while a reduction of the FSR leads to a significantperformance decrease. Although only TR signaling is considered, the general conclusionsare expected to hold for delay-hopped differential signaling or even for energy-detector

5.5. CONCLUSIONS 101

based UWB systems, since both system types can be modelled using FIR Volterra systems.After a brief introduction to the signal model and linear weighting applied on the

I&D samples, a MIMO interpretation of the Volterra system has been presented. Theinterpretation not only explains how linear weighting can suppress non-linear ISI, but alsoexplains the impact of the modulation scheme on the performance of linear weighting withrespect to ISI suppression. It is shown that the number of MIMO inputs depends on themodulation scheme. Since the amount of suppressible ISI depends partly on the numberof MIMO inputs, the modulation scheme will have an impact on the system performance.Based on the model, it can be understood that e.g. scrambling of the RP can have a neg-ative effect on the BER performance of a TR-system under ISI conditions. Furthermore,the system performance deteriorates quickly with increasing channel memory, makinglinear weighting suitable for channels with moderate ISI, but not in case of severe ISI.

Chapter 6

Design of a High-Rate TR UWB

System

6.1 Introduction

In this chapter, the design of a high-rate TR UWB system is presented. The design aimis a TR-UWB PHY supporting a data rate of 100 Mb/s, while occupying a bandwidth of1 GHz. Based on the insight gained in the previous chapters, the design considerationsfor the system are described in Sec. 6.2. A detailed description of the considered systemis presented Sec. 6.3. The performance and complexity of the system is presented inSec. 6.4. Conclusions are drawn in Sec. 6.5.

6.2 Design Considerations for a High-Rate TR UWB

System

6.2.1 Trellis-Based Equalization

To support high data rates, the system inevitably will have to cope with non-linear ISI. Ithas been shown that non-linear ISI can be equalized using linear weighting in moderateISI conditions, if the FSR is sufficiently large, see Sec. 5.4.1. In severe ISI conditions,linear weighting will be performing rather poor, see Sec. 5.4.3. This applies especially ifscrambled QPSK-TR is considered to avoid spectral spikes, see Sec. 6.2.2. Additionally,the linear weighting sees the non-linear ISI terms as interference. However, the non-linearISI also contains information on the transmitted symbols. Taking all these considerationsinto account, linear weighting is not considered for a high data rate TR UWB system.

An alternative to linear equalization is inspired by interpreting a Volterra system asan FSM, see Sec. 4.5.5. As a result, trellis-based equalization can be used to equalize ISIif the FSM structure is known. Regretfully, the complexity of a trellis-based equalizergrows exponentially with the memory of the FSM. In Sec. 4.5.6, a RMDM has beenintroduced to reduces the memory of the FSM, while capturing the essential behaviour ofthe actual/full FSM. This allows to equalize Volterra channels at the expense of (some)performance.

103

104 CHAPTER 6. DESIGN OF A HIGH-RATE TR UWB SYSTEM

6.2.2 Power Spectral Density of TR Signals

To allow for operation under FCC regulation, a smooth PSD is beneficial [75, 25]. In abasic TR UWB signaling scheme, the first pulse of a TR symbol is transmitted unmodu-lated, while the second pulse is modulated. Assuming white, zero mean modulation, theTR UWB signal is thus the superposition of two independent signals, the first one consist-ing of a train of modulated pulses, while the second is a train of reference pulses. Basedon the statistical independence of both signals, the overall PSD will be the superpositionof the PSD of both signals.

The PSD of the train of modulated pulses is easily derived. Since white, zero-meanmodulation is applied on the modulated pulses, the shape of its PSD will be determinedby the squared magnitude of the Fourier transform of a single pulse [37].

As stated before, the reference signal will be a pulse train. It is well-known that thePSD of pulse train consists of a series of spectral spikes and thus anything but smooth [37].In [75], it is shown that time-hopping can be used to smooth the PSD of impulse radiosignals. The resulting PSD will still contain spectral spikes. However, the PSD willconsist of more spikes which are also shorter apart, making the PSD sort of smoother.Hence, time-hopping can also be applied to the TR symbols to smooth the PSD. However,the TR signal will no longer be cyclo-stationary with respect to the symbol period. Thisnot only complicates synchronization, but also results in a time-variant Volterra kernel,making kernel estimation and equalization more complicated. Therefore, time-hoppinghas not been considered.

Another method to smooth the PSD, which does not destroy the cyclo-stationarynature of the TR signal, is to apply non-information-bearing sign modulation on the TRsymbols. In other words, the TR symbols, including the reference pulses, are scrambled.In this fashion, a PSD is obtained of which the shape is determined by the squaredmagnitude of the Fourier transform of the individual pulses. The PSD thus no longercontains spikes, while conserving the cyclo-stationary nature of the TR UWB signal.However, this is only obtained if the modulation applied to the pulses is uncorrelated.This posses a first constraint on the scrambling.

The scrambling can be realized in two fashions. Firstly, the scrambling code can begenerated using a PN sequence or alternatively the scrambling code can be derived fromthe symbol identifiers a[n], see Sec. 4.5.5. When using a PN sequence, the trellis diagramof the FSM describing the Volterra kernel will be time-variant, which complicates itsequalization. To ensure a time-invariant trellis diagram, the modulation applied to bothpulses will be driven by the symbol identifiers, see Sec. 4.5.5. This poses the secondconstraint on the scrambling.

Both constraints on the scrambling code cannot be fulfilled simultaneously for allpossible modulation types. For scrambled QPSK-TR however, a solution has been found,which has been documented in Tab. 6.1. For completeness, the Gray-coding of channelbits on the symbol identifier is presented here as well, where c[n, k] denotes the k-thchannel bit signalled by the n-th TR-symbol.

In Appendix C, it is shown that the modulation of the pulses is indeed uncorre-lated/white. Hence, the PSD of the TR signal will depend only on the pulse shape. Thisproves that a smooth PSD can be obtained using symbol-identifier-driven scrambling incase of QPSK-TR signalling.

6.2. DESIGN CONSIDERATIONS FOR A HIGH-RATE TR UWB SYSTEM 105

Table 6.1: Symbol mapping tablec[n, 0] c[n, 1] a[n] b[n] b[n]

1 1 0 1 j1 -1 1 −1 −j-1 1 2 −1 −1-1 -1 3 1 1

6.2.3 Volterra System Identification

To perform trellis-based equalization, an equalizer needs to know the coefficients of theVolterra system modeling the channel for each fractional sampling position. Due tothe random nature of the radio channel, its coefficients must be estimated at the RXeither blindly or using training-sequences. In [76, 77], it is shown that the identificationof Volterra systems can be conducted blindly. However, the learning times for blindidentification are rather long and therefore not considered.

For training-sequence based identification of the Volterra kernels, two fundamentallydifferent strategies have been considered. The first approach uses the linearity of aVolterra system with respect to kernel elements, which makes the identification simi-lar to the identification of linear systems. Therefore, this technique is referred to as linearVolterra kernel estimation. Instead of aiming to estimate the whole Volterra kernel con-taining (2M + 2)2 elements, the vector notation presented in Sec. 4.5.2 is used to reducethe number of kernel elements to be estimated to Nk. The estimate of the kernel willbe specific for the used modulation type. In practice, the modulation type will not bechanged during transmission of a packet, making this drawback irrelevant. The same al-gorithms used for linear channel estimation, like the Least Mean Square (LMS) algorithmor Least Squares (LS) algorithm, can also be used for Volterra system identification [64].

The other approach regards the Volterra system as an FSM generating state-transition-specific time-invariant outputs. The approach will be referred to as trellis-based systemidentification. The amount of unknown elements to be estimated is equal to the amountof state-transitions, i.e. 2M+1 and 4M+1 for BPSK-TR and QPSK-TR, respectively. Thea-priori knowledge on the Volterra kernel structure is not exploited, which makes thisapproach robust against time-invariant imperfections in the RF front-end.

Roughly speaking, the performance of an estimation algorithm is proportional tothe number of elements to be estimated. In case of scrambled QPSK-TR, the linearVolterra kernel estimator has less unknowns to be estimated and is thus expected toperform better than trellis-based system identification. For every value of M , Nk isnamely smaller than 4M+1, especially for M greater than 2, see Sec. 4.1. Only if Mequals 1, trellis-based system identification is expected to perform slightly better. Basedon these considerations, the linear Volterra kernel estimation approach is favoured, alsobecause it is well-established in literature [64]. Nevertheless, the channel memory shouldbe kept low to allow for shorter training-sequences.


6.2.4 Multiband Transmitted Reference

One way to increase the data rate of a TR system is to increase the pulse rate. Un-avoidably, the channel memory will increase as well. Unfortunately, the complexity oftrellis-based equalization and Volterra system identification grows (approx) exponentiallywith the channel memory, see Sec. 6.2.1 and Sec. 6.2.3. In [78], it is proposed to divide thespectral resource into subbands, where in each subband energy detection based commu-nication is used. In each subband, the data rate will be relatively low, such that little tono ISI will occur, while the accumulated data rate can still be high. Evidently, a trade-offis possible between the amount of ISI and the number of subbands. This concept is notlimited to energy detection, but can be applied to TR signaling as well.

The multiband principle has more distinct advantages. Due to the memory reduction,Nk is reduced for each subband, meaning that less kernel elements need to be estimatedfor each subband. Since the number of kernel-elements grows super linearly with thememory size, the total number of kernel elements to be identified decreases with theintroduction of subbands. Furthermore, it inherently creates parallel structures, such thatthe rate at which the algorithms are operated is reduced. Also the implementation of theTR delay used in each subband is simplified, because its transfer function must only bewell-behaving over a smaller portion of bandwidth. Additionally, the architecture allowsfor the detection and coherent suppression of narrowband interference. An importantadvantage since TR systems are inherently sensitive to interference in general, due totheir non-coherent, non-linear transfer function. Furthermore, the system can easily beextended to support DAA, which may be demanded by the regulation bodies of Europeand Japan to operate UWB [14, 15].

A drawback of multiband is that the TR signal of each subband is more susceptible toSSF, see Chapters 3 and 5. Similar to OFDM, an FEC scheme will be deployed to exploitthe frequency diversity provided by the system bandwidth, where the system bandwidthis defined as the sum of the bandwidths of each subband.

6.2.5 The Role of Forward Error Control

To exploit the frequency diversity provided by the system bandwidth, FEC will be used.However, this will not increase the system complexity significantly. Nowadays, every com-munication system deploys FEC to improve its energy efficiency, such that the coveragearea and/or data rate can be increased. By nature, an FEC scheme is divided over theTX and RX. At the TX, an FEC encoder adds redundant information to the data to betransmitted. The redundant information allows the RX to detect and correct (to someextend) errors introduced by the channel. To what extend depends on the amount andmanner the redundant information is introduced by the FEC encoder. The bits encodedat the TX will be referred to as information bits, while the bits generated by the FECcoder are called channel bits. The channel bits will be allocated to the different subbandsusing a de-multiplexer, which assigns every k-th bit of Nsb subsequent channel bits to sub-band k. To exploit the full potential of the FEC, an interleaver Πc is positioned betweenthe FEC encoder and de-multiplexer to ensure the channel bits are spread randomly overthe subbands.

6.2. DESIGN CONSIDERATIONS FOR A HIGH-RATE TR UWB SYSTEM 107

Demuxinfo bits

channel bits

S0

S1

S0

S1

FSM1/FEC

FSM3/subband 2

S0

S1

FSM2/subband 1

Πc

Figure 6.1: System model of a multiband TR UWB system with two subbands

6.2.6 Principle of Turbo Equalization

Both an FEC encoder and a Volterra system can be modelled using an FSM. Hence,the communication chain starting from the FEC encoder up to the samples generatedat the AcR output at each subband can be modelled as a series of serial and parallelconcatenated FSMs. A graphical representation of the concatenated FSMs for a TRUWB system with two subbands has been depicted in Fig. 6.1.

Optimal Maximum-Likelihood Sequence Detection (MLSD) would be desirable, butthe related algorithm is too complex for implementation. In 1993, the turbo principle wasfirst introduced by Berrou et.al. [79] for parallel concatenated FSMs, where both FSMswere convolutional encoders. Shortly after, the turbo principle was introduced to theequalization of linear ISI channels [80], using the fact that an FEC encoder followed byan ISI radio channel can be seen as a serial concatenation of two FSMs. In [81], it is shownthat iterative decoding using the turbo principle can be seen as a practical implementationof an MLSD. Due to their good performance, both turbo coding and turbo equalizationfor linear ISI channels have been extensively researched in the last decade [82, 83, 84, 85].The application to the equalization of non-linear channels is not as well established, buta few papers have been published on the topic [86, 87]. Nevertheless, the non-linearity ofthe channel has no principle impact on the turbo equalization scheme.

In any turbo scheme, the decoders of each FSM exchange soft decisions on the chan-nel bits, where an equalizer is also considered to be a decoder. Each decoder uses thesoft decisions of the other decoder, the information gathered from the channel and thestructure of the FSM under decoding, to update its soft decisions. These soft decisionsare iterated around to converge to a solution, which is hopefully the correct one.

The iterations of soft decision may also improve the kernel estimates. Due to the finitetraining-sequence length, the Volterra system identification algorithm will provide aninexact description of the Volterra channel FSM, reducing the performance of the system.The soft decisions, e.g. provided by the FEC decoder, can be used in the subsequentiteration to update/improve the estimate of the Volterra channel FSM. Possibly, thisallows for good perform while using shorter training-sequences. In [88], this approach hasbeen proposed for linear channels. Without any fundamental differences, this approach


5.4 5.6 5.8 6 6.2 6.4 6.6−30

−20

−10

0

Eb/N0[dB]

PSD

1−Band

2−Band

4−Band

Figure 6.2: Division of the spectral resources into subbands

can be extended to Volterra channels.

6.3 System Description

In Sec. 6.2, the reasoning behind the design choices have been presented. In this section,a detailed description will be presented of the overall system, including its parameters.

6.3.1 Description of the TX Architecture and RX RF Front-End

The general structure of the transmitter system will be described in this section. Let usassume a block of information bits of length 4098, which will be denoted by b, whereb[n] denotes the n-th information bit. The block b is encoded to a block of channel bitsc, using a terminated rate-1

2FEC coder. The block c will be of length 8200. A more

detailed description of the FEC scheme can be found in Sec. 6.3.2.A channel interleaver Πc is placed between the FEC and de-multiplexer, such that the

channel bits are sent in a pseudo-random order over time and over the subbands. In thisthesis, a random interleaver is deployed. The interleaved channel bits are de-multiplexed,to obtain Nsb sub-blocks with channel bits of length 8200/Nsb, one for each subband. Theblock of channel bits communicated of the i-th subband will be denoted with ci.

Three multi-band systems are considered in this thesis, using respectively 1, 2 and4 subband(s). The system bandwidth is always equal to 1 GHz. Based on the resultsof chapters 3 and 5, 1 GHz of bandwidth will suffice to allow for communication robustagainst SSF. The division of the spectral resources into subbands has been depicted inFig. 6.2.

In each subband, QPSK-TR signaling is deployed, such that 2 channel bits are mappedonto a single TR symbol. For notational convenience, the ci[n, k] will denote the k-thchannel bit signaled using the n-th TR-symbol a[n]. The corresponding scrambling b[n]and modulation b[n] can be found in Tab. 6.1.

The structure of the TR signal is as described in Sec. 4.2.1. The TR-symbol durationin each subband is equal to 10, 20 and 40 nanoseconds and the delay D is equal to 4, 8and 16 nanoseconds for a multiband system with 1, 2 and 4 subband(s), respectively. Inany case, the overall symbol rate will be equal to 100 MHz, i.e. the channel-bit rate is

6.3. SYSTEM DESCRIPTION 109

Demux

info-

ΠcEncoder

TR Mod.

BPFs

+

CV-AcR

CV-AcRTR Mod.

Radio

Modulators BPFs AcRs

FEC InterleaverChannel

I&D-

Channel

Samples

bits

Figure 6.3: Model of the communication chain up to the I&D samples

equal to 200 MHz. Neglecting the termination bits, the information-bit rate will be equalto 100 MHz.

The received signal of each subband is demodulated using a CV AcR with matchingdelay and a fractional sampling rate of two. The sampling phase is not synchronized tothe received signal, i.e. the Analogue to Digital Converter (ADC) are running freely. Theclock-rate of the ADCs and the subsequent Digital Signal Processing (DSP) will be equalto 200, 100 and 50 MHz for a multiband system with 1, 2 and 4 subband(s), respectively,which illustrates that the multiband concept also relieves the demands on the hardware.

A block diagram of the described system from the information bits up to the I&Dsamples has been depicted in Fig. 6.3 for a multiband system with two subbands.

6.3.2 Forward Error Control

In [89], the performance of several FEC schemes has been compared in a turbo equal-ization scheme for linear channels. The best results were obtained using a turbo FECscheme based on recursive systematic convolutional codes. The same FEC scheme will bedeployed in this thesis, in the hope it will also provide good performance on second-orderFIR Volterra channels.

The turbo coder consists of two identical, parallel concatenated, rate-12, Recursive Sys-

tematic Convolutional Codes (RSCCs). Each coder is defined by the polynomials (5, 7),resulting in a memory-two FSM. The first RSCC encoder receives the information bitsdirectly, while the second RSCC encoder encodes information bits that have been passesthrough the interleaver Π. Both coders are terminated to the all-zero state. To obtaina rate 1

2, the systematic output of the second RSCC encoder is punctured continuously.

One out of two non-systematic channel bits of either RSCC encoder is punctured in analternating manner.

A block diagram of the turbo encoder can be found in Fig. 6.4, where the systematicoutput of the second RSCC encoder is not depicted, because it is punctured continuously.All additions are modulo-2. The block diagram shows that a turbo encoder can be seenas two parallel concatenated FSMs.


Π

+

+ +

+

Z−1 Z−1

+

+ +

+

Z−1 Z−1

1 11 00 1

Info bits

Channel

Punct.

Table

bits

Figure 6.4: Block diagram of the turbo encoder

Table 6.2: Relations between probability domain and LLV domain [90]Probability Domain ⇔ LLV domain comments

1 − P(c = 1) ⇔ −L(c)E [c = 1] ⇔ tanh(L(c)/2)

c ⇔ sign (LLV(c))ln(P(c = x)) ⇔ x L(c) − ln(1 + exp(x L(c))) ∀ x ∈ +1,−1

ln(P(c = x)) − ln(P(c = −x)) ⇔ x L(c) ∀ x ∈ +1,−1

6.3.3 Turbo Equalization

As stated before, the decoders of each FSM exchange soft decisions related to the proba-bilities of the channel bits. In this context, an equalizer is also considered to be a decoder.The probabilities themselves can be used as soft decisions, but their logarithmic coun-terparts called Log-Likelihood Values (LLVs) are preferable, because the product of twoprobabilities will become a sum in the logarithmic domain, making the implementationless complex. Furthermore, LLVs are inherently better suited for the representation ofsmall probabilities in finite bit-width.

By definition, the LLV of a bit c ∈ 1,−1 with a probability P(c = 1) and its inverseare defined as

L(c) , ln

(P(c = 1)

P(c = −1)

)

(6.1)

P(c = +1) =

(exp(L(c))

1 + exp(L(c))

)

(6.2)

For completeness, some important relations between the probability domain and LLVdomain are collected in Tab. 6.2.

Each FSM is processed using an algorithm that accepts and generates LLVs, which arereferred to as a-priori and a-posteriori LLVs, respectively. The decoder itself is referred to


L(c)

u

L(bc|u,L(c),F)

L(bi|u,L(c),F)

SISO decoder

F

Figure 6.5: Inputs and outputs of a SISO decoder

as Soft-Input, Soft-Output (SISO) decoder. The input-output diagram of a SISO decoderis depicted in Fig. 6.5.

In a general-purpose SISO decoder, a probabilistic computation is conducted to obtainthe a-posteriori LLVs for both the information bits and channel bits, taking into accountthe I&D samples, the a-priori LLVs and the structure of the FSM under evaluation. Anexample of a trellis diagram is presented in Fig. 6.6, showing the possible state-transition,related input bits and outputs of a Volterra channel. The object describing the FSMstructure will be denoted by F . The information contained in F is

• Trellis structure,

• Input(s) related to each state transition,(denoted as a[n] in Fig. 6.6)

• Output(s) related to each state transition,(denoted as s[n, 1] as s[n, 2] in Fig. 6.6)

• Output specific noise variance,

where the trellis structure includes the number of states, possible state-transitions, num-ber of state transitions and if terminated also the initial state and termination state.

A more detailed description of the internal operation of a SISO decoder will be pre-sented in Sec. 6.3.4.

In our case, the system consists out of Nsb + 1 trellis objects, one for each subbandand the FEC decoder. The trellis object related to the i-th subband will be denoted withKi and the trellis object describing the FEC will be denoted with F .

In practice, not all inputs and outputs of every SISO decoder will be used in turboequalization scheme. For instance, the soft decoder of the sub-channels, i.e. the softequalizers, are unable to compute the LLVs of the information bits, simply because therequired information is not contained in the trellis object describing the channel H. TheFEC soft decoder uses only the a-priori information provided by the soft channel decoders.

When exchanging LLVs, the output LLVs of the first decoder are not directly used asa-priori information by the second decoder. The output LLVs are namely derived usinga-priori information delivered by decoder 2 to decoder 1 in the first place. To ensurethe convergence of the turbo scheme to what is hopefully the MLSD, instead of a self-convincing system, only the information that is foreign to decoder 2 is passed on. Thisinformation is referred to as extrinsic information, where extrinsic means foreign. The


112

102

012

002

n n + 1

0

1

1

1

1 1

1

1

1

0

0

0

0

0

0

0

(0.9, 0.3)

(1.3.1.1)

(0.1, 1.1)

(1.8, 0.9)

(0.8, 0.3)

(0.3, 1.4)

(0.6, 0.5)

(0.2, 0.7)

(0.9, 0.3)

(1.3.1.1)

(0.1, 1.1)

(1.8, 0.9)

(0.8, 0.3)

(0.3, 1.4)

(0.6, 0.5)

(0.2, 0.7)

St[n] St[n + 1] St[n + 2]

if PN-seq.→ Time variant

s[n, 1] s[n, 2]a[n]

112

102

012

002

112

102

012

002

Figure 6.6: Trellis diagram of a FIR Volterra model in the absence of PN scrambling

extrinsic LLVs are defined as,

Le(c) = L(c|u, L(c),H) − L(c). (6.3)

The exchange of extrinsic information in a turbo equalization scheme has been de-picted in Fig. 6.7 for the multiband TR UWB system depicted in Fig. 6.3.

To ensure the extrinsic is presented in the proper order to the soft decoders, the turboscheme conducts the inverse operation of the TX, i.e. the extrinsic information comingfrom the channel decoders to the FEC decoder are multiplexed and de-interleaved by Π−1

c

and vice-versa in the opposite direction.

6.3.4 SISO Decoder Structure

In this section, the trellis-based soft decoding algorithm is presented, assuming the RXhas full knowledge on the trellis object F . To provide for turbo equalization, the equalizerhas to accept LLVs and generate LLVs. Two different classes of algorithms are possible,namely MLSD and symbol-by-symbol Maximum A-Posteriori (MAP) decoding. TheMLSD detection is often performed using the well-known Viterbi algorithm introducedin [91]. A detailed analysis of its operation has been described by Forney in [92]. TheViterbi algorithm itself does not generate soft decisions. In 1989, a Soft-Output ViterbiAlgorithm (SOVA) was introduced by Hagenauer [93].


u1

SISO decoder

K1

u2

SISO decoder

Mux Π−1c SISO decoder

F

Demux Πc

++

−

++

−

++

−

K2

Figure 6.7: Structure of the turbo equalizer

Symbol-by-symbol MAP decoding can be conducted using an Bahl, Cocke, Jelinekand Raviv (BCJR) algorithm or one of its related algorithms. In the original paper,the algorithm was described in terms of probabilities instead of LLVs. To simplify itsimplementation, the BCJR algorithm was translated to the logarithmic domain. In thisdomain, two algorithms were derived, namely a sub-optimal Max-Log-MAP algorithmand an optimal Log-Map algorithm [69, 94].

Although already introduced in 1969, the BCJR algorithm or one of its derivativeswere rarely used. The main reason is that minimizing the sequence error probability wasof more importance for most applications. Furthermore, a Max-Log-MAP or Log-MAPis approx. twice as complex as a Viterbi algorithm [69, 94]. The situation changed withthe introduction of the turbo principle. The convergence of a turbo scheme relies on theexchange of accurate soft decisions between the decoders of the concatenated FSMs. Sincea SOVA is by nature an MLSD algorithm, it is inherently unable to generate accurateLLVs for the individual bits, where a BCJR algorithm is able to compute them.

A Log-MAP computes the a-posteriori LLV for each bit. In case the soft decoder isdecoding a subband channel object Ki, only for the channel bits. In case F is underdecoding, the Log-MAP decoder generates both LLVs of the information bits and thechannel bits. The principle for the generation of the LLVs for either bit type is the same.To allow for a general-purpose description, a general bit is used to identify either a channelbit or information bit, which will be denoted by denoted by q[n]. A Log-MAP algorithmnow computes the LLV of q[n], which is defined as

L(q[n]|u, L(c),H) = ln

(P(q[n] = 1|u, L(c),H)

P(q[n] = −1|u, L(c),H)

)

(6.4)

The object H describes the possible state-transition and the related value for q[n]. Theset of possible state-transitions will be denoted by Stt. This set can be divided intotwo disjoint subsets, where S

+1tt and S

−1tt denote the set of possible state-transition given


q[n] = 1 and q[n] = −1, respectively. A close-to-implementation description of a Log-MAP can be found in Appendix D.

6.3.5 Stop Algorithm

The number of iterations required in a turbo scheme depends on the channel conditions.To limit the power consumption and the load on the hardware, only additional iterationsshould be conducted if there is a good probability that they provide additional informa-tion. Roughly speaking, two situations can be distinguished, where additional iterationsdon’t make sense, namely if the channel is very good or very bad.

If the channel conditions are good, the correct bits are already retrieved after the firstiteration. Evidently, making additional iterations does not make sense. The detectionof this event is not so complicated, assuming the use of a CRC code. If the CRC checkpasses, no additional iterations should be conducted.

If the channel conditions are rather poor, the information retrieved from the channelis corrupted so badly by noise that no convergence will be observed. In other words, theprobability distribution of the channel bits will hardly change from iteration to iteration.The alikeness of two probability distributions can be computed using the so-called cross-entropy [95]. In [90], an approximation for the cross-entropy is proposed as stop criteriafor turbo coding. Later, it was also proposed for turbo equalization in [96, 84]. Denotingthe LLVs and extrinsic information of the channel bits at iteration i as L(i)(c) and L(i)

e (c),respectively, the approximate cross-entropy T (i) is defined as follows

T (i) =∑

n,k

∣∣∣L(i)

e (c[n, k])∣∣∣

2

exp(∣∣∣L(i)(c[n, k])

∣∣∣)

(6.5)

As proposed in [90], no further iterations will be conducted if the T (i) < 10−3T (1). Thisstop-criterion also functions on good channels. However, it requires inherently one moreiteration compared to the CRC-based stop criterion. Both stop criteria will be usedsimultaneously. In addition, no further iterations are conducted if a certain number ofiterations is reached, which will be denoted by imax. If either of these three stop criteriais fulfilled, no further iterations will be conducted. These criteria are used in both theturbo equalization loop and the FEC turbo decoder loop.

6.3.6 Measure of Complexity

Not only is the performance of relevance, but also the required complexity. As measurefor the DSP complexity, the average number of state-transitions summed over all SISOdecoders per information bit is chosen, which will be denoted as NStt

/bi. Due to variablenumber of iterations in the turbo-decoder and equalizer, the complexity will also be afunction of Eb,i/N0. The base-2 logarithm of the complexity will be discussed, because ofthe large complexity differences between the compared schemes.

In the multiband case, it seems that additional DSP complexity is required, since Nsb

equalizers/SISO decoders are operated in parallel. However, each equalizer is processingonly approx. 2/Nsb channel bits per information bit, meaning that they can be clocked

6.4. PERFORMANCE ANALYSIS 115

at a Nsb lower rate compared to a single-band equalizer. Hence, using Nsb equalizers inparallel does not increase the complexity of the DSP in terms of state-transitions perinformation bit.

Operating Nsb equalizers in parallel will be at the expense of Nsb more surface area one.g. an ASIC or FPGA. By using proper signal multiplexing, de-multiplexing and statestoring, a single-equalizer implementation can be obtained, but it has to be operated atapprox. the same rate as an equalizer in a single-band TR system.

6.4 Performance Analysis

6.4.1 Impact of Equalizer Complexity Without Turbo Equaliza-

tion

To reduce their complexity, the SISO decoders operating on the subbands are not pro-vided with a full description of the Volterra channels, but with the related RMDMs, seeSec. 4.5.6. To obtain insight in the trade-off between performance and complexity, thememory of the RMDMs will be varied from one to three, i.e. the number of states, a mea-sure for the complexity, of the equalizer is varied from 4 to 64. The system performancesare evaluated using a pool of NLOS channel realizations described in Sec. 3.3.1.

Before presenting the BER performance, the ability of an RMDM to mimic its FDMis quantified using (4.63) for a single but representative channel realization. The outputSNR of the RMDM, is depicted as a function of the RMDM’s memory N in Fig. 6.8, for aTR UWB system with 1, 2 and 4 subbands, respectively. As reference, the output SNR ofthe FDM, which is an upper-bound for the SNR of a RMDM. The difference between bothSNR-values can not be translated into the Eb/N0-loss in the BER-performance curves.However, it does give an insight in the trade-off between performance and complexity.Please note that the Eb/N0-values are with respect to the channel bits and thereforedenoted as Eb,c/N0.

Fig. 6.8 shows that the RMDM requires less memory to adequately model the FDMwith an increasing number of subbands. In case of a system with four subbands, a 16-stateRMDM (memory N = 2) is able to approximate the FDM for all channel realizations.Only at Eb/N0 > 20 dB, a difference can be observed, which is well above the Eb/N0

working point. In case of two subbands, a 64 states RMDM (N = 3) is required toadequately mimic the FDM for most channel realizations, while in the single band case,256 states (N = 4) are by far not sufficient to mimic the FDM.

To validate the conclusions derived from Fig. 6.8, the channel BER performance hasbeen depicted in Fig. 6.9 as a function of Eb/N0 for the three system architectures. Toobtain insight on its effect on the performance, the equalizer complexity has been variedfrom 4 states to 64 states (N = 1, 2, 3). In case of four subbands, Fig. 6.9 confirmsthat a 16-state equalizer is indeed sufficient to obtain good performance, since almostno further improvement is observed in the channel BER when increasing the equalizercomplexity. In the two-band case, a similar result applies; 64 states are needed to obtaingood performance. In the single-band case, 64 states for the equalizer seems not yetsufficient to extract the complete information available in the received signal; additionalgain seems possible.


6 8 10 12 14 16 180

2

4

6

8

10

12

14

Eb,c/N0[dB]

SN

R[d

B]

N=1,4−Band

N=2,4−Band

N=4,4−Band

FDM,4−Band

N=1,2−Band

N=2,2−Band

N=4,2−Band

FDM,2−Band

N=1,1−Band

N=2,1−Band

N=4,1−Band

FDM,1−Band

Figure 6.8: The average ”overall SNR” of the RMDM as function of its memory

6 8 10 12 14 16 18 2010

−3

10−2

10−1

Eb,c/N0[dB]

P(e

)

N=1,4−Band

N=2,4−Band

N=3,4−Band

N=1,2−Band

N=2,2−Band

N=3,2−Band

N=1,1−Band

N=2,1−Band

N=3,1−Band

Figure 6.9: The average channel BER after the Log-Map equalizer of three multibandTR UWB systems

6.4. PERFORMANCE ANALYSIS 117

Comparing the systems with different number of subbands, the two-band system hasthe best performance with respect to the channel BER. However, the channel BER ofthe different system architectures can not be compared directly. The four-band systemperforms considerably worse than the two-band system at high Eb,c/N0-values, becausethe signal in each subband experiences considerably more fading. It is the task of the FECto exploit the frequency diversity provided by the system bandwidth. To see whether theFEC is able to accomplish this task, the information BER has been depicted in Fig. 6.10.Here, the Eb/N0-values are with respect to the information bits, which will be denotedas Eb,i/N0.

Fig. 6.10 shows that the four-band system performs slightly better than the two-bandsystem—in terms of information BER—, even though its channel BER is considerableworse. Hence, the FEC is indeed able to exploit the full frequency diversity. Further-more, it reveals that an Eb/N0 of approx. 13 dB is needed to obtain virtually error-freecommunication, using the sub-optimal AcR, in the absence of turbo equalization.

Taking only the equalizer complexity into account, Fig. 6.10 shows that the four-bandsystem with N = 2 performs virtually equally well as the two-band system with N = 3.The same performance is obtained using an equalizer that is 4-times less complex withrespect to the former system. Taking into account the FEC, the difference will be less.

In Fig. 6.11, the DSP complexity has been depicted as function of Eb,i/N0. Due tothe large difference in complexity between the system the base-2 logarithm of the numberof state-transitions per information bit has been depicted. First of all, it can be notedthat both on high and low SNR channels, approximately the same amount of complexityis required by the DSP, illustrating the proper functioning of the stop-criterion. This isespecially apparent for low N . Furthermore, when increasing the memory of the RMDM,the equalizers makes the scheme so complex that the FEC-decoder complexity becomesnegligible.

Between both SNR extremes, more turbo iterations are conducted to converge tothe correct solution. Furthermore, it can be noted that the Eb,i/N0 range over whichmore iterations are demanded by the turbo decoder is larger, when employing less sub-bands. Likely, the residual ISI is interfering with the cross-entropy stop criteria, due toits non-Gaussian nature. This suspicion is strengthened by the fact that the single-bandTR system requires more turbo decoder iterations on high-SNR channels, if the turboequalizer memory is low (N = 1). In this case, the residual ISI is namely more dominant.

Comparing the systems at an Eb,i/N0 of 13 dB, the value at which both the two-bandand four-band TR systems accomplish virtually error-free communication, the four-bandsystem requires 2.5 times less complexity with respect to the two-band system.

6.4.2 Benefit of Turbo Equalization

In this section, the performance and complexity are presented of the turbo equalizationscheme. In Fig. 6.12, the information BER has been depicted as a function of Eb,i/N0 forvarious systems with different equalizer complexity, number of subbands and maximumnumber of turbo equalization iterations, assuming perfect kernel side-information.

As one may expect, the performance improves if the maximum number of turbo equal-ization iterations is increased. However, the improvement is rather moderate, with re-


6 7 8 9 10 11 12 13 14 15 16

10−4

10−3

10−2

10−1

Eb,i/N0[dB]

P(e

)

N=1,4−Band

N=2,4−Band

N=3,4−Band

N=1,2−Band

N=2,2−Band

N=3,2−Band

N=1,1−Band

N=2,1−Band

N=3,1−Band

Figure 6.10: The average information BER of the three different systems

6 8 10 12 14 16 18 204

5

6

7

8

9

10

11

Eb,i/N0[dB]

log 2

(NS

tt/b i

)

N=1,4−Band

N=2,4−Band

N=3,4−Band

N=1,2−Band

N=2,2−Band

N=3,2−Band

N=1,1−Band

N=2,1−Band

N=3,1−Band

Figure 6.11: Number of state-transitions as function of Eb,i/N0 of the three differentsystems


spect to the additional complexity invested. The biggest improvement is obtained usingthe single-band system, where a performance improvement of approximately 2 dB canbe observed. Nevertheless, its performance is still considerably worse than those of themultiband systems, even though it is using a 64-state equalizers (N = 3).

The best performance is obtained with a four-band TR system using 4 parallel 16-state equalizers (N = 2) with a maximum of 4 turbo equalization iterations. The Eb/N0

value at which virtually error-free communication is obtained is 12 dB. In the absence ofturbo equalization, error-free communication was obtained at an Eb/N0 value of 13 dB.Turbo equalization then leads to a performance improvement of 1 dB for a four-band TRsystem.

Only a slightly worse performance is obtained using a two-band system using two par-allel 64-state equalizers (N = 3). The improvement obtained by using turbo equalizationis slightly higher for the two-band system under consideration, but using DSP with ahigher complexity.

As to be expected, in every case the performance decreases when the equalizer com-plexity is reduced. However, the performance penalty is rather small. In case of thefour-band system the performance penalty is a mere 0.25 dB, where for the two-bandsystem the penalty is 0.5 dB.

In Fig. 6.13, the complexity of the different schemes is depicted. For any TR-systemthe same behaviour can be observed with respect to the required complexity. Taking theEb,i/N0 value for almost error free communication as reference point, e.g. 12.5 dB for afour-system with N = 1, only slightly more additional complexity is required to improvethe performance. Most blocks of data are error-free after the first iteration, such that theCRC-check passes and no further iterations are conducted. Applying turbo equalizationto those few packets containing errors will in most cases lead to an error-free decodingafter a few iterations. Using an intelligent scheduler to dynamically assign RX hardwareresources to promising packets, potentially improves the performance without the needfor much additional hardware.

6.5 Conclusions

In this chapter, the design of a high-rate TR UWB system has been described, able tosupport a data rate of 100 Mb/s, while occupying a 1 GHz bandwidth. A combinationof trellis-based equalization and the multiband principle has been proposed to allow forhigh data rate UWB communication over multipath radio channels, using non-coherentreceivers. To exploit the frequency diversity provided by the 1 GHz system bandwidth,it is proposed to use FEC for multiband systems. Furthermore, turbo equalization hasbeen considered to improve the performance further.

The performance results reveal that a multiband system performs considerably betterwith respect to a single-band systems at these data rates, using less complex equalizerstructures. It is shown that FEC in combination with a multiband receiver structureprinciple is able to exploit the frequency diversity provided by the system bandwidth.Furthermore, turbo equalization is able to improve the system performance by approxi-mately 1 dB, assuming perfect kernel side information. The improvement is expected tobe larger in the absence of perfect kernel side information, assuming the estimated kernel


8 9 10 11 12 13 14 15 16

10−4

10−3

10−2

10−1

Eb,i/N0[dB]

P(e

) N=1,4−Band,It=1

N=1,4−Band,It=2

N=1,4−Band,It=4

N=2,4−Band,It=1

N=2,4−Band,It=2

N=2,4−Band,It=4

N=2,2−Band,It=1

N=2,2−Band,It=2

N=2,2−Band,It=4

N=3,2−Band,It=1

N=3,2−Band,It=2

N=3,2−Band,It=4

N=3,1−Band,It=1

N=3,1−Band,It=2

N=3,1−Band,It=4

Figure 6.12: The average information BER of three different systems in a turbo equal-ization scheme

6 8 10 12 14 16 18 204

5

6

7

8

9

10

11

Eb,i/N0[dB]

log 2

(NS

tt/b i

)

N=1,4−Band,It=1

N=1,4−Band,It=2

N=1,4−Band,It=4

N=2,4−Band,It=1

N=2,4−Band,It=2

N=2,4−Band,It=4

N=2,2−Band,It=1

N=2,2−Band,It=2

N=2,2−Band,It=4

N=3,2−Band,It=1

N=3,2−Band,It=2

N=3,2−Band,It=4

N=3,1−Band,It=1

N=3,1−Band,It=2

N=3,1−Band,It=4

Figure 6.13: DSP complexity as function of Eb,i/N0 of three different systems in a turboequalization scheme


is updated after each iteration.Taking both complexity and performance into account, a four-band system with a 16-

state (N = 2) equalization is recommended. Not only does it deliver good performance,it also has the potential to equalize channels with larger delay spreads. Furthermore, itis well-suited for a parallel implementation in the digital domain and inherently robustagainst narrowband interference, possibly even extendable to DAA.

The use of turbo equalization is also recommended. The complexity analysis indicatethat an additional 1 dB performance improvement can be obtained, using only slightlymore DSP complexity. Further research is however required to find better stop-criteriato manage the scheduling of packets for additional iterations. Another benefit of turboequalization is that it allows for an improvement of the kernel estimation with eachiteration, which eventually may allow the system to operate well, while using shortertraining-sequences. An interesting option seems to be to use 4-state (N = 1) equaliz-ers during the first turbo iteration. The smaller kernel namely allows for even shortertraining-sequences. During the second iteration, the kernel with N = 2 can be estimatedusing the information gathered during the first iteration.

Chapter 7

Conclusions and Recommendations

Besides the general introduction, the PhD thesis report is structured in three block.The first one consists of Chapter 2 and Chapter 3 and focuses on the diversity of UWBchannels. The second block deals with TR signaling, which is being described in Chapter4 and Chapter 5. Finally, Chapter 6 describes the design of a high-rate TR system.Respecting the structure of the PhD thesis, the conclusions and recommendations havebeen divided in three blocks as well.

Theory and Practise of Fading UWB Channels

In this section, the conclusions and recommendation are presented for Chapter 2 andChapter 3. It is well-known that UWB systems are inherently robust against SSF, due totheir large bandwidth. On the other hand, the implementation of radio systems becomesmore complex when increasing the bandwidth. To accommodate a trade-off betweenboth aspects, a measure is introduced in Chapter 2 to quantify the frequency diversitylevel of radio channels. By assuming uncorrelated scattering, a theoretical model hasbeen developed explaining the relationship between frequency diversity and bandwidth,by decomposing the UWB channel into its principle components. Both for LOS channelsand NLOS channels, the diversity level has been found to scale linearly with the RMS-delay-spread-by-bandwidth product. For NLOS channels specifically, the diversity levelwas found to be twice the RMS-delay-spread-by-bandwidth product. To our knowledge,such mathematical tool for such analysis was not available.

As with any novel model, its ability to model the real world should be validated. Oneof the novelties is that the model decomposes the channel in its PCs to finally predictthe fading statistics of the UWB channel. As a result, no literature was available to usea reference for validations. Therefore, we have validated the model ourselves.

In Chapter 3, the fading model has been verified using measurement data of UWBradio channels with the aim to reveal both the strengths and shortcomings of the model.The linear relationship has been confirmed, but the slope was slightly higher for measuredNLOS channels. Although the PCs of UWB channels are by definition uncorrelated,they are not necessarily independent, which explains the difference between theory andpractice. It is expected that for UWB channels of richer multipath environments, thedifference between both diminishes.

123

124 CHAPTER 7. CONCLUSIONS AND RECOMMENDATIONS

For LOS UWB channels, the difference between theory and practice was significantlylarger. The theoretical model predicts that the LOS component has the same eigenfunc-tion as the largest NLOS component, i.e. they are contained in the same PC. In themodel, the PC will be a Ricean distributed RV with the largest possible variance. In thisrespect, the theoretical model predicts a worst-case scenario. In practice, the dimensionspanned by the LOS component was found to contain considerably less NLOS energy,leading to a Ricean distributed RV with a considerably smaller variance. As a result, theoverall diversity level of LOS radio channels was considerably higher than predicted bythe theoretical model. Unfortunately, the mechanism explaining the behaviour could notbe unveiled and more effort is needed to understand the UWB channel in detail.

The validation results revealed that the model is an oversimplified, but insightful,model of reality. We believe however that the model can be refined to model realitymore accurately, without changing its basic structure. Currently, it is still an hypothesisthat the model under-estimates the diversity level of UWB NLOS channels due to thestatistical dependence between the PCs. For further research, it is therefore recommendedto validate the ability of the model to predict the fading of NLOS UWB channels of richermultipath environments, such that the channel becomes more ”random” and the PCs areindeed less dependent on each other.

Secondly, it may be useful to investigate the reason for the eigenfunction of the LOScomponent to span another subspace than predicted by the theoretical model. The causeis potentially the distortion of the radiated pulse-shape by the TX and RX antenna.Answering this question may provide insight on the shortcomings of the theoretical modelfor LOS UWB channels and allow for an refinement, possibly using another optimizationcriterion for the PC decomposition.

Theory and Analysis of TR Signaling

This section contains the conclusions and recommendation for Chapter 4 and Chapter 5.Chapter 4 starts with a brief introduction of TR signaling including it strengths and

weaknesses with respect to performance and implementation. Afterwards, several novelextensions to the TR principle have been proposed, to relieve some of these shortcomings.Firstly, a fractional-sampling AcR structure has been proposed to relax synchronizationand to allow for weighted autocorrelation, while simplifying the implementation. Theconcept of fractional sampling has been proposed by other, but never with the aim tosuppress more ISI. Secondly, a complex-valued AcR has been proposed to make the systemless sensitive against delay mismatches and to allow for complex-valued modulation ofTR symbols. The usage of multiple AcR branches to overcome delay mismatches hasbeen proposed by other, but the resulting receiver has not before been interpreted as acomplex-valued AcR.

To understand the system’s behaviour, a general-purpose discrete-time equivalent sys-tem model has been developed, taking all extensions into account. It was shown that theI&D samples generated by a fractional sampling AcR in a TR system consist of twoterms with different nature, a signal term and a noise term. The signal term could bemodelled using a SIMO FIR Volterra model. The noise term was shown to consist of twotypes of noise, a Gaussian sub-term with a signal dependent variance and a non-Gaussian

125

sub-term. The discrete-time equivalent system model is one of the first models for TRsignaling, taking the non-linear ISI into account.

Several interpretations for the SIMO FIR Volterra model have been presented, whichallow for more insight in the behaviour of TR systems. Firstly, the Volterra modelhas been written in a vector notation and an extended vector notation, which allowsfor simplified statistical analysis. The extended vector notation also allowed for theinterpretation of the SIMO FIR Volterra model as a linear MIMO model. The linearMIMO model to interpret the SIMO FIR Volterra model has been proposed before. Themodel proposed in this thesis is the first one that explains the role of modulation in theamount of ISI, which can be suppressed using a linear weighting equalizer.

Furthermore, the SIMO FIR Volterra model was modelled as a finite state machine,illustrating that trellis-based algorithms can be used for the equalization of TR systems,which is a well-known in literature. To reduce the trellis-based equalization complexity, areduced-memory system model was introduced that is optimal, in the sense of the MMSEcriterion. The reduced-memory system model mimics the behaviour of TR systems, butwith a significant memory reduction. The reduced-memory FIR model for a secondorder Volterra model has not been reported before in literature. Finally, the statisticalproperties were derived for the signal term as well as for both noise terms. The noisewas shown to be quasi-white, with an output-dependent noise variance. This result isconfirmed by others in literature.

In Chapter 5, the impact of different system parameters on the system performancehas been presented, like FSR, bandwidth, delay, weighting criteria and modulation bothin the absence and presence of ISI. In the absence of ISI, an FSR of two is found tosuffice for close-to-optimal performance. The non-Gaussian noise term was found tohave a significant impact on the system performance, such that small-bandwidth TRsystems perform better, in the absence of fading. Furthermore, it was found that inthe presence of ISI, more ISI can be suppressed using linear weighting if the FSR isincreased. The modulation was found to have a significant impact on the amount ofISI that can be suppressed. The role of the FSR and the modulation on the amount ofsuppressible ISI has been explained using the linear MIMO model for SIMO FIR Volterramodels, presented in Sec. 4.5.4. Most of the results presented in Chapter 5 have beenreported by others. The novelty is that influence of the system parameters on the systemperformance is analyzed, each time using the same basic system set-up. As a result, theresults allow for an improved insight in the behaviour of the system with respect to thesystem parameters. A truly novel contribution to the understanding of TR systems, is theinsight that the amount of non-linear ISI that can be suppressed using linear weightingdepends considerably on the modulation type.

It is recommended to further investigate the potential of linear weighting in the pres-ence of non-linear ISI. In the thesis, a single weighting vector was used, which exploitedthe term depending linearly on the bit under demodulation, considering the other termsas interference. Similar to MIMO systems, additional weighting vectors could be used todemodulate the other non-linear ISI terms, which also contain information on the symbolunder demodulation. The information on the linear and non-linear ISI term potentiallycan be joined in a single decision process on the symbol under demodulation. It is rec-ommended to study the structure, performance and complexity of such an algorithm, to

126 CHAPTER 7. CONCLUSIONS AND RECOMMENDATIONS

analyze its potential for commercial application.

Design of a High-Rate TR UWB System

This section holds the conclusions and recommendations of Chapter 6. Here, the de-sign of a high-rate TR UWB system has been described, able to support a data rate of100 Mb/s, while occupying 1 GHz bandwidth. A combination of trellis-based equaliza-tion and a multiband system architecture has been proposed, to obtain high data rateUWB communication over the multipath radio channel, using non-coherent receivers. Toexploit the frequency diversity provided by the 1 GHz system bandwidth, it is proposedto use FEC for multiband systems. Furthermore, turbo equalization has been consideredto improve the performance further.

The performance results reveal that a multiband system performs considerably bettercompared to a single-band system at the same data rate, using a less complex equalizerstructure. It is shown that FEC in combination with a multiband receiver structure isable to exploit the frequency diversity provided by the system bandwidth. Furthermore,turbo equalization is able to improve the system performance by approximately 1 dB,assuming perfect kernel side information. The improvement is expected to be larger inthe absence of perfect kernel side information, assuming the kernel estimate is updatedwith each iteration.

Taking both complexity and performance into account, a four-band system with fourparallel operating 16-state (N = 2) equalizers is suggested. Not only does it deliver goodperformance, it also has the potential to equalize channels with larger delay spreads.Furthermore, it is well-suited for a parallel implementation in the digital domain andinherently robust against narrowband interference, possibly even extendable to DAA.

The use of turbo equalization is recommended as well. Complexity analysis indicatesthat an additional 1 dB performance improvement can be obtained, using only slightlymore of DSP complexity. It is however critical to find better stop-criteria to manage whatpackets are scheduled for additional iterations. In this respect, further research is neededto obtain better stop-criteria. Another consideration in favour of turbo equalization isthat it allows for improved kernel estimation, which allows the system to operate usingshorter training-sequences. An interesting option seems to be to use 4-state (N = 1)equalizers during the first turbo iteration. The smaller kernel namely allows for evenshorter training sequences. During the second iteration, the kernel with N = 2 can beestimated using the information gathered during the first iteration.

Appendix A

Estimation of the Nakagami-mParameter

In this appendix, a paper on the estimation of the Nakagami-m parameter for FrequencySelective Rayleigh Fading Channels is presented, which has not been published yet.

A.1 Introduction

Probability distributions are often used for the modeling of radio communication channels.For example, the variation of the amplitude gain of flat-fading multipath channels dueto small-scale-fading is often modelled using a Rayleigh distribution or Rice distribution,depending on the absence or presence of a dominant LoS component, respectively. Bothdistributions not only fit well to the measured data, but are also justified by the physicsof multipath radio channels [35]. Bases on this insight, many mathematical tools havebeen developed in communication theory, e.g. for bit error rate analysis.

In the case of frequency selective fading channels (FSFC), by definition, not all fre-quency components of the transmitted signal experience the same channel amplitudegain. Hence, one has to average1 the power attenuation over all frequency componentsand take its square root to obtain the effective amplitude gain (EAG). Hence, the EAGis equal to the square root of the well-known mean power gain, or alternatively, the rootmean square (RMS) value of the channel frequency response (CFR). As in the case offlat fading channels, the EAG is also modelled using random processes. For FSFC ordiversity channels in general, the Nakagami distribution often fits well to measurementdata [97, 98].

The Nakagami distribution is described by two variables, namely Ω and m and occurswhen the RMS value is taken of K independent, identically distributed (i.i.d.) Gaussianrandom variables with a variance σ2. In this case, Ω and m will be Kσ2 and K/2, re-spectively. The Nakagami distribution can be seen as a generalization of the Rayleighdistribution, where m is equal to 1. Also the Nakagami distribution is justified by thephysics of the radio channel. Assuming a system radiating the energy E uniformly overK/2 i.i.d. Rayleigh fading (sub)-channels, its EAG will be a Nakagami distributed RV

1A weighted average can be used if the TX power is not uniformly distributed over the bandwidth

127

128 APPENDIX A. ESTIMATION OF THE NAKAGAMI-M PARAMETER

with Ω and m equal to E and K/2, respectively. Hence, the m-parameter also character-izes the diversity level of FSFCs [16].

The Nakagami parameters are often derived from a set of measured CFR functionsof size N . Normally, the EAG of each measured CFR is computed to obtain a pool ofN EAGs from which the Nakagami parameters can be estimated. Due to its finite size,a residual error will always exist in the estimated Nakagami parameters. Unfortunately,the variance of all known unbiased EAG-based m-parameter estimators is rather high andthe Cramer-Rao lower bound (CRLB) predicts that not much improvement is possible[99, 100, 101, 102].

In this paper, we propose to estimate the m-parameter using an estimate of theCFR covariance matrix. The result is a low-variance, biased estimator. A closed-formapproximation for the bias will be derived, based on which an alternative estimationmethod is derived, which is approximately unbiased. The simulation results reveal asuperior performance for this estimator compared to all known EAG-based estimatorsand their CRLB. Additionally, the performance of a truly unbiased estimator is presented,which is derived using data from simulation results. The CRLB of the proposed algorithmis not investigated in this paper.

A.2 Covariance-based m-parameter estimation

The CFR is often modelled using a complex multivariate, zero-mean, Gaussian randomvector h of length L with a covariance matrix Σ=E

[hhH

]. The channel EAG g= ‖h‖ /

√L

is a RV as well. The Nakagami-m parameter is related to g and Σ (see [103]) accordingto

m=E [g]2

E [g2]=

(∑L

m=1 λm)2

∑Nf

m=1 λ2m

=Tr (Σ)

Tr (ΣΣ), (A.1)

where λm denotes the m-th eigenvalue of Σ. In practice, Σ is unknown a-priori and onehas to estimate it from measurement data. Let’s assume a measurement pool, where thei-th measured CFR vector hi can be seen as the i-th realization of the random vector h.The estimate of Σ, denoted by W, becomes

W =1

N

N∑

i=1

hihHi (A.2)

where N denotes the number of measured CFRs. It is straightforward to obtain anestimate2 for the m-parameter using W, namely

mc =Tr (W)2

Tr (WW). (A.3)

2It is emphasized that the estimator is based on the zero-mean Gaussian assumption, i.e. the proposedmethod is restricted to Rayleigh channels. For channels with a dominant (LOS) component, the methodhas to be modified by repeating the presented derivations for a channel model extended by the dominantpath, i.e. a non-central Nakagami-m distribution.

A.2. COVARIANCE-BASED M -PARAMETER ESTIMATION 129

Let us continue with the derivation of the expectation for mc. To simplify its derivation,(A.3) is re-written to

mc =Tr (W)2

Tr (WW)=

Tr (W)2

Tr (ΣΣ)(

1+ Tr(WW)−Tr(ΣΣ)Tr(ΣΣ)

) . (A.4)

By assuming the estimate Tr (WW) for Tr (ΣΣ) to be reasonably accurate, the divisioncan be approximated by

mc ≈Tr (W)2

Tr (ΣΣ)

(

1 − Tr (WW) − Tr (ΣΣ)

Tr (ΣΣ)

)

≈ 2Tr (W)2

Tr (ΣΣ)− Tr (W)2Tr (WW)

Tr (ΣΣ)2 . (A.5)

Now taking the expectation of both sides, we obtain

E [mc] ≈ 2E[Tr (W)2]

Tr (ΣΣ)− E

[Tr (W)2Tr (WW)

]

Tr (ΣΣ)2 . (A.6)

The derivation of both higher-order moments is rather complex. Several papers have beenpublished on the higher-order moments of Wishart matrices [104, 105]. The followingresults from these publications will be used,

E[Tr (W)2] =Tr (Σ) +

1

NTr (ΣΣ) (A.7)

E[Tr (W)2Tr (WW)

]=Tr (Σ)Tr (ΣΣ) +

1

NTr (Σ)4

+1

N

(Tr (ΣΣ)2 + 4Tr (Σ)Tr (ΣΣΣ)

)+ O

(1

N2

)

. (A.8)

Substituting these results into (A.6) leads to

E [mc] ≈Tr (Σ)

Tr (ΣΣ)

+Tr (ΣΣ)2−Tr (Σ)4−4Tr (Σ)Tr (ΣΣΣ)

NTr (ΣΣ)2 . (A.9)

Now using the fact that m = Tr (Σ)/Tr (ΣΣ), (A.9) can be simplified to

E [mc] ≈ m

(

1 +1

mN− m

N− K(Σ)

N

)

, (A.10)

where

K(Σ) =4Tr (ΣΣΣ)

Tr (Σ)Tr (ΣΣ), (A.11)

which makes it evident that mc is only unbiased for N → ∞.


A.3 Unbiased covariance-based m-parameter estima-

tion

In this section, an unbiased estimate muc is deduced from mc for finite values of N .Therefore, E [mc] will be described as function m and N only. Unfortunately, the K(.)-term does not depend only on E [mc] nor m, but also on the structure of Σ. Simulationresults revealed that the K-term is in practice small compared to the other terms, meaningthat it is often negligible. Alternatively, one can assume a certain Σ of which the structuredepends only on m. Here, we assume h to contain m unit power i.i.d RVs, such that Σ

is an m by m identity matrix. In this case, the K-term will be equal to

K(Σ) = K(Im,m) =4m

m2=

4

m. (A.12)

Substituting (A.12) into (A.10), we obtain

E [mc] = m − m2

N− 3

N, (A.13)

such that the expectation depends only on m and N , which is a second-order equationthat can be inverted. It has only one positive solution, which is

m ≈ 1

2(N −

√

N2 − 4(NE [mc] + 3)). (A.14)

Hence, the approximately unbiased estimate for m is,

muc ≈1

2(N −

√

N2 − 4(Nmc + 3)), (A.15)

which concludes the derivation of the approximately unbiased estimator. In the followingsection, its performance will be presented.

A.4 Simulation Results

In Fig. A.1, the average RMS estimation error of the estimators mc and muc is presentedfor N = 100 synthetically generated realizations of h with Σ = Im,m. As reference, theRMS estimation error of a moment-based estimator and its CRLB have been depictedas well. First of all, it is noted that for small m, both mc and muc perform betterthan the CRLB for EAG-based estimators. Only for increasing m, the RMS estimationerror of mc increases rapidly; this is caused by its bias. It has the tendency to under-estimate the actual m. In this respect, muc has an improved performance. Still it isnot truly unbiased and has the tendency to over-estimate m, which is caused by anincreasing error in the assumption used in (A.5) with an increasing m/N ratio. Byincreasing the amount of terms used for the Taylor series expansion in (A.5) will improvethe performance of muc, but will results in higher-order moments of Wishart matricesmaking its derivation (too) complex. Alternatively, we used a second-order polynomialto describe the relationship between mc and m, assuming Σ = Σ whether it is correct or

A.4. SIMULATION RESULTS 131

0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

rms e

rro

r

m

mc

mm

muc

muc2

CRLB*

Figure A.1: RMS estimation error versus m for the different estimators (N = 100)

not. Hence, muc2 = c2m2c + c1mc + c0, where the coefficients are obtained by polynomial

fitting.

Their values for several values of N can be found in Tab. A.1. Please note the increas-ing dominance of the linear term with increasing N . The performance of muc2 is depictedin Fig. A.1, as well. Overall, the estimator muc2 has the best performance.

In Fig. A.2, the RMS error of the estimators is presented as function of N usingthe same method of generating synthetic h. The figure shows that a covariance-basedm parameter estimator needs considerably less observations N to obtain the same RMSestimation error.

For the previous two figures, the assumption Σ = Σ was valid. To analyze thealgorithm performance on more realistic channels, synthetic CFR data has been generatedfor channels with an exponential power delay profile. The deployed frequency domainautocorrelation function is as follows:

E[hi[k]h∗j [l]

]=

δ(i − j)

1 + j2π((k − l)τ∆f ). (A.16)

where τ represents the channel RMS delay spread. The simulation results are presentedin Fig. A.3. Compared to Fig. A.1, all RMS estimtion error curves have changed, butwithout affecting the derived conclusions.


0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

rms e

rro

r

N

mc

mm

muc

muc2

CRLB*

Figure A.2: RMS estimation error versus N for the different estimators (m = 5)

Table A.1: Poynomial coefficients of muc2

N c(2)2 c

(2)1 c

(2)0 c

(1)1 c

(1)0

10 0.5738 -1.3618 2.4871 3.3603 -5.436620 0.1488 0.3884 0.7753 2.1121 -3.080730 0.0744 0.7207 0.3813 1.7273 -2.183450 0.0339 0.8963 0.1491 1.4296 -1.3898100 0.0133 0.9736 0.0366 1.2124 -0.7326200 0.0058 0.9932 0.0075 1.1056 -0.3775400 0.0027 0.9983 0.0007 1.0526 -0.1917

∗ CRLB applies only to mm

A.5. CONCLUSIONS AND REMARKS 133

0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

rms e

rro

r

m

mc

mm

muc

muc2

CRLB*

Figure A.3: RMS estimation error versus m for the different estimators using realisticsynthetic data (N = 100)

A.5 Conclusions and remarks

A new class of algorithms has been presented for the estimation of the Nakagami-mparameter from coherently measured fading channels. Firstly, a straightforward low-variance, but biased estimator has been presented. Additionally, two alternative, unbiasedestimators have been proposed, both deduced from the biased estimator. The simulationresults show that both unbiased estimators have superior performance compared to othertypes of Nakagami-m parameter estimators.

Appendix B

Complex-Valued AcR

In this appendix, the baseband equivalent model for the Complex-Valued(CV) AcR isderived. Let’s denote the RX passband signal by r(t) and its delay version as y(t) = r(t−D) and their baseband equivalent representation as r(t) and y(t) = r(t−D) exp(−jωcD),respectively. The relation between a bandpass signal r(t) and its baseband equivalentr(t) is given by

r(t) = rr(t) cos(ωct) − ri(t) sin(ωct), (B.1)

where rr(t) and ri(t) denote the real and imaginary part of the signal r(t), respectively.The delayed version of the received signal is expressed in the same manner. The multiplieroutput in the first autocorrelation branch is given by

xr(t) = r(t)y(t). (B.2)

In the absence of a LPF, after substitution of (B.1), leads to the following expression,

xr(t) =rr(t)yr(t) cos2(ωct) − ri(t)yi(t) sin2(ωct)+ (B.3)

(ri(t)yr(t) + yi(t)rr(t)) cos(ωct) sin(ωct).

The LPF-characteristic of the operator after the multiplier will however filter out allterms containing a carrier. Neglecting these terms leads to an equivalent expression forthe multiplier output:

xr(t) =1

2rr(t)yr(t) +

1

2ri(t)yi(t). (B.4)

The multiplier output of the second AcR branch xi(t) is similar, except that y(t) is delayedadditionally with ∆ equal to π/2/ωc, such that

y(t − ∆) = yr(t − ∆) sin(ωct) + yi(t − ∆) cos(ωct). (B.5)

The signal y(t−∆) may be replaced by its zero-th order approximation y(t), if ∆ 1/Bwith B denoting the signal bandwidth, such that

y(t − ∆) ≈ yr(t) sin(ωct) + yi(t) cos(ωct). (B.6)

135

136 APPENDIX B. COMPLEX-VALUED ACR

Hence, the multiplier output of the second AcR branch xi(t) equals

xi(t) = r(t)y(t − ∆) =1

2rr(t)yi(t) −

1

2ri(t)yr(t). (B.7)

Since xr(t) is the real-part of the multiplier output and xi(t) the imaginary part, thecomplex-valued multiplier output x(t) becomes

x(t) , xr(t) − jxi(t) =1

2r(t)r∗(t − D) exp(jωcD). (B.8)

Consequently, the complex-valued AcR output becomes

u[n, α]= exp(jωcD)

∞∫

−∞

h(t−(nL+α)Tclk)r(t)r∗(t−D)dt, (B.9)

where the factor 1/2 has been omitted and h(t) is a rectangular shaped function equal toone for all 0 ≤ t < Tclk and zero otherwise.

The CV AcR is a generalization of the traditional AcR. Therefore, the presentedderivation also applies to a RV AcR. Furthermore, the derivation shows that a modifica-tion of the center frequency only results in a phase shift of the AcR output.

Appendix C

PSD of Scrambled QPSK-TR UWB

Signals

In [25], it is shown that if modulation applied to the pulses is uncorrelated from pulseto pulse, the PSD shape of the radiated signal depends only on the squared Fouriertransform of the individual pulses. In this appendix, it is shown that the modulation isindeed uncorrelated, when deploying scrambled QPSK-TR UWB as defined in Tab. 6.1.Here, it is assumed that each of the four possibly symbol identifiers have the same a-prioriprobability. For completeness, we recall that the reference pulse is modulated with b[n]and the information-bearing pulse with b[n]b[n].

Assuming equally probable symbols, it is straightforward to derive that b[n] and b[n]are both zero mean, such that the signal has no DC component. Assuming independentsymbols, the following correlation properties between the pulses are found. For the pulsetrain of reference pulses, we find that

E[

b[n]b∗[n + k]]

=

1 if k = 0,

0 otherwise(C.1)

which means that this pulse train generates no spectral spikes. Let us continue with thecorrelation properties of the information bearing pulses,

E[

b[n]b[n]b∗[n + k]b∗[n + k]]

=

1 if k = 0,

0 otherwise(C.2)

which means that this pulse train also generates no spectral spikes. Also the cross-correlation between both signals could generate spikes. Therefore, the cross-correlationbetween both TR signals is investigated

E[

b[n]b∗[n + k]b∗[n + k]]

=

0 if k = 0,

0 otherwise.(C.3)

Since the cross-correlation is in any case zero, the resulting cross-PSD will be zero as well.

137

138 APPENDIX C. PSD OF SCRAMBLED QPSK-TR UWB SIGNALS

Appendix D

Derivation of the Log-MAP

Algorithm

In this appendix, the Log-MAP algorithm is derived in the notation used in this thesisand effort has been make the explanation close to implementation.

The Log-MAP algorithm computed the LLV of a bit q[n], which is defined as


(P(q[n] = 1|u, L(c),H)

P(q[n] = −1|u, L(c),H)

)

(D.1)

The object H describes the possible state-transitions and the related values for q[n]. Theset of possible state-transitions will be denoted by Stt. This set can be divided intotwo disjoint subsets, where S

+1tt and S

−1tt denote the set of possible state-transition given

q[n] = 1 and q[n] = −1, respectively. Using these set definitions, (D.1) can be written as,


∑

Stt[n]∈S+1tt

P(Stt[n]|u, L(c),H)

∑

Stt[n]∈S−1tt

P(Stt[n]|u, L(c),H)

(D.2)

Using Bayes’ theorem stating that P (A|B) = P (B|A)P (B)/P (A), (D.3) can be writtenas,


∑

Stt[n]∈S+1tt

P(u, L(c)|Stt[n],H)P(u, L(c)) /P(Stt[n])

∑

Stt[n]∈S−1tt

P(u, L(c)|Stt[n],H)P(u, L(c)) /P(Stt[n])

(D.3)

For different reasons, the probabilities P(u, L(c)) and P(Stt[n]) are the same for all state-transitions: This allows us to simplify (D.3) to


∑

Stt[n]∈S+1tt

P(u, L(c)|Stt[n],H)

∑

Stt[n]∈S−1tt

P(u, L(c)|Stt[n],H)

(D.4)

139

140 APPENDIX D. DERIVATION OF THE LOG-MAP ALGORITHM

To simplify the implementation, the log of the sum of two probabilities P1 and P2 willbe written in another form. Assuming l1, l2 and l1,2 to denote lnP1, ln P2 and ln P1 + P2,respectively. The joint log-probability l1,2 is related to l1 and l2 according to

l1,2 = max(l1, l2) + ln(1 + exp(|l1 − l2|)) (D.5)

A new operator, called the box-plus operator , is now introduced, such that l1,2 = l1l2.A single box-plus operation requires a max-operation, a subtraction, an absolute operationand finally a table-lookup, assuming the function ln(1 + exp(|x|)) is stored in a lookuptable. The box-plus operator is both associative and commutative, i.e. l1 l2 = l2 l1and (l1 l2) l3 = l1 (l2 l3). Another important property of the box-plus operatorfor the implementation reasons is that (l1 + K) (l2 + K) = K + l2 l1.

Using the box-plus operator, the order of the sum and natural logarithm in (D.2) canbe interchanged to obtain

L(q[n]|u, L(c),H) = Stt[n]∈S

+1tt

ln (P(u, L(c)|Stt[n],H))

− Stt[n]∈S

−1tt

ln (P(u, L(c)|Stt[n],H)) (D.6)

Now let us have a close look at the probability P(u, L(c)|Stt[n],H). In [106], it isshown that this probability can be divided in three parts, a pre-cursor part, a on cursorpart and a post-cursor part. The a-priori LLVs and the channel information divided intothese parts are given by

L(c) =[L(c<) L(c[n]) L(c>)

], (D.7)

u =[u< u[n] u>

]. (D.8)

The probability on a given state-transition can now be written as the product of threeprobabilities, the probability on the start state using only pre-cursor information, theprobability of the state-transition using the on-cursor information and the probabilityof the end state using the post-cursor information. The log-probability of a given statetransition is thus described by

ln(P(Stt[n]|u, L(c),H)) = α(St[n]) + γ(Stt[n]) + β(St[n + 1]) (D.9)

where

α(St[n]) = ln (P(u<, L(c<)|St[n],H)) , (D.10)

β(St[n + 1]) = ln (P(u>, L(c>)|St[n + 1],H)) , (D.11)

γ(Stt[n]) = ln (P(u[n], L(c[n])|Stt[n],H)) . (D.12)

In [106], it is shown that both α(St[n]) and β(St[n + 1]) can be written in a recursivemanner. These results in the notation deployed in this thesis give

α(St[n]) = St[n−1]∈S−|St[n]

γ(St[n − 1], St[n]) + α(St[n − 1]) (D.13)

β(St[n + 1]) = St[n+2]∈S+|St[n+1]

γ(St[n + 1], St[n + 2]) + β(St[n + 2]) (D.14)

141

where S−|S and S+|S denote the set with possible preceding states and subsequent states

of the state S, respectively. The required information is contained in the trellis object H.The size of both sets Np equals 2N

b and 2 for the channel and FEC, respectively.The only remaining unknown to be solved is γ(Stt[n]). Assuming u[n] and L(c[n])

to be independent1, the equation for γ(Stt[n]) can be divided into the sum of two log-probabilities,

γ(Stt[n]) = ln(P(u[n]|Stt[n],H)) + ln(P(L(c[n])|Stt[n],H)) (D.15)

Here, the first term contains the a-posteriori information captured from the channel. Thesecond term contains the a-priori information on the channel bits. Both terms will besolved separately. For starters, the a-posteriori term ln(P(u[n]|Stt[n],H)) will be solved.

In chapter 4, the noise was shown to be independent. Using the trellis informationcontained in H, the first right-hand term of (D.15) can be simplified to,

ln(p(u[n]|Stt[n]),H) =L−1∑

α=0

ln(p(u[n, α]|Stt[n])) (D.16)

Now by assuming the noise to be Gaussian distributed with an output-dependent varianceσ2

α independent of the state-transition, the probability

ln(p(u[n, α]|Stt[n]),H) = c1,α − c2,α |u[n, α] − fα(Stt[n])|2 (D.17)

with c1,α = − ln(πσα) and c2,α = 1/σ2α. The expression for ln(p(u[n, α]|Stt[n])) can not be

further simplified.Let us continue with the a-priori information term ln(P(L(c[n])|Stt[n])). From the

a-priori information contained in the trellis object H, the Nb channel bits related tothe given state-transition at time n are known. Assuming a time-invariant trellis, x[k]denotes the k-th channel bit related to the given time-transition. Assuming the LLVs tobe independent,

ln(P(L(c[n])|Stt[n],H)) =

Nb−1∑

k=0

ln(P(L(c[n, k])|x[k]))) (D.18)

Using the result from Tab. 6.2, the log-probability

ln(P(L(c[n, k])|x[k],H))) = x[k] L(c[n, k]) − ln(1 + exp(x[k] L(c[n, k]))) (D.19)

The expression can not be further simplified.In combination, the following expression for γ(Stt[n]) is obtained

γ(Stt[n]) =K1 +L−1∑

α=0

−c2,α |u[n, α] − fα(Stt[n])|2 (D.20)

+

Nb−1∑

k=0

x[k] L(c[n, k]) − ln(1 + exp(x[k] L(c[n, k]))) (D.21)

1In a turbo equalization scheme, the independence assumption is questionable. The use of interleaversmakes the assumption reasonable and allows for a low-complexity algorithm with good performance

142 APPENDIX D. DERIVATION OF THE LOG-MAP ALGORITHM

where K1 ,∑L−1

α=0 c1,α. The equation shows the channel branch metric depends both onthe Euclidian distance between the expected sample values and the actual sample valuesand the noise variance. In fact, the expression can be considered as the computation ofa weighted Euclidian distance. Furthermore, the a-priori term shows that the a-prioriLLVs are summed into the branch metrics.

For the computation of L(q[n]), the term K1 can be neglected, without affecting theresult of (D.6). Adding a constant K2 to γ(Stt[n]), means that the recursive relation forα(St[n]) and β(St[n + 1]) are enlarged by nK2 and (Nst −n− 1)K2, where Nst representsthe number of state transitions in the trellis and n = 0 denotes the first state transition.Hence, (D.9) is increased by a factor NstK2. After the subtraction in (D.6), this additionalterm will be cancelled. Now by selecting K2 = K1 means that the term K1 can indeedbe neglected without affecting the LLVs, while reducing the complexity of the Log-MAP.

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Acknowledgments

The road to the PhD degree has not been an easy one. This PhD could only be completedbecause of the support I have received by many. First of all, I would like to thank myparents, sister, other family members and friends for the emotional support during roughtimes. I would also like to thank both IMST GmbH and the SPSC institute, not only forfacilitating the PhD, but also for the great working atmosphere and understanding. Theaffiliated persons, who have my personal gratitude, are Klaus Witrisal, Norbert Schmidt,Gernot Kubin, Peter Waldow and Birgit Kull. Thanks you so much for your support andfor granting me this opportunity.

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Curriculum Vitae

Jac Romme was born in Breda, The Netherlands, on March 29, 1975. After attending theprimary school ”Onder de Torens” and the secondary school ”Katholieke Scholengemeen-schap Etten-Leur” both in Etten-Leur, The Netherlands, he started Electrical Engineeringat the Eindhoven University of Technology (TU/e) in Eindhoven, The Netherlands, inSeptember 1994. During his education, he conducted two internships. The first one wasat the TU/e, where a comparison was made between the closed-form results and numer-ical results for the radiation diagram and currents of a non-ideal linear antenna. Thesecond internship, he conducted at Alcatel in Antwerp, Belgium on the performance ofTCP/IP over Skybridge Satellite Links. After finishing a graduate project at SiemensICP in Munich on variable-rate convolutional codes, he received the M.Sc. degree inelectrical engineering at the TU/e.

In September 2000, he started at IMST GmbH, Kamp-Lintfort Germany working onradio system design with as main focus UWB communication and localization. Besides hiswork at the IMST, he started as a PhD student at the SPSC-lab at the technical universityof Graz, Austria, in August 2004. The main part of his PhD work was conducted at IMSTGmbH. Furthermore, he has visited the SPSC-lab three times for in total 10 Months. Hismain interests are UWB communication and localization, baseband signal processing,channel coding, equalization, iterative signal processing and non-linear signal processing.

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