Jac Romme - Library of Ph.D. Theses | EURASIP

Channel Fading Statistics and Transmitted-Reference Communication
UWB
Jac Romme
SPSC Dissertation Series

Doctoral Thesis
UWB Channel Fading Statistics and
Transmitted-Reference
Communication
ir. Jac Romme
————————————–
Signal Processing and Speech Communication Laboratory
Graz University of Technology, Austria
Supervisor: Univ.-Prof. Dipl.-Ing. Dr.techn. Gernot Kubin
External Evaluator: Prof. Sergio Benedetto
Co-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 Lévi-Strauss, 1964]

Zusammenfassung
Die Robustheit der Ultra WideBand (UWB) Übertragung gegenüber Small-Scale-Fading
(SSF) in Mehrwegekanälen als Folge der großen Bandbreite ist hinlänglich bekannt. Dennoch
gibt es bislang kein Modell, das die Variation der Empfangssignalstärke in Abhängigkeit
von der Signalbandbreite und genereller Kanaleigenschaften wie dem Leistungsverzögerungs-Profil
beschreibt. Ein solches Modell würde dem Kommunikationsingenieur
erlauben, die Nachteile einer großen Bandbreite wie z.B. steigende Systemkomplexität
gegenüber dem Vorteil einer erhöhten Systemrobustheit abwägen zu können. In dieser
Dissertation wird ein Modell vorgestellt, das diese Analyse erstmals ermöglicht, indem es
die statistischen Eigenschaften von SSF als Funktion der Signalbandbreite und des Leistungsverzögerungs-Profils
beschreibt. Zudem wird eine Berechnung der resultierenden
Bitfehlerrate bei Verwendung von BPSK Modulation vorgestellt.
Die hohe Bandbreite der UWB-Systeme ist zwar vorteilhaft bei der Bekämpfung von
SSF, führt aber im Empfängerdesign zu Problemen. Kohärente Empfängerkonzepte sind
sehr komplex, so dass bereits in 2002 von den Autoren Tomlinson und Hoctor ein alternatives
Konzept vorgeschlagen wurde, das das Transmitted Reference (TR) Verfahren mit
einem Autokorrelationsempfänger kombiniert und eine Kanalschätzung vermeiden kann.
Aufgrund der nichtlinearen Struktur des Empfängers war es bislang schwierig, sein exaktes
Verhalten vorherzusagen. Diese Dissertation gibt nun Einblicke in das prinzipielle
Verhalten des TR-Autokorrelationsempfängers und zeigt zusätzlich Verbesserungen auf,
die es ermöglichen, einige Nachteile des Konzepts abzuschwächen. Weiterhin werden verschiedene
Interpretationen des TR UWB-Prinzips präsentiert, die z.B. den Einfluss von
Intersymbolinterferenz auf das System erklären.
Basierend auf dem theoretischen Verständnis von TR-UWB wird im Anschluss ein
hochratiges Übertragungssystem mit Datenraten im Bereich von einigen 100 Mbps bei
einer Bandbreite von 1 GHz entwickelt. Es verwendet eine Kombination von trellisbasierter
Entzerrung, Turbo Entzerrung, Turbo-Dekodierung und Verarbeitung in mehreren
Bändern, die es erlaubt, die gewünschte Datenrate mit moderater digitaler Signalverarbeitung
zu erzielen. Bereits ein E b /N 0 von 12 dB ist für eine Bitfehlerrate kleiner
als 10 −6 ausreichend.
i

Abstract
It is well known that Ultra WideBand (UWB) transmission is inherently robust against
small-scale-fading (SSF) that arises in multipath scattering environments, due to its large
signal bandwidth. However, no model with a physical interpretation exists that relates
the variations of received signal strength to the signal bandwidth and general channel
parameters, like e.g. the average channel power delay profile. Such a model would be of
relevance 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-scalefading
on the other hand. In this thesis, a model is presented that allows for such a tradeoff
analysis, relating the average power delay profile parameters and signal bandwidth to the
statistical properties of the SSF. Additionally, it is shown how the uncoded and coded
BER of BPSK modulation can be computed in a closed-form for a given average power
delay profile and signal bandwidth.
As stated before, UWB communication is inherently resilient against SSF. Unfortunately,
coherent receivers become rather complex in the UWB case. In 2002, Tomlinson
and Hoctor proposed to combine Transmitted Reference (TR) signaling with an autocorrelation
receiver (AcR) for UWB communications, to dispose of the need for channel
estimation. Due to the non-linear structure of the AcR, little was known with respect to
its behaviour in various situations. This thesis aims to provide better insight in the behaviour
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 drawbacks.
Additionally, novel interpretations for TR UWB systems are presented, which
explain 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-rate
TR UWB system is presented that supports data-rates up to 100 Mb/s, while occupying
1 GHz of bandwidth. Using a combination of trellis-based equalization, multiband processing,
turbo equalization and turbo coding, a system is obtained which is moderately
complex with respect to digital signal processing and requires an E b /N 0 of only 12 dB to
obtain a BER better than 10 −6 .
iii

Contents
Zusammenfassung
Abstract
Acronyms
i
iii
ix
1 General Introduction 1
1.1 Wireless Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Ultra-WideBand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Framework and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Thesis Outline and Contributions . . . . . . . . . . . . . . . . . . . . . . . 6
2 Theory of Fading UWB Channels 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 The Radio Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.2 Radio Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.3 Channel Characterizing Parameters . . . . . . . . . . . . . . . . . . 15
2.1.4 Impact of the Channel on Radio Signals . . . . . . . . . . . . . . . 16
2.2 Frequency Domain Properties of UWB Channels . . . . . . . . . . . . . . . 18
2.2.1 Frequency Domain Correlation . . . . . . . . . . . . . . . . . . . . 19
2.2.2 Eigenvalues and Their Physical Interpretation . . . . . . . . . . . . 20
2.2.3 Asymptotic Behaviour of the Eigenvalues . . . . . . . . . . . . . . . 22
2.3 Diversity of UWB Channels . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.1 The Mean Power Gain . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.2 Statistical Characterization of the NLOS Mean Power Gain . . . . . 26
2.3.3 Generalization of the Statistics to LOS Scenarios . . . . . . . . . . 28
2.3.4 Diversity Level of UWB Channels . . . . . . . . . . . . . . . . . . . 29
2.4 BER on UWB Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.1 BER of BPSK on Fading Channels . . . . . . . . . . . . . . . . . . 31
2.4.2 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Fading of Measured UWB Channels 37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Description of Radio Channel Measurements . . . . . . . . . . . . . . . . . 37
3.3 Overview of Measurement Results . . . . . . . . . . . . . . . . . . . . . . . 38
v

vi
CONTENTS
3.3.1 Delay Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.2 Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 Principal Components of Measured UWB Channels . . . . . . . . . . . . . 43
3.4.1 Estimation of the Eigenvalues and Principal Components . . . . . . 43
3.4.2 Verification of the NLOS Eigenvalues and Principal Components . . 44
3.4.3 Verification of the LOS Eigenvalues and Principal Components . . . 46
3.5 Analysis of the Mean Power Gain . . . . . . . . . . . . . . . . . . . . . . . 46
3.5.1 Estimation of the Diversity Level . . . . . . . . . . . . . . . . . . . 47
3.5.2 Verification of the Diversity Level . . . . . . . . . . . . . . . . . . . 48
3.5.3 Verification of the Mean Power Gain . . . . . . . . . . . . . . . . . 49
3.6 BER Comparison on Measured and Theoretical UWB Channels . . . . . . 51
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 Theory of TR UWB Communications 57
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Principle of Transmitted Reference Communication . . . . . . . . . . . . . 58
4.2.1 Transmitted-Reference Signaling . . . . . . . . . . . . . . . . . . . . 58
4.2.2 Autocorrelation Receiver . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.3 The Drawbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.4 Implementation Considerations . . . . . . . . . . . . . . . . . . . . 61
4.3 Extensions of the TR Principle . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.1 Weighted Autocorrelation and Fractional Sampling AcR . . . . . . 62
4.3.2 Complex-Valued Autocorrelation Receiver . . . . . . . . . . . . . . 65
4.3.3 TR M-ary Phase Shift Keying . . . . . . . . . . . . . . . . . . . . . 67
4.4 Generic TR System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4.2 Continuous-Time System Model . . . . . . . . . . . . . . . . . . . . 68
4.4.3 Discrete-Time Equivalent System Model . . . . . . . . . . . . . . . 69
4.5 Interpretation of the TR System Model . . . . . . . . . . . . . . . . . . . . 72
4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5.2 Vector Notation for Volterra Kernels . . . . . . . . . . . . . . . . . 75
4.5.3 Extension of the Vector Notation . . . . . . . . . . . . . . . . . . . 77
4.5.4 Linear MIMO Model . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5.5 Data Model as Finite State Machine . . . . . . . . . . . . . . . . . 79
4.5.6 Reduced Memory Data Model . . . . . . . . . . . . . . . . . . . . . 82
4.6 Statistical Properties of the TR System Model . . . . . . . . . . . . . . . . 84
4.6.1 Statistics of the Signal Term . . . . . . . . . . . . . . . . . . . . . . 85
4.6.2 Statistics of the Gaussian Noise Term . . . . . . . . . . . . . . . . . 85
4.6.3 Statistics of the Non-Gaussian Noise Term . . . . . . . . . . . . . . 87
4.6.4 Analysis of the Noise Term . . . . . . . . . . . . . . . . . . . . . . . 88
4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5 Analysis of TR UWB Communication 91
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Description of the Linear Weighting . . . . . . . . . . . . . . . . . . . . . . 91
5.3 System Performance in the Absence of ISI . . . . . . . . . . . . . . . . . . 92

CONTENTS
vii
5.3.1 Influence of the Weighting Criteria and Fractional Sampling Rate . 93
5.3.2 Influence of Delay and Fractional Sampling Rate . . . . . . . . . . . 94
5.3.3 Influence of Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3.4 Influence of Modulation . . . . . . . . . . . . . . . . . . . . . . . . 95
5.4 System Performance in the Presence of ISI . . . . . . . . . . . . . . . . . . 97
5.4.1 Influence of the Weighting Criteria and Fractional Sampling Rate . 97
5.4.2 Influence of Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.4.3 Influence of Modulation . . . . . . . . . . . . . . . . . . . . . . . . 98
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6 Design of a High-Rate TR UWB System 103
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2 Design Considerations for a High-Rate TR UWB System . . . . . . . . . . 103
6.2.1 Trellis-Based Equalization . . . . . . . . . . . . . . . . . . . . . . . 103
6.2.2 Power Spectral Density of TR Signals . . . . . . . . . . . . . . . . . 104
6.2.3 Volterra System Identification . . . . . . . . . . . . . . . . . . . . . 105
6.2.4 Multiband Transmitted Reference . . . . . . . . . . . . . . . . . . . 106
6.2.5 The Role of Forward Error Control . . . . . . . . . . . . . . . . . . 106
6.2.6 Principle of Turbo Equalization . . . . . . . . . . . . . . . . . . . . 107
6.3 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.3.1 Description of the TX Architecture and RX RF Front-End . . . . . 108
6.3.2 Forward Error Control . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3.3 Turbo Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3.4 SISO Decoder Structure . . . . . . . . . . . . . . . . . . . . . . . . 112
6.3.5 Stop Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.3.6 Measure of Complexity . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.4 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.4.1 Impact of Equalizer Complexity Without Turbo Equalization . . . . 115
6.4.2 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
2G
3G
4G
ADC
AcR
AMPS
APDP
AWGN
BCJR
BER
BPF
BPSK
CC
CDF
CEPT
CEV
CFR
CIR
CV
CW
First Generation
Second Generation
Third Generation
Fourth Generation
Analogue to Digital Converter
Autocorrelation Receiver
American Advanced Mobile Phone System
Average Power Delay Profile
Additive White Gaussian Noise
Bahl, Cocke, Jelinek and Raviv
Bit Error Rate
Band Pass Filter
Binary-Phase-Shift-Keying
Convolutional Code
Cumulative Distribution Function
European Conference of Postal and Telecommunications
Administrations
Circulant Eigenvalue
Channel Frequency Response
Channel Impulse Response
Complex-Valued
Carrier Wave
ix

x
CONTENTS
DAA
DARPA
DFT
DLL
DS
DSL
DSP
DS-UWB
ECC
ECMA
EM
FCC
FDM
FEC
FER
FH
FIR
HMM
FSFC
FSM
FSR
I&D
IEEE
ISI
ISP
ITU-R
LLV
Detect and Avoid
Defence Advanced Research Projects Agency
Discrete Fourier Transform
Data Link Layer
Direct Sequence
Digital Subscriber Loop
Digital Signal Processing
Direct Sequence - UWB
Electronic Communications Committee
European Computer Manufacturers Association
Electro-Magnetic
Federal Communications Commission
Full Data Model
Forward Error Control
Frame Error Rate
Frequency Hopping
Finite Impulse Response
Hidden Markov Model
Frequency Selective Fading Channel
Finite State Machine
Fractional Sampling Rate
Integrate and Dump
Institute of Electrical and Electronics Engineers
Inter Symbol Interference
Internet Service Provider
International Telecommunication Union Radiocommunication Sector
Log-Likelihood Value

CONTENTS
xi
LMS
LOS
LPF
LS
MAC
MAP
MB-OFDM
MIC
MIMO
MLSD
MMSE
MPG
MRC
NLOS
NMT
OFDM
OSI
PC
PCA
PDF
PDP
PHY
PIAM
PN
PPM
PSD
QAM
Least Mean Square
Line-of-Sight
Low Pass Filter
Least Squares
Multiple Access Layer of the OSI model
Maximum A-Posteriori
Multi-Band Orthogonal Frequency Division Multiplexing
Ministry of Internal Affairs and Communications
Multiple-Input, Multiple-Output
Maximum-Likelihood Sequence Detection
Minimum Mean Square Error
Mean Power Gain
Maximum Ratio Combining
Non-Line-of-Sight
Scandinavian Nordic Mobile Telephone
Orthogonal Frequency Division Multiplexing
Open Systems Interconnection
Principal Component
Principal Component Analysis
Probability Density Function
Power Delay Profile
Physical Layer of the OSI model
Pulse Interval and Amplitude Modulation
Pseudo Noise
Pulse Position Modulation
Power Spectral Density
Quadrature Amplitude Modulation

xii
CONTENTS
QPSK
R&O
RF
RMDM
RMS
RSCC
RV
RX
SIMO
SISO
SNIR
SNR
SOVA
SSF
SVD
TACS
TG3a
TR
TX
UB
US
USB
UWB
VOIP
WPAN
WLAN
WMAN
WWAN
WRAN
Quadrature-Phase-shift-Keying
Report and Order
Radio Frequency
Reduced Memory Data Model
Root Mean Square
Recursive Systematic Convolutional Code
Random Value
Receiver
Single-Input, Multiple-Output
Soft-Input, Soft-Output
Signal-to-Noise-and-Interference Ratio
Signal-to-Noise Ratio
Soft-Output Viterbi Algorithm
Small-Scale Fading
Singular Value Decomposition
British Total Access Communication System
Task Group 3a
Transmitted Reference
Transmitter
Upper Bound
Uncorrelated Scattering
Universal Serial Bus
Ultra-Wideband
Voice over IP
Wireless Personal Area Network
Wireless Local Area Network
Wireless Metropolitan Area Network
Wireless Wide Area Network
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 differential
equations, which together describe the behaviour of electric and magnetic fields, as well
as their interactions with each other and matter. Based on these equations, he predicted
the existence of self-sustaining, oscillating waves composed out of an electric and magnetic
field that travel through space. Nowadays, these waves are referred to as electro-magnetic
waves. 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 antenna
as detector, Heinrich Rudolph Hertz proved in 1886 that a spark at the transmitter can
induce a spark at the receiver, showing that electromagnetic waves can travel through
free space over some distance, as Maxwell predicted.
Fascinated by these results, Lodge, Marconi and Popov began almost simultaneously
transforming radio into a way of wireless communication. In 1896, Marconi and Popov
both sent radio messages over short distances. In 1899, Marconi signalled the first wireless
signal across the English Channel and two years later, he already telegraphed from
England to Newfoundland. These experiments made the world considerably smaller, since
now the transport of information was only limited by the speed of light.
Nowadays, the use of wireless communications for the transport of voice and data
has been integrated into everyday’s life. The penetration of mobile telephony in the
western world is often above 80% and the deployment of wireless local area networks has
become 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 success
of wireless communications started. The 1G mobile radio telephony emerged in the
early eighties, although in its early years the term portable communications was more
appropriate. Due to limitations in technology, these phones deployed analogue modulation
and were still rather big. These systems were developed for voice communication only and
every country used their own frequency bands, disabling the possibility of international
roaming. Nevertheless, these systems allowed for the opening of a mass-market for wireless
voice communication. Some of the most successful 1G systems were American Advanced
Mobile 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 took
place 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. As
a result, devices became significantly smaller, cheaper, allowed for international roaming
and had longer operating time on single charge of battery. All these aspects contributed
to the success of 2G.
The commercial success of 2G together with the Internet boom, caused an explosion of
developments to penetrate mobile communications further into everyday’s life. Nowadays,
many wireless communication systems are in use aside each other, each developed for its
own 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 and
often the terms are used more for market-technical reasons than technical ones. Standards
belonging to different types may therefore compete with each other for the grace of the
customers. Nevertheless, we will make an effort to characterize them coarsely:
A Wireless Personal Area Network (WPAN) is a network technology to interconnect
devices around a workspace or person using wireless radio technology. The typical range
is around 10 meters and often mobility is not supported, meaning that the connection
breaks down when leaving the coverage area.
A Wireless Local Area Network (WLAN) is used to connect computers and other
WLAN enabled devices to the wired network directly via base-stations. The range covered
by 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 covered
by the network of base-stations.
Wireless Metropolitan Area Network (WMAN) is basically an extension of the WLAN
concept to support ranges in the order of 1 km, such that with less base-stations a larger
area 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 an
alternative for cable and Digital Subscriber Loop (DSL). Most likely, access the Internet
is limited to the coverage area of the ISP and the underlying technology supports only
moderate velocities. Nevertheless, WMAN could go in competition with the cellular
network operators, especially now Voice over IP (VOIP) is taking off as well.
The term Wireless Wide Area Network (WWAN) is typically used for standards developed
by/for the cellular network operators. The typical range is 10 km and differs from
WLAN and WMAN, because of the use of cellular network technology. These cellular
technologies provide for nationwide and international access using roaming. In this sense
WWAN provides higher mobility and higher velocities.
A recent development is the Wireless Regional Area Network (WRAN) introduced by
Institute of Electrical and Electronics Engineers (IEEE) standard 802.22, which has as
mandate to develop a standard for a cognitive radio-based PHY/MAC/air interface for
use by license-exempt devices in spectrum allocated to TV Broadcasting. The target application
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 a
single base station is aimed to be up to 100 km to reduce the cost of the infra-structure.

1.2. ULTRA-WIDEBAND 3
∼ 100 km
WRAN
802.22
Standard
∼ 10 km
Under development
max range
WWAN
∼ 1 km
WMAN
∼ 100 m
GSM
GPRS
EDGE
UMTS
802.20
HSDPA
WiMax
3G-LTE
802.16d/e
UWB Standard
WLAN
∼ 10 m
802.11 11b 11a/g
11n
WPAN
ZigBee Bleutooth 802.15.4a WiMedia 802.15.3c
(W-USB)
TerraHz
10 kb/s 100 kb/s 1 Mb/s 10 Mb/s 100 Mb/s 1 Gb/s 10 Gb/s
data rate
100 Gb/s
Figure 1.1: Overview of communication standards
Another way to distinguish standards is with respect to supported data rates. Typically,
recent standards provide higher data rates than the older ones. When violating this
general rule, the new standard will have some distinct benefits with respect to existing
ones, e.g., with respect to cost or added functionality like ranging or localization. An
overview of currently successful standards and promising future standards can be found
in Fig. 1.1, separated with respect to coverage area and data rate. The overview contains
two standards related to the topic of this thesis, namely 802.15.4a and WiMedia. These
will be discussed in more detail in Sec. 1.2. More details on the other radio communication
standards can be found in [1, 2, 3].
1.2 Ultra-WideBand
Although Ultra-Wideband (UWB) is often considered a new radio technology, UWB
technology has been around for many years. In fact, the first wireless transmission experiments
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 of
a pulse that is UWB. Radio communications took another course with the invention
of the Alexanderson radio alternator radio-frequency source, which allowed for Carrier
Wave (CW) communications. Not only because CW allowed for simpler transmitters, but
also 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, radio
regulatory bodies started to assign frequency bands to specific systems, such that they
could 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

4 CHAPTER 1. GENERAL INTRODUCTION
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 communications
in 1973 [4]. In those days, the main applications were radar and positioning, because
of the inherent ability of UWB to resolve objects with a high spatial resolution, and
military communication systems, because of the inherent covertness of UWB signals.
Most developments were therefore conducted in the military or funded by governments
under 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 study
by the Defence Advanced Research Projects Agency (DARPA) in 1990. Despite these
early developments, CW remained to govern commercial wireless radio communications.
The interest in UWB for commercial wireless radio communications revived with a
series of papers by Scholtz and Win [5, 6, 7] and the UWB activities of U.S. based companies
like XtremeSpectrum, Multispectral Solutions and Time Domain. The lobbying
activities of these companies resulted in a Notice of Inquiry by the Federal Communications
Commission (FCC) in September 1998 on the allowance of UWB on an unlicensed
basis 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 a
Power Spectral Density (PSD) of -41.3 dBm/MHz, the same as allowed by part 15 for
unintentional radiators. The main intent of the R&O is to provide re-use of scarce radio
spectrum while enabling high data rate WPAN as well as radar, imaging and localization
systems.
At first, UWB was thought to be a pulse-based system, but the FCC defined UWB in
terms of a transmission from an antenna for which the emitted signal bandwidth exceeds
the lesser of 500 MHz or 20% of the center frequency. This allows Orthogonal Frequency
Division Multiplexing (OFDM) and Direct Sequence (DS) systems to be operated under
the UWB regulation. The opening of several GHz of bandwidth for commercial applications
resulted in an avalanche of academic research and industrial efforts, which eventually
lead to the standardization of UWB for WPAN [10, 11].
The road to standardization has been rather rocky. In December 2002, the IEEE
granted the project authorization request as Task Group 3a (TG3a) part of the 802.15
standards family for WPAN. The aim of TG3a was to specify a standard PHY for
high-data-rate, short-range, low-power, and low-cost wireless networking technology using
UWB. In total 23 UWB PHY specifications were submitted, which quickly merged into
two proposals. The WiMedia Alliance proposed a Multi-Band Orthogonal Frequency
Division 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 Computer
Manufacturers Association (ECMA) released two ISO-based standards for UWB
based 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, the
WiMedia 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 rather
quiet after the Freescale’s departure from the UWB Forum. Therefore, it seems that the
WiMedia Alliance is winning the race.
Besides UWB being considered for high data rate WPAN, also joint low data rate
and localization is considered for WPAN. In March 2004, the IEEE launched task group
802.15.4a for a mandate to develop an alternative PHY as optional extension to the
802.15.4 PHY, which provides low data rate communications and high precision ranging/location
capability, while being low power and low cost. In March 2007, P802.15.4a
was approved as a new amendment to 802.15.4 by the IEEE. Besides the mandatory
DSSS PHY of 802.15.4, one of the two alternative PHYs in 802.15.4a provides UWB in
three frequency bands, allowing for data rates between 110 kb/s up to 27.24 Mb/s and
localization [13].
Following the FCC, the International Telecommunication Union Radiocommunication
Sector (ITU-R) has published a Report and Recommendation on UWB in November
of 2005. National bodies are expected to adopt their regulation to allow UWB. In
September 2005, a draft decision was released by the European Conference of Postal and
Telecommunications Administrations (CEPT). In March 2006, the Electronic Communications
Committee (ECC) decision was issued, allowing UWB for frequencies between
6 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 regulatory
proposal. The foreseen allocated bandwidths are the frequencies between 3.4 until
4.8 GHz and 7.25 until 10.25 GHz, with the same PSD limits as allowed by the FCC. In
contrast to the FCC, the European and Japanese regulation bodies may demand UWB
systems to use so-called Detect and Avoid (DAA) to avoid interference with current and
future wireless services [14, 15].
1.3 Framework and Objectives
The work presented in this thesis is the partial outcome of an objective defined at the
IMST GmbH to develop understanding on UWB technology. Starting in 2000, the objective
was to acquire know-how on the theory and implementation of low-cost UWB
systems for communication and localization. The objective resulted in the participation
in several projects both on a European level as well as on a regional level. The projects
funded by the 5-th and 6-th framework of the IST program of the European Union in a
chronological order are Whyless.com, Europcom and Pulsers 2. The projects funded in
the scope of the Nordrhein-Westfalen Zukunftswettbewerb are Bison and PulsOn
While having many benefits, the implementation of UWB systems is significantly more
complex than those of narrowband systems, since many of the hardware components must
be well-behaving over a larger frequency range. Crudely spoken, more bandwidth more
problems, at least with respect to implementation and cost. On the other hand, one
would like to take advantage of the fundamental benefits of UWB. Hence, during system
design a trade-off is required between both aspects. One of the benefits of UWB
is inherent resilience against small-scale-fading, which allows the Physical Layer of the
OSI model (PHY) to operate with higher energy efficiency. The first aim of this thesis
is to understand and mathematically model the Small-Scale Fading (SSF) behaviour of

6 CHAPTER 1. GENERAL INTRODUCTION
Chapter 1:
General Introduction
Chapter 2:
Theory of fading UWB channels
Chapter 4:
Theory of TR UWB communication
Chapter 3:
Fading of measured UWB channels
Chapter 5:
Analysis of TR UWB communication
Chapter 6:
Design of a high-rate TR UWB system
Figure 1.2: Organization of the thesis
the UWB radio channel to ultimately allow for an educated trade-off between system
performance and complexity. Having low-cost and low-complexity in mind, the second
aim of the work is to model and understand the fundamental behaviour of UWB wireless
communications using Transmitted Reference (TR) signaling and Autocorrelation
Receivers (AcRs). Based on the developed understanding on UWB, SSF and UWB TR
communications, the final aim is to design a low-cost UWB PHY for WPAN operating
at 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 first
branch consists of the subsequent Chapters 2 and 3, which deal with the theory and
practice of SSF on UWB channels, respectively. The second branch deals with the theory
and practice of TR UWB systems in Chapter 4 and 5, respectively. The insight gained
in both branches is used for the design of a high-rate TR-UWB system in Chapter 6. A
graphical 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’s
contributions.
Chapter 2
Chapter 2 relates the statistics of SSF on UWB channels and its dependence on bandwidth
in 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 eigenvalues
are 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) and
Non-Line-of-Sight (NLOS) UWB channels. The diversity level is shown to scale linearly
with respect to the Root Mean Square (RMS)-delay-spread-by-bandwidth product, both
for LOS and NLOS channels.
Finally, upper bounds for the uncoded and coded Bit Error Rate (BER) for ideal UWB
systems will be presented using the eigenvalues of the channel. These bounds allow for a
trade-off analysis between bandwidth and BER performance of UWB systems on NLOS
UWB channels. Assuming a typical RMS delay spread for indoor environments, the
upper bound for the performance of Multiband OFDM systems using frequency hopping
is 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) radio
channels [16].
• Derivation of a lower bound for the diversity level of UWB channels, which converges
to the actual diversity level with increasing bandwidth. The lower bound shows a
linear relationship between the diversity level, bandwidth and RMS-delay-spread,
both for LOS and NLOS channels. This relationship is well-known, but, to our
knowledge, 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 measurement
data of UWB radio channels both emphasizing its strengths and short-comings. Firstly,
the channel measurement campaign is described briefly. The statistical properties of the
model 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 radio
channel have been analyzed. The diversity level as function of bandwidth of measured
radio channels is compared with the theoretical results. Finally, the BER predicted by
theory is compared with the BER on measured channels.
The main contributions are:
• On NLOS channels, the theoretical model was found to be reasonably accurate, but
not exact because the independence assumption of the PC is not valid for the used
measurement data. It is expected that a better prediction is obtained for richer
multipath environments. [to be published].
• For LOS channels, the predicted diversity level of the theoretical model is considerably
lower than for measured LOS channels. In practice, the LOS eigenvalue does
not share a PC-dimension with the largest NLOS eigenvalue, but one which is considerably
smaller. The result is considerably less fading. The mechanism(s) behind
have not been unveiled. [to be published].

8 CHAPTER 1. GENERAL INTRODUCTION
Chapter 4
Firstly, a brief introduction of TR signaling is presented including its strengths and shortcomings
with respect to performance and implementation. To overcome some of these
shortcomings, several extensions of the TR principle are proposed. First, a fractional
sampling AcR structure is proposed to relax synchronization and allow for weighted
autocorrelation, while simplifying the implementation. Second, a complex-valued AcR
is proposed to make the system less sensitive against delay mismatches. Additionally,
complex-valued modulation for TR signaling is proposed. To understand the system’s
behaviour, a general-purpose discrete-time equivalent system model is derived and presented,
where general-purpose means that all extensions are taken into account for. Several
interpretations for the system model are presented, which allow for more insight in
the behaviour of TR systems in various situations. Finally, the statistical properties of
TR UWB system are presented.
The main contributions are:
• Proposal of a fractional sampling autocorrelation receiver to relax synchronization
and allow for weighted autocorrelation demodulation [17].
• Proposal of a complex-valued autocorrelation receiver to relax delay implementation
and allow for complex-valued TR signaling [18].
• Development of a general-purpose model for TR UWB systems, which illustrates
that TR systems in the presence of ISI can be modelled using a second-order FIR
Volterra model [17].
• Development of a linear Multiple-Input, Multiple-Output (MIMO) model for the
second-order FIR Volterra model for TR systems, modulated with finite-alphabet
symbols [19]. The model shows that more ISI in a TR system can be suppressed
with increasing fractional sampling rate [17]. The model explains how the amount
of ISI that can be suppressed is influenced by the TR modulation [19].
• Finite state machine description for the finite-alphabet, second-order FIR Volterra
models, taking reference-pulse scrambling into account. The model shows that
reference-pulse scrambling may lead to a time-variant finite state machine, but
does not complicate a trellis-based equalizer significantly [to be published].
• Derivation of a reduced memory Finite State Machine (FSM) description for finite
alphabet, second-order FIR Volterra models, optimal in the sense of the MMSE
criterion. The model allows for tradeoff analyses between equalizer complexity and
system 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.
The main contributions are:

1.4. THESIS OUTLINE AND CONTRIBUTIONS 9
• Closed-form derivation of the weighting coefficients, optimal in the sense of the
MRC 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 fading
[17].
• In the presence of ISI, more ISI can be suppressed using linear weighting if the FSR
is increased [17].
• In the presence of ISI, the modulation has a significant impact on the amount of
ISI 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 a
TR-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. The
use of trellis-based equalization is considered, to support high data rate. To reduce the
equalizer complexity, the multiband concept, originally proposed for energy detectors,
is applied to TR signaling. The system performance is analyzed taking into account
Forward Error Control (FEC) and using turbo equalization.
The main contributions are:
• Proposal of scrambled QPSK-TR signaling, which avoids spectral spikes, while preserving
the time-invariant character of the FSM describing the Volterra model [to
be published].
• Proposal of multiband TR signaling to reduce the equalizer complexity, while allowing
for higher data rates. Application of the multiband concept allows for an
improvement of 3 dB, while reducing the equalizer complexity by a factor 16 [20].
• Application of turbo equalization to (multiband) TR UWB systems. A performance
improvement of 1.5-3 dB is observed with respect to the Frame Error Rate (FER) [to
be 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 Weighted
Autocorrelation Receivers,” IEEE Transactions on Microwave Theory and Techniques,
Apr. 2006, vol.54, pp.1754-1761, Special Issue on Ultra-Wideband Systems

10 CHAPTER 1. GENERAL INTRODUCTION
Conference Papers
[21] G. Durisi, J. Romme and S. Benedetto, ”A general method for SER computation of
M-PAM and M-PPM UWB systems for indoor multiuser communications,” IEEE Global
Telecommunications Conference (GLOBECOM), Dec. 2003, vol.2, pp.734-738
[22] D. Manteuffel, T.A. Ould-Mohamed and J.Romme, ”Impact of Integration in Consumer
Electronics on the performance of MB-OFDM UWB,” International Conference on
Electromagnetics 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 UWB
systems,” 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 Weighted
Correlation,” Internal Workshop on UWB Systems Joint with Conference on UWB Systems
and Technologies, May 2004, pp.141-145, Kyoto, Japan,
[16] J. Romme and B. Kull, ”On the relation between bandwidth and robustness of indoor
UWB 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 impulse
radio,” 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 Second
Order FIR Volterra Channels in Non-Coherent UWB Systems,” European Signal
Processing Conference (EUSIPCO), Sep. 2006, Florence, Italy, invited paper
[19] J. Romme and K. Witrisal, ”Impact of UWB Transmitted-Reference Modulation on
Linear 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 for
Transmitted-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 Insertion,”
IEEE Global Telecommunications Conference (GLOBECOM), Nov. 2000, vol.2,
pp.830-834, San Francisco, CA

1.4. THESIS OUTLINE AND CONTRIBUTIONS 11
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: Workshop
on UWB for Sensor Networks, Nov. 2005, Lausanne, CH
Unpublished
J. Romme and K. Witrisal, ”Estimation of Nakagami m Parameter for Frequency Selective
Rayleigh Fading Channels,” IEEE Communications Letters, In Preparation

12 CHAPTER 1. GENERAL INTRODUCTION

Chapter 2
Theory of Fading UWB Channels
2.1 Introduction
Understanding the mechanisms behind radio propagation is mandatory for any engineer
evaluating and optimizing the performance of wireless radio communication systems.
This chapter is on the theory of SSF of UWB channels, having in mind indoor data
communication. The goal is to relate the statistical properties of the SSF to general
channel 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 derived
and 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, the
frequency diversity of radio channels in general and UWB channel specifically is quantified
using the eigenvalues of the channel. In Sec. 2.4, the uncoded and coded BER for ideal
UWB systems are presented based on the eigenvalues of the channel, which is useful for
trade-off analyses between bandwidth and BER performance. Finally, conclusions are
drawn in Sec. 2.5.
2.1.1 The Radio Channel
Consider a radio communication system consisting of a transmitter and receiver operating
in an indoor environment. To allow for radio communication, both deploy antennas to
convert electrical signals into radio signals.
In its most elementary form, an antenna consists of two conductive objects, which
are electrically isolated from each other. By applying a time-variant Radio Frequency
(RF) signal to the antenna connectors, electrical and magnetic fields form around the
antenna. The combined fields generate self-sustaining Electro-Magnetic (EM) waves,
allowing energy to ”release” itself from the antenna and to propagate into the surrounding
environment.
In the environment, the EM waves will interact with the objects they encounter. A
typical indoor environment contains many objects, e.g. walls cabinets and chairs. Three
1 Strictly speaking, the radio channel itself has no bandwidth. It is the bandwidth of the transmit
signal 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 and
smooth surface compared to the wavelength. Examples of such objects are a wall or
cabinet. 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 encountered
object are in the order of the wavelength or less and causes the radio signal
to 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 raypath
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-principle
can model the behaviour of radio wave propagation in such scenarios [29, 30]. Since the
object blocks part of the Huygens sources, the radio signal bends around the object. This
phenomenon 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 the
receiver, each with its own delay, direction, distortion and intensity. Each EM wave will
generate a signal in the antenna such that the overall signal at the antenna connectors is
the 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 multipath
radio channel model is introduced. In this model, the radio signal is assumed to
propagate from the transmitter to the receiver along distinct paths, where each path introduces
its own attenuation and delay, see Fig. 2.1. This phenomenon is called multipath
propagation and the channel over which the radio signal propagates is referred to as the
multipath channel. Most often, the propagation environment will vary in time such that
path delays and path attenuations will be a function of time. For instance, the transmitter
and/or the receiver can move. Even if both are static, the environment itself may be
subject to change.
Based on the described mechanisms of indoor radio propagation, a model for the radio
channel can be obtained. Each time-variant path is characterized by a delay τ n (t) and
amplitude gain β n (t), where n identifies the path. Based on this assumption, the received
signal appears as a train of identically shaped transmit pulses, which possibly overlap in
time. The time-variant Channel Impulse Response (CIR) h(τ, t) can thus be formulated
as
h(τ, t) =
N∑
p(t)
n=1
β n (t)δ(τ − τ n (t)), (2.1)
where N p (t) denotes the number of observed multipath components at time t. 2
2 The mathematical representation is both valid for passband and baseband representations of passband
channels. In the baseband case, β n (t) is complex-valued and its phase is related to the path delay
τ n (t) according to arg(β n (t)) = 2πf c τ n (t)[rad], where f c denotes the center frequency

2.1. INTRODUCTION 15
Scatterer
Scatterer
0000 1111
0000 1111
0000 1111
0000 1111
0000 1111
0000 1111
TX Antenna
0000 1111
0000 1111
0000 1111
0000 1111
0000 1111
0000 1111
RX Antenna
Scatterer
Figure 2.1: The multipath radio channel
Following the discussion in the previous section, it is evident that the multipath channel
model is an oversimplification of reality. For instance, the ray-model of (2.1) does not
include diffraction. Nevertheless, the assumption is widely accepted, because the resulting
model is intuitive, practical and, more importantly, the results closely resembles reality
for narrowband channels. Although yet to be proven for UWB channels, the multipath
model 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. The
Power Delay Profile (PDP) is defined as the power of the CIR as a function of τ. The
CIR h(τ, t) has a PDP given by
P(τ, t) = |h(τ, t)| 2
= ∑ n
|β n (t)| 2 δ (τ − τ n (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

16 CHAPTER 2. THEORY OF FADING UWB CHANNELS
of the PDP, i.e.
∞∫
τ d (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 excess
delay using the normalized path energies as a weighting function.
The RMS delay spread is often averaged over space. In this manner, it does no
longer characterize a single CIR, but a certain propagation environment. The average
RMS delay spread is an important measure to characterize radio channels and used to
model the Average Power Delay Profile (APDP). An exponential decay model is a widely
accepted model for the APDP in NLOS environments for UWB and radio channels in
general [32, 33, 34]. This model is described by the equation,
E [ {
|h(τ)| 2] A 2
σ
=
exp ( )
− τ ∀ τ ≥ 0,
σ
(2.5)
0 ∀ τ < 0.
where E[.] denotes a mathematical expectation and the parameters σ and A 2 allow the
model to mimic specific NLOS radio environments and should be chosen such that σ = τ d
and A 2 = ∑ N p
n=1 |β n| 2 .
The model can be generalized to include LOS scenarios, by adding an additional
component to the APDP,
E [ |h(τ)| 2] =
{
A 2 K
δ(τ) + A2 exp( )
− τ for all τ ≥ 0,
(K+1) σ(K+1) σ
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-defined
to
σ = τ d
K + 1
√
2K + 1
. (2.7)
These parameters will be used throughout this thesis report as characterization of the
radio 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 not
only to radio communication engineers. Anyone who listens to their car radio is likely
to have observed the following phenomenon. While stopping at a traffic light, first the
reception is very poor, but by moving the car only slightly the audio signal quality
improves drastically. This phenomenon is referred to as fading.
In case of a narrowband signal y(t) with a center frequency f c , the impact of the radio
channel can be well approximated by a scalar multiplication, such that the received signal
will be
r(t) ≈ H(f c , t)y(t). (2.8)

2.1. INTRODUCTION 17
In this case, the channel is referred to as flat fading, since all frequency components of
y(t) are scaled equally [31].
The scalar multiplication factor H(f, t) is the Channel Frequency Response (CFR) at
time t, which is equal to the Fourier transform of h(τ, t) with respect to τ, i.e.
H(f, t) =
N∑
p(t)
n=1
β n (t) exp (j2πfτ n (t)) (2.9)
The equation shows that each radio path has its own distinct phase. Since H(f, t) is the
summation of all paths, the paths can interfere destructively with each other. By moving
slightly, the number of paths and the path amplitude gains will not change. However
the phase of each path can change significantly. Hence, the interference between paths
is 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, the
variation of H(f, t) as function of time is often modelled as a complex-valued 3 Gaussian
distributed 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 β n (t) is truly
dominant. 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 the
statistics of measured channel gain with good accuracy.
If one of the rays is dominant, which is often the case in LOS environments, a generalization
of the Rayleigh distribution, called the Rice distribution, accurately models the
statistics of measured channel gain [31]. More on the Rice distribution will follow in the
remainder 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 vary
extensively. For 1 percent of time, the received signal power will be 20 dB lower than its
average. To complicate matters, the received power can vary rapidly and unpredictably,
making it difficult for the transmitter to compensate for the variations using power control.
4 Therefore, radio communication systems often use large fading margins, which
inevitably reduces the system’s energy efficiency.
Fortunately, one can reduce the probability of such deep fades and waste less TX power
on fading margins. If the information is communicated over two or more independently
faded channels, evidently the probability that all channels are in a deep fade simultaneously
becomes smaller. This probability decreases with every additional channel used.
The principle described here is referred to as diversity and the amount of independently
fading channels is called the diversity level. Diversity can be found in three directions of
the radio channel, namely space, time and frequency. 5
The availability of independent fading channels is not sufficient. To exploit the diversity,
it should be ensured that the radiated energy related to a single unit of information
3 Assuming a baseband notation.
4 Assuming a return channel to inform the transmitter on the channel state.
5 In 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 on
frequency diversity.

18 CHAPTER 2. THEORY OF FADING UWB CHANNELS
1
p(|H(ω 0
)| = r)
0.5
10 0
0
−30 −25 −20 −15 −10 −5 0 5 10
20log10(r)
E[|H(ω 0
)| ≤ r]
10 −1
10 −2
10 −3
−30 −25 −20 −15 −10 −5 0 5 10
20log10(r)
Figure 2.2: The Rayleigh distribution
is spreads over multiple and at best all available fading channels. The drawback is that
parts of the TX signal are communicated over independent fading channels and thus affected
differently. Inherently, the receiver has to conduct signal processing on the received
signal in order to exploit the diversity. This type of signal processing is referred to as
diversity combining.
Several signal processing techniques for diversity combining exist, each with its own
performance and complexity. Assuming Gaussian noise and the absence of Inter Symbol
Interference (ISI), Maximum Ratio Combining (MRC) is the optimal one with respect to
both the Signal-to-Noise Ratio (SNR) and BER. Other techniques are Minimum Mean
Square Error (MMSE) combining, switched combining, selective combining and equalgain
combining. More information on diversity and diversity combining can be found in
literature [36, 31].
Due to their large bandwidth, UWB systems inherently allow for a large amount of frequency
diversity, explaining the large interest of both industry and academic society. The
focus 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 the
second chapter, the frequency diversity is analyzed using radio channel measurements to
validate 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 frequency
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. These
results are not only of statistical relevance, but also explain the mechanism of frequency
selective fading channels. Furthermore, the eigenvalues of UWB US channels are derived
in 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 complexvalued,
zero-mean Gaussian function. Evidently, the CFR at two distinct frequencies
f 1 and f 2 at the same time instant t will be correlated if the two frequencies are close
together. To capture the statistical properties of CFR, we introduce the correlation
function of the frequency response
For the US case, the result is well-known [37], namely
φ(f 1 , f 2 )E[H(f 1 )H ∗ (f 2 )]. (2.10)
φ(f 1 , f 2 ) =
∫ ∞
E [ |h(τ)| 2] e −j2π∆ fτ dτ (2.11)
−∞
where ∆ f is defined equal to f 1 −f 2 . Since its value depends only on the frequency difference,
φ(f 1 , f 2 ) is inherently Hermitian and Toeplitz. Furthermore, E[|h(τ)| 2 ] is definition
the APDP as defined in Sec. 2.1.3.
Substitution of the NLOS APDP model of 2.5 into (2.11) leads to the following expression
for the frequency correlation,
where
φ(f 1 , f 1 − ∆ f ) = ρ(τ d ∆ f ) (2.12)
ρ(x) =
A 2
1 + j2πx
(2.13)
This result is easily generalized to include LOS scenarios, by adding a constant. For
illustrative purposes, the magnitude of ρ(x) has been depicted in Fig. 2.3.
The frequency correlation function is closely related to the coherence bandwidth. No
generally accepted definition exists for the coherence bandwidth, but in most cases, it is
defined as the frequency separation ∆ f for which ρ(τ d ∆ f ) equals 1/2. This definition will
also be used in this thesis. The coherence bandwidth in case of the APDP model is
B coh = √ 3/(2πτ d ) ≈ 0.28
τ d
. (2.14)
Hence, the analytical model states a reciprocal relation between B coh and τ d . The reciprocal
relationship between the RMS delay spread and coherence bandwidth is confirmed
by measurements (see [37]).

20 CHAPTER 2. THEORY OF FADING UWB CHANNELS
1
0.9
0.8
0.7
|ρ(∆f)|
0.6
0.5
0.4
0.3
0.2
0.1
0 0.2 0.4 0.6 0.8 1
τ d ∆ 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 channel
as ”experienced” by a UWB signal or in fact by any radio signal can be obtained from its
eigenvalues. 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 B
around a center frequency f c , such that
|Y (f)| 2 =
{
Pt
B
for |f − f c | ≤ 1 2 B,
0 otherwise.
(2.15)
The advantage of this definition is that the channel properties can be investigated without
any influence of the TX signal spectrum, except for the influence of bandwidth and center
frequency. To simplify the derivations, unit transmit power is assumed, i.e. P t = 1. Without
having impact on R(f), H(f) may be assumed to be zero outside the spectral mask
of Y (f) as well. Consequently, the two-dimensional autocorrelation function φ(f 1 , f 2 ) is
defined for a finite square area from f c − B/2 to f c + B/2 in both dimensions f 1 and f 2 ,
and zero otherwise.
Using Principal Component Analysis (PCA), the bandwidth-limited function φ(f 1 , f 2 )
can be decomposed into the most efficient set of eigenfunctions and eigenvalues, giving
us information on the uncorrelated random processes driving the CFR. In [38] PCA is
described as follows:
“The central idea of principal component analysis is to reduce the dimension-

2.2. FREQUENCY DOMAIN PROPERTIES OF UWB CHANNELS 21
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 principal
components, which are uncorrelated, and which are ordered so that the
first 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 range
under consideration. More information on PCA can be found in [39, 38].
Using PCA and the fact that φ(f 1 , f 2 ) is Hermitian, φ(f 1 , f 2 ) can be decomposed into
the following form,
1
B φ(f 1, f 2 ) =
∞∑
λ[k]G k (f 1 )G k (f 2 ), (2.16)
k=1
where λ[k] and G k (f) denotes the k-th eigenvalue and its eigenfunction, respectively. The
division by B in (2.16) ensures that the eigenvalues and eigenfunctions are dimensionless
and simplifies derivations later on. Since φ(f 1 , f 2 ) is Hermitian, the eigenfunctions are
orthogonal with respect to each other.
Although the summation index k theoretically goes to infinity, it can be truncated
to N without losing much accuracy by choosing N sufficiently large. The low-passcharacteristic
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 that
the truncated summation represents the best possible approximation using only N components.
The principal components can rarely be found in closed-form, except for some asymptotic
cases, see Sec. 2.2.3. Fortunately, numerical tools exist to obtain them, like Singular
Value Decomposition (SVD). In Fig. 2.4, the eigenvalues are depicted obtained using
SVD for different RMS-delay-spread-by-bandwidth products. It shows that the number
of significant eigenvalues increases with an increasing RMS-delay-spread-by-bandwidth
product. 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 interpretation
of radio channels. Any band-limited radio channel can be thought to be decomposed
by the eigenfunctions into N sub-channels, such that
H(f) =
N∑
u[k]G k (f). (2.17)
k=1
where u[k] is by definition equal to the inner-product < H(f), G k (f) >. Assuming H(f)
to be a complex-valued Gaussian distributed random function, u[k] will be a complexvalued
Gaussian distributed RV with a variance λ[k], which will be referred to as the k-th
PC of the channel. Using the orthogonality of the eigenfunctions, it can be shown that
u[k] is independent of u[l] if k is unequal to l. Hence, the radio channel can be seen as
the sum of N parallel independent fading channels.
Furthermore, the radio channel can be thought to decompose the transmit signal y(t)

22 CHAPTER 2. THEORY OF FADING UWB CHANNELS
80
70
60
τ d B = 0.5
τ d B = 2
τ d B = 5
50
λ[k]
40
30
20
10
0
5 10 15 20 25
index k
Figure 2.4: Dependence of the eigenvalue distribution on the τ d B product
into N sub-signals using a filter bank, since
R(f) = H(f)Y (f) =
k-th sub-signal
N∑ { }} {
u[k] G k (f) Y (f). (2.18)
} {{ }
k-th filter
k=1
where the k-th sub-signal is multiplied with the RV u[k], i.e. all sub-signals experience flatfading.
A graphical representation of this interpretation for the time-domain is presented
in 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 eigenvalues
could not be obtained in closed form. Analytical expressions however often lead to more
insight in the behaviour of the system with respect to its parameters. In this section, a
closed form approximate relationship will be presented between the channel eigenvalues
on 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(f 1 , f 2 ) was shown to have a Toeplitz structure. Furthermore,
φ(f 1 , f 2 ) is a banded function in the UWB case, i.e. the significant values are around
the main diagonal of φ(f 1 , f 2 ) and virtually zero otherwise. An illustration of a UWB
φ(f 1 , f 2 ) 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 of
banded Toeplitz functions exists. However, for a special case of Toeplitz functions the

2.2. FREQUENCY DOMAIN PROPERTIES OF UWB CHANNELS 23
u[N]
(y ∗ g N )(t)
y(t)
.
(y ∗ g 2 )(t)
u[2]
+
r(t)
u[1]
(y ∗ g 1 )(t)
Figure 2.5: Physical interpretation of the eigenfunctions and eigenvalues of radio channels
φ(f 1 , f 2 )
φ c (f 1 , f 2 )
1
1
0.8
0.8
|φ(f 1
,f 2
)| [dB]
0.6
0.4
0.2
|φ(f 1
,f 2
)| [dB]
0.6
0.4
0.2
0
5
0
5
0
5
0
5
0
0
τ d
(f 2
−f c
)
−5
−5
τ d
(f 1
−f c
)
τ d
(f 2
−f c
)
−5
−5
τ d
(f 1
−f c
)
Figure 2.6: Comparison between φ(f 1 , f 2 ) and φ c (f 1 , f 2 ) of a UWB channel

24 CHAPTER 2. THEORY OF FADING UWB CHANNELS
eigenvalues can be derived in closed-form, namely for circulant functions. Furthermore,
Gray has proven that every banded Toeplitz matrix has a circulant counterpart that
is asymptotically identical, such that their eigenvalues are asymptotically identical as
well [42].
Hence, the strategy is to derive a circulant function φ c (f 1 , f 2 ), which is asymptotically
identical to φ(f 1 , f 2 ). Like all Toeplitz functions, the value of φ(f 1 , f 2 ) and φ c (f 1 , f 2 )
depends only on the difference between both arguments. In case of φ(f 1 , f 2 ), its value is
determined by ρ(f 1 −f 2 ). To obtain a Toeplitz function φ c (f 1 , f 2 ), which is asymptotically
identical to φ(f 1 , f 2 ), we defined
φ c (f 1 , f 1 − ∆ f ) = ρ c (∆ f )
= ρ(∆ f ) + ρ ∗ (B − ∆ f ) (2.19)
To illustrate their relation, a comparison of φ c (f 1 , f 2 ) with φ(f 1 , f 2 ) 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 λ A [k] of a circulant matrix A must
be equal to the sum of λ B [k] and λ D [k], if B and D are also circulant matrices and
λ B [k] and λ D [k] their k-th eigenvalue, respectively.
Applying property 1 to the circulant function φ c (f 1 , f 2 ), the k-th Circulant Eigenvalue
(CEV) will be equal to
∫ 1
λ c [k] = ρ c (xB) exp (−j2πkx)dx (2.20)
0
This equation shows that the eigenvalues λ c [k] can be seen as the weights of the Fourier
series of the frequency domain autocorrelation function ρ c (xB), where the eigenfunctions
exp (−j2πkx) are the Fourier modes [43]. Substitution of (2.19) into (2.20) and using the
fact that ρ(x) is an Hermitian function, this result can be further simplified to
∫ 1
λ c [k] = (ρ(xB) + ρ ∗ (B − xB)) exp (−j2πkx)dx (2.21)
0
∫ 1
= ρ(xB) exp (−j2πkx)dx
−1
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.
λ c [k] =
∫ ∞
ρ(xB) exp (−j2πkx)dx
−∞
= F{ρ(xB)}
A 2
= F{
1 + j2πτ d xB }
= A2
τ d B exp(− k
τ d B ) (2.22)
Hence, the eigenvalues drop exponentially with increasing k in the asymptotic UWB case
at a pace inverse proportional to both B and τ d .
It has already been mentioned that by adding a constant to (2.13), also LOS scenarios
can be modelled. Since a constant function is also circulant and using property 2 of
circulant matrices on page 24, it can be understood that the eigenvalues for LOS scenarios
are equal to
with
λ c [k] = λ c,L [k] + λ c,N [k] (2.23)
A2 K
λ c,L [k] = δ[k]
(K + 1)
(2.24)
A 2
λ c,N [k] =
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 the
largest eigenvalue/variance of the NLOS part of the APDP. Since the eigenvalue of this
PC decreases with increasing bandwidth, the LOS component asymptotically has its own
dimension.
2.3 Diversity of UWB Channels
The gain of the radio channel is a valuable measure of the signal quality. Detailed knowledge
on its statistical properties is relevant for any system engineer, not only to predict
the average BER performance, but also how the BER will vary in time/space. Previously
in this thesis, the radio channel was modelled as a random process. Since the received
power depends on the radio channel, it will be modelled as random process as well. In
this section, the statistical properties of the power gain of the channel will be derived in
closed form as experienced by a UWB signal with bandwidth B.

26 CHAPTER 2. THEORY OF FADING UWB CHANNELS
2.3.1 The Mean Power Gain
Given a Transmitter (TX) signal with a PSD |Y (f)| 2 , the received signal power will be
equal to
P r =
∫ ∞
−∞
|H(f)| 2 |Y (f)| 2 df (2.26)
Using the definition of the TX PSD |Y (f)| 2 of (2.15), the received power will be
P r = P t
B
∫
f c+B/2
f c−B/2
|H(f)| 2 df. (2.27)
To isolate the transmit power from channel properties, let us define the Mean Power
Gain (MPG) of the channel as follows,
g c = 1 B
∫
f c+B/2
|H(f)| 2 df (2.28)
such that
f c−B/2
P r = g c P t (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
g c = 1 f c+B/2
∫
2
N∑
N∑ N∑
u[k]G k (f)
df = u[k]u ∗ [l] 1 f c+B/2
∫
G k (f)G ∗ l (f)df (2.30)
B ∣ ∣
B
f c−B/2
k=0
k=0
l=0
f c−B/2
Due to the orthonormality of the eigenfunctions, this simplifies to
g c =
N∑
|u[k]| 2 (2.31)
k=0
which shows that the MPG of the channel is related one-on-one to the value of the RVs
driving 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, since
they give information on the behaviour of the radio channel. Here, the relationship
between the MPG and the RVs driving the CFR simplifies the derivations greatly and is
therefore used as starting point. Hence,
[ N
]
∑
E[g c ] = E |u[k]| 2 , (2.32)
k=0

2.3. DIVERSITY OF UWB CHANNELS 27
since the RVs driving the radio channel u[k] are uncorrelated. This simplifies to
E[g c ] =
N∑
E [ |u[k]| 2] =
k=0
∞∑
λ[k] (2.33)
k=0
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 λ c [k]. Otherwise,
the approximation for the eigenvalues will be inaccurate. However, the sum over all
eigenvalues λ c [k] will be identical to the sum over all λ[k] independent of the bandwidth.
The sum over all eigenvalues λ[k] and λ c [k] is namely equal to trace of the function
φ(f 1 , f 2 ) and φ c (f 1 , f 2 ), respectively. Since the functions φ(f 1 , f 2 ) and φ c (f 1 , f 2 ) have
identical main diagonals, their trace and thus their sum over all eigenvalues are inevitably
equal to each other. Hence, λ[k] can be substituted by λ c [k] without affecting the result.
The expected MPG will thus be
E[g c ] =
∞∑
λ c [k] (2.34)
k=0
without the need to make assumptions regarding the channel bandwidth nor the environment.
In the UWB case, the function for the eigenvalues λ c [k] changes slowly for consecutive
k’s. Hence, the summation over λ c [k] can be approximated by an integration over λ c (ϑ)
if λ c (ϑ) = λ c [ϑ]. Because k is incremented with unit steps, no step-size factor is required,
such that
E[g c ] =
∫ ∞
0
λ c (ϑ)dϑ =
∫ ∞
0
A 2
τ d B exp(− ϑ [
τ d B )dϑ = −A 2 exp(− ϑ ] ∞
τ d B ) 0
= A 2 (2.35)
Hence, the expected MPG is equal to the accumulated path powers. This is not surprising,
since E[|H(f)| 2 ] = ∑ N p
n=1 β2 n 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 LOS
environments.
Let us continue with the derivation of the variance of the MPG, i.e.
[ N
]
∑
Var[g c ] = Var |u[k]| 2 (2.36)
Without making any additional assumptions, no further simplifications are possible. Although
the RVs u[k] are due to the PCA ensured to be uncorrelated, the variance of the
MPG 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
k=0
Var[g c ] =
N∑
Var [ |u[k]| 2] (2.37)
k=0

28 CHAPTER 2. THEORY OF FADING UWB CHANNELS
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 ] = λ 2 [k],
such that
∞∑
Var[g c ] = λ 2 [k]. (2.38)
Hence, the variability of the MPG depends on the distribution of the eigenvalues, which
on their turn depend on the RMS delay spread and bandwidth. For any bandwidth, it
can be proven that
k=0
∞∑
λ 2 [k] ≤
k=0
∞∑
λ 2 c[k] (2.39)
k=0
To obtain φ c (f 1 , f 2 ), additional correlation terms were introduced in φ(f 1 , f 2 ). These
additional correlation terms unavoidably lead to an increase of the sum over the squared
eigenvalues. Nevertheless, they are asymptotically identical for an RMS-delay-spread-bybandwidth
product going to infinity. Hence, the following upper-bound can be derived
for the variance of the MPG,
Var[g c ] ≤
∞∑
λ 2 c[k]. (2.40)
k=0
As stated before, λ c [k] changes slowly for consecutive k’s in the UWB case, such that the
summation can be replaced by an integration without altering the result, i.e.
∞∑
λ 2 c[k] =
k=0
≈
≈
∞∑
( A
2
τ d B exp(− k ) 2 ∞∑
τ d B ) =
k=0
∫ ∞
A 4
0
τ 2 d
B2
exp(−2
ϑ
τ d B )dϑ ≈ [
τ 2 k=0 d
A 4
− A4
B2
exp(−2
k
τ d B )
2τ d B exp(−2 ϑ ] ∞
τ d B ) 0
A4
2τ d B . (2.41)
In other words, the variance of the MPG is smaller or equal to A4
2τ d
, which shows a
B
reciprocal relation between the MPG variance and the RMS-delay-spread-by-bandwidth
product.
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) also
applies to LOS channels. However, the variance of the MPG for both channel types will
be 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 = 0
with the largest NLOS RV. The power of the LOS PC and the power of the NLOS PC
have been found to be λ c,L and λ c,N , respectively. As in the NLOS case, the NLOS PC
is assumed to be a circular zero mean complex Gaussian distributed RV and the LOS

2.3. DIVERSITY OF UWB CHANNELS 29
component is modelled as a circular complex RV with a random phase and constant magnitude.
This corresponds to the traditional model for LOS flat-fading channels. In other
word, the resulting PC, obtained from the superposition of the constant magnitude RV
and the Gaussian distributed RV, will have a magnitude that is Ricean distributed. The
Rice 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 to
the power contribution of the non-LOS PC, i.e. κ = λ c,L /λ c,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 Rayleigh
distributed. As a result, the MPG will be the superposition of a squared Rice distributed
RV and N − 1 squared Rayleigh distributed RVs of which the variance will be smaller
than
Var[g c ] ≤
(
A 4
∞
2K
(K + 1) 2 σB + ∑
k=0
(
1
σ 2 B exp −2 k ) ) (2.42)
2 σB
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[g c ] ≤
A4 4K + 1
2σB (K + 1) 2. (2.43)
Similar to the NLOS case, the variance is found to be reciprocal with respect to
bandwidth. Looking at the impact of the Ricean K-factor, a remarkable insight can be
obtained. First, let us consider the case that K = 0, which actually relates to a NLOS
scenario. Realizing that σ will be equal to τ d , 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 MPG
will increase and a maximum is obtained at K = 1/2 at which the variance will be 4/3
times the variance at K = 0. Only from there on, the variance starts to decrease and
ultimately 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 the
MPG. 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 of
the eigenvalues, which in turn depends on RMS-delay-spread-by-bandwidth product and
the accumulated path powers A 2 . The dependency on A 2 makes it less useful as measure
for the frequency diversity of the radio channel. To obtain such an objective measure, we
consider,
m = E[g c] 2
(2.44)
Var[g c ]
6 Here the Ricean κ factor defines the ratio of the LOS component gain with respect to overall
channel gain in the first dimension only. The Ricean K-factor as defined in (2.6) is the ratio of the LOS
component gain with respect to gain of the complete APDP, i.e. with respect to the channel gain over
all dimensions, see (2.35).

30 CHAPTER 2. THEORY OF FADING UWB CHANNELS
which will be referred to as diversity level or diversity order. An intuitive explanation for
the diversity level can be given as well. A channel with a diversity level m has the same
diversity 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 the
signal 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 be
computed 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 as
special case. Later on the results will be simplified to the NLOS scenarios. The diversity
level computed using the CEVs will be denoted by m c , where
m c =
2K
+ ∑ ∞
σB
k=0
(K + 1) 2
1
σ 2 B 2 exp ( −2 k
σB
)
(2.45)
The second lower-bound for the diversity level is obtained by approximating the summation
by an integration using the UWB assumption. Using the closed-form expression
for the asymptotic UWB case of the eigenvalues, the m value can be approximated. The
diversity level computed using the UWB assumption will be denoted by m UWB , where
(K + 1)2
m uwb = 2σB
4K + 1
(2.46)
which shows that in the UWB case both in LOS and NLOS scenarios the diversity level
is proportional to the bandwidth. For NLOS scenarios, the result further simplifies to
m uwb = 2τ d B, which is a rather intuitive result already. Both m c and m uwb can be used
as lower-bound for the actual diversity level m. Since the variance of the MPG is less or
equal to the sum of squared eigenvalues λ c [k], it is evident that m c is a lower-bound for
m.
In Fig. 2.7, both lower-bounds for the diversity level of NLOS scenarios are compared
with 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 constantly
equal to 1, which means that the signal experiences a flat-fading channel. For a
bandwidth 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 a
slope equal to two. Furthermore, the diversity level comes close to the lower-bound if the
bandwidth is approx. 5 − 10 times the coherence bandwidth. This linear increase of the
diversity level with the bandwidth is confirmed by analyses of UWB channel measurement
data, 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 only
marginally increased starting from zero, which is rather counter-intuitive. When increasing
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 2 τ d B [ ]
K=10
K=0
K=0.5
m [ ]
10 1
10 0
10 −1 10 0 10 1
m c
m uwb
τ d B coh
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 perform
optimal detection [37]. The system does not suffer from ISI. Taking the fading into
account, the average uncoded BER of BPSK over a frequency selective Rayleigh fading
channel, denoted by Q f (.), is given by
Q f
(
Eb,TX
N 0
)
=
∫ ∞
−∞
Q
(√ )
2Eb,TX g c
p (g c )dg c , (2.47)
N 0
where E b,TX denotes the transmitted energy per bit and N 0 the noise power spectral
density. For performance analysis it is common to express the BER as function of the
average received energy per bit E b over the noise spectral density. The variable E b is
related to the MPG according to
E b = E b,TX E[g c ] (2.48)

32 CHAPTER 2. THEORY OF FADING UWB CHANNELS
so that the average uncoded BER of BPSK over a frequency selective Rayleigh fading
channel is equal to
Q f
(
Eb
N 0
)
=
∫ ∞
−∞
Q
(√
2 E b g c
N 0 E[g c ]
)
p (g c )dg c , (2.49)
The PDF of the MPG is dictated by the eigenvalues of the radio channel. In [45], a closed
form expression has been derived for the BER of BPSK as function of E b /N 0 on diversity
channels that requires only the eigenvalues of the diversity channel. Hence, the function
Q f (.) is defined as follows,
( )
Eb
Q f = 1 ∑L−1
ϱ k I(λ n [k], m[k]) (2.50)
N 0 2
k=0
where m[k] and ϱ k denote the number of occurrence and k-th residue in the partialfraction
expansion of the k-th normalized eigenvalue λ n [k], respectively. The normalized
eigenvalue λ n [k] is equal to λ[k] normalized as follows
λ n [k] =
E b
∑
N L
(2.51)
0 k=0
λ[k]λ[k],
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
ϱ k =
L−1
∏
l=0,l≠k
λ n [k]
λ n [k] − λ n [l]
(2.52)
and
( 1
I(c, m) =
2 − 1 2
√ ) m m−1
c
1 + c
∑
k=0
( m − 1 + k
k
) ( 1
2 + 1 2
√ ) k
c
(2.53)
1 + c
which concludes the derivation of the closed-form expression of the average bit error rate
using the eigenvalues. This expression will be used to quantify the impact of the diversity
level on the BER.
Using the union bound, it is straightforward to obtain an upper bound (UB) for the
average coded BER on FSFCs from the uncoded one. Equivalent to the coded BER
on AWGN channels (see [31]), the coded BER on Frequency Selective Fading Channels
(FSFCs) is
∞∑
( ) dEb
P b ≤ a d Q f (2.54)
N 0
d=d f
where d f denotes the free distance of the deployed code and a d denotes the number
of corrupted information bits accumulated over all erroneous paths with an Euclidean
distance of d.

2.4. BER ON UWB CHANNELS 33
An approximation (AP) for the average coded BER of FSFC, which is accurate at
high E b /N 0 -values can also be derived, namely
( )
df E b
P b ≈ a df Q f . (2.55)
N 0
If the approximation is close to the upper bound, it is known that both are close to the
actual BER.
2.4.2 Performance Analysis
In this subsection, the BER performance is presented for BPSK modulation on a frequency
selective Rayleigh fading channel as function of the RMS delay-spread-by-bandwidth
product, using the eigenvalues of the radio channel. The eigenvalues are obtained in
two manners. Firstly, by applying SVD on the discrete equivalent of the autocorrelation
function φ(f 1 , f 2 ), which represents the actual eigenvalues of the channel. Secondly, the
CEVs are used, which are obtained in closed form in Sec. 2.2.3. These eigenvalues converge
to the actual eigenvalues with increasing RMS-delay-spread-by-bandwidth product.
Additionally, the BER on an Additive White Gaussian Noise (AWGN) channel has been
depicted. In [31], the BER performance of an infinite bandwidth signal on a frequency
selective fading channel is proven to be identical to the BER performance on an AWGN
channel. Hence, the AWGN performance can be used as lower-bound for the average
BER on any FSFC.
In Fig. 2.8, the uncoded BER performance is depicted. As expected, an enlargement of
the RMS-delay-spread-by-bandwidth product leads to an improvement of the performance
in terms of the BER as function of the E b /N 0 . Furthermore, the BER curve computed
using the analytical eigenvalues converges indeed to the actual BER. If the RMS-delayspread-by-bandwidth
product is larger or equal to 5, the CEVs can be used for BER
analysis without introducing any significant error.
In Fig. 2.9 and Fig. 2.10, the Upper Bound (UB) and the approximation (AP) are
presented for the coded BER of (UWB) radio systems using convolutional coding on
frequency selective Rayleigh fading channels. The convolutional codes (CCs) of rate 1/2
and 1/3 used in WiMedia standard are assumed. The rate 1/3 CC has the generator
polynomials g 0 = 133 8 , g 1 = 165 8 , g 2 = 171 8 . The rate 1/2 CC is obtained by puncturing
the second output g 1 . Due to the absence of ISI, both the diversity gain and coding gain
are assumed to be fully exploited. Hence, the lower-bounds apply to all systems deploying
the 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 a
CC 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 than
the upper-bound using the actual eigenvalues. Hence, it is a useful bound for the BER
performance analysis on FSFC, although the bound is rather loose if the RMS-delayspread-by-bandwidth
product is small. In case of an RMS-delay-spread-by-bandwidth
product of 2, the CEV upper-bound is approx. 1 dB more conservative than the upperbound
computed from the actual CEVs for both code rates. If the product is equal to
5, the CEV upper-bound is at most 0.2 dB more conservative than the upper-bound
computed using the actual CEVs.

34 CHAPTER 2. THEORY OF FADING UWB CHANNELS
10 0 E b /N 0 [dB]
10 −1
10 −2
SVD,τ d B = 0.5
SVD,τ d B = 1
SVD,τ d B = 2
SVD,τ d B = 5
AWGN
CEV,τ d B = 0.5
CEV,τ d B = 1
CEV,τ d B = 2
CEV,τ d B = 5
BER
10 −3
10 −4
−5 0 5 10 15 20 25
Figure 2.8: The BER of BPSK modulation on frequency selective Rayleigh fading channels
with different RMS-delay-spread-by-bandwidth products
10 0 E b /N 0 [dB]
BER
10 −1
10 −2
UB,SVD,τ d B = 2
UB,SVD,τ d B = 5
UB,AWGN
UB,CEV,τ d B = 2
UB,CEV,τ d B = 5
UB,CEV,τ d B = 15
AP,SVD,τ d B = 2
AP,SVD,τ d B = 5
AP,AWGN
AP,CEV,τ d B = 2
AP,CEV,τ d B = 5
AP,CEV,τ d B = 15
10 −3
10 −4
−5 0 5 10 15
Figure 2.9: Bounds for the rate 1/2 CCd BER of BPSK modulation on frequency selective
Rayleigh fading channels with different RMS-delay-spread-by-bandwidth products

2.5. CONCLUSIONS 35
10 0 E b /N 0 [dB]
BER
10 −1
10 −2
UB,SVD,τ d B = 2
UB,SVD,τ d B = 5
UB,AWGN
UB,CEV,τ d B = 2
UB,CEV,τ d B = 5
UB,CEV,τ d B = 15
AP,SVD,τ d B = 2
AP,SVD,τ d B = 5
AP,AWGN
AP,CEV,τ d B = 2
AP,CEV,τ d B = 5
AP,CEV,τ d B = 15
10 −3
10 −4
−5 0 5 10 15
Figure 2.10: Bounds for the rate 1/3 CCd BER of BPSK modulation on frequency selective
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 the
actual lower-bound. By nature, this makes little sense and therefore useless as lowerbound
for BER performance analysis. However, if the RMS-delay-spread-by-bandwidth
product 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 five
apply to systems with a bandwidth of approx. 500 MHz on channels with an RMS delayspread
of 10 ns, e.g. multiband OFDM systems without frequency hopping. Using the
BER bounds for an AWGN channel as reference, and using the fact that the UB are close
to the actual performance at low BER, an energy efficiency gain of 3.1 dB at a BER of
10 −4 is possible.
The BER performance for an RMS-delay-spread-by-bandwidth product equal to fifteen,
apply to systems with a bandwidth of approx. 1.5 GHz on channels with an RMS
delay-spread of 10 ns, e.g. multiband OFDM systems with frequency hopping. Due to
numerical stability problems with the eigenvalues obtained using SVD, only the BER
bounds are presented using the CEV. In case of frequency hopping, the energy efficiency
can be improved only by 1.1 dB at a BER of 10 −4 . Here, losses in terms of energy
efficiency 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 presented
for the fading statistics of UWB radio channels both for LOS and NLOS channels.

36 CHAPTER 2. THEORY OF FADING UWB CHANNELS
The model describes in closed form the relationship between the eigenvalue distribution
of UWB radio channels, the signal bandwidth and the RMS-delay-spread. The NLOS
eigenvalues are found to follow an exponential curve of which the decay-factor depends
solely on the RMS-delay-spread-by-bandwidth product. Additionally, the LOS component
was found to be contained in the largest eigenvalues together with a component
resulting from the NLOS part of the PDP.
A single, insightful measure was proposed for the diversity level of fading channels and
closed-form under-bounds were derived for UWB fading channels both for the LOS and
NLOS case. In both cases, the diversity level was found to scale linearly with the RMSdelay-spread-by-bandwidth
product. Based on UWB radio channel measurements, the
same linear relationship has already been observed in [46, 41], but also sub-linear scaling
has been reported [40]. Furthermore, the theoretical model predicts that the presence of
an LOS component will increase the fading, if the Ricean K-factor has a value less than
two.
Additionally, upper bounds for the uncoded and coded BER for ideal UWB systems
were presented using the eigenvalues of the channel. These bounds are shown to be
accurate and useful for trade-off analyses between bandwidth and BER performance of
UWB systems on NLOS frequency selective Rayleigh fading channel. Assuming a typical
RMS delay spread for an indoor environment, the upper bound for the performance of
Multiband 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 only
needing to describe the APDP of the channel. Using the model, the performance of UWB
systems can be evaluated in closed-form up to the coded BER. In chapter 3, the fading
model will be verified using measurement data of UWB radio channels both emphasizing
its 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 was
derived. By definition, a model is a representation of a system that allows for investigation
of the (statistical) properties of the system. To achieve this goal, a model makes a series
of simplifying assumptions from which it deduces how the system will behave. It is a
deliberate simplification of reality. For a proper use of the model, the strengths and
weaknesses of the model have to be known by its user.
To accommodate these needs, the fading model of Chapter 2 will be verified in this
chapter using measurement data of UWB radio channels both emphasizing its strengths
and short-comings. The outline of the chapter is as follows. Firstly, the channel measurement
campaign is introduced in brief in Sec. 3.2, followed by a discussion of the indoor
UWB radio channel in the time and frequency domain in Sec. 3.3. The statistical properties
of the PCs of the measured UWB radio channel are analyzed in Sec. 3.3.1 and used
for 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 measurement
campaign conducted at the premises of IMST GmbH in Kamp-Lintfort [47]. Using a
vector network analyzer, the complex CFR was measured for the frequency range from
f 1 = 1 GHz to f 2 = 11 GHz. Two identical bi-conical horn antennas were used with a
gain of approx. 1 dBi at both the transmitter and receiver, that is approx. constant over
the whole frequency range. Both antennas were positioned at a height of 1.5 m above
floor level.
The RX antenna was mounted on a tripod and positioned at various positions within
the environment. The TX antenna was mounted on a rail and moved along the rail in
steps of 1 cm over a distance of 150 cm. During the measurement, the rail was moved
to obtain parallel tracks spaced 1 cm apart. As a result, the CFRs were obtained for
a 150 cm by 30 cm rectangular grid, where H i (f) denotes the i-th CFR measured at
frequency 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 environment
c○ J.Kunisch, IMST GmbH
the same position are gathered in a vector h[i] of length F. The frequency step size of
the measurements is 6.25 MHz, such that F = 1601.
The 1 cm spatial grid ensures that the spatial resolution is better than half a wavelength
over the complete measurement frequency band, to allow for the analysis of the
channel response as a function of space. During the channel measurements, the environment
was ensured to remain static, allowing for a time-invariant characterization of the
radio channel.
The deployed measurements were performed in an office of approx. 5 m by 5 m with a
height of 2.6 m. Within the office, positions were selected to obtain two different visibility
conditions, namely line-of-sight (LOS) and non-line-of-sight (NLOS). The positions of TX
grid and the RX during the LOS measurement were TxC and RxB, respectively. During
the NLOS measurements, the TX grid and RX are positioned respectively at TxA and
RxA. The LOS path has been blocked using a metal cabinet of size 1.78 m x 0.42 m x
1.96 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, its
behaviour in the delay domain is of relevance since both domains are related by the
Fourier transform.
All CIRs presented in the thesis are in the baseband to provide for a better view
on the radio paths, due to the absence of a carrier. Additionally, all CIRs have been
compensated for the propagation delay of the LOS component, i.e. the LOS component

3.3. OVERVIEW OF MEASUREMENT RESULTS 39
−65
−70
−75
−80
LOS
NLOS
Single CIR
Path Enh. CIRs
|h(τ)| 2 [dB(10GHz)]
−85
−90
−95
−100
−105
−110
Dense Multipath
−115
−10 0 10 20 30 40 50 60 70 80
excess delay [ns]
Figure 3.2: Example of local PDP in the LOS office environment
arrives at τ = 0. For illustrative reasons, the PDP 1 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 from
a small geometric area. Due to the limited size of the geometric area, distinct paths
present in each CIR arrive more or less with the same delay. Nevertheless, only the LOS
component truly adds up coherently. To ensure that the magnitude of the NLOS paths
has the proper relation to the LOS magnitude, an additional Gaussian filter has been
applied over the delay domain with a width of approx. 0.5 ns. The resulting APDP is
also 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 on
the walls. In this respect the UWB indoor radio channel differs from narrowband indoor
radio channels. In a typical indoor environment, the rays of different radio paths
arrive shortly after each other. Hence, only (ultra) wideband signals allow for the separation/identification
of these distinct radio paths.
Besides the presence of distinct radio paths, dense multipath can be identified in the
PDP. After the arrival of the LOS, the power contained in the dense multipath first
rapidly increases, achieves its maximum value after approx. 8 ns and, then follows an
exponential decay. The dense multipath is caused by the interaction of a radio signal with
objects like walls, which is more complex than a mere reflection. Although a significant
part is reflected, part of the ray energy is at first absorbed by an object to be released
at a later time instant. As a result, each reflection is followed by a tail with decaying
1 Every PDP is normalized by the measurement bandwidth to ensure independence of the measurement
bandwidth

40 CHAPTER 3. FADING OF MEASURED UWB CHANNELS
magnitude.
For the LOS as well as the NLOS measurements, the Ricean APDP model parameters
K and τ d have been extracted by [48]. Their results are listed in Tab. 3.1. These param-
Table 3.1: APDP Model parameters [48]
Scenario K τ d [ns] σ [ns]
LOS 1.26 8.75 10.54
NLOS 0 11.2 11.2
eters will be used when comparing the analytical results of Chapter 2 with measurement
data.
3.3.2 Frequency Domain
In Sec. 2.2, the CFR has been statistically characterized by the function φ(f 1 , f 2 ). As in
any measurement campaign, the CFR was measured at distinct frequencies only. Therefore
a discrete equivalent matrix Φ of φ(f 1 , f 2 ) is introduced. In this case h[i] can be
seen as the i-th realization of a random vector h, which is statistically characterized by
Φ. Due to the finite number of measured realizations, only an estimate for Φ can be
obtained, which will be denoted as W. The estimate W is as follows,
where M denotes the number of used CFRs and
W = 1 M HHH (3.1)
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 of
the 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, which
is equivalent to the main diagonal of the auto-covariance matrix W. Using the whole
measurement grids, W is computed and its main diagonal is presented for both LOS and
NLOS scenario in Fig. 3.3.
For the measurements, bi-conical antennas have been used, which have approx. a
constant gain. For constant gain antennas, the Friis transmission equation predicts a
6 dB gain loss with each doubling of the frequency [29]. Therefore, the measured channel
power gain decreases with increasing frequency. Due to the logarithmic scaling of both
axis, the frequency gain follows an approx. linear curve with a slope of −7 dB with each
doubling of the frequency. Additionally, spectral spikes can be observed above 8 Ghz
in both the LOS and NLOS scenario, most likely caused by interferers present during
measurement that are mixed to these frequencies. As a consequence, all measurement
data above 8 GHz will be considered less trust-worthy.
Additionally, the function φ(f 1 , f 2 ) was assumed to be banded. Let us verify whether
this 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 environment
(lower dashed curve) and a LOS environment (upper solid curve).
average CFR, its correlation-coefficient matrix C will be presented instead. The element
at 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 scenario
both using the complete measurement grid. Both results are presented in Fig. 3.4.
As expected for the NLOS environment, the measured correlation-coefficient matrix
is indeed a band-limited matrix. Somewhat larger out of band cross-correlations are
observed at frequencies lower than 3.5 GHz. The finite grid size in combination with the
slower spatial de-correlation of the CFRs at lower frequencies reduces the effective number
of uncorrelated observations, which is possibly causing the larger correlation coefficients
at lower frequencies.
The same analysis has been conducted for the LOS scenario. Based on the Ricean
APDP model, one expects the correlation to approach a certain floor with increasing
frequency separation whose amplitude depends on the Ricean K-factor. The measurement
results confirm the validness of the model.
To obtain another view on the banded character of W, the correlation coefficients have
been averaged over a frequency range from 1 until 3 GHz as function of the frequency
difference. This procedure is repeated several times while increasing the center frequency
in 2 GHz steps. The results have been depicted in Fig. 3.5.

42 CHAPTER 3. FADING OF MEASURED UWB CHANNELS
(a)
(b)
Figure 3.4: Estimated Frequency domain correlation function of the CFR in a NLOS and
LOS scenario in subplot (a) and (b), respectively.
1
0.9
0.8
LOS
|ρn(∆f)|
0.7
0.6
0.5
0.4
0.3 1−3 [GHz]
3−5 [GHz]
0.2 5−7 [GHz]
7−9 [GHz]
0.1 9−11 [GHz]
Theory
0
10 0 10 1 10 2 10 3
∆ f [MHz]
NLOS
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 frequency
range, indicating that the correlation indeed depends mostly on the frequency difference
and less on the center frequency. Starting from a frequency difference equal to zero, first a
fast 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 coherence
bandwidth equals approx. 20 MHz. A complete de-correlation cannot be expected due
to the finite-sized measurement pool. Taking this into account, a rather good match is
observed with respect to the theoretical model, even though the parameters for the model
presented 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 difference
that a correlation floor exists. Again a good match is observed between the measurement
data and the theoretical model. However, the floor is slightly higher for the frequency
range from 1-3 GHz, indicating that the optimal APDP parameters are weakly frequency
dependent.
3.4 Principal Components of Measured UWB Channels
Following the same structure as in Chapter 2, the PC of the radio channel measurements
are analyzed and compared with the theoretical results. Firstly, the algorithms
are described to extract the desired information from the measurements, followed by a
comparison 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 only
those elements corresponding to the frequency range under evaluation. For notational
convenience, the sub-matrix will be denoted by ˜H with ˜H ∈ C ˜F,M , where ˜F denotes the
number of frequency point in the frequency range under evaluation.
An estimate for the eigenvalues of the PCs is obtained by applying SVD on ˜F −1 ˜W =
V ˆΛV H , where V is a unitary matrix with eigenvectors and ˆΛ is a diagonal matrix
containing the eigenvalues. Although the eigenvalues are exact with respect to ˜F −1 ˜W,
they are only estimates of the actual eigenvalues of the channel. Therefore, the k-th
estimate of the channel eigenvalues will be denoted by ˆλ[k]. Without loss of generality,
the eigenvalues are assumed to have a descending order. The division by ˜F can be seen
as 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 as
follows,
U = 1˜F V H H, (3.4)
where the element at row k and column l denotes the l-th realization of the PC u[k]. For
notational convenience, all realizations of u[k] are gathered in the vector u[k].

44 CHAPTER 3. FADING OF MEASURED UWB CHANNELS
In the previous chapter, the PCs are assumed to be complex-valued Gaussian distributed
RVs, except for the PC containing the LOS component. To analyze whether this
assumption applies to the measurement data, the kurtosis of each PC is computed. The
kurtosis of a zero-mean, complex-valued RV x is defined as 2
k(x) = ˆµ 4(x)
− 2, (3.5)
ˆµ 2 2(x)
where x contains the realizations of x and ˆµ m (x) is an estimate of the m-th order moment
of x. For a zero-mean RV, an estimate for the m-th order moment is given by
ˆµ m (x) 1 N
N∑
|x[n]| m (3.6)
n=1
where N denotes the length of the vector x. The closer the magnitude of the kurtosis is
to zero, the better the validity of the Gaussian assumption.
In Sec. 2.2.3, the largest eigenvalue λ c [0] in LOS scenarios was shown to be the superposition
of the eigenvalues λ c,L [0] and λ c,N [0]. To accommodate its validation, a procedure
will be presented for the division of the estimate of the largest eigenvalue ˆλ[0] in its two
components.
The procedure consists of two steps. Firstly, the Ricean κ factor is estimated using a
method of moments [49], where the Ricean κ estimate for a RV x is shown to be obtained
by
ˆκ = −2ˆµ2 2(x) + ˆµ 4 (x) − ˆµ 2 (x) √ 2ˆµ 2 2(x) − ˆµ 4 (x)
, (3.7)
ˆµ 2 2(x) − ˆµ 4 (x)
where x denotes the vector containing the observations of x. Using the estimate ˆκ, both
components are obtained as follows
λ c,L [0] =
ˆκ
ˆκ + 1ˆλ[0] (3.8)
λ c,N [0] = 1
ˆκ + 1ˆλ[0], (3.9)
which concludes the description of the procedure to separate the largest eigenvalue in its
two components.
3.4.2 Verification of the NLOS Eigenvalues and Principal Components
The eigenvalues and kurtosis of all PCs are depicted in Fig. 3.6 for the NLOS scenario
for UWB signals with a bandwidth of 1 GHz and a center frequency of 4.5 GHz. As
reference, the theoretical eigenvalues and kurtosis have been depicted as well using the
APDP parameters presented in Tab. 3.1.
2 A value of two has been subtracted, to ensure that a complex-valued Gaussian RV has a kurtosis
equal 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
−60
2
λ est
[k]
−65
λ c
[k]
k(u[k])
k(u c
[k])
1.5
−70
1
−75
0.5
Gain [dB]
−80
0
kurtosis
−85
−0.5
−90
−1
−95
−1.5
−100
0 10 20 30 40 50
−2
60
Index k
Figure 3.6: The eigenvalues and kurtosis of the PCs of a NLOS UWB channel with a
bandwidth of 1 GHz
0 10 20 30 40 50 60
−60
2
λ est
[k]
−65
−70
λ c
[k]
(NLOS)
λ est [0]
λ c
(NLOS) [0]
k(u[k])
k(u c
[k])
1.5
1
−75
0.5
Gain [dB]
−80
0
kurtosis
−85
−0.5
−90
−1
−95
−1.5
−100
0 10 20 30 40 50
−2
60
Index k
Figure 3.7: The eigenvalues and kurtosis of the PCs of a LOS UWB channel with a
bandwidth of 1 GHz.

46 CHAPTER 3. FADING OF MEASURED UWB CHANNELS
It is seen that the eigenvalues of the PCs do not follow exactly an exponential decay
as expected based on the theoretical eigenvalues, possibly caused by the use of frequency
dependent gain antennas. Nevertheless, the measured eigenvalues match well with the
theoretical 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]| is
approx. Rayleigh distributed for all k in the NLOS case. Overall, a reasonably good
match can be observed between the theory and practice.
3.4.3 Verification of the LOS Eigenvalues and Principal Components
The same analysis has been repeated for the LOS measurement, again using the theoretical
eigenvalues and kurtosis as reference, see Fig. 3.7. Remember that the largest PC is
assumed to be a Ricean distributed RV, while all other PCs are assumed to be Rayleigh
distributed. Again a rather good match is found between theory and practice, although in
the LOS case the eigenvalues are shifted in weight towards the eigenvalues with a smaller
index.
Nevertheless, all is not as it seems. Fig. 3.7 reveals a 4 dB difference between the
expected and measured NLOS eigenvalues denoted by λ (NLOS)
c [k] and λ (NLOS)
est [k], respectively.
The theoretical model predicts the LOS component to share its dimension with the
PC 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. The
kurtosis is however not very sensitive in the vicinity of −1 for Ricean distributed RVs
and explains why the difference is so small. This discrepancy has a significant impact on
the 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 CFR
for a signal with center frequency f c and bandwidth B is defined as,
g c [i] = 1˜F
∥
∥˜h[i] ∥ 2 (3.10)
where the ˜h[i] contains only those elements within the corresponding frequency range
[
fc − 1 2 B, f c + 1 2 B] . Since frequency-domain oversampling is applied to H i (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 small
bandwidth of 10 MHz. Furthermore, the spatial separation between local maxima and
minima 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)
(b)
−60
−60
−65
−65
−70
−70
G i,j
−75
G i,j
−75
−80
−80
−85
−85
−90
30
−90
30
25
20
15
j
10
5
0
0
5
10
i
15
20
25
30
25
20
15
j
10
5
0
0
5
10
i
15
20
25
30
Figure 3.8: The MPG for a signal with B = 10 MHz,f c = 4.6 GHz in (a) and B = 1 GHz,
f c = 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 frequency
diversity 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 UWB
channel data. The first estimate is applied to both LOS and NLOS channels, while the
second 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 MPGs
measured in a local area, i.e.
ˆm m =
ˆµ 2 1(g c )
ˆµ 2 (g c ) − ˆµ 2 1(g c ) . (3.11)
Hence, this estimate makes no assumptions regarding the statistical properties of the
PCs, 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 are
independent 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
( ˆλ[k]) ∑N 2
k=0
ˆm R =
∑ N
k=0 ˆλ 2 [k]
. (3.12)
The third estimate is based on the LOS fading model of Sec. 2.3.1, which is referred
to as the Rice-Rayleigh fading model. Here, the largest PC is assumed to be a Ricean
distributed RV, while all others are assumed to be Rayleigh distributed. In contrast to the
Rayleigh distribution, the Rice distribution has an additional shape parameter κ, i.e. its
Probability Density Function (PDF) is not fully described by the estimated eigenvalues.

48 CHAPTER 3. FADING OF MEASURED UWB CHANNELS
The κ-parameter has been estimated using the method described in Sec. 3.4.1. Based on
the Rice-Rayleigh (RR) model, the estimate for the diversity level becomes,
( ˆλ[k]) ∑N 2
k=0
ˆm RR =
∑ ) , (3.13)
N ˆκ
k=0
(1 − δ[k] 2 ˆλ2 [k]
(1+ˆκ) 2
where the estimate ˆκ is obtained using the method of moments described by (3.7).
The estimate ˆm m 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 of
the estimates will be high, but also a bias will be present, which is proportional to the
actual diversity level.
Due to the experimental nature of the PCs based estimates ˆm R and ˆm RR , little is
known regarding their behaviour for finite measurement sets. In Appendix A, an analytical
evaluation is presented for the estimate m R . It was found to have a superior
performance with respect to the estimation variance compared to the moment based
estimate m m , in case the PCs are indeed independent Rayleigh distributed RVs. Furthermore,
the m R is found to be asymptotically unbiased. For finite set-sized, it is found to
produce downwards biased. One can compensate for this bias if the number of independent
observations is known. Due to spatial correlation, this is not the case. Therefore, no
effort has been made to compensate for any bias. Due to their similar nature, it is likely
that the estimate m RR 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 obtain
accurate estimates for the diversity level. Unfortunately, the luxury of large data sets
inherently does not apply to SSF analyses. The local area over which the diversity level
is estimated may not be to large. If chosen too large, the probability that distinct radiopaths
will appear and/or vanish becomes too high and by definition one can no longer
speak of SSF. Additionally, the data set of a single local-area will not contain uncorrelated
observations/measurements, due to spatial correlation. As a result, the effective area-size
will be reduced. Taking both aspects into consideration, the local-areas are limited to
a 30-by-50 cm rectangular area. Hence, the 150-by-30 cm measurement grids could be
divided into 3 adjacent local areas.
Furthermore, the diversity level is independent of the center frequency, at least from
a theoretical point of view. The validity of this assumption is partially covered by the
results of Sec. 3.3.2, where it is shown that the frequency domain correlation depends
mainly 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 this
manner, in total 12 subsets are obtained and used to extract the diversity level of the
measured channels.
The estimates for the diversity level are presented for both the NLOS and LOS channel
in 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 bandwidth,
as expected based on (2.45). Nevertheless, a difference can be observed between
the two estimates. At small bandwidths, the difference is still rather small, because the
underlying assumption of the NLOS fading model that the PCs are approx. independent
is valid. The diversity level m c is too small, because the circulant approximation for
φ(f 1 , f 2 ) is not accurate for such small RMS-delay-spread-by-bandwidth products.
With increasing bandwidth, the estimate m R under-estimates ˆm m . This is caused by
the presence of distinct NLOS paths reported in Sec. 3.3.1, which are more and more
resolved with increasing bandwidth. As a result, the PCs remain uncorrelated but are
no longer independent, explaining the discrepancy between both estimates. Finally, m R
converges to m c . It is expected that in environments with richer multipath, like e.g.
industrial environments, the difference between theory and practice is smaller. In any
case, m c was found to lower-bound ˆm m , possibly making it a useful conservative estimate
for the actual diversity level of NLOS channels. Whether this observation is universally
valid has not been determined.
In the LOS case, both estimates agree again rather well for small bandwidths. With
increasing bandwidth, a similar behaviour is observed as in the NLOS case; the estimate
m RR under-estimates ˆm m . Hence, the same reasoning can be applied as in the NLOS
case. However, the theoretical model m c constantly under-estimates the diversity level
by far. Even at higher bandwidth, where the model is expected to be accurate. The
discrepancy can be explained as follows. The theoretical model namely predicts the LOS
component to share a dimension with the largest NLOS eigenvalue, which leads to the
smallest diversity level. In combination with any other eigenvalue, the diversity level will
be higher, i.e. it represents the worst-case and can therefore be used as lower-bound for
the diversity level. In Fig. 3.7, the measured NLOS eigenvalue is shown to be considerably
smaller than the expected NLOS eigenvalue
The difference is responsible for the difference between the theoretical and measured
diversity level. 3 A possible cause is the alteration of the pulse-shape due to the frequency
dependency of the antennas, causing the LOS component to occupy another dimension
then the one expected assuming a frequency independent antenna gain. To incorporate
this phenomenon, an extension of the theoretical model is mandatory. When correcting
for this phenomena using the measurement data, one obtains the estimate m RR since it
uses ˆκ. This explains why m RR performs considerably better.
3.5.3 Verification of the Mean Power Gain
The differences between theory and practice with respect to ˆm m should not be overvalued.
At higher diversity levels the Cumulative Distribution Function (CDF) of the
MPG 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 using
the different estimates for the diversity level, assuming the square-root of the MPG to
be a Nakagami distributed RV. This distribution is not only selected because its CDF
3 In contrast to the legendary words of W.C. Jakes:”Nature is seldom kind.”, the UWB fading appears
to be one of those rare exceptions [35].

50 CHAPTER 3. FADING OF MEASURED UWB CHANNELS
40
m m
35
m c
m R
30
25
m
20
15
10
5
0
0 100 200 300 400 500 600 700 800 900 1000
B [MHz]
Figure 3.9: Comparison of the different diversity level estimates as function of bandwidth
in the NLOS scenario
60
m m
m c
50
m RR
40
m
30
20
10
0
0 100 200 300 400 500 600 700 800 900 1000
B [MHz]
Figure 3.10: Comparison of the different diversity level estimates as function of bandwidth
in 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 are
available for the BER on Nakagami m fading channels. Furthermore, a Nakagami fading
model 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 for
three 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 rather
well to the measured CDF, indicating the validity of the Nakagami m distribution. Those
estimates that produce a bad-fit are m c in the case of a 200 MHz bandwidth (N)LOS
channel and a 500 MHz bandwidth LOS channel, and m RR for all LOS channels. The
reasons for these bad fits have already been explained when discussing the diversity level
in 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 on
UWB 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 previous
sections 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 all
MPGs within that area. Both the average uncoded and the UB for the coded BERs are
presented for both the NLOS and LOS scenario. The convolutional code of rate 1/3 is
used as presented in Sec. 2.4. Two bandwidths have been considered, namely 200 MHz
and 500 MHz.
In Fig. 3.12, the obtained BERs are presented for the NLOS case. In Fig. 3.12(a) on
a 200 MHz bandwidth channel, the theoretical model is approx, 2 dB more conservative
than the BER based on the measured MPG at a BER of 1e −4 for reasons already presented
in Sec. 3.5.2. In the coded cases, depicted in Fig. 3.12(b), the differences becomes 5.5 dB
due to nature of coding. On good channels, the coding ensures practically error-free
communication, but when the signal comes below a certain SNR-threshold, the BER
rapidly becomes poor.
When increasing the bandwidth to 500 MHz, the differences become significantly
smaller. In the uncoded case, depicted in Fig. 3.12(a), the difference between the theoretical
MPG model and the measured MPGs is merely 0.7 dB at a BER of 1e −4 . The
difference will decrease further with increasing bandwidth. When comparing both in
the scenario with FEC, the difference will increase to 1.8 dB, which is still significantly
smaller than in the 200 MHz case. Both in the uncoded and coded case of Fig. 3.12, the
theoretical model performs approx. equally well as the model based on the estimated
diversity level m RR , indicating the validity of the model.
In Fig. 3.13, the obtained BERs are presented for the LOS case. In all cases, the

52 CHAPTER 3. FADING OF MEASURED UWB CHANNELS
(a)
1
0.9
0.8
CDF [P(MPG¡Abscissa)]
0.7
0.6
0.5
0.4
0.3
NLOS
LOS
Meas
m m
0.2
m c
0.1
m RR
0
−70 −68 −66 −64 −62 −60 −58 −56
Abscissa [dB]
m R
1
(b)
0.9
0.8
CDF [P(MPG¡Abscissa)]
0.7
0.6
0.5
0.4
0.3
NLOS
LOS
Meas
m m
0.2
m c
0.1
m RR
0
−70 −68 −66 −64 −62 −60 −58 −56
Abscissa [dB]
m R
(c)
1
0.9
0.8
CDF [P(MPG¡Abscissa)]
0.7
0.6
0.5
0.4
0.3
NLOS
LOS
Meas
m m
0.2
m c
0.1
m RR
0
−70 −68 −66 −64 −62 −60 −58 −56
Abscissa [dB]
m R
Figure 3.11: Comparison of the CDF of the measured MPG with the CDFs using the estimated
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 estimated
diversity level m RR , the principle validity of the Rice-Rayleigh fading channel model is
confirmed.
3.7 Conclusions
In this chapter, the fading model derived in the previous chapter was verified using
measured channels to obtain insight in the strength and weaknesses of the model. After
a short description of the radio channel measurement set-up, the typical behaviour has
been 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 channels
have been compared with the expectations based on the theoretical model. Several
algorithms have been described to obtain estimates for the eigenvalues, the PCs and
the kurtosis from the measurement data. The resulting estimates were compared with
expectation 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 model
and practice. Also for the LOS scenario, the estimates for the eigenvalues and kurtosis
matched reasonably well with theory. However, a 4 dB difference was observed between
the expected and measured NLOS part of the largest PC. When validating the diversity
level, this discrepancy was found to have a significant impact in the LOS scenario. In
both scenarios, the theoretical model was found to accurately describe the change in the
eigenvalue distribution of channel with increasing bandwidth.
For UWB NLOS scenarios, the diversity of the MPG predicted with the theoretical
model fitted rather well to the measured diversity. Both reveal a linear increase with
bandwidth. However, the theoretical model consistently under-estimated the diversity
level slightly. With increasing bandwidth, more and more distinct radio paths are resolved,
such that the PCs are no longer independent. It is expected that in environments
with richer multipath, like e.g. industrial environments, the difference between theory
and practice becomes smaller. In any case, the theoretical model was found to be a
conservative estimate for the actual diversity of UWB NLOS channels.
For UWB LOS scenarios, the theoretical model consistently under-estimated the actual
diversity level by far. In this case, the theoretical model predicts the LOS component
to share a dimension with the largest NLOS eigenvalue, which leads to the smallest diversity
level. The 4 dB smaller measured NLOS eigenvalue with respect to the theoretical
one, leads to a significant larger diversity level in practise. When compensating for this
discrepancy, the theoretical model is performing considerably better, indicating the basic
validity of the Rice-Rayleigh fading model. This gives hope that the theoretical fading
model for LOS channels can be refined.
Also the CDF of the MPG has been presented for signal with different bandwidths
in both a LOS and NLOS case. To convert the previously described estimates for the
diversity level, the MPG was assumed to be Nakagami distributed RV, where the diversity
level was used as shape parameter. The Nakagami distribution was shown to accurately
describe the CDF of the actual MPG.
Finally, the diversity level was used to obtains estimates for both the uncoded and

54 CHAPTER 3. FADING OF MEASURED UWB CHANNELS
(a)
MPG,200MHz
m ,200MHz
m
m ,200MHz
10 −1 cir
10 −2
m R
,200MHz
MPG,500MHz
m m
,500MHz
m cir
,500MHz
m R
,500MHz
10 −3
10 −4
0 2 4 6 8 10 12 14 16
(b)
MPG,200MHz
m ,200MHz
m
m ,200MHz
10 −1 cir
10 −2
m R
,200MHz
MPG,500MHz
m m
,500MHz
m cir
,500MHz
m R
,500MHz
10 −3
10 −4
0 2 4 6 8 10 12 14 16
Figure 3.12: Comparison of average BER on measured and modelled UWB NLOS channel
with different bandwidths both coded and uncoded, in sub-figure (a) and (b), respectively.

3.7. CONCLUSIONS 55
(a)
MPG,200MHz
m ,200MHz
m
m ,200MHz
10 −1 cir
10 −2
m RR
,200MHz
MPG,500MHz
m m
,500MHz
m cir
,500MHz
m RR
,500MHz
10 −3
10 −4
0 2 4 6 8 10 12 14 16
(b)
MPG,200MHz
m ,200MHz
m
m ,200MHz
10 −1 cir
10 −2
m RR
,200MHz
MPG,500MHz
m m
,500MHz
m cir
,500MHz
m RR
,500MHz
10 −3
10 −4
0 2 4 6 8 10 12 14 16
Figure 3.13: Comparison of average BER on measured and modelled UWB LOS channel
with different bandwidths both coded and uncoded, in sub-figure (a) and (b), respectively.

56 CHAPTER 3. FADING OF MEASURED UWB CHANNELS
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 the
actual BER performance, although the theoretical model in the NLOS case is fully defined
by the RMS delay spread only. For UWB LOS channels, the theoretical model was overly
conservative, making it less useful for BER analysis. Therefore, it is recommended to
refine/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 its
parameters are incorrect.
In general, it is concluded that the theoretical model accurately describes the statistical
behaviour of NLOS UWB channels. For a LOS UWB channels, the theoretical model
does not match well to reality and a refinement of the model is needed. The analysis
showed that the Rice-Rayleigh distribution is able to accurately describe the statistical
nature of measured LOS channels. The distribution parameters, obtained using the
theoretical 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. Unfortunately,
this advantage does not come without a price. A coherent receiver as used in
current 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 rake
fingers, due the richness of the UWB channel, i.e the large number of resolvable radio
paths [7, 51]. Additionally, each finger has to be synchronized to a distinct radio path
with high accuracy, due to the large signal bandwidth and the channel gain of each path
has to be estimated. To complicate matters further, each path distorts a UWB signal
differently [52], such that the template waveform at each rake-finger has to be adaptable
in order to be optimal.
Tomlinson and Hoctor proposed to combine TR signaling with an AcR for UWB
communications, to dispose of the need for channel estimation, while still capturing the
complete pulse energy [53]. Furthermore, its simple structure may sustain the promise of
UWB technology to bring low-cost wireless communication. As result, the UWB society
showed a great interest in this concept, resulting in many scientific studies, one of them
being presented in this thesis.
The aim of this chapter is to provide better insight in the behaviour of TR UWB
systems in various situations. Firstly, the principle of TR UWB communication will
be introduced, including a discussion of its pro’s and con’s with respect to performance
and implementation. Several extensions of the TR principle will be proposed. Firstly, a
fractional sampling AcR structure will be proposed to relax synchronization and allow
for weighted autocorrelation, while simplifying the implementation. Secondly, a complexvalued
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 scheme
to complex-valued modulation.
To understand the system’s behaviour, a general-purpose discrete-time equivalent system
model will be derived, where general-purpose means that all extensions are taken into
account. 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 statistical
properties of TR UWB system will be presented.
4.2 Principle of Transmitted Reference Communication
4.2.1 Transmitted-Reference Signaling
A TR symbol in its essence consists of two identically-shaped pulses p(t) transmitted
with a predefined time separation in between. The first pulse is left unmodulated, while
the second pulse is data modulated with b[n]. The time-separation in seconds D is short
compared to the coherence time of the channel, such that both pulses are distorted equally
by the channel. The TR UWB TX signal y(t) in a mathematical notation is as follows
∞∑
y(t) = p(t − nT s ) + b[n]p(t − nT s − D), (4.1)
n=−∞
where T s denotes the symbol duration in seconds, such that Ts
−1 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 RX
signal 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 pulses
is the same for all symbols. In this example, the time-interval D is 10 ns and the symbol
duration T s 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 a
reference for the demodulation of the second pulse, by computing essentially the shortterm
autocorrelation of the received signal at delay lag D. Similar to a matched filter, the
first pulse is used as reference for the demodulation of the modulated second pulse. Since
both 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 most
simple case, a single sample is generated for each TR symbol, which is further processed
using digital circuitry. The rate at which the digital circuitry operates is thus no longer
dictated by the bandwidth of the TR signal, such that the digital sampling and clock
rates can be significantly lower than the Nyquist rate. This allows for a reduction in cost
and power consumption for the digital circuitry.
In an AcR, the demodulation is performed in several stages. In the first stage, bandpass
filtering is applied to the received signal to mitigate out-of-band noise and interference.
The signal after the RX BPF r(t) will consist out of the desired signal and
noise
∞∑
r(t) = q(t − nT s ) + b[n]q(t − nT s − D) + n(t). (4.2)
n=−∞

4.2. PRINCIPLE OF TRANSMITTED REFERENCE COMMUNICATION 59
Figure 4.1: Signals at different stages in TR system

60 CHAPTER 4. THEORY OF TR UWB COMMUNICATIONS
BPF
∫ b
a .dt
DSP
Delay
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 f rx (t), i.e. q(t) = (p∗h∗f rx )(t). Assuming white Gaussian noise on
the channel with a double-sided spectral density N 0 /2, the noise after the BPF will be
coloured Gaussian noise with an autocorrelation function
r nn (τ) = N 0
2
∫ ∞
−∞
f rx (t + τ)f rx (t)dt (4.3)
In stage two, r(t) is divided over two parallel branches. The first branch leaves the
signal unaltered, while the second delays the signal by D seconds. The RX signal and its
delayed version are depicted in Fig. 4.1(b). Please notice that the modulated pulse and
reference pulse on the parallel branches now overlap in time.
In stage three, the output of both branches are multiplied with each other. The
multiplier output is depicted as a solid line in Fig. 4.1(c). By integrating the multiplier
output over the proper interval in stage four, the computation of the autocorrelation is
completed and the integrator output can be sampled. The block diagram of the described
AcR is depicted in Fig. 4.2.
An illustration of the signals in stage 3 can be found in Fig. 4.1(c). Here, the dotted
line represents the output of the integrator and the start and duration of integration
interval are denoted by means of a box. After ending the integration, the integrator
output is stable, i.e. it can be sampled for further processing by the digital circuitry. The
value of the received signal will be
u[n] =
∫ nTs+T end
nT s+T start
r(t)r(t − D)dt (4.4)
Afterwards, the integrator will be reset to zero and ready for the next TR-symbol. In the
absence of pulse-overlapping, noise and assuming appropriate integration intervals, the
value of the n-th sample u[n] will be equal to
u[n] =
∫ nTs+T end
nT s+T start
r(t)r(t − D)dt (4.5)
∫ nTs+T end
= b[n]q 2 (t − nT s − D)dt (4.6)
nT s+T start
= b[n]E q (4.7)

4.2. PRINCIPLE OF TRANSMITTED REFERENCE COMMUNICATION 61
where E q denotes the energy of the pulse q(t). The sign of u[n] will be equal to the BPSK
modulation b[n] applied, allowing for a simple threshold detection scheme in the digital
circuitry.
4.2.3 The Drawbacks
Nothing comes without a price and TR UWB communication is no exception. The
use of TR UWB leads to a loss of at least 6 dB compared to an ideal matched filter
receiver; 3 dB due to noise contained within the reference and 3 dB due to usage of two
pulses per bit, instead of one. The loss can be even higher. If the integration interval
duration is set too long (which is not the case in Fig. 4.1), additional noise is accumulated
during the integration. These noise terms can be identified in the multiplier output in
Fig. 4.1(c) in the intervals 〈10, 20〉, 〈50, 60〉, 〈110, 120〉 and 〈150, 160〉, where all values
are in nanoseconds. Furthermore, an additional noise signal exists not present in linear
RXs, 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. This
effect can be identified in the multiplier output in Fig. 4.1(c) in the interval from 60 ns
until 100 ns. A performance loss of at least 6 dB is rather high, but compared to more
realistic, sub-optimal rake receivers, equipped with only a limited amount of fingers and
imperfect channel state information, the difference diminishes [51].
4.2.4 Implementation Considerations
Although the TR principle itself is rather straight-forward, its implementation has several
open issues. For instance, the implementation of UWB analog delays is not straightforward
[54, 55]. Although this thesis is not on the design of analog circuitry, like delaylines,
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 of
bandwidth 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 and
Tomlinson has not been considered, since it requires long delays [53]. Therefore, the
focus is on the most elementary TR signaling scheme as described in this section, using
only two pulses per symbol and a single delay. Multi-user access functionality should be
provided by one of the other OSI-layers, for instance by the Data Link Layer (DLL) using
an Aloha-like access scheme.
Besides having to select an appropriate value for the delay, the delay unavoidably will
vary from one device to another and from time to time due to variations in the production
process, 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 kept
small using sophisticated delays, but will increase the cost of the devices. Taking the
delay variation into account during system design is therefore a must to obtain a low-cost
system. In Sec. 4.3.2, a complex-valued AcR is proposed, which not only decreases the
system’s sensitivity to delay mismatches, but also allows for an increase in data rate, see
Sec. 4.3.3.
Another topic of this thesis is the proper setting of the start and duration of the inte-

62 CHAPTER 4. THEORY OF TR UWB COMMUNICATIONS
BPF
r(t)
∫ α[n]
.dt
SIGN
ˆb[n]
Delay
c
Reset
DSP
w(t − nT s )
Figure 4.3: Block diagram of a weighted AcR
gration interval. In a practical implementation, most likely both variables will be under
the control of the digital circuitry. This requires both additional DSP and additional
interfaces between the RF front-end and the digital circuitry, increasing both complexity
and cost. In Sec. 4.3.1, a fractional sampling AcR is proposed, which allows for synchronization
to be obtained in the digital domain, reducing the power consumption and cost
of 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 receiver
signal is multiplied with a delayed version of itself, followed by an integration of the
multiplier output. Hence, no weighting is applied to multiplier output signal, although its
Signal-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, to
improve the performance of the AcR receiver. The weighting function is also used to
synchronize the RX to the received signal. In addition, it is proposed to add a constant c
to 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 − nT s )r(t)r(t − D)dt + c. (4.8)
Assuming BPSK modulation, the sign of α[n] can be used as decision for the transmitted
symbol. If neither delay-hopping nor time-hopping is used, the TR signal will be cyclostationary,
such that the weighting function can be the same for every symbol. As the
weighting is applied in the analog domain of the receiver, the proposed AcR is called an
analog weighted AcR. The proposed structure is depicted in Fig. 4.3. For simplicity, a
real-valued AcR is assumed, but the principle can be applied to complex-valued AcRs as
well, see Sec. 4.3.2.
In order to be optimal, the weighting function must be adapted to the conditions on the
channel, like channel impulse response, SNR and delay. A likely implementation would be
to 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 will
lead to sub-optimal results if the TR symbols overlap in time.
To overcome these shortcomings, it is proposed to restrict the degrees-of-freedom of
the weighting function w(t) at the cost of some performance. Concretely, the weighting
function w(t) is restricted to the following general expression with ML degrees of freedom
w(t) =
M∑ ∑L−1
w[k, α]h clk (t − (α/L + k)T s ), (4.9)
k=0 α=0
where h clk was defined in Sec. 4.4.2 as a rectangular function, which equals one for 0 ≤
t < T s /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∑ ∑L−1
w[k,α]h clk (t − (α/L + k + n)T s )r(t)r(t − D)dt + c
−∞
k=0 α=0
M∑ ∑L−1
w[k,α]
k=0 α=0
∫ ∞
h clk (t − (α/L + k + n)T s )r(t)r(t − D)dt +c. (4.10)
−∞
} {{ }
= u[n + k, α]
After changing the order of the summations and the integration, the samples generated
by a so-called fractional sampling AcR u[n, α] can be identified. This allows us to write
the value of the decision statistic at time n as a weighted sum of fractional samples of an
AcR. In other words,
α[n] =
M∑ ∑L−1
w[k,α]u[n + k, α] + c (4.11)
k=0 α=0
with
u[n, α] =
((α+1)/L+n)T
∫ s
(α/L+n)T s
r(t)r(t − D)dt. (4.12)
This illustrates that the analog weighted AcR with limited degree of freedom defined
by (4.9) is mathematically equivalent to applying weighting to the fractional samples of
an AcR. From an implementation point-of-view, applying weighting in the digital domain
is simpler and allows for overlapping weighting functions for consecutive symbols.
In essence, a fractionally sampled AcR divides the symbol period into several integration
intervals, where one sample is generated per interval. For simplicity, all intervals are
of equal duration T clk , which is an integer fraction of the symbol duration, i.e. T clk = T s /L
with L ∈ N, such that L samples are generated per TR symbol. To simplify the implementation,
these intervals are by no means synchronized to the received signal.
Regarding the implementation of fractional sampling AcRs, two mathematically equivalent
schemes are possible. For example, an integrator with reset can be used. After

64 CHAPTER 4. THEORY OF TR UWB COMMUNICATIONS
Figure 4.4: Signals in a fractional sampling AcR
BPF
I&D
DSP
Delay
Figure 4.5: Block diagram of a fractional sampling AcR
receiving a reset signal, the integrator output is sampled, forced to zero and starts integrating
again until the next reset is received. This operation is often referred to as an
Integrate and Dump (I&D). The block diagram of a fractional sampling AcR using an
I&D is depicted in Fig. 4.5. Alternatively, the integrator can be replaced by a Low Pass
Filter (LPF) with a rectangular impulse response of duration T clk . The output of the LPF
, i.e. again L samples are taken per symbol. The mathematical
equivalence of both AcR implementations is depicted in Fig. 4.4. The dashed line represents
the integration value in a I&D sampler, where the dot-dashed line represents the
LPF output. In both cases, the markers identify the sample moment and value. Please
note 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 integrator
is that after receiving the reset signal, the integrator will be shortly insusceptible
to the input signal. The LPF based implementation is at all time susceptible to the
input signal, but the implementation of a LPF with a rectangular impulse response is
impossible. The appropriate choice depends on the application scenario.
Applying adaptive weighting has several distinct advantages. Firstly, the synchronization
process can fully take place in the digital domain. Secondly, it implicitly controls the
effective 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
is sampled at a rate of T −1
clk

4.3. EXTENSIONS OF THE TR PRINCIPLE 65
several authors, including the author of this thesis [24, 56, 57]. The main novelty of
this 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 interference
is considered. However, the introduction of ISI leads to effects, which are fundamentally
different 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 D rx must be
well-matched to the TX delay D tx . Any delay mismatch δ means that R q (τ) is sampled
at lag δ instead of lag zero, where R q (x) denotes the autocorrelation of the RX pulse q(t)
defined as
R q (τ) =
∫ ∞
−∞
q(t + τ)q(t)dt (4.13)
Assuming BPSK modulation, the Euclidean distance between both symbols 2|R q (δ)| will
decrease with any delay mismatch, making the system more susceptible to noise. The
Euclidian distance has been depicted as a solid line in Fig. 4.8 for R q (0) = 1. In case of
a normal AcR, the figure shows that a delay-mismatch of 1/(8f c ) ≈ 31 ps already results
in a 3 dB loss in the system’s energy efficiency and a delay-mismatch of 1/(4f c ) = 62.5 ps
will make communication completely impossible. Note that the multipath channel has
no impact on the sensitivity of TR systems to delay mismatches [59].
To increase the robustness of the system against delay mismatches, we propose to use
a Complex-Valued (CV) AcR. In addition to the autocorrelation branch used in a normal
AcR, the CV AcR has a second autocorrelation branch, which samples the autocorrelation
function at lag D rx +1/(4f c ). Hence, the autocorrelation function is sampled at two lags,
R p (δ) and R p (δ + 1/(4f c )). In Appendix B, it is shown that the proposed AcR computes
the short-term complex-valued autocorrelation of the received signal at delay lag D and
is therefore referred to as such. Its operation in a baseband notation is as follows,
∫ ((α+1)/L+n)Ts
u[n, α] = exp(jω c D) r(t)r ∗ (t − D)dt. (4.14)
(α/L+n)T s
A block-diagram of the proposed receiver architecture can be found in Fig. 4.6. Since a
delay of 1/(4f c ) 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 functions
has been set out against each other as function of the delay-mismatch δ. The mismatch
value is given in picoseconds within the figure. The resulting trajectory resembles a
damped spiral, meaning that both BPSK constellation points are rotated around the
origin and the Euclidean distance between both points slowly decreases with an increasing
mismatch. This rotation must be compensated for, before a decision on the symbol
value 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 as
function of the delay mismatch. In case of a CV AcR, the decrease of Euclidean distance
depends on the pulse envelope and thus the bandwidth. Fig. 4.8 shows that the Euclidian

66 CHAPTER 4. THEORY OF TR UWB COMMUNICATIONS
I&D
D
90˚
0˚
T clk
DSP
I&D
Figure 4.6: QPSK-TR receiver architecture, where 90 ◦ denotes the delay 1/(4f c )
1
60
R q
(δ+1/(4f c
))
0.8
0.6
0.4
0.2
0
−0.2
90
−150
360
120 −390
−120
390
−360
−180
330
−420
300
−210
−450
−480
480
30
270
−240
240
−270
0
−0.4
−0.6
−0.8
150
−90
420 −330 450−300
210
180 −60
−30
−1
−1 −0.5 0 0.5 1
R q
(δ)
Figure 4.7: IQ-diagram shift as function of the delay mismatch in picoseconds for a 1 ns
pulse with a rectangular envelope and a 4.0 GHz carrier frequency

4.3. EXTENSIONS OF THE TR PRINCIPLE 67
Figure 4.8: Euclidian distance as function of the delay mismatch in picoseconds for a 1 ns
pulse with a rectangular envelope and a 4.0 GHz carrier frequency demodulated using
either 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 instead
of the carrier frequency, decreasing its sensitivity by an order of magnitude.
The use of multiple AcR branches to overcome delay mismatches has previously been
described in [61, 62]. Our proposal is fundamentally different, since it exploits the bandpass
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 the
authors referred to as TR Pulse Interval and Amplitude Modulation (PIAM). A CV
AcR is able to demodulate certain types of TR PIAM signaling, without the need to
extend the RF front-end of the AcR. Assuming the pulse time-interval can assume two
distinct values, and if these are equal to D and D + 1/(4f c ). The so-called pulse interval
modulation factor T D equals 1/(4f c ) and allows for the translation of the time-shift of the
modulated 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 assume
four 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 is
straight-forward to extend this concept to higher-order PSK modulation or Quadrature
Amplitude Modulation (QAM). Only BPSK and QPSK modulation are considered in
this thesis.
Independently, the combination of higher order Pulse Position Modulation (PPM) and
additional AcR branches to overcome delay mismatches has been proposed simultaneously
in [62].

68 CHAPTER 4. THEORY OF TR UWB COMMUNICATIONS
4.4 Generic TR System Model
4.4.1 Introduction
In this section, a baseband time-discrete equivalent system model for TR signaling demodulated
using AcRs will be derived. The model has been kept general, to include all
extensions proposed in section 4.3. The discrete-time equivalent notation is used, since
it 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 time
observations to characterize the signal, leading to shorter computation and simulation
times. The term observation is used instead of sample to emphasize that the signal is not
actually sampled by the RX.
4.4.2 Continuous-Time System Model
Assuming TR signaling in which a single TR symbol is transmitted each T s seconds. In
this case, the time-line is divided in frames/symbol periods of T s seconds, where each
frame contains a single TR symbol. On its turn, a frame is again divided into N h chips
of duration T c . In each frame, two identically shaped pulses are transmitted; a reference
pulse followed by a modulated one. Let us focus on the waveform transmitted within the
n-th frame. To obtain a general-purpose system model, both pulses are allowed to be
modulated. 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], while
the delayed pulse in the n-th frame is modulated by both the scrambling factor and the
information bearing symbol ˜b[n]b[n]. The term scrambling factor is used to emphasize
that ˜b[n] is not used for the signaling of information. The scrambling code ˜b[n] may be
generated using a PN generator, but could depend on the information to be transmitted
as well.
The reference pulse will be transmitted in the c-th chip and the modulated pulse
is transmitted d chips later. To allow for a compact mathematical representation, the
position of the reference pulse within the frame is represented by the column vector
s = [s i ], where s i = δ[i −c] for i = 1, 2, ...N h and δ[i] denotes the Kronecker delta.
Similarly, the position of the modulated pulse can be denoted by the column vector
˜s = [˜s i ] where ˜s i = δ[i−c−d] for i = 1, 2, ...N h . The received signal after the RX bandpass
filter (BPF) in a baseband notation can be written as
r(t) = ∑ n
q (t, nT s ) T S˜b[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) is
the 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 +T c ), ...,q(t, τ +(N h −1)T c )] T (4.16)
with q(t, τ) = q m (t − τ) exp(−jω c τ), where ω c = 2πf c and q m (t) represents the envelope
signal of the bandlimited UWB signal q(t). Due to the RX BPF, the complex-valued
Gaussian noise signal n(t) is coloured with an autocorrelation function r nn (τ).

4.4. GENERIC TR SYSTEM MODEL 69
In the original notation, the matrix S was labelled with symbol-index index n to
obtain a generic model, which includes time-hopping and delay hopping. It has also
been implemented in the simulation environment, but finally not used. Therefore, the
symbol-index n has been omitted.
As stated in Sec. 4.1, the usage of two autocorrelation branches is proposed, where
the first one is matched to a lag D and the second to a lag D + 1/(4f c ). Without loss
of generality, D is chosen equal to dT c with d ∈ N. Each branch is sampled L times
per symbol, such that the sampling period T clk = T s /L and L denotes the fractional
sampling ratio (FSR) with L ∈ N. Hence, the AcR generates two parallel real-valued
sample streams, which can be seen as a single complex-valued sample stream. Appendix
4.3.2 shows the input-output relation of the proposed AcR in a complex-valued baseband
notation is
where α ∈ {1, 2, ...,L}.
u[n, α]= exp(jω c D)
∫
((α+1)/L+n)T s
(α/L+n)T s
r(t)r ∗ (t−D)dt, (4.17)
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 be
developed by taking an observation of r(t) every T r seconds, where T r will be chosen to
fulfill the Nyquist criterion. The analog received signal is modelled using its discrete-time
equivalent.
Since neither delay-hopping nor time-hopping is used, the received signal is cyclostationary
with period T s . In this case, a finite interval [nT s , (n+1)T s 〉 is sufficient to fully
characterize the received signal, i.e. only N ob observations with N ob = T s /T r are needed.
The vector containing these N ob observations will be denoted by r[n]. Without loss of
generality, 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 identical
to the relationship between b[n − 1] and r[n]. Because of the finite duration of q(t), a
finite number of symbols M + 1 can influence the observation interval, independently of
whether the ISI is caused by a reference pulse, a modulated pulse or both. Based on
causality, 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] = QŠd[n] + W 1n[n], (4.18)
where the column-vector d[n] contains the modulation applied to both pulses of the TR
symbol with time index n and the M previous TR symbols. Hence, d[n] will have 2M +2
elements constructed according to
d[n] =
[˜b[n − M] T , ˜b[n − M + 1] T , ..., ˜b[n]
] T T . (4.19)
The matrix Š contains the positioning of the pulses within each symbol period and
related to S as follows,
Š = I M+1 ⊗ S, (4.20)

70 CHAPTER 4. THEORY OF TR UWB COMMUNICATIONS
where ⊗ denotes the Kronecker product, such that Š ∈ {0, 1}N h(M+1),2M+2 .
The matrix Q is the channel matrix Q = [q kl ] with q kl = q(kT r , N l T s − lT c ), for
k +1 = 1, 2, ...,N ob samples and l = 0, 1, ...,(M +1)N h −1 chips, such that the channel
matrix Q ∈ C N ob,(M+1)N h.
The noise vector n[n] contains complex-valued, identically distributed, zero-mean,
Gaussian random variables (RVs), characterized by the discrete autocorrelation function
r nn [k − l] = 1 2 N 0R nn ((k − l)T r ). (4.21)
The noise vector has N ob +N d elements, where N d denotes the delay in samples N d = D/T r .
Hence, it contains more elements than the vector r[n]. The purpose of these N d additional
elements becomes apparent when introducing the delayed version of the received signal.
The matrix W 1 is constructed such that W 1 n[n] equals the last N ob values of n[n] in the
proper order. Consequently,
W 1 = [ 0 Nob ,N d
I Nob
]
. (4.22)
Using the same methodology, the discrete equivalent signal of the delayed version of the
received signal r(t − D), denoted by r d [n], can be written as
r d [n] = QDŠd[n] + W 2n[n]. (4.23)
The signal components have been delayed by introducing a delay matrix D, such that
[
01,(M+1)Nh −1 0
D = exp(−jω c D)
I (M+1)Nh −1 0 (M+1)Nh −1,1
] d
. (4.24)
The matrix W 2 is constructed such that W 2 n[n] is a column-vector containing the first
N ob elements of n[n]. In this fashion, the system model takes into account that the noise
contained in both r[n] and r d [n] originates from the same noise process. The matrix W 2
is constructed as
W 2 = [ I Nob 0 Nob ,N d
]
. (4.25)
In the third stage of an AcR, the received signal is multiplied with its delayed version
to form the multiplier output, see Fig. 4.2. In a vector notation, this multiplication will
be modelled using an element-wise multiplication of r[n], r d [n]. Therefore, the diagonal
operator 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])r d [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 the
n-th symbol interval u[n, α] is equal to,
where the k-th element of h[α] equals h(kT r − αT clk ).
u[n, α] = h[α] T Λ(r[n])r ∗ d[n], (4.26)

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- and
inter-symbol-interference terms and η[n, α] is a noise term. A more detailed derivation of
the system model can be found in [18].
The signal term can be represented by the following structure,
s[n, α] = d[n] H K α d[n], (4.28)
which is known in literature as an FIR, second-order Volterra system [64]. The matrix
K α is defined as
⎡
h T αΛ ( ) ⎤
Q ∗ D ∗ Še 1 Q Š
h T αΛ ( )
Q ∗ D ∗ Še 2 Q Š
K α = ⎢
⎥
(4.29)
⎣ .
h T αΛ ( ⎦
)
Q ∗ D ∗ Še L Q Š
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] = η g [n, α] + η z [n, α], (4.30)
The term denoted by η g is called the Gaussian noise term, while the term denoted by
η z will be referred to as the non-Gaussian noise term. The terminology has been chosen
for the following reasons. The Gaussian noise term is a superposition of two Gaussian
sub-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,
η g1 [n, α] = d[n] H L α,1 n[n], (4.31)
η g2 [n, α] = d[n] T L α,2 n[n] ∗ . (4.32)
Hence, both η g1 [n, α] and η g2 [n, α] are the superposition of the Gaussian distributed RVs
contained in n[n]. Hence, both terms and η g are all Gaussian distributed. Although not
independent, both Gaussian noise sub-terms are uncorrelated, since the cross-correlation
between the noise vector and its conjugate is zero. The matrices L α,1 and L α,2 are
structured as follows
⎡
h T αΛ ( ) ⎤
Q ∗ D ∗ Še 1 W1
h T αΛ ( )
Q ∗ D ∗ Še 2 W1
L α,1 = ⎢
⎥
(4.33)
⎣ .
h T αΛ ( ⎦
)
Q ∗ D ∗ Še L W1
⎡
h T αΛ ( QŠe ⎤
1)
W2
h T αΛ (
L α,2 = ⎢
QŠe 2)
W2
⎥
(4.34)
⎣ .
h T αΛ ( ⎦
QŠe )
L W2

72 CHAPTER 4. THEORY OF TR UWB COMMUNICATIONS
The noise term η z is independent of the TX signal y(t) and is given by the following
equation,
η z [n, α] = h T αz[n]. (4.35)
where the non-Gaussian noise vector z[n] is related to the Gaussian noise vector according
to
z[n] = Λ(W 1 n[n])W 2 n[n] ∗ . (4.36)
Hence, the elements in z[n] are the product of two Gaussian RVs, which are possibly correlated,
and thus themselves not Gaussian distributed. Therefore, η z [n, α] is not Gaussian
distributed either and referred to as such. The actual distribution depends on system parameters
like bandwidth, and integration duration. Based on the central limit theorem,
it can be understood that the distribution of η z [n, α] converges towards a Gaussian one
with an increasing product of bandwidth and integration duration, where a product larger
than 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 transmitted
symbols and the I&D output is described by an Finite Impulse Response (FIR) secondorder
Volterra system. Volterra systems are widely used for the modeling of non-linear
systems. However, Volterra systems of TR-UWB systems differ to some extent from those
used e.g. for the modeling of analog components. The difference is not so much caused
by the Volterra systems themselves, but in the way they are excited. When used for
the modeling of analog components, Volterra systems are typically excited by continuous
valued signals. In our case, the Volterra system is excited using a digitally modulated
signal which is by nature finite alphabet. The difference in excitation allow for alternative
interpretations of the Volterra system. In this section, some of those interpretations will
be presented to provide insight in the behaviour of TR systems. More information on
non-linear system modeling can be found in [64]. An extensive bibliography on non-linear
system modeling and other aspects can be found in [67].
In (4.28), the Volterra system describing the relationship between the fractional samples
and the modulation was shown to be
s[n, α] = d[n] H K α d[n],
where the vector d[n] contains both the modulation applied to the reference pulses as well
to the information bearing pulses. Furthermore, the elements in d[n] are shifted by two
positions with each increment of the time-index n. In this respect, the Volterra system
as defined here differs from the typical definition for Volterra systems [64].
However, an alternative equivalent interpretation of the system model, presented in
Sec. 4.4.3, can be obtained. The introduction of the decimator known from multi-rate
systems allows us to use the default definition of Volterra system as described in [64]. In

4.5. INTERPRETATION OF THE TR SYSTEM MODEL 73
Constant
a[n]
Options
ã[n]
˜b[n]
Mod.
b[n]
PN sequence
K 1 2
P
d[n]
S
K L 2
u[n, 1]
u[n, L]
Figure 4.9: Block diagram of the SIMO FIR Volterra model
Fig. 4.9, its block diagram is depicted describing the complete system model including
the modulation. Please be aware that the decimator undoes the rate increase introduced
by the parallel-to-serial conversion, such that the overall system generates L I&D samples
for each symbol. Hence, a fractional sampled AcR can be seen as a single-input, multipleoutput
FIR Volterra system. The scrambling applied to the reference pulse ˜b[n] is not
interpreted as input, since it is either fully described by the TX symbol b[n] or a Pseudo
Noise (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 parameters
like the delay, channel, FSR, BPFs, symbol rate etc. The memory in the Volterra system
is determined primarily by the radio channel and symbol-rate, i.e. increasing either the
channel 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 the
transmitted symbols. In this respect, TR communication differs from ”ordinary” communication
systems, where the ISI is modelled using a FIR structure. To illustrate the
non-linear ISI and its dependency on the data rate, several constellation diagrams have
been depicted in Fig. 4.10 of TR systems using fractional sampled CV-AcR with the FSR
equal to twice the symbol rate, deployed in an indoor environment. QPSK-TR signaling
is 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. As
a result, only one of the constellation diagrams contains the four constellation points of
the QPSK modulation.
Moderate ISI can be observed when the data rate is increased to 20 Mb/s. In the
left-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 the
10 Mb/s case, both outputs contain information regarding the transmitted symbol. The
nature 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. Secondly,
an FIR structure inherently results in a constellation diagram, which is rotational

74 CHAPTER 4. THEORY OF TR UWB COMMUNICATIONS
(a)
1
1
0.5
0.5
Imag
0
0
Imag
−0.5
−0.5
−1
−1 −0.5 0 0.5
−1
1 −1 −0.5 0 0.5 1
Real
Real
1
(b)
1
0.5
0.5
Imag
0
0
Imag
−0.5
−0.5
−1
−1 −0.5 0 0.5
−1
1 −1 −0.5 0 0.5 1
Real
Real
1
(c)
1
0.5
0.5
Imag
0
0
Imag
−0.5
−0.5
−1
−1 −0.5 0 0.5
−1
1 −1 −0.5 0 0.5 1
Real
Real
Figure 4.10: IQ diagrams at both outputs of a TR-QPSK system, operating in a multipath
environment at 10, 20 and 40 Mb/s, in sub-figure (a),(b) and (c) respectively

4.5. INTERPRETATION OF THE TR SYSTEM MODEL 75
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 dense
cloud 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] T k α . (4.37)
In [64] the vectors ˜d[n] and k α are defined to be equal to vec ( d[n]d[n] H) and vec (K α ),
respectively. The operator vec (K) creates a column vector by stacking the columns
of 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 same
for 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 N k = rank(A) is less than (2M + 2) 2 , it means
that 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 N k uncorrelated variables.
Assuming these variables to be gathered in ˜d[n], a linear transformation matrix T exists,
which fulfils the following two criteria:
T˜d[n] = vec ( d[n]d[n] H) (4.39)
E
[˜d[n]˜d[n]
H]
= I Nk ,N k
. (4.40)
The fact that all components in ˜d[n] are uncorrelated, unit power RVs makes the vector
notation powerful for statistical analysis.
Assuming T to be available, ˜d[n] and k α can be obtained by
˜d[n] = T † vec ( d[n]d[n] H) , (4.41)
k α = T H vec (K α ). (4.42)
For all modulation types considered in this thesis, the composition of A was rather
straight-forward. The elements of vec ( d[n]d[n] H) are either fully correlated or uncorrelated.
In other words, the matrix A contains only zero- and one-valued elements, making
the identification of identical rows and columns relatively easy as well as the construction
of transformation matrix T.

76 CHAPTER 4. THEORY OF TR UWB COMMUNICATIONS
Table 4.1: Dependence of N k 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 81 118
BPSK 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 by
applying SVD on A = EΛE H , where E is the unitary matrix of eigenvectors and Λ is a
diagonal matrix of eigenvalues. The transformation matrix T is then obtained from
T = E NZ
√
ΛNZ (4.43)
where the zero-eigenvalues and the corresponding eigenvectors are skipped, as denoted
by 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, leading
to incorrect results.
Without loss of generality, some assumptions are made with respect to the composition
of ˜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 the
remaining positions. How many additional elements ˜d[n] has depends on the statistical
properties of d[n], i.e. the applied modulation. To summarize, ˜d[n] is assumed to have
the 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 statistical
properties,
]
E
[˜d[n] = e 1 (4.45)
E
[˜d[n]˜d[n]
H]
= I Nk ,N k
(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]
∗]
e 2+m ∀m ∈ {0, 1, ...,M},
=
(4.47)
0 otherwise.
As stated before, N k depends on the applied modulation. In Tab. 4.1, the number
of uncorrelated elements is presented as function of the modulation scheme and channel
memory M. In the case of a memory-less channel, i.e. in the absence of ISI, N k is equal to
two. The first one is the desired term, while the second is a constant to model DC offsets

4.5. INTERPRETATION OF THE TR SYSTEM MODEL 77
in the constellation diagram. Only in the case of QPSK modulation, an additional term
exists, because intra-symbol interference is still possible. In the presence of ISI, N k is
super linear with respect to the channel memory. Assuming a constant channel memory,
N k is reduced if the modulation can assume less values, i.e. if the degree of freedom of
the 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 if
the symbol decision is delayed. Similar to how I&D samples at time index n are influenced
by M preceding symbols, the symbol b[n] will influence samples with a time index between
n 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 with
memory M and taking only information theoretical consideration into account and no
implementation aspects.
Taking only the signal part into account, these samples are assumed to be gathered
in 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 ⊗I M+1,M+1 (4.51)
where K = [ k 1 k 2 . .. k L
]
. In this case the elements in ˘d[n] are surely correlated if M
is larger than zero, where N i denotes the number of uncorrelated elements. By applying
the same mathematical trick as in Sec. 4.5.2, an alternative definition is obtained such
that ˘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 = ˘T H (K ⊗I M+1,M+1 ), (4.52)
where the matrix ˘T is obtained in the same manner as T using the autocorrelation matrix
of ˘d[n] as defined in (4.50).
Similar to N k , also the relationship between N i , M and modulation has been computed.
The results are gathered in Tab. 4.2. Similar to N k , the number of uncorrelated
elements N i decreases if the degree of freedom of the modulation is reduced. In case the
Volterra model has no memory, the value for N k and N i are equal, because samples with
a 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

78 CHAPTER 4. THEORY OF TR UWB COMMUNICATIONS
Table 4.2: Dependence of N i 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 60 120 201
BPSK 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 in
˘d[n]. Assuming the same memory and modulation, N i is thus bounded to be equal or
larger than N k . This result of reasoning is confirmed by Tab. 4.2. Furthermore, it shows
that N i grows considerably faster than N k 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 this
section, 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, the
MIMO model regards only uncorrelated, modulated inputs as different inputs, where the
MIMO 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 BER
performance in the presence of ISI, see Sec. 5.4.3.
Eq.(4.49) shows that s[n] is a superposition of N i vectors, which are gathered in ˘K,
that are modulated by N i uncorrelated RVs gathered in ˘d[n]. In other words, the system
can be interpreted as a MIMO system with N i uncorrelated inputs. The number of
outputs is ML or 2ML for a RV AcR and a CV AcR, respectively. However, the first
element 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] is
zero mean, i.e. that ˘k 1 is an all zero vector, where ˘k n denotes the n-th column of ˘K. The
MIMO model has now N i − 1 uncorrelated inputs, which modulate N i − 1 vectors in an
multi-dimensional linear vector space. A simplified representation of the linear vector is
given in Fig. 4.11.
The RX can apply linear weighting on s[n] to form a decision statistic based on which
a decision is made on the value of b[n]. Assuming MMSE weighting in the absence of
noise, the MMSE solution will suppress all ISI if ˘k 2 , which describes the linear relationship
between b[n] and s[n], has a component that is perpendicular to the space spanned by
the remaining interfering terms. In other words, if
{
(I − P ISI ) ˘k
true: full ISI suppression possible,
2 ≠ 0 (4.53)
otherwise: no full ISI suppression possible.
where P ISI denotes the space spanned by the ISI terms, e.g. obtained using GramSchmidt
orthonormalization. Furthermore, if ˘k 2 is orthogonal with respect to P ISI , the ISI can be
suppressed without increased sensitivity to noise. On the other hand, if the ISI projection
matrix P ISI is full-rank, no linear weighting vector exists, which fully suppress the ISI.

4.5. INTERPRETATION OF THE TR SYSTEM MODEL 79
dim 2
˘k 1
˘k 3
dim 1
˘k 4
dim 3
˘k 2
Figure 4.11: Vector space spanned by the MIMO kernel ˘K
The actual rank of P ISI is upper-bounded by the number of interfering MIMO inputs
N i − 2 and secondly depends on on the composition of ˘K. Since, ˘K is a random matrix
due to the radio channel, it’s impact on the system performance has a random nature
as well. Nevertheless, it is likely that the ISI can be fully suppressed if the number of
MIMO inputs N i is smaller than the number of output ML, such that P ISI can never be
full-rank. With all other parameters being the same, more ISI can be suppressed with
decreasing N i and/or increasing number of MIMO outputs.
The number of MIMO outputs can be changed by increasing the FSR or by using
additional autocorrelation branches. For instance, extending a real-valued AcR to a
complex-valued AcR means that the effective number of MIMO outputs is increased by
a factor 2.
The number of MIMO inputs changes when modifying the modulation scheme. In
Fig. 4.12, the dependence of the constellation diagram on the modulation type before
and after MMSE weighting in the absence of noise. Fig. 4.12 shows that the constellation
diagram the MMSE weighting is able to suppress more ISI if the reference pulse is left
unscrambled, 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, since
some of the constellation points belonging to the TR-QPSK symbol in quadrant 1 fall in
quadrant 2.
4.5.5 Data Model as Finite State Machine
The FIR Volterra system is in fact a Hidden Markov Model (HMM), resulting from the
radio channel. After the RX identifies the HMM, the HMM reduces to a Markov model
or FSM with a non-linear relationship between state-transitions and outputs. In this

80 CHAPTER 4. THEORY OF TR UWB COMMUNICATIONS
(a)
1
Scrambled QPSK−TR
Unscrambled QPSK−TR
1
Scrambled QPSK−TR
Unscrambled QPSK−TR
0.5
0.5
Imag
0
0
Imag
−0.5
−0.5
−1
−1 −0.5 0 0.5 1
Real
−1
−1 −0.5 0 0.5 1
Real
(b)
1.5
Scrambled QPSK−TR
Unscrambled QPSK−TR
1
0.5
Imag
0
−0.5
−1
−1.5
−1.5 −1 −0.5 0 0.5 1 1.5
Real
Figure 4.12: Dependence of the constellation diagram(s) of the modulation before (a) and
after MMSE weighting (b).

4.5. INTERPRETATION OF THE TR SYSTEM MODEL 81
Memory
Memory-less
n
PN seq
∈ 2 N p
N Sp =2 N pM
S 0
˜b[n], Sp [n]
constant
S 1
Sz=2 (N p+N b )(M+1)
η[n,α]
bits
a[n]
Synchrone
Table f α
+
s[n,α]
u[n,α]
S 0
∈ 2 N b
S 1
b[n], S t [n]
Random Entries
N St =2 N bM
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 be
applied with the same complexity as needed for equalization of linear channels, assuming
the same channel memory for both. Also the role of scrambling of reference pulses on the
identification and equalization of FIR Volterra systems will be discussed, showing that
scrambling does not significantly increases the equalizer complexity.
To obtain a generic system model, the modulation applied to both pulses has been
kept general so far. For the description of the system as FSM, it is required to introduce
a more formal description of the modulation. In practice, it may be assumed that an
integer amount of (channel) bits N b are mapped on a single TR-symbol. For notational
convenience, the k-th (channel) bit mapped on the n-th TR symbol will be denoted by
c[n, k] and the vector c[n] gathers all bits with time-index n. Assuming these bits to
be i.i.d. RVs in B = {0, 1}, the n-th TR-symbol is identified by i.i.d. RV a[n], where
a[n] ∈ {0, 1, ...,2 N b − 1} with equal probability, where the modulation b[n] depends only
on a[n] and thus the bits to be transmitted. In the same fashion, an identifier ã[n] is
defined, 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 memory
size depends on the symbol rate and CIR. Evidently, the radio channel does not distinguish
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 same
memory depth, one driven by the symbol identifiers a[n] and the other one driven by the
scrambling identifiers ã[n]. The state of both FSMs at time n will be denoted by S t [n]
and S p [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-transition
probabilities are unknown a-priori. Based on the states and inputs of both FSMs, a
memory-less relationship exists for each output f α (a[n], S t [n],˜b[n], S p [n]). In Fig. 4.13,
the structure of the system has been depicted.

82 CHAPTER 4. THEORY OF TR UWB COMMUNICATIONS
Memory
Memory-less
bits
a[n]
S 0
S 1
b[n], S t [n]
Sz=2 (N b)(M+1)
η[n,α]
Table f α +
s[n,α]
u[n,α]
N St =2 N bM
Random Entries
Figure 4.14: Simplified FSM description for a FIR Volterra model in the absence of PN
scrambling
If the scrambling code is driven by a PN sequence, the RX can reconstruct the state
S p [n] at all time, assuming the RX is aware of the initial state, phase and structure of the
PN generator. In this case, the input-output relationship of the system can be described
by a time-variant function,
s[n, α] = f α (a[n], S t [n], n) (4.54)
The number of state transitions, an important measure for the complexity of a trellisbased
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 sequence.
In this case, the table function becomes time-invariant f α (a[n], S t [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] = ã[n] for all values of
n. In this case, both FSMs will be running synchronously, i.e. S p [n] is fully describing
S t [n]. The system model of Fig. 4.13 is simplified to a single FSM. Additionally, the table
function becomes time-invariant f α (a[n], S t [n]) containing at most 2 N b(M+1) entries as in
the unmodulated case, but since the reference pulse is scrambled, the PSD will be smooth
if appropriate modulation is applied. The simplified block diagram has been depicted in
Fig. 4.14.
4.5.6 Reduced Memory Data Model
The usage of trellis-based algorithms for the equalization of FIR Volterra channels is
a promising technique, since the information contained in non-linear ISI terms is taken
into account. Unfortunately, the complexity of a trellis is proportional to the number
of channel-states, i.e. it is exponentially proportional to the channel memory. As a
result, trellis-based equalization becomes quickly too complex for practical application,
if the full channel memory is taken into account. In this subsection, a Reduced Memory
Data 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 equalize
the 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 the
fact that the incoming symbols are delayed by m and the memory N of the RMDM is
less or equal to the FDM’s memory M. In a vectorial notation, the output of the RMDM

4.5. INTERPRETATION OF THE TR SYSTEM MODEL 83
is defined as
ǔ[n, α] =ď[n − m]H ǩ α + ˇη[n, α]. (4.55)
where ď[n] is of length N r and contains a subset of the elements in ˜d[n].
It is the challenge to find the optimal combination of kernel ǩα and delay m, such
that
[
∑ L [m,ǩα] =argmin E[ ∣∣∣˜d[n] H k α − ď[n − x]H k∣ 2]] (4.56)
x∈N,k∈C Nr,1
α=1
where N and C denote the sets of non-negative integers and complex numbers, respectively.
To our knowledge, (4.56) cannot be solved in closed form. Therefore, a divideand-conquer
approach is applied to the problem. Firstly, the MMSE solution for ǩα is
derived for a single given output and delay x, denoted by ǩ(x) α , such that
[
ǩ α
(x) = argmin E[ ∣∣∣˜d[n] H k α − ď[n − x]H k∣ 2]] (4.57)
k∈C Nr,1
Since both ˜d[n] and ď[n − x] contain per definition only uncorrelated variables, it is easy
to prove that the optimal kernel in the sense of the MMSE criterion is given by
ǩ (x)
α = Ck α (4.58)
with
CE
[ď[n − x]˜d[n]
H]
(4.59)
where C ∈ {0, 1} Nr,N k . The under-modeling error, i.e. the average squared difference
between the RMDM and the FDM, for the delay under evaluation and output σu,α(x)
2
equals
[ ) ] 2
σu,α(x)E
(˜d[n] 2 H k α − ď[n − x]H ǩ (x)
α
= k H α
(
I − C H C ) k α (4.60)
Using the previously obtained result, the MMSE delay m is selected using
[ L
]
∑
[m] = argmin σu,α(x)
2
x∈{0,...,M}
α=1
(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 RX
signal. On the contrary, the unused part will have a noise-like effect from the equalizer’s
point of view, deteriorating the system performance. Hence, the noise variance at the
α-th output of the RMDM can be written as,
σ 2ˇη,α = σ 2 η,α + σ 2 u,α(m). (4.62)

84 CHAPTER 4. THEORY OF TR UWB COMMUNICATIONS
Figure 4.15: Constellation diagram at the outputs of an FDM and its RMDM. The lefthand
plot refers to α = 1 and the right-hand plot to α = 2
where ση,α 2 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 shown
in Sec. 4.6. However, the under-modeling-error σu,α(m) 2 is most likely not white. Hence, a
closed-form expression for the performance degradation due to the under-modeling is not
easily obtained. An indication for the performance is obtained by computing the ”overall
SNR” seen from the equalizer’s perspective as,
∥ ∥ L∑ ∥ǩ α 2
SNR RMDM =
ση,α 2 + σu,α(m) . (4.63)
2
α=1
The ability of an RMDM to mimic its FDM can be visualized by comparing their
constellation diagrams. In Fig. 4.15, the constellation diagram is depicted of a memoryfour
FDM describing a QPSK-TR system, together with the constellation diagram of
its memory-one RMDM. The RMDM’s constellation diagram resembles the constellation
diagram of the FDM reasonably well, in the sense that the general structure of the FDM’s
constellation is preserved. Nevertheless, the number of states is reduced by a factor of
64, 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 term
s[n, α], a Gaussian noise term η g [n, α] and a non-Gaussian noise term η z [n, α]. Although
they are statistically dependent, it is straight-forward to prove that they are uncorrelated
and as only the first and second order moments of the I&D samples are derived, they can
be 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 following
three sub-sections. In the final section (Sec. 4.6.4), some claims regarding the noise term
are validated.
4.6.1 Statistics of the Signal Term
Although ordinary rather complicated, deriving the statistical properties of the signal
term has become rather straight-forward, due to the extended linear model presented in
Sec. 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]] = ˘K T e 1 . (4.65)
The correlation between s[n] and b[n] is by definition as follows
E[s[n]b ∗ [n]] = ˘K
]
T E[˘d[n + m]b ∗ [n] . (4.66)
Using (4.46), it is evident that,
E[s[n]b ∗ [n]] =
{ ˘KT e 2+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],s H [n]] ] [
= C ˘KT ˘d[n], ˘dH [n] ˘K
]
∗ (4.68)
= ˘K
] T C[˜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 equal
to
C [ s[n],s[n] H ] ] = ˘K T ( I − e 1 e T 1
) ˘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 η g will be computed.
As stated before, the term η g is in fact the superposition of two uncorrelated
Gaussian noise terms, η g1 and η g2 . Since both terms are uncorrelated and since the interest
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 η g1 [n, α] and

86 CHAPTER 4. THEORY OF TR UWB COMMUNICATIONS
η g2 [n, α] depend linearly on n[n], it is evident both Gaussian noise terms η g1 [n, α] and
η g2 [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[η g1 [n, α], b[n]] = E [ b ∗ [n]d[n] H] L α,1 E[n[n]] . (4.71)
Since the expectation of the noise vector is equal to zero, the first Gaussian noise is not
correlated to b[n]. Using a similar derivation, it is straight-forward to show that the same
also applies to the second Gaussian noise term η g2 .
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 transmitted
symbols d[n]. Fortunately, both processes are independent, which allows us to compute
the covariance in two consecutive steps. Firstly the covariance of both terms will be
computed conditioned on the transmitted symbols, after which the statistical properties
of the transmitted symbols are taken into account. The covariance between η g1 [n, α] and
η g1 [m, β] conditioned on d[n] and d[m] is given by
where
C[η g1 [n, α], η g1 [m, β]|d[n],d[m]] = d[n] H L α,1 C [ n[n],n[m] H] L H β,1d[m] (4.72)
= d[n] H L α,1 N nm L H β,1d[m] (4.73)
N nm E [ n[n]n[m] T] , (4.74)
N nm [k,l] = r nn ((n − m)T s + (k − l)T r ) (4.75)
In the same manner, the conditional covariance of the second Gaussian noise term
η g2 [n, α] is found to be
C[η g2 [n, α], η g2 [m, β]|d[n],d[m]] = d[n] T L α,2 N ∗ nmL H β,2d[m] ∗ (4.76)
In practice, the integration duration is long compared to the correlation time of the
noise. As a result, the noise matrix N nm is approx. an all zeros-matrix if n is unequal
to m. To simplify both derivation and the noise model, it will be assumed that they are
fully uncorrelated for n ≠ 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 a
single Gaussian η g noise term of which the variance can be described using a quadratic
Volterra model. The Volterra description for the Gaussian noise terms has been first
reported for traditional AcR receivers in [70, 71]. In our case, the model has the following
form,
{
d[n] H H α,β d[n] if n = m,
C[η g [n, α], η g [m, β]|d[n]] =
(4.77)
0 otherwise.
with
H α,β = L α,1 N nn L H β,1 + (L α,2 N ∗ nnL H β,2) T . (4.78)

4.6. STATISTICAL PROPERTIES OF THE TR SYSTEM MODEL 87
To simplify the statistical derivation, the linear model is also applied to the Volterra
noise model, i.e.
where
C[η g [n, α], η g [m, β]|d[n]] = ˜d[n]h α,β (4.79)
h α,β = T H vec (H α,β ). (4.80)
This notation greatly simplifies the derivation of unconditional covariance. In the unconditional
case, the covariance becomes,
]
C[η g [n, α], η g [m, β]] = δ[n − m]E
[˜d[n] h α,β (4.81)
Using statistical property 1 of ˜d[n] given by (4.45), the unconditional covariance of the
joint Gaussian noise terms is found to be
C[η g [n, α], η g [m, β]] = δ[n − m]e T 1 h α,β , (4.82)
which concludes the derivation of the first and second order moments of the Gaussian
noise terms.
4.6.3 Statistics of the Non-Gaussian Noise Term
Before deriving its statistical properties, the non-Gaussian noise term will be re-written
to simplify interpretation. Previously, the non-Gaussian noise term was defined as
η z [n, α] = h T αz[n] (4.83)
with z[n] equal to Λ(W 1 n[n])W 2 n[n] ∗ . The matrices W 1 and W 2 are however blockselection
matrices, containing only binary entries with only one-valued elements on its
main diagonal. Using this structure, the k-th element of the vector z[n], which will be
denoted as z[n, k], can be written as.
z[n, k] = n[n + k]n ∗ [n + k − N d ]. (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,
which means that
E[z[n, k]] = E[n[n + k]n ∗ [n + k − N d ]] r nn (D), (4.85)
E[η[n, α]] = h T α1r nn (D), (4.86)
which in turn means that the expectation of the non-Gaussian noise term depends on the
impulse response of the RX BPF, the delay and the integration interval duration only. In
practise, the delay duration will be larger than the duration of the impulse response of

88 CHAPTER 4. THEORY OF TR UWB COMMUNICATIONS
the BPF, i.e. r nn (D) = 0, so that it is reasonable to assume that the non-Gaussian noise
term is zero mean.
Since it does not depend on b[n], it is evident that the non-Gaussian noise term is not
correlated 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[η z [n, α], η z [m, β]] = C [ h T αz n ,z H mh β
]
= h
T
α Z n,m h β , (4.87)
where Z n,m = C [ z n ,z H m]
. 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 the
complex-valued case,
Z n,m [k,l] = |r nn ((n − m)T s − (k − l)T r )| 2 . (4.88)
where Z n,m [k,l] denotes the element at position k,l in the matrix Z n,m . Using the same
reasoning as applied to N n,m , in practise Z n,m will be virtually an all zero matrix. The
validity of this assumption will be verified in Sec. 4.6.4. This also concludes the derivation
of 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 the
samples n n,α and n m,β are potentially correlated, when the difference between n and m is
small. In the derivation of the statistical properties of the noise in the previous subsections
4.6.2 and 4.6.3, they were assumed to be uncorrelated. In this subsection, the validity of
this assumption will be shown.
To compute the covariance between samples related to different outputs, the L outputs
are multiplexed into a single sample stream. The covariance matrix of the resulting
cyclo-stationary sample stream with period L is investigated. The assumed scenario is a
residential NLOS environment in which the TR system is operated at 10 and 80 Mb/s and
four times oversampling. This represents two extreme scenarios, one without ISI and with
severe ISI, respectively. Both the Gaussian and the non-Gaussian term can dominate the
overall noise term, depending on the value of E b /N 0 . Therefore, the covariance of both
noise terms will be analyzed separately.
In Fig. 4.16, the covariance matrix of the Gaussian noise term is depicted at both data
rates. 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 between
two consecutive samples is well below 0.25. At lower data rates, this correlation is even
less. Strictly speaking, the Gaussian noise term is not truly white, but the approximation
error will be small when assuming it to be white.
The same procedure is repeated for the non-Gaussian noise term. In Fig. 4.17, the
covariance of this noise term is presented. As expected, every element on the main
diagonal has the same value, due to the stationary nature of this noise term. Comparing
both data rates, the variance is 8 times higher at 10 Mb/s compared to the 80 Mb/s
case, because the integration duration is 8 time longer as well. In either case, the decorrelation
is rapid and zero if one or more samples are in between the samples under

4.6. STATISTICAL PROPERTIES OF THE TR SYSTEM MODEL 89
(a)
(b)
C[ng[x],ng[y]]
4
3
2
1
[n+1,3] 0
[n+1,1]
1
0.8
0.6
0.4
0.2
[n+1,3] 0
[n+1,1]
C[ng[x],ng[y]]
y
[n,3]
y
[n,3]
[n,1]
[n,0] [n,1] [n,2] [n,3] [n+1,0] [n+1,1] [n+1,2] [n+1,3]
x
[n,1]
[n,0] [n,1] [n,2] [n,3] [n+1,0] [n+1,1] [n+1,2] [n+1,3]
x
Figure 4.16: The covariance matrix of Gaussian noise in the sample-stream of a four
times fractionally sampled AcR operating at 10 and 80 Mb/s in sub-figure (a) and (b),
respectively
(a)
(b)
C[nz[x],nz[y]]
10
8
6
4
2
[n+1,3] 0
[n+1,1]
1
0.8
0.6
0.4
0.2
[n+1,3] 0
[n+1,1]
C[nz[x],nz[y]]
y
[n,3]
y
[n,3]
[n,1]
[n,0] [n,1] [n,2] [n,3] [n+1,0] [n+1,1] [n+1,2] [n+1,3]
x
[n,1]
[n,0] [n,1] [n,2] [n,3] [n+1,0] [n+1,1] [n+1,2] [n+1,3]
x
Figure 4.17: The covariance matrix of Non-Gaussian noise in the sample-stream of a four
times fractionally sampled AcR operating at 10 and 80 Mb/s in sub-figure (a) and (b),
respectively

90 CHAPTER 4. THEORY OF TR UWB COMMUNICATIONS
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 compared
to the integration duration at this data rate. Nevertheless, the correlation between two
different samples is always lower than 0.15, making this process quasi-white. This result
is inline with the conclusions drawn in [58].
4.7 Conclusions
In this chapter, the principle of UWB Transmitted-Reference communication was introduced,
including a discussion of its pro’s and con’s with respect to performance and
implementation. Furthermore, several extensions of the TR principle were proposed.
Firstly, a fractional sampling AcR structure was proposed to allow for synchronization and
weighted autocorrelation, which also simplifies the implementation. Secondly, a complexvalued
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 scheme
to complex-valued modulation.
A general-purpose discrete-time equivalent system model was presented for the analysis
of TR systems, where general-purpose means that all extensions are accounted for. It
was shown that the I&D samples generated by a fractional sampling AcR in a TR system
consist of two contributions with a different nature, a signal term and a noise term. The
signal term could be modelled using a SIMO FIR Volterra model. The noise terms was
shown to consist of two types of noise, a Gaussian noise sub-term and a non-Gaussian
noise sub-term.
Several interpretations for the SIMO FIR Volterra model have been presented, which
allow for more insight in the behaviour of TR systems. Firstly, the Volterra model
has been written in a normal vector notation and extended vector notation, to allow
for simplified statistical analysis. The extended vector notation also allowed for the
interpretation 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 that
trellis-based algorithms can be used for the equalization of TR systems. In this line of
reasoning, a reduced-memory system model was introduced, which mimics the behaviour
of TR systems, but with a significant memory reduction.
Finally, the statistical properties were derived of the signal term as well as both
noise terms and the noise was shown to be quasi-white, with an output dependent noise
variance.

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 many
system parameters on the system performance was left undefined, but required to obtain a
solid system design based on the TR UWB communication. In this chapter, the impact of
different parameters on the system performance will be analyzed. The evaluated system
parameters are FSR, bandwidth, delay, weighting criteria and modulation both in the
absence and presence of ISI. One of the main contributions presented in this chapter is
that 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 this
section, the weighting applied to the I&D samples and the decision process for the detected
bits is presented. For notational convenience, the samples on which linear weighting is
applied are assumed to be gathered in a vector denoted by u[n]. Its composition is as
follows,
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] = w T u[n] + c, (5.2)
where w is a vector containing the weighting coefficients. Due to the random nature of
the channel, both w and c should be adaptable. Therefore, adaptive weighting algorithms
will 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, namely
MRC 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 weighting
vector is given by,
{
R −1
w =
uup MMSE criterion
(5.3)
p MRC criterion
where
R uu = C[u[n],u[n]] , (5.4)
p = C[u[n], b n ] .
In either case, the offset equalization factor will be equal to
c = −wE[u[n]] . (5.5)
The expressions of R uu , p and E[u[n]] have been presented in Sec. 4.6, such that all the
mathematical tools are available to compute the weighting vector in a closed form.
The value of the detected b[n], denoted by ˆb[n], is based on the sign of α[n], such that
ˆb[n] = sign (α[n]). (5.6)
The probability of an erroneous decision for b[n] can be computed in closed-form. A
detailed 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 and
modulation 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, except
for the system parameter(s) under evaluation.
The general system set-up is as follows; to allow for reference with work by others, a
TR signaling scheme using BPSK signaling is assumed, demodulated using a traditional
real-valued AcR. The symbol rate is chosen equal to 10 MHz, which in case of BSPK
modulation 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 assumed
to match perfectly. The bandwidth of the TX signal is approx. 500 MHz with a center
frequency of 4.5 GHz. The RX BPF is matched to the TX signal. The FSR L is chosen
equal 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 the
I&D samples, so that the weighting vector is optimal. However, no time-synchronization
is assumed between TX and RX, i.e. due to the cyclo-stationarity of the TX signal with
period T s , the time-offset has been modelled as a RV with an uniform distribution over
the interval [0, T s 〉. The propagation environment is the NLOS environment as described
in Sec. 3.2. To obtain better insight on the role of the system parameters, SSF has not
been taken into account. The role of bandwidth with respect to SSF has already been
thoroughly investigated in chapters 2 and 3.

5.3. SYSTEM PERFORMANCE IN THE ABSENCE OF ISI 93
10 0 E b /N 0 [dB]
10 −1
10 −2
FSR=1,MMSE
FSR=1,MRC
FSR=2,MMSE
FSR=2,MRC
FSR=4,MMSE
FSR=4,MRC
FSR=8,MMSE
FSR=8,MRC
P(e)
10 −3
10 −4
5 10 15 20 25
Figure 5.1: Influence of the weighting criteria and FSR on the system performance in the
absence of ISI
5.3.1 Influence of the Weighting Criteria and Fractional Sampling
Rate
In Sec. 5.2, two weighting principles have been proposed, MMSE combining and MRC. In
this subsection, the difference in performance between both principles and the role of the
FSR will be investigated. Four different FSRs have been considered, namely 1, 2, 4 and
8. Additionally, the performance results also show the ability of the RX to synchronize
to 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 it
cannot influence the time-offset of the integration interval. As a result, the I&D samples
may contain information not only regarding the symbol under demodulation, but also
of other symbol. In other words, the I&D samples suffer from ISI, not caused by pulseoverlapping,
but because the integration interval is gathering information of multiple
symbols. Secondly, the AcR is accumulating more noise due to the long integration
duration. Comparing both weighting principles, MMSE combining is coping better with
ISI than MRC weighting.
If the FSR is increased, both weighting principles obtain more information from the
channel, so that the problems occurring in a system without fractional sampling can be
resolved by the RX. As a result, MRC has approx. the same performance as its MMSE
combining 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 single
sample contains the information on b[n]. In this case, the MRC and MMSE weighting

94 CHAPTER 5. ANALYSIS OF TR UWB COMMUNICATION
10 0 E b /N 0 [dB]
10 −1
FSR=2,D=40ns
FSR=2,D=10ns
FSR=4,D=40ns
FSR=4,D=10ns
FSR=8,D=40ns
FSR=8,D=10ns
10 −2
P(e)
10 −3
10 −4
5 10 15 20 25
Figure 5.2: Influence of the delay and FSR on the system performance in the absence of
ISI
vector will be virtually the same, explaining the similar performance. At (very) low E b /N 0
values, the stationary, non-Gaussian noise term is dominant and the SNR of the samples
in u[n] becomes proportional to p, so that the MMSE combining vector converges to the
MRC vector.
As to be expected, increasing the FSR will improve the system performance, but the
improvement cannot justify the additional complexity. The avoidable part of Gaussian
noise namely falls largely in the samples that precede and follow the sample containing
the desired information term, even when the FSR is equal to two. This is caused by the
delay, 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 the
system.
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 pulse
and modulated pulse arrive overlapped, the variance of the Gaussian noise terms will
increase 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 combining
is considered and the delay is decreased to 10 ns. The resulting performances have been
depicted 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
10 0 E b /N 0 [dB]
10 −1
BW=250MHz
BW=500MHz
BW=1GHz
10 −2
P(e)
10 −3
10 −4
5 10 15 20 25
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 analyzed.
In linear systems, the BER performance on AWGN channels does not depend on
the bandwidth but only on the E b /N 0 ratio [31]. This rule however does not apply to
AcRs. Their non-linear structure leads to the presence of the non-Gaussian noise term
with a variance proportional to the RX BPF bandwidth.
To obtain insight in the impact of this additional noise term on the overall performance,
the performance of TR systems is compared for three different bandwidths in
Fig. 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 impact
on the overall system performance. As to be expected, the system with the smallest
bandwidth outperforms the others. At a BER of 10 −2 , the performance decreases approximately
1 dB with each doubling of the bandwidth. The distance between the BER
curves has the tendency to decrease with increasing E b /N 0 -ratio. However, the curves are
still 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 modulation
was introduced. The modulation has been shown to influence the number of MIMO

96 CHAPTER 5. ANALYSIS OF TR UWB COMMUNICATION
Table 5.1: Properties of the considered TR systems to analyze the role of modulation
System mod(˜b[n]) mod(b[n]) Receiver L
1 1 BPSK Real-Valued AcR 4
2 1 BPSK Complex-Valued AcR 4
3 1 QPSK Complex-Valued AcR 4
4 BPSK QPSK Complex-Valued AcR 4
10 0 E b /N 0 [dB]
10 −1
BPSK,RV−AcR
BPSK,CV−AcR
unscrambled QPSK,CV−AcR
scrambled QPSK,CV−AcR
10 −2
P(e)
10 −3
10 −4
5 10 15 20 25
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, the
number of MIMO inputs will be unaffected by the modulation. Therefore, it is expected
that the modulation has little impact on the performance of the TR system in the absence
of ISI. This expectation will be validated in this section.
The performance of 4 TR systems will be compared. The first TR system used BPSK
modulation and a real-valued AcR. Two other systems employ QPSK-TR, one applies
no scrambling on the TR waveform, while other one does. Since in case of complexvalued
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 a
complex-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 a
fair analysis of the role of modulation. An overview of the properties of the 4 TR systems
can be found in Tab. 5.1. The performance of the four systems in the absence of ISI has
been depicted in Fig. 5.4.
In the absence of ISI, all modulation schemes have the same Euclidian distance between
the symbols, such that approximately the same BER performance is expected for

5.4. SYSTEM PERFORMANCE IN THE PRESENCE OF ISI 97
10 0 E b /N 0 [dB]
10 −1
10 −2
P(e)
10 −3
10 −4
FSR=1,MMSE
FSR=1,MRC
FSR=2,MMSE
FSR=2,MRC
FSR=4,MMSE
FSR=4,MRC
5 10 15 20 25
Figure 5.5: Influence of the weighting criteria and FSR on the system performance in the
presence of ISI
all four systems. As expected, Fig. 5.4 shows that all four systems have approximately
the same performance, indicating that the modulation scheme has little to no influence
on 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 for
comparison, all system parameters are the same as in the previous section, except that
the 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 Sampling
Rate
In Sec. 5.3.1, it was concluded that the weighting criteria and FSR had hardly any influence
on the system performance in the absence of ISI, provided the FSR is at least equal
to 2. In this section, it will be investigated whether this conclusion holds in the presence
of 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 efficiency
of the system. Furthermore, MRC performs considerable worse than MMSE combining,
since it incorrectly presumes the noise and ISI to be stationary. This in contrast to
ISI-free conditions, where both weighting principles performed almost equally well. To
illustrate the effect of the FSR, it is varied from 1, 2 to 4. In any case, the performance

98 CHAPTER 5. ANALYSIS OF TR UWB COMMUNICATION
improved when increasing the FSR. In case of MRC weighting, the BER floor decreases
only slightly with increasing FSR, while MMSE combining is able to use the additional
degree of freedom for weighting more effectively, leading to a significant performance
improvement.
This does not explain why increasing the FSR improved the performance in the presence
of ISI. For traditional linear narrowband systems, the Nyquist criteria states that a
sampling rate larger than twice the symbol-rate will not provide the RX more information
on the channel and thus not on the transmitted symbol. Hence, increasing the FSR
above 2 will not lead to a performance improvement in this case. The explanation for the
performance improvement in the case of FSR TR systems can be given using the MIMO
interpretation presented in Sec. 4.5.4.
In the MIMO model, both linear and non-linear ISI terms were modelled as additional
inputs, where all inputs are fed using symbols with the same first- and second-order
statistical properties as the symbol under demodulation b[n]. Without changing the
modulation 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 an
increasing ratio of outputs with respect to the inputs, i.e. more ISI will be suppressed
with an increasing FSR L. 1
5.4.2 Influence of Bandwidth
Previously, its was shown that with increasing bandwidth, the amount of non-Gaussian
noise 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 results
have been shown in Fig. 5.6.
Firstly, in the E b /N 0 region in which the noise is dominant, i.e. at lower E b /N 0 -
values, the system with the smallest bandwidth still outperforms the other. However,
with increasing E b /N 0 the noise becomes less significant and the system becomes ISI
limited. In this case, Fig. 5.6 shows that large bandwidth TR systems suffer less from ISI
than their narrowband counterparts. The system’s sensitivity to ISI is namely related to
the 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 systems
eventually outperform their smaller bandwidth counterparts with an increasing E b /N 0 .
5.4.3 Influence of Modulation
In Sec. 5.4.1, it was shown that with an increasing FSR more ISI can be suppressed. Not
so obvious is however the role of modulation in the presence of ISI. In this section, the
BER performance of four TR systems is compared under ISI conditions, to show that
1 These insights are expected to hold for delay-hopped differential signaling and for systems deploying
on/off keying in combination with an energy detector. All these systems can namely be modelled using
a Single-Input, Multiple-Output (SIMO) FIR Volterra system, assuming the related detectors to be
fractionally sampled.

5.4. SYSTEM PERFORMANCE IN THE PRESENCE OF ISI 99
10 0 E b /N 0 [dB]
10 −1
10 −2
P(e)
10 −3
10 −4
MMSE,BW=250MHz
MRC,BW=250MHz
MMSE,BW=500MHz
MRC,BW=500MHz
MMSE,BW=1GHz
MRC,BW=1GHz
5 10 15 20 25
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 systems
as 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 E b /N 0 is
depicted. Comparing the performance of both BPSK TR systems, one can see that the
use of a complex-valued AcR can improve the system performance significantly. In the
complex-valued case, in fact, the sampling rate is increased by a factor 2, meaning that
the MMSE weighting vector has twice as much degrees of freedom to suppress ISI. Using
the linear MIMO system interpretation, one can say that the amount of MIMO inputs is
unaltered, but the number of outputs is increased by a factor 2, such that more ISI can
be suppressed.
Comparing the three TR systems using a complex-valued AcR, the performance decreases
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, which
on its turn means that the ISI is spread over more dimensions in the space spanned by
the vector u[n]. As a result, the probability becomes larger if more ISI interferes with the
desired term, such that linear weighting can suppress less ISI. On the same channel, it
can be expected that more ISI can be suppressed when the modulation is more restrictive
without needing to reduce the symbol rate.
This result also means that scrambling of the reference pulses, i.e. to avoid spectral
peaks in the PSD of the TX signal, may decrease the system performance depending on
the channel conditions. Furthermore, with increasing N t , the number of MIMO inputs
is super linear with respect to M, while the number of outputs is linearly related to

100 CHAPTER 5. ANALYSIS OF TR UWB COMMUNICATION
10 0 E b /N 0 [dB]
10 −1
10 −2
P(e)
10 −3
10 −4
BPSK,RV−AcR
BPSK,CV−AcR
unscrambled QPSK,CV−AcR
scrambled QPSK,CV−AcR
5 10 15 20 25
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 deteriorate
with increasing channel memory, making linear weighting only suitable for channels with
moderate ISI.
5.5 Conclusions
An alternative AcR has been proposed for basic TR signaling, that allows for synchronization
using DSP and is able to suppress more ISI if MMSE combining is deployed. A
statistical characterization of the system has been presented, which allows for the computation
of the weighting vector. To analyze the performance, a method to compute
the BER has been described, which is exact with respect to ISI, but assumes Gaussian
distributed noise at the demodulator output. Simulation results for several TR systems
have been compared.
In the absence of ISI, a TR system with a smaller bandwidth will outperform one
with a larger bandwidth, not taking SSF into account. MRC results in approximately
the same performance as MMSE combining.
In the presence of ISI, large bandwidth TR systems are inherently less sensitive to
ISI. However, the performance loss can be partly compensated by increasing the FSR,
illustrating that with proper linear filtering and fractional sampling, non-linear ISI can
be suppressed. Specifically, the 40 Mb/s, 250 MHz system with a FSR of 4 and MMSE
combining performs reasonably well, while a reduction of the FSR leads to a significant
performance decrease. Although only TR signaling is considered, the general conclusions
are 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. The
interpretation not only explains how linear weighting can suppress non-linear ISI, but also
explains the impact of the modulation scheme on the performance of linear weighting with
respect to ISI suppression. It is shown that the number of MIMO inputs depends on the
modulation scheme. Since the amount of suppressible ISI depends partly on the number
of 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 negative
effect on the BER performance of a TR-system under ISI conditions. Furthermore,
the system performance deteriorates quickly with increasing channel memory, making
linear weighting suitable for channels with moderate ISI, but not in case of severe ISI.

102 CHAPTER 5. ANALYSIS OF TR UWB COMMUNICATION

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 aim
is a TR-UWB PHY supporting a data rate of 100 Mb/s, while occupying a bandwidth of
1 GHz. Based on the insight gained in the previous chapters, the design considerations
for the system are described in Sec. 6.2. A detailed description of the considered system
is presented Sec. 6.3. The performance and complexity of the system is presented in
Sec. 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. It
has been shown that non-linear ISI can be equalized using linear weighting in moderate
ISI 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 if
scrambled 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-linear
ISI also contains information on the transmitted symbols. Taking all these considerations
into 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 as
an FSM, see Sec. 4.5.5. As a result, trellis-based equalization can be used to equalize ISI
if the FSM structure is known. Regretfully, the complexity of a trellis-based equalizer
grows exponentially with the memory of the FSM. In Sec. 4.5.6, a RMDM has been
introduced to reduces the memory of the FSM, while capturing the essential behaviour of
the 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 a
basic TR UWB signaling scheme, the first pulse of a TR symbol is transmitted unmodulated,
while the second pulse is modulated. Assuming white, zero mean modulation, the
TR UWB signal is thus the superposition of two independent signals, the first one consisting
of a train of modulated pulses, while the second is a train of reference pulses. Based
on the statistical independence of both signals, the overall PSD will be the superposition
of the PSD of both signals.
The PSD of the train of modulated pulses is easily derived. Since white, zero-mean
modulation is applied on the modulated pulses, the shape of its PSD will be determined
by 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 the
PSD 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 radio
signals. The resulting PSD will still contain spectral spikes. However, the PSD will
consist 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. This
not only complicates synchronization, but also results in a time-variant Volterra kernel,
making kernel estimation and equalization more complicated. Therefore, time-hopping
has not been considered.
Another method to smooth the PSD, which does not destroy the cyclo-stationary
nature of the TR signal, is to apply non-information-bearing sign modulation on the TR
symbols. 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 squared
magnitude of the Fourier transform of the individual pulses. The PSD thus no longer
contains 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 be
generated using a PN sequence or alternatively the scrambling code can be derived from
the symbol identifiers a[n], see Sec. 4.5.5. When using a PN sequence, the trellis diagram
of the FSM describing the Volterra kernel will be time-variant, which complicates its
equalization. To ensure a time-invariant trellis diagram, the modulation applied to both
pulses will be driven by the symbol identifiers, see Sec. 4.5.5. This poses the second
constraint on the scrambling.
Both constraints on the scrambling code cannot be fulfilled simultaneously for all
possible 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 channel
bits on the symbol identifier is presented here as well, where c[n, k] denotes the k-th
channel bit signalled by the n-th TR-symbol.
In Appendix C, it is shown that the modulation of the pulses is indeed uncorrelated/white.
Hence, the PSD of the TR signal will depend only on the pulse shape. This
proves that a smooth PSD can be obtained using symbol-identifier-driven scrambling in
case of QPSK-TR signalling.

6.2. DESIGN CONSIDERATIONS FOR A HIGH-RATE TR UWB SYSTEM 105
Table 6.1: Symbol mapping table
c[n, 0] c[n, 1] a[n] ˜b[n] b[n]
1 1 0 1 j
1 -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 the
Volterra system modeling the channel for each fractional sampling position. Due to
the random nature of the radio channel, its coefficients must be estimated at the RX
either blindly or using training-sequences. In [76, 77], it is shown that the identification
of Volterra systems can be conducted blindly. However, the learning times for blind
identification are rather long and therefore not considered.
For training-sequence based identification of the Volterra kernels, two fundamentally
different strategies have been considered. The first approach uses the linearity of a
Volterra system with respect to kernel elements, which makes the identification similar
to the identification of linear systems. Therefore, this technique is referred to as linear
Volterra kernel estimation. Instead of aiming to estimate the whole Volterra kernel containing
(2M + 2) 2 elements, the vector notation presented in Sec. 4.5.2 is used to reduce
the number of kernel elements to be estimated to N k . The estimate of the kernel will
be specific for the used modulation type. In practice, the modulation type will not be
changed during transmission of a packet, making this drawback irrelevant. The same algorithms
used for linear channel estimation, like the Least Mean Square (LMS) algorithm
or 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-transitionspecific
time-invariant outputs. The approach will be referred to as trellis-based system
identification. The amount of unknown elements to be estimated is equal to the amount
of state-transitions, i.e. 2 M+1 and 4 M+1 for BPSK-TR and QPSK-TR, respectively. The
a-priori knowledge on the Volterra kernel structure is not exploited, which makes this
approach robust against time-invariant imperfections in the RF front-end.
Roughly speaking, the performance of an estimation algorithm is proportional to
the number of elements to be estimated. In case of scrambled QPSK-TR, the linear
Volterra kernel estimator has less unknowns to be estimated and is thus expected to
perform better than trellis-based system identification. For every value of M, N k is
namely smaller than 4 M+1 , especially for M greater than 2, see Sec. 4.1. Only if M
equals 1, trellis-based system identification is expected to perform slightly better. Based
on these considerations, the linear Volterra kernel estimation approach is favoured, also
because it is well-established in literature [64]. Nevertheless, the channel memory should
be kept low to allow for shorter training-sequences.

106 CHAPTER 6. DESIGN OF A HIGH-RATE TR UWB SYSTEM
6.2.4 Multiband Transmitted Reference
One way to increase the data rate of a TR system is to increase the pulse rate. Unavoidably,
the channel memory will increase as well. Unfortunately, the complexity of
trellis-based equalization and Volterra system identification grows (approx) exponentially
with the channel memory, see Sec. 6.2.1 and Sec. 6.2.3. In [78], it is proposed to divide the
spectral resource into subbands, where in each subband energy detection based communication
is used. In each subband, the data rate will be relatively low, such that little to
no ISI will occur, while the accumulated data rate can still be high. Evidently, a trade-off
is possible between the amount of ISI and the number of subbands. This concept is not
limited to energy detection, but can be applied to TR signaling as well.
The multiband principle has more distinct advantages. Due to the memory reduction,
N k is reduced for each subband, meaning that less kernel elements need to be estimated
for each subband. Since the number of kernel-elements grows super linearly with the
memory size, the total number of kernel elements to be identified decreases with the
introduction of subbands. Furthermore, it inherently creates parallel structures, such that
the rate at which the algorithms are operated is reduced. Also the implementation of the
TR delay used in each subband is simplified, because its transfer function must only be
well-behaving over a smaller portion of bandwidth. Additionally, the architecture allows
for the detection and coherent suppression of narrowband interference. An important
advantage since TR systems are inherently sensitive to interference in general, due to
their non-coherent, non-linear transfer function. Furthermore, the system can easily be
extended to support DAA, which may be demanded by the regulation bodies of Europe
and Japan to operate UWB [14, 15].
A drawback of multiband is that the TR signal of each subband is more susceptible to
SSF, see Chapters 3 and 5. Similar to OFDM, an FEC scheme will be deployed to exploit
the frequency diversity provided by the system bandwidth, where the system bandwidth
is 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 communication
system deploys FEC to improve its energy efficiency, such that the coverage
area and/or data rate can be increased. By nature, an FEC scheme is divided over the
TX and RX. At the TX, an FEC encoder adds redundant information to the data to be
transmitted. The redundant information allows the RX to detect and correct (to some
extend) errors introduced by the channel. To what extend depends on the amount and
manner the redundant information is introduced by the FEC encoder. The bits encoded
at the TX will be referred to as information bits, while the bits generated by the FEC
coder are called channel bits. The channel bits will be allocated to the different subbands
using a de-multiplexer, which assigns every k-th bit of N sb subsequent channel bits to subband
k. To exploit the full potential of the FEC, an interleaver Π c is positioned between
the FEC encoder and de-multiplexer to ensure the channel bits are spread randomly over
the subbands.

6.2. DESIGN CONSIDERATIONS FOR A HIGH-RATE TR UWB SYSTEM 107
FSM 2 /subband 1
channel bits
S 0
FSM 1 /FEC
S 1
info bits
S 0
S 1
Π c
Demux
FSM 3 /subband 2
S 0
S 1
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 generated
at the AcR output at each subband can be modelled as a series of serial and parallel
concatenated FSMs. A graphical representation of the concatenated FSMs for a TR
UWB system with two subbands has been depicted in Fig. 6.1.
Optimal Maximum-Likelihood Sequence Detection (MLSD) would be desirable, but
the related algorithm is too complex for implementation. In 1993, the turbo principle was
first introduced by Berrou et.al. [79] for parallel concatenated FSMs, where both FSMs
were convolutional encoders. Shortly after, the turbo principle was introduced to the
equalization of linear ISI channels [80], using the fact that an FEC encoder followed by
an ISI radio channel can be seen as a serial concatenation of two FSMs. In [81], it is shown
that iterative decoding using the turbo principle can be seen as a practical implementation
of an MLSD. Due to their good performance, both turbo coding and turbo equalization
for 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, but
a few papers have been published on the topic [86, 87]. Nevertheless, the non-linearity of
the channel has no principle impact on the turbo equalization scheme.
In any turbo scheme, the decoders of each FSM exchange soft decisions on the channel
bits, where an equalizer is also considered to be a decoder. Each decoder uses the
soft decisions of the other decoder, the information gathered from the channel and the
structure of the FSM under decoding, to update its soft decisions. These soft decisions
are 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 finite
training-sequence length, the Volterra system identification algorithm will provide an
inexact 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 subsequent
iteration to update/improve the estimate of the Volterra channel FSM. Possibly, this
allows for good perform while using shorter training-sequences. In [88], this approach has
been proposed for linear channels. Without any fundamental differences, this approach

108 CHAPTER 6. DESIGN OF A HIGH-RATE TR UWB SYSTEM
0
PSD
−10
−20
1−Band
2−Band
4−Band
−30
5.4 5.6 5.8 6 6.2 6.4 6.6
E b /N 0 [dB]
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 us
assume a block of information bits of length 4098, which will be denoted by b, where
b[n] denotes the n-th information bit. The block b is encoded to a block of channel bits
c, using a terminated rate- 1 FEC coder. The block c will be of length 8200. A more
2
detailed description of the FEC scheme can be found in Sec. 6.3.2.
A channel interleaver Π c 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 this
thesis, a random interleaver is deployed. The interleaved channel bits are de-multiplexed,
to obtain N sb sub-blocks with channel bits of length 8200/N sb , one for each subband. The
block of channel bits communicated of the i-th subband will be denoted with c i .
Three multi-band systems are considered in this thesis, using respectively 1, 2 and
4 subband(s). The system bandwidth is always equal to 1 GHz. Based on the results
of chapters 3 and 5, 1 GHz of bandwidth will suffice to allow for communication robust
against SSF. The division of the spectral resources into subbands has been depicted in
Fig. 6.2.
In each subband, QPSK-TR signaling is deployed, such that 2 channel bits are mapped
onto a single TR symbol. For notational convenience, the c i [n, k] will denote the k-th
channel 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 duration
in each subband is equal to 10, 20 and 40 nanoseconds and the delay D is equal to 4, 8
and 16 nanoseconds for a multiband system with 1, 2 and 4 subband(s), respectively. In
any case, the overall symbol rate will be equal to 100 MHz, i.e. the channel-bit rate is

6.3. SYSTEM DESCRIPTION 109
Modulators BPFs
BPFs AcRs
FEC
Encoder
Channel
Interleaver
Π c
infobits
TR Mod.
Radio
Channel
CV-AcR
Demux
+
I&D-
Samples
TR Mod.
CV-AcR
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 equal
to 100 MHz.
The received signal of each subband is demodulated using a CV AcR with matching
delay and a fractional sampling rate of two. The sampling phase is not synchronized to
the received signal, i.e. the Analogue to Digital Converter (ADC) are running freely. The
clock-rate of the ADCs and the subsequent Digital Signal Processing (DSP) will be equal
to 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&D
samples 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 equalization
scheme for linear channels. The best results were obtained using a turbo FEC
scheme based on recursive systematic convolutional codes. The same FEC scheme will be
deployed in this thesis, in the hope it will also provide good performance on second-order
FIR Volterra channels.
The turbo coder consists of two identical, parallel concatenated, rate- 1 , Recursive Systematic
Convolutional Codes (RSCCs). Each coder is defined by the polynomials (5, 7),
2
resulting in a memory-two FSM. The first RSCC encoder receives the information bits
directly, while the second RSCC encoder encodes information bits that have been passes
through the interleaver Π. Both coders are terminated to the all-zero state. To obtain
a rate 1 , the systematic output of the second RSCC encoder is punctured continuously.
2
One out of two non-systematic channel bits of either RSCC encoder is punctured in an
alternating manner.
A block diagram of the turbo encoder can be found in Fig. 6.4, where the systematic
output 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 seen
as two parallel concatenated FSMs.

110 CHAPTER 6. DESIGN OF A HIGH-RATE TR UWB SYSTEM
Info bits
Π
+
+
+
Punct.
Table
⎡
Z −1 Z −1 ⎣ 1 1 ⎤
1 0⎦
0 1
+ +
+
Z −1 Z −1
Channel
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)
ĉ ⇔ 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 probabilities
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 counterparts
called Log-Likelihood Values (LLVs) are preferable, because the product of two
probabilities will become a sum in the logarithmic domain, making the implementation
less complex. Furthermore, LLVs are inherently better suited for the representation of
small probabilities in finite bit-width.
By definition, the LLV of a bit c ∈ {1, −1} with a probability P(c = 1) and its inverse
are defined as
( ) P(c = 1)
L(c)ln
(6.1)
P(c = −1)
( ) exp(L(c))
P(c = +1) =
(6.2)
1 + exp(L(c))
For completeness, some important relations between the probability domain and LLV
domain are collected in Tab. 6.2.
Each FSM is processed using an algorithm that accepts and generates LLVs, which are
referred to as a-priori and a-posteriori LLVs, respectively. The decoder itself is referred to

6.3. SYSTEM DESCRIPTION 111
L(c)
u
SISO decoder
L(b c |u,L(c), F)
L(b i |u,L(c), F)
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 decoder
is depicted in Fig. 6.5.
In a general-purpose SISO decoder, a probabilistic computation is conducted to obtain
the a-posteriori LLVs for both the information bits and channel bits, taking into account
the I&D samples, the a-priori LLVs and the structure of the FSM under evaluation. An
example 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 FSM
structure 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, number
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 presented
in Sec. 6.3.4.
In our case, the system consists out of N sb + 1 trellis objects, one for each subband
and the FEC decoder. The trellis object related to the i-th subband will be denoted with
K i 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 turbo
equalization scheme. For instance, the soft decoder of the sub-channels, i.e. the soft
equalizers, are unable to compute the LLVs of the information bits, simply because the
required information is not contained in the trellis object describing the channel H. The
FEC 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 as
a-priori information by the second decoder. The output LLVs are namely derived using
a-priori information delivered by decoder 2 to decoder 1 in the first place. To ensure
the convergence of the turbo scheme to what is hopefully the MLSD, instead of a selfconvincing
system, only the information that is foreign to decoder 2 is passed on. This
information is referred to as extrinsic information, where extrinsic means foreign. The

112 CHAPTER 6. DESIGN OF A HIGH-RATE TR UWB SYSTEM
n
00 2 00 2 00 2
S t [n] S t [n + 1] n + 1 S t [n + 2]
11 2
1 (0.9, 0.3)
11 2
1
(0.9, 0.3)
11 2
0
(1.3.1.1)
0
(1.3.1.1)
1
1
10 2
(0.1, 1.1) 10 2
(0.1, 1.1) 10 2
0
0
(1.8, 0.9)
(0.8, 0.3)
(1.8, 0.9)
(0.8, 0.3)
1
1
01 2
(0.3, 1.4)
01 2
(0.3, 1.4)
01 2
0
0
1
(0.6, 0.5) 1
(0.6, 0.5)
0
(0.2, 0.7) 0
(0.2, 0.7)
a[n]
s[n, 1] s[n, 2]
if PN-seq.→ Time variant
Figure 6.6: Trellis diagram of a FIR Volterra model in the absence of PN scrambling
extrinsic LLVs are defined as,
L e (c) = L(c|u, L(c), H) − L(c). (6.3)
The exchange of extrinsic information in a turbo equalization scheme has been depicted
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 turbo
scheme conducts the inverse operation of the TX, i.e. the extrinsic information coming
from 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 RX
has full knowledge on the trellis object F. To provide for turbo equalization, the equalizer
has to accept LLVs and generate LLVs. Two different classes of algorithms are possible,
namely MLSD and symbol-by-symbol Maximum A-Posteriori (MAP) decoding. The
MLSD detection is often performed using the well-known Viterbi algorithm introduced
in [91]. A detailed analysis of its operation has been described by Forney in [92]. The
Viterbi algorithm itself does not generate soft decisions. In 1989, a Soft-Output Viterbi
Algorithm (SOVA) was introduced by Hagenauer [93].

6.3. SYSTEM DESCRIPTION 113
u 1
SISO decoder
−
+
+
Demux
Π c
−
+
+
K 1
Mux
Π −1
c
SISO decoder
SISO decoder
−
+
+
F
u 2
K 2
Figure 6.7: Structure of the turbo equalizer
Symbol-by-symbol MAP decoding can be conducted using an Bahl, Cocke, Jelinek
and 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 its
implementation, the BCJR algorithm was translated to the logarithmic domain. In this
domain, two algorithms were derived, namely a sub-optimal Max-Log-MAP algorithm
and an optimal Log-Map algorithm [69, 94].
Although already introduced in 1969, the BCJR algorithm or one of its derivatives
were rarely used. The main reason is that minimizing the sequence error probability was
of more importance for most applications. Furthermore, a Max-Log-MAP or Log-MAP
is approx. twice as complex as a Viterbi algorithm [69, 94]. The situation changed with
the introduction of the turbo principle. The convergence of a turbo scheme relies on the
exchange of accurate soft decisions between the decoders of the concatenated FSMs. Since
a SOVA is by nature an MLSD algorithm, it is inherently unable to generate accurate
LLVs 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 is
decoding a subband channel object K i , only for the channel bits. In case F is under
decoding, the Log-MAP decoder generates both LLVs of the information bits and the
channel 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 channel
bit or information bit, which will be denoted by denoted by q[n]. A Log-MAP algorithm
now computes the LLV of q[n], which is defined as
( )
P(q[n] = 1|u, L(c), H)
L(q[n]|u, L(c), H) = ln
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]. The
set of possible state-transitions will be denoted by S tt . This set can be divided into
two disjoint subsets, where S +1
tt and S −1
tt denote the set of possible state-transition given

114 CHAPTER 6. DESIGN OF A HIGH-RATE TR UWB SYSTEM
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 iterations
should be conducted if there is a good probability that they provide additional information.
Roughly speaking, two situations can be distinguished, where additional iterations
don’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 first
iteration. Evidently, making additional iterations does not make sense. The detection
of this event is not so complicated, assuming the use of a CRC code. If the CRC check
passes, no additional iterations should be conducted.
If the channel conditions are rather poor, the information retrieved from the channel
is corrupted so badly by noise that no convergence will be observed. In other words, the
probability 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 crossentropy
[95]. In [90], an approximation for the cross-entropy is proposed as stop criteria
for turbo coding. Later, it was also proposed for turbo equalization in [96, 84]. Denoting
the 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) = ∑ ∣
∣L (i)
e (c[n, k]) ∣ 2
∣
n,k exp( ∣L (i) ∣
(6.5)
(c[n, k]) ∣)
As proposed in [90], no further iterations will be conducted if the T(i) < 10 −3 T(1). This
stop-criterion also functions on good channels. However, it requires inherently one more
iteration compared to the CRC-based stop criterion. Both stop criteria will be used
simultaneously. In addition, no further iterations are conducted if a certain number of
iterations is reached, which will be denoted by i max . If either of these three stop criteria
is fulfilled, no further iterations will be conducted. These criteria are used in both the
turbo 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 measure
for the DSP complexity, the average number of state-transitions summed over all SISO
decoders per information bit is chosen, which will be denoted as N Stt /b i . Due to variable
number of iterations in the turbo-decoder and equalizer, the complexity will also be a
function of E b,i /N 0 . The base-2 logarithm of the complexity will be discussed, because of
the large complexity differences between the compared schemes.
In the multiband case, it seems that additional DSP complexity is required, since N sb
equalizers/SISO decoders are operated in parallel. However, each equalizer is processing
only approx. 2/N sb channel bits per information bit, meaning that they can be clocked

6.4. PERFORMANCE ANALYSIS 115
at a N sb lower rate compared to a single-band equalizer. Hence, using N sb equalizers in
parallel does not increase the complexity of the DSP in terms of state-transitions per
information bit.
Operating N sb equalizers in parallel will be at the expense of N sb more surface area on
e.g. an ASIC or FPGA. By using proper signal multiplexing, de-multiplexing and state
storing, a single-equalizer implementation can be obtained, but it has to be operated at
approx. 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 Equalization
To reduce their complexity, the SISO decoders operating on the subbands are not provided
with a full description of the Volterra channels, but with the related RMDMs, see
Sec. 4.5.6. To obtain insight in the trade-off between performance and complexity, the
memory of the RMDMs will be varied from one to three, i.e. the number of states, a measure
for the complexity, of the equalizer is varied from 4 to 64. The system performances
are 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 FDM
is quantified using (4.63) for a single but representative channel realization. The output
SNR of the RMDM, is depicted as a function of the RMDM’s memory N in Fig. 6.8, for a
TR UWB system with 1, 2 and 4 subbands, respectively. As reference, the output SNR of
the FDM, which is an upper-bound for the SNR of a RMDM. The difference between both
SNR-values can not be translated into the E b /N 0 -loss in the BER-performance curves.
However, it does give an insight in the trade-off between performance and complexity.
Please note that the E b /N 0 -values are with respect to the channel bits and therefore
denoted as E b,c /N 0 .
Fig. 6.8 shows that the RMDM requires less memory to adequately model the FDM
with an increasing number of subbands. In case of a system with four subbands, a 16-state
RMDM (memory N = 2) is able to approximate the FDM for all channel realizations.
Only at E b /N 0 > 20 dB, a difference can be observed, which is well above the E b /N 0
working point. In case of two subbands, a 64 states RMDM (N = 3) is required to
adequately 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 has
been depicted in Fig. 6.9 as a function of E b /N 0 for the three system architectures. To
obtain insight on its effect on the performance, the equalizer complexity has been varied
from 4 states to 64 states (N = 1, 2, 3). In case of four subbands, Fig. 6.9 confirms
that a 16-state equalizer is indeed sufficient to obtain good performance, since almost
no further improvement is observed in the channel BER when increasing the equalizer
complexity. In the two-band case, a similar result applies; 64 states are needed to obtain
good performance. In the single-band case, 64 states for the equalizer seems not yet
sufficient to extract the complete information available in the received signal; additional
gain seems possible.

116 CHAPTER 6. DESIGN OF A HIGH-RATE TR UWB SYSTEM
SNR[dB]
14
12
10
8
6
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
4
2
0
6 8 10 12 14 16 18
E b,c /N 0 [dB]
Figure 6.8: The average ”overall SNR” of the RMDM as function of its memory
10 −1 E b,c /N 0 [dB]
P(e)
10 −2
10 −3
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
6 8 10 12 14 16 18 20
Figure 6.9: The average channel BER after the Log-Map equalizer of three multiband
TR UWB systems

6.4. PERFORMANCE ANALYSIS 117
Comparing the systems with different number of subbands, the two-band system has
the best performance with respect to the channel BER. However, the channel BER of
the different system architectures can not be compared directly. The four-band system
performs considerably worse than the two-band system at high E b,c /N 0 -values, because
the signal in each subband experiences considerably more fading. It is the task of the FEC
to exploit the frequency diversity provided by the system bandwidth. To see whether the
FEC is able to accomplish this task, the information BER has been depicted in Fig. 6.10.
Here, the E b /N 0 -values are with respect to the information bits, which will be denoted
as E b,i /N 0 .
Fig. 6.10 shows that the four-band system performs slightly better than the two-band
system—in terms of information BER—, even though its channel BER is considerable
worse. Hence, the FEC is indeed able to exploit the full frequency diversity. Furthermore,
it reveals that an E b /N 0 of approx. 13 dB is needed to obtain virtually error-free
communication, 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-band
system 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 with
respect 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 E b,i /N 0 . Due to
the large difference in complexity between the system the base-2 logarithm of the number
of state-transitions per information bit has been depicted. First of all, it can be noted
that both on high and low SNR channels, approximately the same amount of complexity
is required by the DSP, illustrating the proper functioning of the stop-criterion. This is
especially apparent for low N. Furthermore, when increasing the memory of the RMDM,
the equalizers makes the scheme so complex that the FEC-decoder complexity becomes
negligible.
Between both SNR extremes, more turbo iterations are conducted to converge to
the correct solution. Furthermore, it can be noted that the E b,i /N 0 range over which
more iterations are demanded by the turbo decoder is larger, when employing less subbands.
Likely, the residual ISI is interfering with the cross-entropy stop criteria, due to
its non-Gaussian nature. This suspicion is strengthened by the fact that the single-band
TR system requires more turbo decoder iterations on high-SNR channels, if the turbo
equalizer memory is low (N = 1). In this case, the residual ISI is namely more dominant.
Comparing the systems at an E b,i /N 0 of 13 dB, the value at which both the two-band
and four-band TR systems accomplish virtually error-free communication, the four-band
system 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 equalization
scheme. In Fig. 6.12, the information BER has been depicted as a function of E b,i /N 0 for
various systems with different equalizer complexity, number of subbands and maximum
number of turbo equalization iterations, assuming perfect kernel side-information.
As one may expect, the performance improves if the maximum number of turbo equalization
iterations is increased. However, the improvement is rather moderate, with re-

118 CHAPTER 6. DESIGN OF A HIGH-RATE TR UWB SYSTEM
10 −1 E b,i /N 0 [dB]
10 −2
P(e)
10 −3
10 −4
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
6 7 8 9 10 11 12 13 14 15 16
Figure 6.10: The average information BER of the three different systems
11
10
9
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
log 2 (NStt/bi)
8
7
6
5
4
6 8 10 12 14 16 18 20
E b,i /N 0 [dB]
Figure 6.11: Number of state-transitions as function of E b,i /N 0 of the three different
systems

6.5. CONCLUSIONS 119
spect to the additional complexity invested. The biggest improvement is obtained using
the single-band system, where a performance improvement of approximately 2 dB can
be observed. Nevertheless, its performance is still considerably worse than those of the
multiband 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 E b /N 0
value at which virtually error-free communication is obtained is 12 dB. In the absence of
turbo equalization, error-free communication was obtained at an E b /N 0 value of 13 dB.
Turbo equalization then leads to a performance improvement of 1 dB for a four-band TR
system.
Only a slightly worse performance is obtained using a two-band system using two parallel
64-state equalizers (N = 3). The improvement obtained by using turbo equalization
is slightly higher for the two-band system under consideration, but using DSP with a
higher complexity.
As to be expected, in every case the performance decreases when the equalizer complexity
is reduced. However, the performance penalty is rather small. In case of the
four-band system the performance penalty is a mere 0.25 dB, where for the two-band
system the penalty is 0.5 dB.
In Fig. 6.13, the complexity of the different schemes is depicted. For any TR-system
the same behaviour can be observed with respect to the required complexity. Taking the
E b,i /N 0 value for almost error free communication as reference point, e.g. 12.5 dB for a
four-system with N = 1, only slightly more additional complexity is required to improve
the performance. Most blocks of data are error-free after the first iteration, such that the
CRC-check passes and no further iterations are conducted. Applying turbo equalization
to those few packets containing errors will in most cases lead to an error-free decoding
after a few iterations. Using an intelligent scheduler to dynamically assign RX hardware
resources to promising packets, potentially improves the performance without the need
for much additional hardware.
6.5 Conclusions
In this chapter, the design of a high-rate TR UWB system has been described, able to
support a data rate of 100 Mb/s, while occupying a 1 GHz bandwidth. A combination
of trellis-based equalization and the multiband principle has been proposed to allow for
high data rate UWB communication over multipath radio channels, using non-coherent
receivers. To exploit the frequency diversity provided by the 1 GHz system bandwidth,
it is proposed to use FEC for multiband systems. Furthermore, turbo equalization has
been considered to improve the performance further.
The performance results reveal that a multiband system performs considerably better
with respect to a single-band systems at these data rates, using less complex equalizer
structures. It is shown that FEC in combination with a multiband receiver structure
principle is able 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 in the absence of perfect kernel side information, assuming the estimated kernel

120 CHAPTER 6. DESIGN OF A HIGH-RATE TR UWB SYSTEM
10 −1 E b,i /N 0 [dB]
10 −2
P(e)
10 −3
10 −4
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
8 9 10 11 12 13 14 15 16
Figure 6.12: The average information BER of three different systems in a turbo equalization
scheme
log 2 (NStt/bi)
11
10
9
8
7
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
6
5
4
6 8 10 12 14 16 18 20
E b,i /N 0 [dB]
Figure 6.13: DSP complexity as function of E b,i /N 0 of three different systems in a turbo
equalization scheme

6.5. CONCLUSIONS 121
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, it
is well-suited for a parallel implementation in the digital domain and inherently robust
against narrowband interference, possibly even extendable to DAA.
The use of turbo equalization is also recommended. The complexity analysis indicate
that an additional 1 dB performance improvement can be obtained, using only slightly
more DSP complexity. Further research is however required to find better stop-criteria
to manage the scheduling of packets for additional iterations. Another benefit of turbo
equalization is that it allows for an improvement of the kernel estimation with each
iteration, which eventually may allow the system to operate well, while using shorter
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 even shorter
training-sequences. During the second iteration, the kernel with N = 2 can be estimated
using the information gathered during the first iteration.

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

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 UWB
channels. The second block deals with TR signaling, which is being described in Chapter
4 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 have
been 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 and
Chapter 3. It is well-known that UWB systems are inherently robust against SSF, due to
their large bandwidth. On the other hand, the implementation of radio systems becomes
more complex when increasing the bandwidth. To accommodate a trade-off between
both aspects, a measure is introduced in Chapter 2 to quantify the frequency diversity
level of radio channels. By assuming uncorrelated scattering, a theoretical model has
been developed explaining the relationship between frequency diversity and bandwidth,
by decomposing the UWB channel into its principle components. Both for LOS channels
and NLOS channels, the diversity level has been found to scale linearly with the RMSdelay-spread-by-bandwidth
product. For NLOS channels specifically, the diversity level
was 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. One
of the novelties is that the model decomposes the channel in its PCs to finally predict
the fading statistics of the UWB channel. As a result, no literature was available to use
a reference for validations. Therefore, we have validated the model ourselves.
In Chapter 3, the fading model has been verified using measurement data of UWB
radio 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 measured
NLOS channels. Although the PCs of UWB channels are by definition uncorrelated,
they are not necessarily independent, which explains the difference between theory and
practice. It is expected that for UWB channels of richer multipath environments, the
difference between both diminishes.
123

124 CHAPTER 7. CONCLUSIONS AND RECOMMENDATIONS
For LOS UWB channels, the difference between theory and practice was significantly
larger. The theoretical model predicts that the LOS component has the same eigenfunction
as the largest NLOS component, i.e. they are contained in the same PC. In the
model, the PC will be a Ricean distributed RV with the largest possible variance. In this
respect, the theoretical model predicts a worst-case scenario. In practice, the dimension
spanned 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, the
overall diversity level of LOS radio channels was considerably higher than predicted by
the theoretical model. Unfortunately, the mechanism explaining the behaviour could not
be 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 reality
more accurately, without changing its basic structure. Currently, it is still an hypothesis
that the model under-estimates the diversity level of UWB NLOS channels due to the
statistical dependence between the PCs. For further research, it is therefore recommended
to validate the ability of the model to predict the fading of NLOS UWB channels of richer
multipath environments, such that the channel becomes more ”random” and the PCs are
indeed less dependent on each other.
Secondly, it may be useful to investigate the reason for the eigenfunction of the LOS
component to span another subspace than predicted by the theoretical model. The cause
is 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 model
for LOS UWB channels and allow for an refinement, possibly using another optimization
criterion 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 novel
extensions to the TR principle have been proposed, to relieve some of these shortcomings.
Firstly, a fractional-sampling AcR structure has been proposed to relax synchronization
and to allow for weighted autocorrelation, while simplifying the implementation. The
concept of fractional sampling has been proposed by other, but never with the aim to
suppress more ISI. Secondly, a complex-valued AcR has been proposed to make the system
less sensitive against delay mismatches and to allow for complex-valued modulation of
TR symbols. The usage of multiple AcR branches to overcome delay mismatches has
been proposed by other, but the resulting receiver has not before been interpreted as a
complex-valued AcR.
To understand the system’s behaviour, a general-purpose discrete-time equivalent system
model has been developed, taking all extensions into account. It was shown that the
I&D samples generated by a fractional sampling AcR in a TR system consist of two
terms with different nature, a signal term and a noise term. The signal term could be
modelled using a SIMO FIR Volterra model. The noise term was shown to consist of two
types of noise, a Gaussian sub-term with a signal dependent variance and a non-Gaussian

sub-term. The discrete-time equivalent system model is one of the first models for TR
signaling, taking the non-linear ISI into account.
Several interpretations for the SIMO FIR Volterra model have been presented, which
allow for more insight in the behaviour of TR systems. Firstly, the Volterra model
has been written in a vector notation and an extended vector notation, which allows
for simplified statistical analysis. The extended vector notation also allowed for the
interpretation of the SIMO FIR Volterra model as a linear MIMO model. The linear
MIMO model to interpret the SIMO FIR Volterra model has been proposed before. The
model proposed in this thesis is the first one that explains the role of modulation in the
amount 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, a
reduced-memory system model was introduced that is optimal, in the sense of the MMSE
criterion. The reduced-memory system model mimics the behaviour of TR systems, but
with a significant memory reduction. The reduced-memory FIR model for a second
order Volterra model has not been reported before in literature. Finally, the statistical
properties were derived for the signal term as well as for both noise terms. The noise
was shown to be quasi-white, with an output-dependent noise variance. This result is
confirmed by others in literature.
In Chapter 5, the impact of different system parameters on the system performance
has been presented, like FSR, bandwidth, delay, weighting criteria and modulation both
in the absence and presence of ISI. In the absence of ISI, an FSR of two is found to
suffice for close-to-optimal performance. The non-Gaussian noise term was found to
have a significant impact on the system performance, such that small-bandwidth TR
systems perform better, in the absence of fading. Furthermore, it was found that in
the presence of ISI, more ISI can be suppressed using linear weighting if the FSR is
increased. The modulation was found to have a significant impact on the amount of
ISI that can be suppressed. The role of the FSR and the modulation on the amount of
suppressible ISI has been explained using the linear MIMO model for SIMO FIR Volterra
models, presented in Sec. 4.5.4. Most of the results presented in Chapter 5 have been
reported by others. The novelty is that influence of the system parameters on the system
performance is analyzed, each time using the same basic system set-up. As a result, the
results allow for an improved insight in the behaviour of the system with respect to the
system parameters. A truly novel contribution to the understanding of TR systems, is the
insight that the amount of non-linear ISI that can be suppressed using linear weighting
depends considerably on the modulation type.
It is recommended to further investigate the potential of linear weighting in the presence
of non-linear ISI. In the thesis, a single weighting vector was used, which exploited
the term depending linearly on the bit under demodulation, considering the other terms
as interference. Similar to MIMO systems, additional weighting vectors could be used to
demodulate the other non-linear ISI terms, which also contain information on the symbol
under demodulation. The information on the linear and non-linear ISI term potentially
can be joined in a single decision process on the symbol under demodulation. It is recommended
to study the structure, performance and complexity of such an algorithm, to
125

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 design
of a high-rate TR UWB system has been described, able to support a data rate of
100 Mb/s, while occupying 1 GHz bandwidth. A combination of trellis-based equalization
and a multiband system architecture has been proposed, to obtain high data rate
UWB communication over the multipath radio channel, using non-coherent receivers. To
exploit the frequency diversity provided by the 1 GHz system bandwidth, it is proposed
to use FEC for multiband systems. Furthermore, turbo equalization has been considered
to improve the performance further.
The performance results reveal that a multiband system performs considerably better
compared to a single-band system at the same data rate, using a less complex equalizer
structure. It is shown that FEC in combination with a multiband receiver structure is
able 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 in
the absence of perfect kernel side information, assuming the kernel estimate is updated
with each iteration.
Taking both complexity and performance into account, a four-band system with four
parallel operating 16-state (N = 2) equalizers is suggested. Not only does it deliver good
performance, 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 and
inherently robust against narrowband interference, possibly even extendable to DAA.
The use of turbo equalization is recommended as well. Complexity analysis indicates
that an additional 1 dB performance improvement can be obtained, using only slightly
more of DSP complexity. It is however critical to find better stop-criteria to manage what
packets are scheduled for additional iterations. In this respect, further research is needed
to obtain better stop-criteria. Another consideration in favour of turbo equalization is
that it allows for improved kernel estimation, which allows the system to operate using
shorter 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 even
shorter training sequences. During the second iteration, the kernel with N = 2 can be
estimated using the information gathered during the first iteration.

Appendix A
Estimation of the Nakagami-m
Parameter
In this appendix, a paper on the estimation of the Nakagami-m parameter for Frequency
Selective 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 due
to 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. Both
distributions not only fit well to the measured data, but are also justified by the physics
of multipath radio channels [35]. Bases on this insight, many mathematical tools have
been developed in communication theory, e.g. for bit error rate analysis.
In the case of frequency selective fading channels (FSFC), by definition, not all frequency
components of the transmitted signal experience the same channel amplitude
gain. Hence, one has to average 1 the power attenuation over all frequency components
and take its square root to obtain the effective amplitude gain (EAG). Hence, the EAG
is equal to the square root of the well-known mean power gain, or alternatively, the root
mean square (RMS) value of the channel frequency response (CFR). As in the case of
flat fading channels, the EAG is also modelled using random processes. For FSFC or
diversity channels in general, the Nakagami distribution often fits well to measurement
data [97, 98].
The Nakagami distribution is described by two variables, namely Ω and m and occurs
when the RMS value is taken of K independent, identically distributed (i.i.d.) Gaussian
random variables with a variance σ 2 . In this case, Ω and m will be Kσ 2 and K/2, respectively.
The Nakagami distribution can be seen as a generalization of the Rayleigh
distribution, where m is equal to 1. Also the Nakagami distribution is justified by the
physics of the radio channel. Assuming a system radiating the energy E uniformly over
K/2 i.i.d. Rayleigh fading (sub)-channels, its EAG will be a Nakagami distributed RV
1 A 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 characterizes
the diversity level of FSFCs [16].
The Nakagami parameters are often derived from a set of measured CFR functions
of size N. Normally, the EAG of each measured CFR is computed to obtain a pool of
N 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 and
the Cramér-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 the
CFR covariance matrix. The result is a low-variance, biased estimator. A closed-form
approximation for the bias will be derived, based on which an alternative estimation
method is derived, which is approximately unbiased. The simulation results reveal a
superior performance for this estimator compared to all known EAG-based estimators
and 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 algorithm
is not investigated in this paper.
A.2 Covariance-based m-parameter estimation
The CFR is often modelled using a complex multivariate, zero-mean, Gaussian random
vector h of length L with a covariance matrix Σˆ=E [ hh H] . The channel EAG g ˆ= ‖h‖ / √ L
is a RV as well. The Nakagami-m parameter is related to g and Σ (see [103]) according
to
mˆ= E[g]2
E[g 2 ] = (∑ L
m=1 λ m) 2
∑ Nf
m=1 λ 2 m
=
Tr (Σ)
Tr (ΣΣ) ,
(A.1)
where λ m denotes the m-th eigenvalue of Σ. In practice, Σ is unknown a-priori and one
has to estimate it from measurement data. Let’s assume a measurement pool, where the
i-th measured CFR vector h i 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∑
h i h H i
i=1
(A.2)
where N denotes the number of measured CFRs. It is straightforward to obtain an
estimate 2 for the m-parameter using W, namely
ˆm c =
Tr (W)2
Tr (WW) .
(A.3)
2 It is emphasized that the estimator is based on the zero-mean Gaussian assumption, i.e. the proposed
method is restricted to Rayleigh channels. For channels with a dominant (LOS) component, the method
has to be modified by repeating the presented derivations for a channel model extended by the dominant
path, 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 ˆm c . To simplify its derivation,
(A.3) is re-written to
Tr (W)2
ˆm c =
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 division
can be approximated by
(
)
Tr (W)2 Tr (WW) − Tr (ΣΣ)
ˆm c ≈ 1 −
Tr (ΣΣ) Tr (ΣΣ)
Tr (W)2
≈ 2
Tr (ΣΣ) − Tr (W)2 Tr (WW)
Tr (ΣΣ) 2 . (A.5)
Now taking the expectation of both sides, we obtain
E[ ˆm c ] ≈ 2 E[ Tr (W) 2]
Tr (ΣΣ)
− E[ Tr (W) 2 Tr (WW) ]
Tr (ΣΣ) 2 . (A.6)
The derivation of both higher-order moments is rather complex. Several papers have been
published on the higher-order moments of Wishart matrices [104, 105]. The following
results from these publications will be used,
E [ Tr (W) 2] =Tr (Σ) + 1 Tr (ΣΣ)
N (A.7)
E [ Tr (W) 2 Tr (WW) ] =Tr (Σ)Tr (ΣΣ) + 1 Tr (Σ)4
N
+ 1 (
Tr (ΣΣ) 2 + 4Tr (Σ)Tr (ΣΣΣ) ) ( ) 1
+ O . (A.8)
N
N 2
Substituting these results into (A.6) leads to
E[ ˆm c ] ≈
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[ ˆm c ] ≈ m 1 + 1
mN − m N − K(Σ) )
, (A.10)
N
where
K(Σ) =
4Tr (ΣΣΣ)
Tr (Σ)Tr (ΣΣ) ,
(A.11)
which makes it evident that ˆm c is only unbiased for N → ∞.

130 APPENDIX A. ESTIMATION OF THE NAKAGAMI-M PARAMETER
A.3 Unbiased covariance-based m-parameter estimation
In this section, an unbiased estimate ˆm uc is deduced from ˆm c for finite values of N.
Therefore, E[ ˆm c ] will be described as function m and N only. Unfortunately, the K(.)-
term does not depend only on E[ ˆm c ] nor m, but also on the structure of Σ. Simulation
results revealed that the K-term is in practice small compared to the other terms, meaning
that it is often negligible. Alternatively, one can assume a certain Σ of which the structure
depends 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(I m,m ) = 4m
m 2 = 4 m .
(A.12)
Substituting (A.12) into (A.10), we obtain
E[ ˆm c ] = m − m2
N − 3 N ,
(A.13)
such that the expectation depends only on m and N, which is a second-order equation
that can be inverted. It has only one positive solution, which is
m ≈ 1 2 (N − √ N 2 − 4(NE[ ˆm c ] + 3)).
(A.14)
Hence, the approximately unbiased estimate for m is,
ˆm uc ≈ 1 2 (N − √ N 2 − 4(N ˆm c + 3)),
(A.15)
which concludes the derivation of the approximately unbiased estimator. In the following
section, its performance will be presented.
A.4 Simulation Results
In Fig. A.1, the average RMS estimation error of the estimators ˆm c and ˆm uc is presented
for N = 100 synthetically generated realizations of h with Σ = I m,m . As reference, the
RMS estimation error of a moment-based estimator and its CRLB have been depicted
as well. First of all, it is noted that for small m, both ˆm c and ˆm uc perform better
than the CRLB for EAG-based estimators. Only for increasing m, the RMS estimation
error of ˆm c increases rapidly; this is caused by its bias. It has the tendency to underestimate
the actual m. In this respect, ˆm uc has an improved performance. Still it is
not truly unbiased and has the tendency to over-estimate m, which is caused by an
increasing error in the assumption used in (A.5) with an increasing m/N ratio. By
increasing the amount of terms used for the Taylor series expansion in (A.5) will improve
the performance of ˆm uc , but will results in higher-order moments of Wishart matrices
making its derivation (too) complex. Alternatively, we used a second-order polynomial
to describe the relationship between ˆm c and m, assuming Σ = ˜Σ whether it is correct or

A.4. SIMULATION RESULTS 131
3.5
m c
3
m m
m uc
2.5
m uc2
CRLB *
rms error
2
1.5
1
0.5
0
0 5 10 15 20
m
Figure A.1: RMS estimation error versus m for the different estimators (N = 100)
not. Hence, ˆm uc2 = c 2 ˆm 2 c + c 1 ˆm c + c 0 , 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 increasing
dominance of the linear term with increasing N. The performance of ˆm uc2 is depicted
in Fig. A.1, as well. Overall, the estimator ˆm uc2 has the best performance.
In Fig. A.2, the RMS error of the estimators is presented as function of N using
the same method of generating synthetic h. The figure shows that a covariance-based
m parameter estimator needs considerably less observations N to obtain the same RMS
estimation error.
For the previous two figures, the assumption Σ = ˜Σ was valid. To analyze the
algorithm performance on more realistic channels, synthetic CFR data has been generated
for channels with an exponential power delay profile. The deployed frequency domain
autocorrelation function is as follows:
E [ h i [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 presented
in Fig. A.3. Compared to Fig. A.1, all RMS estimtion error curves have changed, but
without affecting the derived conclusions.

132 APPENDIX A. ESTIMATION OF THE NAKAGAMI-M PARAMETER
1
rms error
0.9
0.8
0.7
0.6
0.5
0.4
m c
m m
m uc
m uc2
CRLB *
0.3
0.2
0.1
0
0 50 100 150 200 250 300 350 400
N
Figure A.2: RMS estimation error versus N for the different estimators (m = 5)
∗ CRLB applies only to m m
Table A.1: Poynomial coefficients of ˆm uc2
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 -5.4366
20 0.1488 0.3884 0.7753 2.1121 -3.0807
30 0.0744 0.7207 0.3813 1.7273 -2.1834
50 0.0339 0.8963 0.1491 1.4296 -1.3898
100 0.0133 0.9736 0.0366 1.2124 -0.7326
200 0.0058 0.9932 0.0075 1.1056 -0.3775
400 0.0027 0.9983 0.0007 1.0526 -0.1917

A.5. CONCLUSIONS AND REMARKS 133
3.5
m c
3
m m
m uc
2.5
m uc2
CRLB *
rms error
2
1.5
1
0.5
0
0 5 10 15 20
m
Figure A.3: RMS estimation error versus m for the different estimators using realistic
synthetic data (N = 100)
A.5 Conclusions and remarks
A new class of algorithms has been presented for the estimation of the Nakagami-m
parameter from coherently measured fading channels. Firstly, a straightforward lowvariance,
but biased estimator has been presented. Additionally, two alternative, unbiased
estimators have been proposed, both deduced from the biased estimator. The simulation
results show that both unbiased estimators have superior performance compared to other
types of Nakagami-m parameter estimators.

134 APPENDIX A. ESTIMATION OF THE NAKAGAMI-M PARAMETER

Appendix B
Complex-Valued AcR
In this appendix, the baseband equivalent model for the Complex-Valued(CV) AcR is
derived. Let’s denote the RX passband signal by ˜r(t) and its delay version as ỹ(t) = ˜r(t−
D) and their baseband equivalent representation as r(t) and y(t) = r(t−D) exp(−jω c D),
respectively. The relation between a bandpass signal ˜r(t) and its baseband equivalent
r(t) is given by
˜r(t) = r r (t) cos(ω c t) − r i (t) sin(ω c t),
(B.1)
where r r (t) and r i (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 multiplier
output in the first autocorrelation branch is given by
x r (t) = ˜r(t)ỹ(t).
(B.2)
In the absence of a LPF, after substitution of (B.1), leads to the following expression,
x r (t) =r r (t)y r (t) cos 2 (ω c t) − r i (t)y i (t) sin 2 (ω c t)+
(r i (t)y r (t) + y i (t)r r (t)) cos(ω c t) sin(ω c t).
(B.3)
The LPF-characteristic of the operator after the multiplier will however filter out all
terms containing a carrier. Neglecting these terms leads to an equivalent expression for
the multiplier output:
x r (t) = 1 2 r r(t)y r (t) + 1 2 r i(t)y i (t).
(B.4)
The multiplier output of the second AcR branch x i (t) is similar, except that y(t) is delayed
additionally with ∆ equal to π/2/ω c , such that
ỹ(t − ∆) = y r (t − ∆) sin(ω c t) + y i (t − ∆) cos(ω c t).
(B.5)
The signal y(t −∆) may be replaced by its zero-th order approximation y(t), if ∆ ≪ 1/B
with B denoting the signal bandwidth, such that
ỹ(t − ∆) ≈ y r (t) sin(ω c t) + y i (t) cos(ω c t).
(B.6)
135

136 APPENDIX B. COMPLEX-VALUED ACR
Hence, the multiplier output of the second AcR branch x i (t) equals
x i (t) = ˜r(t)ỹ(t − ∆) = 1 2 r r(t)y i (t) − 1 2 r i(t)y r (t).
(B.7)
Since x r (t) is the real-part of the multiplier output and x i (t) the imaginary part, the
complex-valued multiplier output x(t) becomes
x(t) x r (t) − jx i (t) = 1 2 r(t)r∗ (t − D) exp(jω c D).
(B.8)
Consequently, the complex-valued AcR output becomes
u[n, α]= exp(jω c D)
∫ ∞
h(t−(nL+α)T clk )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 to
one for all 0 ≤ t < T clk and zero otherwise.
The CV AcR is a generalization of the traditional AcR. Therefore, the presented
derivation also applies to a RV AcR. Furthermore, the derivation shows that a modification
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 pulse
to pulse, the PSD shape of the radiated signal depends only on the squared Fourier
transform of the individual pulses. In this appendix, it is shown that the modulation is
indeed 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-priori
probability. 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 independent
symbols, the following correlation properties between the pulses are found. For the pulse
train 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 the
correlation properties of the information bearing pulses,
{
]
E[˜b[n]b[n]˜b∗ [n + k]b ∗ 1 if k = 0,
[n + k] =
(C.2)
0 otherwise
which means that this pulse train also generates no spectral spikes. Also the crosscorrelation
between both signals could generate spikes. Therefore, the cross-correlation
between 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 thesis
and 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)
L(q[n]|u, L(c), H) = ln
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]. The
set of possible state-transitions will be denoted by S tt . This set can be divided into
two disjoint subsets, where S +1
tt and S −1
tt 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,
⎛
⎜
L(q[n]|u, L(c), H) = ln ⎝
∑
S tt[n]∈S +1
tt
∑
S tt[n]∈S −1
tt
⎞
P(S tt [n]|u, L(c), H)
⎟
⎠
P(S tt [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 written
as,
⎛ ∑
⎞
P(u, L(c)|S tt [n], H)P(u, L(c))/P(S tt [n])
⎜S tt[n]∈S
L(q[n]|u, L(c), H) = ln
+1
tt
⎝ ∑
⎟
⎠ (D.3)
P(u, L(c)|S tt [n], H)P(u, L(c))/P(S tt [n])
S tt[n]∈S −1
tt
For different reasons, the probabilities P(u, L(c)) and P(S tt [n]) are the same for all statetransitions:
This allows us to simplify (D.3) to
⎛
⎜
L(q[n]|u, L(c), H) = ln ⎝
∑
S tt[n]∈S +1
tt
∑
S tt[n]∈S −1
tt
139
⎞
P(u, L(c)|S tt [n], H)
⎟
⎠
P(u, L(c)|S tt [n], H)
(D.4)

140 APPENDIX D. DERIVATION OF THE LOG-MAP ALGORITHM
To simplify the implementation, the log of the sum of two probabilities P 1 and P 2 will
be written in another form. Assuming l 1 , l 2 and l 1,2 to denote lnP 1 , ln P 2 and lnP 1 + P 2 ,
respectively. The joint log-probability l 1,2 is related to l 1 and l 2 according to
l 1,2 = max(l 1 , l 2 ) + ln(1 + exp(|l 1 − l 2 |))
(D.5)
A new operator, called the box-plus operator ⊞, is now introduced, such that l 1,2 = l 1 ⊞l 2 .
A single box-plus operation requires a max-operation, a subtraction, an absolute operation
and finally a table-lookup, assuming the function ln(1 + exp(|x|)) is stored in a lookup
table. The box-plus operator is both associative and commutative, i.e. l 1 ⊞ l 2 = l 2 ⊞ l 1
and (l 1 ⊞ l 2 ) ⊞ l 3 = l 1 ⊞ (l 2 ⊞ l 3 ). Another important property of the box-plus operator
for the implementation reasons is that (l 1 + K) ⊞ (l 2 + K) = K + l 2 ⊞ l 1 .
Using the box-plus operator, the order of the sum and natural logarithm in (D.2) can
be interchanged to obtain
L(q[n]|u, L(c), H) =
⊞
S tt[n]∈S +1
tt
−
⊞
S tt[n]∈S −1
tt
ln (P(u, L(c)|S tt [n], H))
ln (P(u, L(c)|S tt [n], H))
(D.6)
Now let us have a close look at the probability P(u, L(c)|S tt [n], H). In [106], it is
shown that this probability can be divided in three parts, a pre-cursor part, a on cursor
part and a post-cursor part. The a-priori LLVs and the channel information divided into
these 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 three
probabilities, the probability on the start state using only pre-cursor information, the
probability of the state-transition using the on-cursor information and the probability
of the end state using the post-cursor information. The log-probability of a given state
transition is thus described by
where
ln(P(S tt [n]|u, L(c), H)) = α(S t [n]) + γ(S tt [n]) + β(S t [n + 1])
α(S t [n]) = ln (P(u < , L(c < )|S t [n], H)),
β(S t [n + 1]) = ln (P(u > , L(c > )|S t [n + 1], H)),
γ(S tt [n]) = ln (P(u[n], L(c[n])|S tt [n], H)).
(D.9)
(D.10)
(D.11)
(D.12)
In [106], it is shown that both α(S t [n]) and β(S t [n + 1]) can be written in a recursive
manner. These results in the notation deployed in this thesis give
α(S t [n]) = ⊞
S t[n−1]∈S − |S t[n]
γ(S t [n − 1], S t [n]) + α(S t [n − 1])
β(S t [n + 1]) = ⊞ γ(S t [n + 1], S t [n + 2]) + β(S t [n + 2])
S t[n+2]∈S + |S t[n+1]
(D.13)
(D.14)

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 N p equals 2 N b and 2 for the channel and FEC, respectively.
The only remaining unknown to be solved is γ(S tt [n]). Assuming u[n] and L(c[n])
to be independent 1 , the equation for γ(S tt [n]) can be divided into the sum of two logprobabilities,
141
γ(S tt [n]) = ln(P(u[n]|S tt [n], H)) + ln(P(L(c[n])|S tt [n], H))
(D.15)
Here, the first term contains the a-posteriori information captured from the channel. The
second term contains the a-priori information on the channel bits. Both terms will be
solved separately. For starters, the a-posteriori term ln(P(u[n]|S tt [n], H)) will be solved.
In chapter 4, the noise was shown to be independent. Using the trellis information
contained in H, the first right-hand term of (D.15) can be simplified to,
∑L−1
ln(p(u[n]|S tt [n]), H) = ln(p(u[n, α]|S tt [n]))
α=0
(D.16)
Now by assuming the noise to be Gaussian distributed with an output-dependent variance
σ 2 α independent of the state-transition, the probability
ln(p(u[n, α]|S tt [n]), H) = c 1,α − c 2,α |u[n, α] − f α (S tt [n])| 2
(D.17)
with c 1,α = − ln(πσ α ) and c 2,α = 1/σ 2 α. The expression for ln(p(u[n, α]|S tt [n])) can not be
further simplified.
Let us continue with the a-priori information term ln(P(L(c[n])|S tt [n])). From the
a-priori information contained in the trellis object H, the N b channel bits related to
the 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 to
be independent,
ln(P(L(c[n])|S tt [n], H)) =
N∑
b −1
k=0
Using the result from Tab. 6.2, the log-probability
ln(P(L(c[n, k])|x[k])))
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.18)
(D.19)
The expression can not be further simplified.
In combination, the following expression for γ(S tt [n]) is obtained
∑L−1
γ(S tt [n]) =K 1 + −c 2,α |u[n, α] − f α (S tt [n])| 2
+
N b −1
α=0
∑
x[k] L(c[n, k]) − ln(1 + exp(x[k] L(c[n, k])))
k=0
(D.20)
(D.21)
1 In a turbo equalization scheme, the independence assumption is questionable. The use of interleavers
makes the assumption reasonable and allows for a low-complexity algorithm with good performance

142 APPENDIX D. DERIVATION OF THE LOG-MAP ALGORITHM
where K 1 ∑ L−1
α=0 c 1,α. The equation shows the channel branch metric depends both on
the Euclidian distance between the expected sample values and the actual sample values
and the noise variance. In fact, the expression can be considered as the computation of
a weighted Euclidian distance. Furthermore, the a-priori term shows that the a-priori
LLVs are summed into the branch metrics.
For the computation of L(q[n]), the term K 1 can be neglected, without affecting the
result of (D.6). Adding a constant K 2 to γ(S tt [n]), means that the recursive relation for
α(S t [n]) and β(S t [n + 1]) are enlarged by nK 2 and (N st − n − 1)K 2 , where N st represents
the number of state transitions in the trellis and n = 0 denotes the first state transition.
Hence, (D.9) is increased by a factor N st K 2 . After the subtraction in (D.6), this additional
term will be cancelled. Now by selecting K 2 = K 1 means that the term K 1 can indeed
be 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 completed
because of the support I have received by many. First of all, I would like to thank my
parents, sister, other family members and friends for the emotional support during rough
times. I would also like to thank both IMST GmbH and the SPSC institute, not only for
facilitating the PhD, but also for the great working atmosphere and understanding. The
affiliated 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 and
for granting me this opportunity.
153

154 BIBLIOGRAPHY

Curriculum Vitae
Jac Romme was born in Breda, The Netherlands, on March 29, 1975. After attending the
primary school ”Onder de Torens” and the secondary school ”Katholieke Scholengemeenschap
Etten-Leur” both in Etten-Leur, The Netherlands, he started Electrical Engineering
at the Eindhoven University of Technology (TU/e) in Eindhoven, The Netherlands, in
September 1994. During his education, he conducted two internships. The first one was
at the TU/e, where a comparison was made between the closed-form results and numerical
results for the radiation diagram and currents of a non-ideal linear antenna. The
second internship, he conducted at Alcatel in Antwerp, Belgium on the performance of
TCP/IP over Skybridge Satellite Links. After finishing a graduate project at Siemens
ICP in Munich on variable-rate convolutional codes, he received the M.Sc. degree in
electrical engineering at the TU/e.
In September 2000, he started at IMST GmbH, Kamp-Lintfort Germany working on
radio system design with as main focus UWB communication and localization. Besides his
work at the IMST, he started as a PhD student at the SPSC-lab at the technical university
of Graz, Austria, in August 2004. The main part of his PhD work was conducted at IMST
GmbH. Furthermore, he has visited the SPSC-lab three times for in total 10 Months. His
main interests are UWB communication and localization, baseband signal processing,
channel coding, equalization, iterative signal processing and non-linear signal processing.
155