Ultra-wideband based Communications and

Ultra-wideband based Communications and
Ultra-wideband based Communications and
Localization in Wireless Sensor Networks
Vom Fachbereich Elektrotechnik und Informatik der
Universität Siegen
zur Erlangung des akademischen Grades
Doktor der Ingenieurwissenschaften
(Dr.-Ing.)
genehmigte Dissertation
von
M.Sc. Tech. Gustave Franck Tchere
1. Gutachter: Prof. Dr.-Ing. habil. O. Loffeld
2. Gutachter: Prof. Dr.-Ing. H. Roth
Vorsitzender: Univ.-Prof. Dr. Christoph Ruland
Tag der mündlichen Prüfung: 28.10.2010
Acknowledgments
This work was conducted under the framework of the International Postgraduate Programme (IPP) Multi Sensorics of the Center for Sensorsystems (ZESS) and the Research
Center for Multidisciplinary Analysis and Applied System Optimization (FOMAAS) at
the University of Siegen.
I would like to take this opportunity to express my greatest thanks to all of you who
have supported me in various ways. It is a great pleasure for me to express my sincere
gratitude to Prof. Dr.-Ing. habil. Otmar Loffeld, who supervised this work. During the last
three years, I have been inspired by his broad knowledge of applied estimation theory, his
brilliant insights and his scientific rigorousness. I am also grateful to Prof. Dr.-Ing. Hubert Roth, who is my second supervisor. I have benefitted from his useful comments
and advice throughout my thesis work. I would like to extend my gratefulness to Dr.Ing. Stefan Knedlik for his continuous encouragement and support during the wireless
sensor network related activities. It was a pleasure for me working with him.
I am grateful to the thesis committee for their careful reading of this manuscript.
I would like to express my gratitude to all colleagues in the Center for Sensor Systems (ZESS) and members of International Postgraduate Programme (IPP), especially to
M.Sc. Pakorn Ubolkosold and M.Sc. Miao Zhang for numerous discussions on both scientific
and non-scientific matters and their friendly helps with various problems. I am grateful to
Silvia Niet-Wunram, Renate Szabó and Ira Dexling, and for their excellent administrative
services.
Finally, I dedicate this thesis to my parents and friends, whose unconditional love and
support has been a great source of inspiration to me.
i
Kurzfassung
Die Hauptbeweggründe für das Verwenden von Ultrabreitband (UWB) Kommunikationssysteme ist ihre Fähigkeit für das Bereitstellen drahtloser Kommunikationen mit hoher Kapazität, sowie die Verwendbarkeit der Technologie, um UWB Signale mit verhältnismäßig
niedriger Komplexität einzuführen und zu erzeugen. Nicht kohärente Empfänger ohne
die Schätzung des Kanals sind vorgeschlagen worden, um die UWB Technologie attraktiv
für die Anwendungen zu machen, in denen niedrige Kosten und niedrige Stromaufnahme
eine wichtige Rolle spielen, wie es bei drahtlosen Sensor-Netzen (WSNs) der Fall ist. Die
Signalübermittlung der Sendereferenz (TR), in Verbindung mit einem Autokorrelations Empfänger (AcR) sind in diesem Kontext besonders verwendbar. Die große Bandbreite der
UWB Technologie liefert nicht nur die Möglichkeit, um mit einer sehr hohen Datenrate zu
übertragen, sondern auch sehr genaue zeitliche und räumliche Informationen, die für exakte
Schätzung der Zeitverschiebung verwendet werden können. Das UWB Impuls-Radio (IR)
verwendet Sub-Nanosekunden - Impulse, die eine hoch auflösende Fähigkeit in der Zeitdomäne liefern und ist dadurch attraktiv für die genaue drahtlose Lokalisation. In der Tat
macht seine Fähigkeit, Mehrwegbestandteile zu beheben möglich, genaue Positionsbestimmung ohne die Notwendigkeit komplizierter Schätzungsalgorithmen zu erhalten. Dieses
erleichtert viele Anwendungen wie das Position-bewußte Sensor-Netzwerk.
Diese Doktorarbeit trägt zur Entwicklung der UWB Technologie für Kurzstrecken
niedriger bis zu durchschnittlicher Datenraten der WSN Anwendungen bei. TR-UWB
Empfänger niedriger Komplexität werden zunächst unter Annahme Gauß’scher Rauschmodelle analysiert, danach werden äquivalente nicht lineare System- und Rauschmodelle
betrachtet. Beide Ansätze führen zu der Leistungsanalyse der Bit Fehler, die auch in
dieser Dissertation dargestellt werden. Die TR-UWB Systeme wurden als Polynome nichtlinearer Systeme modelliert. Da die Statistiken des Funkkanals in der Systemmodellierung
sowie in der Leistungsanalyse verwendet werden, werden passende Kanalmodelle für UWB
Impuls-Radiosysteme in dieser Doktorarbeit besprochen.
Weiter wird die Synchronisierung angesprochen. Zu diesem Zweck wird eine neue
datengestützte Zeiterfassungstechnik für die Rahmen-Niveau Synchronisierung der DTRUWB Systeme vorgeschlagen. Sie basiert auf dem Einfügen der parallelen Integration-anddump Schaltkreise innerhalb der Impuls-Paare der Korrelatorzweige, um die Energieerfassung bei der Präsenz der Zeitverschiebung beträchtlich zu verbessern. Außerdem wird ein
einfacher Algorithmus für die Feinsynchronisierung der TR-UWB Systeme, keine Intersymbolstörung annehmend, vorgeschlagen. Er verwendet die Energie, die mit der Symbolrate
iii
gesammelt wird und verringert so beträchtlich die Implementierungskomplexität. Wenn
der Taktzeitfehler bekannt ist, sind Schätzungsprobleme der Zeitverschiebung analog zu
Entfernungsmessungsproblemen, und der vorgeschlagene Algorithmus kann leicht als neue
Abstandmaßtechnik verwendet werden. Es wird in dieser Dissertation gezeigt, dass der
Entfernungsansatz eine Lokalisierungsauflösung im Zentimeterbereich mit TR-UWB Systemen bei Datenraten bis zu 5 Mb/s erlaubt.
Executive Summary
The key motivations for using ultra-wideband (UWB) communication systems are their
capability for providing high capacity wireless communications, as well as the availability of technology to implement and generate UWB signals with relatively low complexity.
Non-coherent receivers with no channel estimation have been proposed to make the UWB
technology attractive for applications where low cost and low power consumption are playing an important role, as in the case of wireless sensor networks (WSNs). Transmitted
Reference (TR) signaling, in combination with an autocorrelation receiver (AcR) is especially suitable in this context. The large bandwidth of UWB technology does not only
provide the possibility to transmit at a very high data rate, but also provides very accurate temporal and spatial information that can be used for precise timing offset estimation.
UWB Impulse Radio (IR) uses sub-nanosecond pulses which provide a high resolution capability in the time domain, making it attractive for accurate wireless localization. Indeed,
its ability to resolve multipath components makes it possible to obtain accurate location
estimates without the need for complex estimation algorithms. This facilitates many applications such as location-aware sensor networking.
This thesis presents several contributions towards developing UWB technology for
short-range low to medium data rate WSN applications. First, different low complexity
TR-UWB receivers are thoroughly analyzed, using the Gaussian approximation on the
noise terms in the receiver statistics, and then using the equivalent system and noise models of these receivers. Both approaches lead to the bit error performance analysis, which
is also presented in this thesis. The TR-UWB systems were modelled as polynomial nonlinear systems. Since the statistics of the radio channel are used in the system modelling as
well as in the performance analysis, appropriate channel models for UWB impulse radio
systems are discussed in this thesis.
Further, synchronization issues are addressed. To this end, a novel data-aided timing
acquisition technique for frame-level synchronization of DTR-UWB systems is suggested.
It is based on incorporating parallel integration-and-dump circuits within pulse-pair correlator branches to improve considerably the energy capture in the presence of timing offset.
Moreover, a simple algorithm for fine synchronization of low complexity TR-UWB systems,
assuming no inter-symbol interference, is proposed. It uses energy collected at the symbol
rate, thus reducing considerably the implementation complexity. If the hardware clock timing error is known, timing offset estimation problems are analogical to ranging problems,
and the proposed algorithm can readily be used as a new distance measurement technique.
It is shown, in this thesis, that the proposed ranging approach allows localization accuracy
in the centimeter range using TR-UWB systems with data rates up to 5 Mb/s.
Contents
Acknowledgements
i
Kurzfassung
ii
Executive Summary
iv
Contents
v
List of Figures
viii
List of Tables
x
Abbreviations
xi
Mathematical Symbols
xiv
1 Introduction
1.1 Ultra-Wideband Technology . . . . . . . . . . . . . . . . . . .
1.1.1 UWB History and Regulatory Issues . . . . . . . . . .
1.1.2 Basics of UWB Techniques . . . . . . . . . . . . . . . .
1.1.3 Open Research Problems . . . . . . . . . . . . . . . . .
1.2 Ultra-Wideband Wireless Sensor Networks . . . . . . . . . . .
1.2.1 Wireless Sensor Network Requirements . . . . . . . . .
1.2.2 Ultra-Wideband Approach to Wireless Sensor Networks
1.2.3 Conceptual Design of the Envisaged Application . . . .
1.3 Goal of the Thesis . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Organization of the Thesis . . . . . . . . . . . . . . . . . . . .
1.5 Contributions of the Thesis . . . . . . . . . . . . . . . . . . .
1.5.1 Systems Modelling . . . . . . . . . . . . . . . . . . . .
1.5.2 Performance Analysis . . . . . . . . . . . . . . . . . . .
1.5.3 BER derivation . . . . . . . . . . . . . . . . . . . . . .
1.5.4 Coarse Synchronization . . . . . . . . . . . . . . . . . .
1.5.5 Fine Synchronization . . . . . . . . . . . . . . . . . . .
1.5.6 Distance and Location Estimation . . . . . . . . . . . .
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1
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2 Polynomial Nonlinear Systems: Overview
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Homomorphic Systems . . . . . . . . . . . . .
2.1.2 Order Statistic Filters . . . . . . . . . . . . .
2.1.3 Morphological Filters . . . . . . . . . . . . . .
2.1.4 Neural Networks . . . . . . . . . . . . . . . .
2.1.5 Polynomial Filters . . . . . . . . . . . . . . .
2.2 Volterra Series Expansion . . . . . . . . . . . . . . .
2.2.1 Continuous-time Systems . . . . . . . . . . . .
2.2.2 Discrete-time Systems . . . . . . . . . . . . .
2.2.3 Algebraic Representation of Quadratic Filters
2.2.4 Realization of Quadratic Filters . . . . . . . .
2.3 Wiener & Hammerstein Models . . . . . . . . . . . .
2.3.1 Hammerstein-Wiener Model . . . . . . . . . .
2.3.2 Hammerstein Model . . . . . . . . . . . . . .
2.3.3 Wiener Model . . . . . . . . . . . . . . . . . .
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . .
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18
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3 Channel Characterization
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Characterization of the Mobile Radio Channel . . . . . .
3.2.1 Components of a Multipath Channel Model . . .
3.2.2 Channel Impulse Response . . . . . . . . . . . . .
3.2.3 Channel Parameters . . . . . . . . . . . . . . . .
3.2.4 Channel Description . . . . . . . . . . . . . . . .
3.3 Standardized Channel Models for UWB Communications
3.3.1 Overview . . . . . . . . . . . . . . . . . . . . . .
3.3.2 The IEEE 802.15.4a Channel Model . . . . . . . .
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . .
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33
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46
4 Transmitted Reference Systems
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
4.2 TR-UWB Signal and Systems Models . . . . . . . .
4.2.1 TR-UWB Transmission Schemes . . . . . .
4.2.2 TR-UWB Receiver Front-End . . . . . . . .
4.3 Equivalent TR-UWB Systems Models . . . . . . . .
4.3.1 System with ISI: Volterra Equivalent Model
4.3.2 Systems without ISI . . . . . . . . . . . . .
4.4 Bit Error Rate Performance Analysis . . . . . . . .
4.4.1 Gaussian Approximations approach . . . . .
4.4.2 System Modelling Approach . . . . . . . . .
4.4.3 Derivation of Average BER . . . . . . . . .
4.4.4 Simulation Results . . . . . . . . . . . . . .
4.5 Optimization of the Integration Time Interval TI .
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . .
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vii
5 Synchronization of Transmitted Reference Systems
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Data-Aided Timing Acquisition . . . . . . . . . . . . .
5.2.1 Timing Acquisition Technique . . . . . . . . . .
5.2.2 Simulation Results . . . . . . . . . . . . . . . .
5.3 Timing Offset Estimation . . . . . . . . . . . . . . . .
5.3.1 Normalization of the decision variable . . . . . .
5.3.2 Classical Approach to Timing Offset Estimation
5.3.3 Bayesian Approach to Timing Offset Estimation
5.3.4 Determination of Es , A and B . . . . . . . . . .
5.3.5 Simulation Results . . . . . . . . . . . . . . . .
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . .
6 Localization with Transmitted Reference
6.1 Introduction . . . . . . . . . . . . . . . .
6.2 Localization Systems . . . . . . . . . . .
6.2.1 Ranging-based Systems . . . . . .
6.2.2 Directionality-based Systems . . .
6.2.3 System Architecture . . . . . . .
6.3 Location Estimation . . . . . . . . . . .
6.3.1 The LS algorithm . . . . . . . . .
6.3.2 The SI Method . . . . . . . . . .
6.3.3 Simulation results . . . . . . . . .
6.4 Summary . . . . . . . . . . . . . . . . .
Systems
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7 Conclusions and Outlook
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7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.2 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.3 Discussion and Recommendations . . . . . . . . . . . . . . . . . . . . . . . 107
Appendices
109
Publications
123
Bibliography
125
List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
1.7
FCC spectral mask for indoor commercial applications.
Proposed spectral mask for Europe. . . . . . . . . . . .
Gaussian pulses and spectra . . . . . . . . . . . . . . .
UWB modulation schemes. . . . . . . . . . . . . . . . .
Typical TH-PPM waveform. . . . . . . . . . . . . . . .
Transmit PSD corresponding to PPM and THPPM . .
UWB based wireless sensor network. . . . . . . . . . .
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2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
A homomorphic filter employing a logarithmic nonlinearity.
An artificial neuron. . . . . . . . . . . . . . . . . . . . . .
Nonlinearities commonly employed in neural networks. . .
A quadratic filter . . . . . . . . . . . . . . . . . . . . . . .
Realization of a quadratic operator using Simulink. . . . .
The Hammerstein-Wiener model. . . . . . . . . . . . . . .
The Hammerstein model. . . . . . . . . . . . . . . . . . . .
The Wiener model. . . . . . . . . . . . . . . . . . . . . . .
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3.1
3.2
3.3
3.4
3.5
3.6
Model of the average power delay profile . . . . . . .
Typical channel realization . . . . . . . . . . . . . . .
Channel realizations from the IEEE 802.15.4a model
Average power delay profile. . . . . . . . . . . . . . .
RMS delay spreads for different CIR realizations. . .
Number of significant paths. . . . . . . . . . . . . . .
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36
38
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45
45
46
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
Diagram of the transmitted and received signals in DTR-UWB. . . .
Diagram of transmitted signals for LDC DTR-UWB systems . . . . .
Received signal for a DP TR-UWB system. . . . . . . . . . . . . . . .
Receiver front-end for a DTR-UWB system. . . . . . . . . . . . . . .
2nd order Volterra equivalent system model of the DTR-UWB system.
Simulated receiver output 1 . . . . . . . . . . . . . . . . . . . . . . .
Simulated receiver output 2 . . . . . . . . . . . . . . . . . . . . . . .
Normalized average delay-dependent noise-free decision variable . . .
Average and variance of the output noise versus delay . . . . . . . . .
Diagram of the received signals in DTR-UWB (without IFI). . . . . .
Simulated noise autocorrelation function. . . . . . . . . . . . . . . .
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50
51
51
52
57
58
59
59
60
62
63
viii
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ix
4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19
Autocorrelation function of the received prototype
Noise autocorrelation function Rν (κ). . . . . . . .
Probability density function of zτ [i]. . . . . . . . .
BER performance of the DTR-UWB system 1 . .
BER performance of the DTR-UWB system 2 . .
BER performance of the LDC DTR-UWB system
BER performance of the DP TR-UWB system . .
Optimal choice for TI . . . . . . . . . . . . . . . .
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65
65
68
72
73
73
74
74
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
Illustration of the timing acquisition technique . . . . .
Illustration of the outputs Yk for different timing offsets
Normalized timing acquisition MSE vs. SNR 1 . . . . .
Normalized timing acquisition MSE vs. SNR 2 . . . . .
Probability of detection vs. SNR . . . . . . . . . . . .
Average BER vs. SNR . . . . . . . . . . . . . . . . . .
Normalized timing offset estimation MSE vs. SNR 1 .
Normalized timing offset estimation MSE vs. SNR 2 .
Normalized timing offset estimation MSE vs. SNR 3 .
Timing offset estimation rmse vs. SNR . . . . . . . . .
Timing offset estimation rmse vs. delay . . . . . . . . .
Biasedness of timing offset estimator 1 . . . . . . . . .
Biasedness of timing offset estimator 2 . . . . . . . . .
Biasedness of timing offset estimator 3 . . . . . . . . .
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78
79
80
81
81
82
87
87
88
88
89
89
90
90
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
TOA localization principle. . . . . . . . . . . . . . . . . . . . . . .
TDOA localization principle. . . . . . . . . . . . . . . . . . . . . .
Normalized average distance-dependent noise-free decision variable
AOA localization principle. . . . . . . . . . . . . . . . . . . . . . .
Illustration of the spherical interpolation approach. . . . . . . . .
Distance estimation rmse vs. distance TX-RX . . . . . . . . . . .
Distance estimation variance vs. distance TX-RX . . . . . . . . .
2D localization scenario. . . . . . . . . . . . . . . . . . . . . . . .
Location estimation with DTR-UWB systems. . . . . . . . . . . .
Location estimation with LDC DTR-UWB systems. . . . . . . . .
Location estimation with DP TR-UWB systems. . . . . . . . . . .
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94
94
95
96
99
101
101
102
103
103
104
1
2
3
4
Experimental design of wireless sensor nodes
Algorithm flowchart . . . . . . . . . . . . . .
Interaction application-Home RF functions .
Relation DLL functions-HomeRF functions .
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118
120
121
122
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pulse.
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List of Tables
1.1
FCC spectral mask for UWB systems (EIRP in dBm). . . . . . . . . . . .
3
3.1
3.2
Simulation values of the channel parameters. . . . . . . . . . . . . . . . . .
Channel parameters for indoor office environment. . . . . . . . . . . . . . .
38
44
4.1
4.2
DTR-UWB system and simulation parameters . . . . . . . . . . . . . . . .
TR-UWB systems and simulation parameters. . . . . . . . . . . . . . . . .
57
72
5.1
TR-UWB systems and simulation parameters . . . . . . . . . . . . . . . .
86
6.1
Localization accuracy of different TR-UWB systems. . . . . . . . . . . . . 104
7.1
Performance of different TR-UWB systems. . . . . . . . . . . . . . . . . . 108
x
Abbreviations
AcR
ADC
A-GPS
APDP
AOA
ASIC
BER
BPF
BPPM
BPSK
CEPT
CIR
CMOS
CRC
CRLB
DAA
dB
DDP
DLL
DP
DTI
DTR
DOA
e.g.
EIRP
ETSI
FBO
FCC
FIFO
FIR
FT
GHz
GNSS
GPS
Autocorrelation Receiver
Analog-to-Digital Converter
Aided Global Positioning System
Average Power Delay Profile
Angle of Arrival
Application Specific Integrated Circuit
Bit Error Rate
Band Pass Filter
Biorthogonal Pulse Position Modulation
Binary Phase Shift Keying
European Conference of Postal and Telecommunications Administrations
Channel Impulse Response
Complementary MetalOxideSemiconductor
Cyclic redundancy check
Cramér-Rao Lower Bound
Detection-and-Avoidance
Decibel
Dominant direct path
Dynamic-link Library
Dual-Pulse
Discrete Time-invariant
Differential Transmitted Reference
Direction of Arrival
Abbreviation for exempli gratia (for example)
Equivalent Isotropically Radiated Power
European Telecommunications Standards Institute
Feedback Block-oriented
Federal Communications Commission
First In, First Out
Finite Impulse Response
Fourier Transform
Gigahertz
Global Navigation Satellite System
Global Positioning System
xi
xii
Hz
I&D
i.e.
IF
IFI
IIR
IMU
IPP
IR
ISI
kb/s
kHz
LDC
LOS
LS
LSE
LTI
MA
MAC
MAI
Mb/s
MHz
ML
MSE
MU
NBI
NDDP
NLOS
NMEA
OOK
PAM
PC
pdf
PDP
PPM
PSD
PVT
RF
RMS
RSS
SI
SNR
TAIP
Hertz
Integrate-and-dump
Abbreviation for id est (that is)
Intermediate Frequency
Infinite Impulse Response
Inter-frame Interference
Inertial Measurement Unit
International Postgraduate Programme
Impulse Radio
Inter-symbol Interference
Kilobit per second
Kilohertz
Low-Duty-Cycle
Line-of-sight
Least Square
Least Square Error
Linear Time Invariant
Multi-user Access
Medium Access Control
Multiple Access Interference
Megabit per second
Megahertz
Maximum Likelihood
Mean Square Error
Multiuser
Narrowband Interference
Non-dominant Direct Path
Non-line-of-sight
National Marine Electronics Association
On-off Keying
Pulse Amplitude Modulation
Personal Computer
Probability Density Function
Power Delay Profile
Pulse Position Modulation
Power Spectral Density
Position, Velocity and Time
Radio Frequency
Root Mean Square
Received Signal Strength
Spherical-interpolation
Signal-to-Noise Ratio
Trimble ASCII Interface Protocol
xiii
TDOA
TH
TOA
TR
TSIP
UART
UDP
UWB
WPAN
w.r.t.
WSN
ZESS
Time-difference of Arrival
Time Hopping
Time of Arrival
Transmitted Reference
Trimble Standard Interface Protocol
Universal Asynchronous Receiver/Transmitter
Undetected Direct Path
Ultra-Wideband
Wireless Personal Area Network
With respect to
Wireless Sensor Network
Center for Sensor Systems
Mathematical Symbols
|·|
⌊·⌋
⌈·⌉
(·)T
∀
∝
⊗
!
!!
7→
cjk
∂
δ(·)
diag(· · · )
E{·}
e(·) , exp(·)
Eb
Es
f
fc
H
h(t)
iN
IN
L{(·)}
log(·)
ln(·)
Ncr
Nf
n(t), ν(t)
Q(·)
Rν (t)
r(t)
Absolute value
Floor function
Ceiling function
Transpose
For all
Proportional to
Kronecker product
factorial
double factorial
maps to
TH code sequence of the k th user
Partial differentiation
Dirac delta function
Diagonal matrix
Expectation
Exponential
Bit energy
Symbol energy
Frequency
Center frequency
Homomorphism
Channel impulse response
All-ones vector of length N
Identity matrix of size N × N
Linear operator
Logarithm to base 10
Natural logarithm
Number of pulse-pair correlators
Number of frames in a symbol
White Gaussian noise
Complementary Gaussian cumulative distribution function
Noise autocorrelation function
Received signal
xiv
xv
σ2
si (t)
t
τ
τm
tm
Tf
Tm
Ts
Tω
Ts
var{·}
ω(t)
x
X
z[i]
Z
Noise variance
Transmitted signal containing the ith symbol
Time
Delay
Optimal modulation index in M -ary PPM
Pulse parameter
Frame duration
Channel maximum excess delay
Symbol duration
Pulse duration
Symbol duration
Variance
Gaussian pulse
Vector
Matrix
Decision variable
Field of integers
Chapter 1
Introduction
1.1
1.1.1
Ultra-Wideband Technology
UWB History and Regulatory Issues
The term Ultra-WideBand or UWB signal has come to signify a number of synonymous
terms such as impulse, carrier-free, baseband, time domain, nonsinusoidal, orthogonal
function and large-relative-bandwidth radio/radar signals. According to modern definition
[1], UWB characterizes transmission systems with instantaneous spectral occupancy in
excess from 500 MHz, or fractional bandwidth of more than 20 %. The fractional bandwidth
is defined as B/fc , where B = fH − fL denotes the -10 dB bandwidth and center frequency
fc = (fH + fL )/2 with fH being the upper frequency of the -10 dB emission point and fL
the lower frequency of the -10 dB emission point. UWB systems with fc > 2.5 GHz need
to have a -10 dB bandwidth of at least 500 MHz, while UWB systems with fc < 2.5 GHz
need to have a fractional bandwidth of at least 0.20.
Contributions to the development of a field addressing UWB radio frequency (RF) signals commenced in the late 1960’s with the pioneering contributions of Harmuth at the
Catholic University of America, Ross and Robbins at Sperry Rand Corporation, Paul van
Etten at the USAF’s Rome Air Development Center and in Russia [2]. The Ross and
Robbins (R&R) patents (1972-1987) pioneered the use of UWB signals in a number of application areas, including communications and radar and also using coding schemes. After
the 1970s, the only innovations in the UWB field came from improvements in particular
instantiations of subsystems, but not in the overall system concept itself, nor even in the
overall subsystems’ concepts. The basic components were known, e.g., pulse train generators, pulse train modulators, switching pulse train generators, detection receivers and
wideband antennas.
In 1994, T.E. Mc Ewan invented the Microwave Impulse Radar (MIR) which provided,
for the first time a UWB operating at ultra low power, besides being extremely compact
and inexpensive [3]. This was the first UWB radar to operate on only microwatts of battery
drain.
The UWB radar and communications approach is a shift in emphasis with respect to the
use of the available time-bandwidth-power product. As the signal duration decreases, the
1
Chapter 1. Introduction
2
bandwidth increases and the signal-to-noise ratio (SNR) per frequency decreases. Moreover, the SNR per frequency decreases below the threshold of frequency selective receivers,
which is a major argument made by UWB proponents that UWB systems are able to
operate in the presence of frequency selective receivers without interference. The methods
used to reliably receive a UWB signal with such low SNR per frequency are:
• a high sampling rate receiver to capture in a non-synchronous way all the signal energy in a minimum number of sampling bins, summing across all the signal bandwidth
which implies a receiver front-end open to that instantaneous ultra-wide bandwidth
and thus also open to noise; or
• signal averaging or matched filtering which lowers the data rate; or
• counteracting the low power per frequency by increasing to high signal transmit
power, which implies interference to other receivers.
As in the case of more conventional communications systems, the UWB wireless system
designer must balance trade-offs among high bandwidth efficiency, low transmission peak
power, low complexity, flexibility in supporting multiple rates and reliable performance as
expressed in bit error rates. A UWB radar system is more unconventional than a UWB
communications system, not only in system components, but also in the physics involved
in the signal-target interactions. Indeed, conventional radars use signals longer in distance
length than the distance length of the target, thus the target is a point scatterer. In the
UWB radar case, the transmitted signal is shorter in distance length than the target, so
the target is not a point scatterer.
Even though the concept on which UWB transmission is based is the overlay of lowpower signals over existing spectrum and the peaceful coexistence with narrow-band systems already in operation, UWB devices will emit radiation as any other radio system.
In order to minimize the level of interference and to coordinate its ordered deployment, a
set of regulations is necessary to indicate limits and restrictions in the implementation of
this technology and introduction into the real market. The Federal Communications Commission (FCC) in the United States released a huge frequency band 3.1 - 10.6 GHz with
equivalent isotropically radiated power (EIRP) below -43 dBm, where UWB radios overlaying coexistent RF systems can operate using low-power ultra-short information bearing
pulses. The applications of UWB technology approved by the FCC today include communications (short-range very high speed broadband communications, covert communication
links), vehicular radar systems, imaging systems and distance measurement systems (localization at centimeter-level accuracy, precision navigation and asset tracking). The FCC’s
spectral masks assigned to these applications are listed in Tab. 1.1.
In particular, the FCC’s assigned bandwidth and spectral mask for indoor communications is illustrated in Fig. 1.1. Far from being universally accepted and welcomed by
everyone, the FCC’s initial work raised the discussion levels also in Europe, where very
important coexistence studies started in early 2000, first in the CEPT (European Conference of Postal and Telecommunications Administrations) working group SE24 and ETSI
(European Telecommunications Standards Institute) working group TG31a, and later also
in the newly formed working group TG3. The recommended spectral mask for Europe
Chapter 1. Introduction
Frequency
GHz
0.96 - 1.61
1.61 - 1.9
1.9 - 1.99
1.99 - 3.1
3.1 - 10.6
10.6 - 22
22 - 29
29 - 31
> 31
Indoor
Comm.
-75.3
-53.3
3
Hand held
Comm.
-75.3
-63.3
-61.3
-51.3
-41.3
Low Frq.
Imaging
-65.3
-53.3
-41.3
High Frq.
Imaging
-65.3
-53.3
Med. Frq.
Imaging
-53.3
-51.3
Vehic.
Radar
-75.3
-51.3
-41.3
-41.3
-61.3
-51.3
-51.3
-41.3
-51.3
-61.3
-51.3
-51.3
-61.3
Table 1.1: FCC spectral mask for UWB systems (EIRP in dBm).
−40
UWB EIRP Emission Level in dBm
−45
−50
3.1
1.99
−55
−60
10.6
Indoor Limit
−65
GPS
Band
−70
−75
1.61
0.96
0
10
1
Frequency in GHz
10
Figure 1.1: FCC spectral mask for indoor commercial applications.
is shown in Fig. 1.2, where power levels are equivalent to those of the FCC regulations
in the frequency band 3.1 - 10.6 GHz. However, European regulators have introduced
the concept of “Detection-and-Avoidance” (DAA) technology to ensure coexistence with
existing wireless communications technologies [4]. It means that UWB devices operating
in the 3.1-4.2 GHz band should be fitted with DAA systems, which automatically search
for nearby broadband wireless signals and then switch frequency to prevent interference.
DAA is not required in the 4.2-4.8 GHz band until 2010, and no DAA is required in the
6-10.6 GHz band.
In parallel to the regulatory efforts, another fundamental part of the introduction of this
new technology into the market was the definition of an industry inter-operability standard
Chapter 1. Introduction
4
−40
3.1
UWB EIRP Emmission Level in dBm
−50
−60
−70
2.8
−80
2.1
−90
10.6
1.5
0.5
−100
0
10
1
Frequency in GHz
10
Figure 1.2: Proposed spectral mask for Europe.
based on the best properties of UWB technology. Given the nature of UWB systems, which
can be suitable for high-speed links, low to medium data rate and positioning applications,
standardization efforts were carried out in parallel in two international standardization
groups: the IEEE 802.15.3a (late 2001) and in the IEEE 802.15.4a (late 2003).
1.1.2
Basics of UWB Techniques
Shape and Spectrum of UWB Signals
Impulse Radio (IR) based UWB systems rely on ultra-short sub-nanosecond waveforms
that can be free of sine-waves carriers and do not require Intermediate Frequency (IF)
processing because they can operate at baseband. Generally adopted spectrum shapers
ω(t) for UWB communications include the Gaussian pulse, the Gaussian monocycle (first
derivative of Gaussian pulse), and the Gaussian doublet (second derivative of the Gaussian
pulse), as depicted in Fig. 1.3, along with their Fourier Transforms (FT). The reason behind
the popularity of these pulses is twofold:
• Gaussian pulses come with the smallest possible time-bandwidth product of 0.5,
which maximizes range-rate resolution and
• the Gaussian pulses are readily available from the antenna pattern [5].
Let Tω be the duration of the transmitted pulse ω(t). With Tω at the sub-nanosecond
scale, ω(t) occupies an UWB with bandwidth B ≈ 1/Tω . The mathematical expression of
Chapter 1. Introduction
5
10
Gaussian doublet
1
Gaussian monocycle
0
Gaussian pulse
0.8
−10
0.6
−20
0.2
Power [dB]
Amplitude [dB]
0.4
0
−0.2
−30
−40
−0.4
Gaussian doublet
−50
−0.6
Gaussian monocycle
−0.8
Gaussian pulse
−60
−1
0
0.1
0.2
0.3
0.4
Time [ns]
0.5
0.6
0.7
0.8
−70
1
Frequency (GHz)
10
(b)
(a)
Figure 1.3: (a) Adopted pulse shapes in UWB communications; (b) Fourier transform of the
pulse shapes. Pulse duration: Tω = 0.7 ns.
the Gaussian pulse, Gaussian monocycle and the Gaussian doublet are given, respectively,
as
ωg (t) = K1 e−2π(t/tm ) ,
2
ωg1 (t) = dωg (t)/dt = K1
and
−4πK1
ωg2 (t) = d ωg (t)/dt =
t2m
2
2
(1.1)
−4πt −2π(t/tm )2
e
,
t2m
(
4πt2
1− 2
tm
(1.2)
)
e−2π(t/tm ) ,
2
(1.3)
where tm is the pulse parameter. It can be seen that by utilizing one of the properties of
the pulses (i.e., by differentiating or integrating them) another pulse can be created, with
the order of the pulse being one more or less than the original pulse, respectively. In this
thesis, the energies associated with the Gaussian pulse and its derivatives are expressed
analytically as
K12 tm
,
2
(1.4)
K12 π
=
,
tm
(1.5)
Eg =
Eg1
and
Eg2 =
6π 2 K12
.
t5m
(1.6)
Chapter 1. Introduction
6
The derivations can be found in Appendix II.
UWB Modulation Techniques
Being real, baseband UWB transmissions neither have to entail frequency modulation nor
phase modulation. Consequently, symbol values can be transmitted by modulating the
position and/or the amplitude of ω(t).
{ In M -ary Pulse
} Position Modulation (PPM), M distinctly delayed pulses are employed,
M −1
ω(t − τm )m=0 , each representing one symbol value. Generally, the modulation indices
τm are chosen such that τm = mτ with τ ≥ Tω , which corresponds to orthogonal
PPM. In
∫
binary PPM, the delay τ can also be chosen to minimize the correlation ω(t)ω(t − τ )dt [6].
As bandwidth efficiency drops with increasing modulation size M , PPM is suitable for
power-limited applications. In fact, PPM was almost exclusively adopted in the early
development of UWB radios because negating ultrashort pulses were difficult to implement.
Biorthogonal pulse position modulation (BPPM) was introduced in [7]. It is shown that
N-ary BPPM has better performance than N-ary PPM with the same throughput and half
the computational complexity.
Another modulation scheme that does not require pulse negation is the so termed OnOff Keying (OOK), where symbol “1” is represented by transmitting a pulse, and “0”
by transmitting nothing. The main advantage of the OOK scheme is its ease of physical
implementation.
As pulse negation became easier to implement, Pulse Amplitude Modulation (PAM)
attracted more attention. In particular, when M = 2, antipodal pulses are used to represent
binary symbols, as in Binary Phase Shift Keying (BPSK) or bipolar signaling. The bit error
rate (BER) performance of the BPSK scheme is superior to that of the OOK since OOK has
a smaller symbol separation (transmitted symbols 0 and 1) compared to BPSK (transmitted
symbols 1 and -1). The different modulations schemes are illustrated in Fig. 1.4 for a bit
sequence b = {1, 0, 1}.
Multiple Access Techniques
To allow for multi-user access (MA) to the UWB channel, Time Hopping (TH) was introduced [6]. With TH, each pulse is positioned with each frame duration Tf according
to a user-specific TH sequence ckj . Specifically, dividing each frame into Nc chips each of
duration Tc , the kth user’s TH code ckj ∈ [0, Nc − 1] corresponds to a time shift of ckj Tc
during the kth frame [8]. A typical TH-PPM transmitted waveform from the kth user is
given by
sk (t) =
+∞
∑
j=0
ω(t − jTf − ckj Tc − τ bk⌊j/N ⌋ ),
f
(1.7)
where Nf is the number of pulses transmitted per symbol and ⌊x⌋ is the floor function of a
real number x. It is defined as ⌊x⌋ = max{n ∈ Z|n ≤ x}. With TH codes, MA is achieved
by altering the pulse position from frame to frame, according to the sequence ckj .
Chapter 1. Introduction
7
1
1
BPPM
PPM
0.5
0.5
0
0
−0.5
0
2
4
Time [ns]
6
1
−0.5
0
2
1
OOK
4
6
4
6
BPSK
0.5
0.5
Time [ns]
0
0
−0.5
−0.5
0
2
4
Time [ns]
6
−1
0
2
Time [ns]
Figure 1.4: UWB modulation schemes.
Fig. 1.5 depicts a TH-PPM transmitted waveform, where Nf = 10, Tf = 10 ns, Tc = 0.1 ns
and the code sequence is given by cj = {0, 5.3, 0.4, 3, 7.3, 0.8, 5, 1.6, 1.8, 0}, j = {0,1,· · · ,9}.
1.5
15.3 20.4
Amplitude [dB]
1
33
47.3 50.8
65
71.6
81.8
90
0.5
0
−0.5
−1
0
20
40
time [ns]
60
80
Figure 1.5: Typical TH-PPM waveform.
100
Chapter 1. Introduction
8
In addition to facilitating multiple access, spreading codes also shape the transmit
spectrum, as shown in Fig. 1.6.
TH−PPM
40
40
20
20
0
0
Power [dB]
Power [dB]
PPM
−20
−20
−40
−40
−60
−60
1
Frequency [GHz]
(a)
10
1
Frequency [GHz]
10
(b)
Figure 1.6: Transmit power spectral density (PSD) corresponding to (a) no spreading code, (b)
random TH codes.
Applications
• Wireless Personal Area Networks (WPANs): Also known as in-home networks, WPANs
address short-range (within 10 - 20 m) ad hoc connectivity among portable consumer
electronics and communication devices. UWB technology emerges as a promising
physical layer candidate for WPANs, because it offers high-rates over short-range,
with low cost, high power efficiency and low duty cycle.
• Wireless Sensor Networks (WSNs): WSNs consist of a large number of nodes spread
across a geographical area. Key requirements for WSNs operating in challenging
environments include low cost, low power, and multi-functionality. Emerging applications of UWB are foreseen for sensor networks as well. Such networks combine low
to medium rate communications with positioning capabilities. UWB signaling is especially suitable in this context because it allows centimeter accuracy in ranging, as well
as low-power and low-cost implementation of communication systems. These features
allow a new range of applications, including logistics (package tracking), security applications (localizing authorized persons in high-security areas), medical applications
(monitoring of patients), family communications/supervision of children, search and
rescue missions (communications with fire fighters, or avalanche/earthquake victims),
control of home appliances, and military applications [9]. High data-rate UWB communication systems are well motivated for gathering and exchanging a vast quantity
of sensory data in a timely manner.
Chapter 1. Introduction
9
• Radar Systems: UWB-based sensing has the potential to improve the resolution of
conventional proximity and motion sensors. Relying on the high ranging accuracy and
target differentiation capability enabled by UWB, intelligent collision-avoidance and
cruise-control systems may be envisioned. These systems can also improve airbag
deployment and adapt suspension/braking systems depending on road conditions.
UWB technology can also be integrated into vehicular entertainment and navigation
systems by downloading high-rate data from airport off ramp, road-side gas station
UWB transmitters [10].
• Imaging Systems: Different from conventional radar systems where targets are typically considered as point scatterers, UWB radar pulses are shorter than the target
dimensions. UWB reflections off the target exhibit not only changes in amplitude
and time shift but also changes in the pulse shape. As a result, UWB waveforms
exhibit pronounced sensitivity to scattering relative to conventional radar signals.
This property has been readily adopted by radar systems [2] and can be extended
to additional applications, such as underground, through-wall and ocean imaging, as
well medical diagnostics and border surveillance devices [11].
1.1.3
Open Research Problems
UWB is an emerging new technology that is expected to enable low-cost and low-power
devices. As already mentioned in 1.1.2, streams of ultra-short pulses (< 1 ns) are used for
wireless data transmission instead of a modulated sinusoidal carrier, yielding signals of huge
bandwidths (> 1 GHz) but at very low power densities. In principle, the baseband nature of
the signal transmission makes the UWB technology suitable for low-cost implementations in
standard CMOS (complementary metal-oxide-semiconductor) technology. Indeed, unlike
conventional radio systems, the transmitted pulse is able to propagate without the need for
an additional RF mixing stage. Then the UWB receiver can bypass the need for complex
phase tracking loops. However, before UWB systems can be produced at a large scale and
low cost, there are numerous open research issues to be solved.
UWB systems have opened up new dimensions of antenna design. Antennas have
become an organic part of RF systems, providing filtering and other custom-designed
frequency-dependent properties. The wide bandwidths of UWB antennas present new
challenges for design, simulation, and modelling. Optimizing UWB antennas to meet the
demands of UWB propagation channels is similarly challenging. Designers are meeting
these challenges with novel antenna designs, novel materials and using concepts like polarization diversity, directivity arrays, and electric-magnetic element combinations.
Still more research at fundamental and applied levels is needed to make cheap and
power-efficient UWB chips available, going beyond the existing state-of-the-art solutions in
several areas of transceiver architecture design and signal processing. Due to the extremely
large bandwidth of UWB systems, which prevents direct sampling of the received signal at
sufficient accuracy, it is expected that a straight-forward downscaling of signal processing
algorithms for conventional receivers will not lead to practical solutions for UWB devices.
This implies new algorithms for channel estimation, synchronization and other typical
receiver tasks that have to be developed for UWB devices. The derivation of appropriate
Chapter 1. Introduction
10
system models including the accurate modelling of the multi-path radio channel is as well
necessary. Additionally, the shared nature of the medium means that UWB receivers
must contend with a variety of interference sources. Traditional interference mitigation
techniques are not amenable to UWB due to the complexity of straight-forward translations
to UWB bandwidths. Thus, multi-user signal detection, multiple access interference (MAI)
cancellation, narrowband interference (NBI) detection and cancellation are open research
issues that must be met in order to exploit the potential benefits of UWB systems.
UWB radio technology is a well-suited physical layer candidate for the robust, low power
and ubiquitous data communication requirements of WSNs. However, the fundamental role
of UWB technology in WSNs is still open and a wide range of research questions continue to
present challenges. The characterization of the UWB radio channel in industrial areas and
large open spaces has to be addressed. The design of signaling schemes that make efficient
use of the channel capacity at low complexity and low power consumption is necessary, as
well as efficient medium access control (MAC) protocols for a large number of distributed
sensor nodes. The derivation of algorithms to provide location information of wireless
sensor nodes and geographic information about the environment has equal interest.
1.2
Ultra-Wideband Wireless Sensor Networks
In this section, the basic challenges in designing WSNs are revisited, followed by the
introduction of the UWB approach for radio communications among the sensor nodes in
a WSN, that addresses each one of the design challenges. Finally, the overview of our
envisaged application is presented.
1.2.1
Wireless Sensor Network Requirements
WSNs have emerged as a new wireless networking paradigm allowing distributed sensing
and cooperative communication strategies. They also represent a new generation of realtime embedded systems with different significant communication constraints when compared with traditional networked systems. The main design challenges in WSNs can be
categorized into the following areas:
• Scalability: As the number of sensor nodes in a WSN increases, scalability imposes
difficulties in transferring data. In order to send information to far away nodes,
signals with higher transmission power should be employed which can cause internode interference or a multi-hop approach need to be considered.
• Power conservation: The nodes in WSNs have limited energy resources, so to extend the lifetime of the entire network power conservation in individual nodes is of
significant importance. In WSNs, radio communications is the major consumer of
energy. Hence, minimizing the radio transmission power or avoiding unnecessary
communications can considerably save power in sensor nodes.
• Synchronization: During radio communications between different wireless sensor
nodes of a WSN, nodes continuously listen to transmissions and consume power
Chapter 1. Introduction
11
if they are not time synchronized with each other. While global synchronization is
unrealistic, especially when there is large sensor population, node-by-node synchronization becomes a necessity in WSN design.
• Channel estimation: Channel estimation plays a critical role in WSNs, since sensor
nodes communicate over wireless channels and have to overcome the effects of a
wireless link, such as noise, multipath effect, intentional jamming and inter-node
interference. Estimating the wireless link between a specific transmitter and receiver
pair provides directionality and reliable data transfer between the nodes.
• Self-organization: When large number of sensor nodes are deployed in remote environments, the ability of individual sensor nodes to self-organize is vital. Selforganization should be done in a way to improve the performance while reducing
the power consumption of the entire sensor network.
1.2.2
Ultra-Wideband Approach to Wireless Sensor Networks
WSNs are characterized by devices with low complexity that have limitations on processing
power and memory, and severe restrictions on power consumption. By the very nature of
the application, traffic in WSNs is often bursty with long periods of no activity (low duty
cycle). For event detection operations, a device may remain idle for long periods, then
suddenly be required to send significant amounts of information when an event occurs. For
devices deployed in the field, this has significant implications for the design of efficient MAC
protocols, radio communications technology, and the reliability of information transfer.
For devices involved in continuous monitoring, the flow of traffic will be more stable.
However, efficient multiple access techniques, reliability, and battery life are still major
considerations.
Since 2002, interest in UWB-based applications has increased greatly, especially concerning the use of UWB for WSNs. IR-based UWB technology has a number of inherent
properties that are well suited for WSN applications. In particular, UWB-IR systems have
potentially low complexity and low cost. This arises from the essentially baseband nature of the signal transmission. Unlike conventional radio systems, the UWB transmitter
produces a very short time domain pulse that is able to propagate without the need for
an additional RF mixing stage. The RF mixing stage translates the signal to a frequency
that has desirable propagation characteristics. The signal will propagate well without need
for additional upconversion and amplification. The UWB receiver also does not require
the reverse process of downconversion. Again, this means a local oscillator in the receiver
can be omitted, which means the removal of associated complex delay and phase tracking
loops.
Using UWB technology for inter-node communication in WSNs will offer not only low
complexity and cost factors, but also provide high performance for communication over
the wireless channels in spite of multipath distortions. Furthermore, transmission of short
duration UWB pulses requires much lower power compared to strong narrowband signal
transmission. In UWB-based WSNs, nodes can only communicate with their close-by
Chapter 1. Introduction
12
neighbors due to low transmission power and avoid the inter-node interference issue that
exists in narrowband techniques.
Despite all the benefits that UWB technology offers to the design of WSNs, it can create
a unique set of challenges too. Employing the low powered UWB pulses for inter-node
communications, introduces the scalability problem in WSNs. As the distance between
nodes or the number of nodes increases, weak UWB pulses cannot transfer information
between nodes in a reliable manner. Also, the short duration of UWB pulses introduces
a major challenge in time synchronization for sensor nodes in a wireless network. In
order to synchronize sub-nanosecond pulses, very high-speed analog-to-digital converter
(ADC) components are needed. Another problem with using UWB technology for WSNs
is the performance degradation due to interference from strong narrowband signals that
share the spectrum with low powered UWB pulses. Moreover, detection of UWB pulses is
commonly performed using classical matched filtering technique, where the received signal
is correlated with a UWB pulse template. Thus, wireless channel multipath effects on
the received signal can significantly degrade the detection process due to low correlation
between the predefined template and the distorted received signal.
The challenge posed by synchronization and channel degradation to narrow and low
powered UWB pulses has been addressed by the transmitted-reference (TR) signaling
which is extensively presented in this thesis. TR receivers correlate the received signal
with a delayed replica of itself. Using TR-UWB transceivers for radio communication of
WSNs offers simplicity, low transmission power, and capability to reduce the stringent
UWB synchronization requirements, as well as channel estimation. On the other hand the
performance of TR receivers is considerably limited by the severity of noise-noise products
introduced by the correlation of noise in the received signal. Therefore, scalability becomes
a major issue in designing WSNs with TR-UWB transceivers. On the other hand, channel
estimation is not required with TR-UWB transceivers.
Finally, UWB-IR systems have a very good time domain resolution, allowing for localization
and tracking applications [12].
1.2.3
Conceptual Design of the Envisaged Application
The ranging network presented in this subsection can be viewed as a WSN where the
physical parameter to be sensed is location. The radio module of each node is based on
low-power low-complexity TR-UWB transceivers since sensor nodes are inherently resource
constrained: limited processing capabilities, storage capacity, and power consumption.
Moreover, they need to be designed with low complexity to meet the low cost requirements.
Within the overall objective of investigating the communication requirements in WSNs
our primary focus was the design of some experimental sensor nodes, which are capable
of a tremendous diversity of functionality such as sensing capabilities, signal processing,
network protocol functions, short-range radio communication at low to medium bit rates
(up to 1 Mbps) and localization capabilities. The experimental design of the ZESS sensor
nodes, as well as the asynchronous transfer of GPS data to those sensor nodes are described
in Appendix VI.
An overview of the different activities involved in our envisaged WSN, with communi-
Chapter 1. Introduction
13
Beacon
Nodes
Central
station
GPS
sensor
Serial interface
GPS - MCU
MCU
UWB
Radio
Extraction of position
information
Sensor
Node
GPS
sensor
sensor
MCU
DSP
MCU
UWB
Radio
Dissection of GPS
message string
Radio
Channel
UWB
Radio
GPS
sensor
Timing offset
estimation
Decision
Decoder
MCU
UWB
Radio
Text
Image
Video
Coarse
Acquisition
LOS/NLOS
detection
Central
station
Location
estimation
Distance
estimation
Estimated
Distance
Estimated
Location
Figure 1.7: UWB based wireless sensor network.
cation and localization capabilities using IR TR-UWB systems, is depicted in Fig. 1.7. In
this application, the sensor node communicates with different beacon nodes. The beacon
nodes are assumed to be aware of their position by interfacing them with GPS sensors and
allowing them to be mobile, or by keeping them at fixed known positions. The signals
received at the beacon nodes are processed in order to localize the sensor node. Using a
centralized tightly-coupled system architecture, the beacon nodes are connected to a central station, through a wire or wirelessly, where the location of the sensor node is estimated.
Tracking of a mobile sensor node is also possible using appropriate tracking algorithms.
The UWB radio module in Fig. 1.7 contains two different subsystems: the data subsystem and the distance measurement subsystem. All the different blocks in both subsystems, that is, coarse acquisition, timing offset estimation, symbol decision, decoding, and
distance estimation are thoroughly addressed in this thesis. The line-of-sight (LOS)/non
line-of-sight (NLOS) detection, which is beyond the scope of this thesis, is briefly addressed
hereafter.
The radio channel presented in Fig. 1.7 may generally have three different profiles.
Chapter 1. Introduction
14
The first one is the dominant direct path (DDP) case, in which the strongest path of the
channel estimate corresponds to the LOS. Those channel profiles, where the first path
is not the strongest, however still detectable by an appropriate receiver architecture, are
called non-dominant direct path (NDDP). Finally and only regarded as NLOS, in this
thesis, are the situations in which the receiver architecture is not able to detect the direct
path anymore. This case has been defined as undetected direct path (UDP) and leads
to ranging errors. Consequently, NLOS channels are one of the major drawbacks for
accurate ranging and localization with UWB technology. To mitigate this drawback, many
LOS/NLOS detection algorithms are proposed in the literature. In [13] a simple estimator
has been proposed for NLOS detection using the running variance on N subsequent range
estimates dˆn . Another LOS/NLOS detection solely uses the estimate of one channel impulse
response and therefore introduces no additional delay. The so called confidence metric, as
defined in [12] and used for LOS/NLOS detection, is based on the idea that later multipath
components should have less power than the direct component in the LOS case and vice
versa in the NLOS case. This reasoning is valid for distinguishing between DDP and UDP
channel profiles. Some authors have stated that the root mean square (RMS) delay spread
could be used as an LOS/NLOS detection method [14], however, they have not investigated
the proposal further since no threshold was proposed. Finally, LOS/NLOS detection could
be based on the assumption that a sudden decrease of signal-to-noise ratio (SNR) could
indicate the movement from a LOS into an NLOS condition, and vice versa. In this case,
changes in the power of subsequent maximum paths are detected and normalized. Many
of the described algorithms are difficult to apply with real UWB localization systems.
For a suitable hypothesis testing of ranging error distributions, many range estimations
must be evaluated at the same position to create an error distribution estimate, and valid
error distributions for the LOS and NLOS case must be available. Both demands are
rather seldom in reality. In addition, the necessity of a statistical significance contrasts the
need for a small latency of detection for a robust localization system. Similar obstacles
occur when channel estimates are to be tested against predefined channel models. Even if
these methods work in real environments, the existence of enough LOS channels in realistic
indoor applications is probably quite seldom. Consequently, distance measuring techniques
which are less prone to LOS/NLOS conditions are useful.
In this thesis, a novel distance measurement approach based on the modelling of the
energy collected at the output of the TR-UWB receivers as a function of delay is proposed.
Since the total energy is the sum of signal strengths of each individual path in a multipath
environment, this measurement technique takes advantage of the multipath diversity in the
channel. Furthermore, the timing requirement in our method is less rigorous than when
using time-based distance measuring techniques, and it is more tolerant in UDP cases.
This implies that the NLOS/NLOS detection can be omitted while using our distance
measurement technique, at the expense of loosing some accuracy in distance measurement.
This is the tradeoff complexity/accuracy.
Chapter 1. Introduction
1.3
15
Goal of the Thesis
The main objective of this doctoral thesis is to present the conceptual design of UWB-IR
transceivers that have dual functionality in providing robust, low power data communication and accurate location information in WSNs. The intended application would be
the monitoring and localization of wireless sensor nodes within a closed area (indoor and
outdoor environments).
The emphasis of this study lays on the physical layer: modelling and analysis of the
radio transceivers, simulation of receiver algorithms and characterization of the physical
radio link. Wireless transmission using different TR-UWB IR systems allows peak data
rates up to 5 Mbit/s and location estimation with around 30 cm accuracy for a transmitterreceiver separation within 10 m. The TR signalling was used due to its simple receiver
structure.
1.4
Organization of the Thesis
This doctoral thesis is structured as follows:
• In Chapter 1 the UWB technology was presented highlighting the history behind it
and the regulatory issues. The basic techniques in UWB systems were introduced and
an insight was provided on the open research issues in this new emerging technology.
The envisaged application related to this work was presented. This chapter ends
with emphasis on the goal and contributions of this thesis.
• Chapter 2 deals with polynomial nonlinear systems. The Volterra model is also
presented in this chapter. Further, the Hammerstein and Wiener models, which are
the simplified forms of the Volterra model, are addressed as well.
• Chapter 3 presents an important tool for the design and analysis of communication
systems: the radio channel model. Different channel models for UWB applications
are revisited.
• Different TR systems are introduced in chapter 4. System models are derived for
evaluating more effectively the BER performance of the low complexity TR-UWB
systems considered through simulation.
• Chapter 5 addresses the synchronization issues. Coarse and fine synchronization
algorithms are developed and analyzed.
• Based on the results from chapter 5, the performance of different location estimation
algorithms is evaluated in chapter 6.
• Finally, general conclusions, recommendations and possible directions for future
works are summarized in chapter 7.
Chapter 1. Introduction
1.5
16
Contributions of the Thesis
This thesis presents several contributions towards developing UWB-IR technology for
short-range low to medium data rate WSN applications. The contributions can be classified into two classes. The first class of contribution concerns the design through simulation
of low complexity TR-UWB systems capable of providing data rates up to 5 Mbit/s. Based
on previous work done on modelling TR-UWB systems, delay-dependent systems models
for different low complexity TR-UWB systems are proposed. These models are aimed in
facilitating the performance analysis of the various systems considered, designing efficient
detectors and developing new algorithms. The second class of contribution consists of the
development of simple algorithms for synchronization and distance estimation using the
proposed delay-dependent systems models.
1.5.1
Systems Modelling
In [15] a discrete-time Volterra equivalent system model was derived for DTR-UWB systems in a multipath channel, where both inter-frame interference (IFI) and inter-symbol
interference (ISI) are present. Using this model, delay-dependent discrete time equivalent
system and noise models for different low complexity TR-UWB systems are proposed in
this thesis.
1.5.2
Performance Analysis
The different TR-UWB receiver statistics are analyzed, first, using the Gaussian approximation on the noise terms in the receiver statistics, then, using the equivalent system and
noise models of the different TR-UWB receivers.
1.5.3
BER derivation
Both perspectives for receiver statistics analysis allow for bit error rate (BER) performance
analysis to be performed. The derivation of an analytic expression for the average BER is
provided in this thesis using the concept of minimum distance receiver.
1.5.4
Coarse Synchronization
In this thesis, a novel data-aided timing acquisition technique for differential TR-UWB
systems is presented. It achieves efficient multipath energy collection even in presence of
timing offset and noise.
1.5.5
Fine Synchronization
Another contribution of this thesis in the field of synchronization concerns a simple timing
offset estimator for low complexity TR-UWB systems. The proposed algorithm uses energy
collected at the symbol rate, thus reducing considerably the implementation complexity.
Chapter 1. Introduction
1.5.6
17
Distance and Location Estimation
Assuming coarse synchronization achieved and hardware clock timing error known, the
timing offset estimation algorithm can readily be used for distance and location estimation.
To my knowledge, the distance measuring approach proposed in this thesis has never been
dealt with before. It is similar to the signal strength measurement technique, but instead
of using a path loss model, the modelling of the energy collected at the output of the UWB
receivers as a function of delay is used.
Chapter 2
Polynomial Nonlinear Systems:
Overview
2.1
Introduction
All linear systems obey the superposition principle, which implies that the output of a
linear combination of input signals to a linear system is the same linear combination of the
outputs of the system corresponding to the individual components. That is,
L{αx1 (n) + βx2 (n)} = αL{x1 (n)} + βL{x2 (n)},
(2.1)
where L{(·)} denotes the output of the linear system, x1 (n) and x2 (n) are two different
input signals, and α and β are two arbitrary constants. The superposition principle is a
powerful mechanism that allows the study of all linear systems in a unified manner. Nonlinear systems do not satisfy the superposition principle. Furthermore, since every system
that does not satisfy the superposition principle is nonlinear, it is impossible to develop
a framework that is applicable to all nonlinear systems. Consequently, the traditional approach for modelling and studying nonlinear systems is to consider one or more classes of
such systems and to develop a theory for analysis, design and realization as well as applications of such classes individually. In the next subsections several classes of nonlinear
systems are briefly described [16].
2.1.1
Homomorphic Systems
A homomorphic system satisfies the generalized superposition principle stated as
H{x1 (n) ∗ x2 (n)} = H{x1 (n)} + H{x2 (n)},
(2.2)
where “∗” and “+” are two operations defined on the input signal and the output signal,
respectively.
As an example, the intensity of light reflected from an object can be modelled as the
product of the intensity of the light that shines on the object at each location and the
reflection coefficient of the object [17], i.e., an image x(n1 , n2 ) may be expressed as
18
Chapter 2. Polynomial Nonlinear Systems: Overview
19
x(n1 , n2 ) = I(n1 , n2 )r(n1 , n2 ),
(2.3)
where I(n1 , n2 ) is the light intensity that falls on the object at location (n1 , n2 ) and r(n1 , n2 )
denotes the reflection coefficient of the object at the same location. Homomorphic filters
are particularly useful in situations, where the information-bearing function r(n1 , n2 ) needs
to be separated from the image, or when it is necessary to process the two components in
different ways using an appropriate nonlinear transformation and a linear filter following
the nonlinearity. Fig. 2.1 shows the block diagram of a homomorphic filter, where the first
block transforms the input signal using a logarithmic nonlinearity. The transformed signal
is then processed by a linear filter. The output of the logarithmic nonlinearity is given by
y(n1 , n2 ) = ln (x(n1 , n2 )) = ln (I(n1 , n2 )) + ln (r(n1 , n2 )) .
(2.4)
Thus, the linear filter will see the multiplicative components as additive components at
its input. Consequently, linear filtering techniques can be used to discriminate the two
components.
x(n)
Log
Nonlinearity
Linear
Filter
y(n)
Figure 2.1: A homomorphic filter employing a logarithmic nonlinearity.
2.1.2
Order Statistic Filters
Order statistic filters are employed in applications in which the input signal is corrupted
by impulsive noise. The class of order statistic filters includes the median filter and its
generalizations. A one-dimensional (2K+1)-point median filter is defined by the inputoutput relationship
y(n) = median{x(n + K), x(n + K − 1), . . . , x(n − K)},
(2.5)
where the median of the 2K+1 samples within the curly brackets is the (K + 1)th value
among the samples rearranged in the ascending or descending order of magnitude.
A generalization of the median filter computes the output signal as a linear combination
of samples arranged in ascending or descending order of magnitude. The output of such a
filter may be expressed as
y(n) =
N
−1
∑
hi xi (n),
(2.6)
i=0
where xi (n) is the ith sample in a rearrangement of the set x(n), x(n − 1), . . . , x(n − N + 1)
in the ascending order of magnitude and hi denotes the coefficients of xi (n). This type
Chapter 2. Polynomial Nonlinear Systems: Overview
20
of generalization provides the filter with a combination of properties associated with the
linear filters and the median filters. Other types of order statistic filters include weighted
median filters, finite impulse response (FIR)-median hybrid filters, weighted order statistic
filters, rank selection filters and stack filters [18] [19].
2.1.3
Morphological Filters
In applications such as pattern recognition and robotics, it is necessary to partition an
image into segments on the basis of the geometric properties of the objects depicted in the
image. This can be achieved through the use of mathematical transformations of the input
images using operators known as morphological transforms. Signal processing systems
that employ morphological transforms are known as morphological filters [20]. In addition
to pattern and shape recognition and image decomposition using skeletal representation,
morphological filters have also been used in image compression systems [21].
2.1.4
Neural Networks
Neural networks model nonlinear systems using interconnections of simple nonlinear devices
known as artificial neurons. Artificial neurons are typically multiple-input, single-output
systems with input-output relationship in the form
{ N
}
∑
y(n) = f
wi xi (n) − θ ,
(2.7)
i=1
where wi is the weight associated with the ith input xi (n) to the device and θ is a constant
term that controls the operating point of the nonlinearity f . An artificial neuron with this
characteristic is depicted in Fig. 2.2.
w1
x1(n)
w2
x2(n)
...
f( .)
y(n)
wN
xN(n)
-
Figure 2.2: An artificial neuron.
Many different types of nonlinear functions have been employed, and some of the common ones are shown in Fig. 2.3.
Chapter 2. Polynomial Nonlinear Systems: Overview
(a) Hard limiter
(b) Threshold logic
21
(c) Sigmoid
Figure 2.3: Nonlinearities commonly employed in neural networks.
In typical applications, the weights wi used in the network are selected by training the
neural network on data that are representative of what the network will encounter in normal
applications. The training is accomplished using an adaptation algorithm such as the back
propagation algorithm. Algorithms for neural network design and their applications are
thoroughly described in [22].
The advantage of artificial neural networks is their ability to model most nonlinear systems.
However, in order to perform the modelling accurately, the network might require a very
large number of artificial neurons. Another disadvantage of neural networks is that global
convergence of training algorithms such as back propagation is not guaranteed.
2.1.5
Polynomial Filters
Causal, discrete-time polynomial filters satisfy the input-output relationship of the form
y(n) =
P
∑
fi [x(n), x(n − 1), . . . , x(n − N ), y(n − 1), . . . , y(n − M )] ,
(2.8)
i=0
where the function fi [. . . ] is an ith-order polynomial in the variables within the bracket.
This definition contains the linear filters as special case since the system in (2.8) is linear
if fi [. . . ] = 0 for all i ̸= 1. If the system in (2.8) is stable in the bounded-input boundedoutput sense, it admits a convergent Volterra series expansion of the form
y(n) = h0 +
+
N∑
1 −1
h1 (m1 )x(n − m1 )+
m1 =0
N∑
1 −1 N∑
2 −1
m1 =0 m2 =0
...
N∑
r −1
N∑
1 −1 N∑
2 −1
h2 (m1 , m2 )x(n − m1 )x(n − m2 ) + . . .
m1 =0 m2 =0
hr (m1 , m2 , . . . , mr )x(n − m1 )x(n − m2 ) . . . x(n − mr )
mr =0
+ . . .,
(2.9)
where hr (m1 , m2 , . . . , mr ) denotes the r -order Volterra kernel of the nonlinear system.
The term involving the rth -order kernel looks like an r-dimensional convolution. Consequently, polynomial systems are considered as generalizations of linear systems.
th
Chapter 2. Polynomial Nonlinear Systems: Overview
2.2
22
Volterra Series Expansion
Volterra series expansions form the basis of the theory of polynomial nonlinear systems.
In this section, the Volterra series expansions are introduced and their properties are
discussed.
2.2.1
Continuous-time Systems
A system is defined mathematically as a rule for transformation of an input signal x into
another signal y by means of an operator S so that
y = S{x}.
(2.10)
The input and output signals are usually functions of one or more independent variables
such as position or time. If they are functions only of time t, and the variable t is defined
over a continuous range of values, (2.10) defines a continuous one-dimensional system whose
input-output relationship can be represented as
y(t) = S{x(t)}.
(2.11)
Linear Shift-invariant Systems
A shift-invariant system is characterized by the invariance of its output with respect to
a shift in the independent variable. As an example, a time-invariant system satisfies the
relationship
y(t + τ ) = S{x(t + τ )}
(2.12)
for all values of τ . A linear system is one that satisfies the superposition principle. This
implies that
S{αx1 (t) + βx2 (t)} = αS{x1 (t)} + βS{x2 (t)}
(2.13)
for all arbitrary constants α and β and arbitrary x1 (t) and x2 (t).
The output of a continuous time linear and time-invariant (LTI) system is related to the
input signal through the convolution integral
∫∞
h(τ )x(t − τ )dτ ,
y(t) =
(2.14)
−∞
where h(t) is the impulse response of the system. Equation (2.14) implies that the impulse
response completely characterizes an LTI system. A system is said to be causal if its
output at any given time does not depend on the future values of its input. An LTI system
is causal if and only if
h(t) = 0 for t < 0.
(2.15)
Chapter 2. Polynomial Nonlinear Systems: Overview
23
The unit impulse response signal represents the memory of the LTI system since the
contribution to the current value of the output signal from the value of the input signal T
seconds prior to the present time is determined by h(T ). The output of a memoryless LTI
system is given by
y(t) = c(x(t)),
(2.16)
which is obtained by convolving the input x(t) with cδ(t), where δ(t) represents the Dirac
function and c is a constant that determines the extent of amplification or attenuation of
the input signal at the output of the system.
Volterra Series Expansion for Nonlinear Systems
A nonlinear system without memory can be often described by means of an appropriate
series expansion. A power series expansion such as the Taylor series expansion may be
used to describe the output of such systems as
y(t) =
∞
∑
cp xp (t).
(2.17)
p=0
A nonlinear system with memory can be represented by means of an extension of this
expression. Such an extension, known as the Volterra series expansion [23], relates the
input and output signals of the system as
y(t) = h0 +
+
∫∞
∫∞
−∞
−∞
...
h1 (τ1 )x(t − τ1 )dτ1 +
∫∞
−∞
∫∞ ∫∞
−∞ −∞
h2 (τ1 , τ2 )x(t − τ1 )x(t − τ2 )dτ1 dτ2 + . . .
hp (τ1 , τ2 , . . . , τp )x(t − τ1 )x(t − τ2 ) . . . x(t − τp )dτ1 . . . dτp .
(2.18)
The nonlinear system represented by a Volterra series expansion is completely characterized
by the multidimensional functions hp (t1 , t2 , . . . , tp ), called the Volterra kernels. The zeroth order kernel h0 is a constant. The higher-order kernels can be assumed, without loss
of generality, as symmetric functions of their arguments so that any of the p! possible
permutations of t1 , t2 , . . . , tp leaves hp (t1 , t2 , . . . , tp ) unchanged. This symmetry is the direct
result of the invariance of the products of the delayed input functions {x(t−τi ), i = 1, . . . , p}
with respect to their permutations. The system is causal if and only if
hp (t1 , . . . , tp ) = 0 for any ti < 0 and i = 1, . . . , p.
(2.19)
The lower limits in the integrals in (2.18) are therefore set to zero for causal nonlinear
systems. The upper limits of the integrals in (2.18) given as ∞ indicate that the system
may have infinite memory. If the upper limits are all finite, the system possesses finite
memory. Each integral in (2.18) has the form of a multidimensional convolution. By
defining the pth -order Volterra operator h̄p [x(t)] as
Chapter 2. Polynomial Nonlinear Systems: Overview
∫∞
h̄p [x(t)] =
∫∞
hp (τ1 , τ2 , . . . , τp )x(t − τ1 )x(t − τ2 ) . . . x(t − τp )dτ1 . . . dτp .
...
−∞
24
(2.20)
−∞
Equation (2.18) may be written more compactly as
y(t) = h0 +
∞
∑
h̄p [x(t)].
(2.21)
p=1
A truncated Volterra series expansion is obtained by setting the upper limit of the summation in (2.21) to a finite integer value P . The parameter P is called the order, or the degree,
of the Volterra series expansion. (2.21) reveals the similarity of Volterra series expansions
with the Taylor series expansions. When the input signal is multiplied by a constant factor
c, the output y(t) becomes
y(t) = h0 +
∞
∑
h̄p [cx(t)] = h0 +
p=1
∞
∑
cp h̄p [x(t)].
(2.22)
p=1
which is a power series expansion in c.
Limitations of Volterra Series Expansions
As a consequence of its power series characteristic, there are some limitations associated
with the application of the Volterra series expansion to nonlinear system modeling. These
are:
• Convergence issues: Because of its close relationship with Taylor series expansions,
Volterra series expansions exhibit convergence problems when the nonlinear systems
to be modeled include strong nonlinearities such as saturating elements. Therefore,
the Volterra series approach can be applied with good results only to systems with
mild nonlinearities.
• Nonlinear systems not completely charaterized by the impulse response function: In
contrast to the LTI systems, the impulse response function does not fully characterize
a nonlinear system. Indeed, the response of the nonlinear system to the impulse cδ(t)
is given by
h(t) = h0 + ch1 (t) + c2 h2 (t, t) + . . . + cp hp (t, . . . , t) + . . . .
(2.23)
This indicates that the impulse response is determined only by diagonal values of he
kernel hp , that is for t = t1 = t2 = . . . = tp . Thus, it does not completely specify the
pth -order Volterra kernels of higher order than 1.
• Multiple-valued nonlinearities: Since the Volterra series expansion is a single-valued
representation, it cannot be used to represent multiple-valued nonlinearities.
Chapter 2. Polynomial Nonlinear Systems: Overview
2.2.2
25
Discrete-time Systems
Consider a system for which the input and output values are known only at given time
instants t = nT , where n is an integer variable. By assuming a normalized time interval
T = 1, (2.11) becomes
y(n) = S{x(n)}.
(2.24)
This equation describes a general one-dimensional discrete-time system. Many classifications of discrete-time systems can be made in a manner similar to those of continuous-time
systems.
A discrete time shift-invariant system is characterized by the property that it is invariant
to shifts in the independent variable:
y(n + m) = S{x(n + m)},
(2.25)
where m is an integer shift. A discrete time shift-invariant system is said to be discrete
time-invariant (DTI) when the independent variable is time. A discrete-time is said to be
linear if it obeys the superposition principle:
S[αx1 (n) + βx2 (n)] = αS[x1 (n)] + βS[x2 (n)],
(2.26)
for all arbitrary constants α and β and arbitrary input signals x1 (n) and x2 (n). If a discretetime system is both linear and time-invariant its output can be evaluated by convolving
its input signal with its unit impulse response function hn as
y(n) =
∞
∑
h(m)x(n − n).
(2.27)
m=−∞
A discrete-time LTI system is causal if and only if
h(n) = 0 for n < 0.
(2.28)
DTI Nonlinear Systems
In a manner similar to continuous case, it is possible to describe DTI nonlinear systems
with memory by means of the discrete-time Volterra series expansion
y(n) = h0 +
∞
∑
h̄p [x(n)],
(2.29)
p=1
where y(n) and x(n) are the output and input signals, respectively, and
h̄p [x(n)] =
∞
∑
m1 =−∞
...
∞
∑
mp =−∞
hp (m1 , . . . , mp )x(n − m1 )x(n − m2 ) . . . x(n − mp ),
(2.30)
Chapter 2. Polynomial Nonlinear Systems: Overview
26
where hp (m1 , . . . , mp ) is the pth -order Volterra kernel of the system. If hp (m1 , . . . , mp ) = 0
for all mi < 0, and i = 1, . . . , p, the DTI nonlinear system is causal, and (2.30) becomes
h̄p [x(n)] =
∞
∑
...
m1 =0
∞
∑
hp (m1 , . . . , mp )x(n − m1 )x(n − m2 ) . . . x(n − mp ).
(2.31)
mp =0
The discrete-time Volterra kernels can be interpreted in a manner similar than the continuous-time systems. The constant h0 is an offset term, h1 (m1 ) is the impulse response
of a discrete-time LTI filter, and the pth -order kernel hp (m1 , . . . , mp ) can be considered
as a generalized pth -order impulse response characterizing the nonlinear behavior of the
system. The upper limit in the summations in (2.31) given as infinity indicates that the
discrete system may have infinite memory. The difficulties that arise because of the infinite
summations in (2.31) may be avoided by using recursive polynomial system models. This
approach is similar to the use of infinite impulse response (IIR) filters in applications
involving linear system models.
Usually, the memory required to adequately approximate a nonlinear system is finite.
In such situations, a Volterra series expansion involving only the input signal is sufficient
to model the system. This simpler class of nonlinear system models is derived by limiting
all the summations in (2.31) to finite values. In this case, h1 (m1 ) represents the impulse
response of a finite impulse response (FIR) filter, and the effect of the nonlinearity on
the output depends only on the present and past values of the input signals defined on
the extent of the filter support. If the discrete Volterra series expansion is truncated by
limiting the summation in (2.29) to a finite value P , finite-order expansion is obtained as
y(n) = h0 +
P
∑
h̄p [x(n)],
(2.32)
p=1
where
h̄p [x(n)] =
N
−1
∑
m1 =0
...
N
−1
∑
hp (m1 , . . . , mp )x(n − m1 )x(n − m2 ) . . . x(n − mp ).
(2.33)
mp =0
Equation (2.32) represents a general discrete-time Volterra model of order P and memory
length N . The upper limits in all the summations of (2.33) are made identical only for
convenience. They may be set to arbitrary values to obtain a more general expression. The
simplest polynomial filter of this class is the quadratic filter obtained by choosing P = 2
in (2.32). This type of filter is discussed in the next two subsections.
2.2.3
Algebraic Representation of Quadratic Filters
A causal, discrete and time-invariant quadratic filter with finite support for its kernels is
described by means of the first three terms of the Volterra series expansion as
Chapter 2. Polynomial Nonlinear Systems: Overview
y(n) = h0 +
N
−1
∑
h1 (m1 )x(n − m1 ) +
m1 =0
N
−1 N
−1
∑
∑
27
h2 (m1 , m2 )x(n − m1 )x(n − m2 ),
(2.34)
m1 =0 m2 =0
where h0 represents a constant offset term, {h1 (m1 ); 0 ≤ m1 ≤ N − 1} denotes the coefficients of a linear time-invariant FIR filter and {h2 (m1 , m2 ); 0 ≤ m1 , m2 ≤ N −1} represents
the coefficient of a homogenous quadratic filter. The quadratic filter can be realized as a
parallel combination of these three components as shown in Fig. 2.4.
h0
h1 (m)
x (n)
y (n)
h2 ( m 1 ,m 2 )
Figure 2.4: A quadratic filter realized as a parallel combination of three components.
Matrix-Vector Representation
It is usual to arrange the input values and the filter coefficients of one-dimensional linear
FIR filters into two vectors x(n) and h1 defined as
x(n) =
[
and
h1 =
x(n) x(n − 1) · · · x(n − N + 1)
[
h1 (0) h1 (1) · · · h1 (N − 1)
]T
]T
.
(2.35)
(2.36)
h̄1 [x(n)] is expressed as
h̄1 [x(n)] = xT (n)h1 = hT1 x(n).
(2.37)
In similar way, the input-output relation of a quadratic filter can be written as
h̄2 [x(n)] = xT (n)H2 x(n),
(2.38)
where H2 is an N × N matrix in which the coefficients of the quadratic kernel h2 (m1 , m2 )
are arranged as
Chapter 2. Polynomial Nonlinear Systems: Overview

h2 (0, 0)
h2 (0, 1)
 h2 (1, 0)
h2 (1, 1)
H2 = 
..
..

.
.
h2 (N − 1, 0) h2 (N − 1, 1)
2.2.4

···
h2 (0, N − 1)
···
h2 (1, N − 1) 
.
..
...

.
· · · h2 (N − 1, N − 1)
28
(2.39)
Realization of Quadratic Filters
The homogeneous quadratic term in (2.34) may be equivalently represented using a triangular kernel as
h̄2 [x(n)] =
N
−1
∑
N
−1
∑
h2t (m1 , m2 )x(n − m1 )x(n − m2 ),
(2.40)
m1 =0 m2 =m1
where

m1 = m2
 h2 (m1 , m2 );
h2 (m1 , m2 ) + h2 (m2 , m1 ); m1 < m2
h2t (m1 , m2 ) =
 0;
otherwise.
(2.41)
h̄2 [x(n)] can be implemented by means of a nonlinear combiner, a number of multipliers,
and a summing bus as shown in Fig. 2.5. The nonlinear combiner computes all the necessary
products of the input samples.
2.3
Wiener & Hammerstein Models
One of the most frequently studied classes of polynomial nonlinear models are the so
called block-oriented models, which consist of the interconnection of LTI systems and static
(memoryless) nonlinearities. Within this class, four of the more common model structures
are the:
• Hammerstein model, which consists of the cascade connection of a static nonlinearity
followed by a LTI system (see for instance [24] for a review on identification of
Hammerstein models)
• Wiener model, in which the order of the linear and the nonlinear blocks in the cascade
connection is reversed (see for instance [25] and [26] for different methods for the
identification of Wiener models)
• LNL cascade model, which is represented by a combination of Hammerstein and
Wiener systems. Indeed, it is a cascade of a LTI filter followed by a static nonlinearity, followed by another LTI filter. In [27] a three steps adaptive algorithm for
identification of the LNL model is outlined.
Chapter 2. Polynomial Nonlinear Systems: Overview
1
[x(n)]
1
z
input signal
z
Unit Delay1
x(n)*x(n)
29
Unit Delay2
x(n)*x(n-1)
x(n)*x(n-2)
K
h_2(0,2)
K
h_2(0,0)
Unit Delay3
y_out
Add1
Add3
1
Add4
Add2
z
K
K
h_2(1,1)
Unit Delay4
1
Add5
Unit Delay5
1
h_2(0,1)
z
K
z
K
h_2(1,2)
h_2(2,2)
Figure 2.5: Realization of a quadratic operator using Simulink.
y(n)
Chapter 2. Polynomial Nonlinear Systems: Overview
30
• Feedback block-oriented (FBO) model, which consists of a static nonlinearity in the
feedback path around a LTI system (see for instance [28] for an identification algorithm for this type of model).
These models have been successfully used to represent nonlinear systems in a number of
practical applications in the areas of signal processing [29], communications and control
theory [30]. In the next subsections, the Hammerstein and Wiener models will be discussed.
2.3.1
Hammerstein-Wiener Model
The Hammerstein-Wiener structure models dynamic systems using up to two static nonlinear blocks in series with a linear block. The input signal passes through the first nonlinear
block, a linear block, and a second nonlinear block to produce the output signal, as shown
in Fig. 2.6.
x(n)
Input
Nonlinearity
u(n)
Linear
Block
w(n)
Output
Nonlinearity
y(n)
Figure 2.6: The Hammerstein-Wiener model.
This model structure represents a nonlinear system as a linear system that is modified
by static input and output nonlinearities. Thus, the linear model provides a reference point
for estimating the nonlinear contributions in a system.
The following general equation describes the Hammerstein-Wiener structure:
u(n) = f (x(n))
−1 )
w(n) = B(q
u(n)
−1
A(q )
y(n) = g (w(n))
(2.42)
where x(n) and y(n) are the input and output for the system, respectively. f and g are
nonlinear functions corresponding to the input and output nonlinearities, respectively. For
multiple inputs and multiple outputs, f and g are defined independently for each input
and output channel. u(n) and w(n) are internal variables. u(n) has the same dimension
as x(n), whereas w(n) has the same dimension as y(n). B(q −1 ) and A(q −1 ) in the linear
dynamic block are polynomials in the unit delay operator q −1 . For multiple inputs and
outputs, the linear block is a transfer function matrix.
If only the input nonlinearity is present, the model is called the Hammerstein model.
If only the output nonlinearity is present, the model is called the Wiener model.
2.3.2
Hammerstein Model
The Hammerstein model of a nonlinear system is shown in Fig. 2.7.
Chapter 2. Polynomial Nonlinear Systems: Overview
31
f(u) Taylor series expansion
x(n)
u(n)
H(q -1)
y(n)
LTI filter
Static nonlinearity
Figure 2.7: The Hammerstein model.
It consists of a static nonlinear mapping or gain followed by a linear dynamic block.
The input signal x(n) goes through the nonlinear static mapping block and gives an unobservable intermediate signal u(n). Then u(n) goes through the linear dynamic block to give
the output signal y(n), with f [x(n)] denoting the nonlinear static gain function and the
dynamics being modelled by a linear discrete transfer function H(z). The Hammerstein
model models the nonlinear effects as an input-dependent gain nonlinearity. If the static
nonlinear function is assumed to be approximated by a finite polynomial expansion, the
Hammerstein model can be described by the following equations
y(n) + a1 y(n − 1) + . . . + ak y(n − k) = b1 u(n − 1) + . . . + bk u(n − k)
(2.43)
and
u(n) = γ1 x(n) + γ2 x2 (n) + . . . + γm xm (n).
(2.44)
The intermediate variable u(n) cannot be measured, but it can be eliminated from the
equations. Substituting (2.44) into (2.43) we get
y(n) + a1 y(n − 1) + . . . + ak y(n − k) =
b1 γ1 x(n − 1) + . . . + b1 γm xm (n − 1)
+b2 γ1 x(n − 2) + . . . + b2 γm xm (n − 2)
. (2.45)
..
.
+bk γ1 x(n − k) + . . . + bk γm xm (n − k)
Equation (2.45) can be written in operator form as
( m
)
m
∑
∑
B(q )
γi xi (n) = H(q −1 )
γi xi (n)
y(n) =
−1
A(q ) i=1
i=1
−1
(2.46)
where the polynomials A(q −1 ) and B(q −1 ) are expressed as
A(q −1 ) = 1 + a1 q −1 + . . . + ak q −k
.
B(q −1 ) = b1 q −1 + . . . + bk q −k
(2.47)
The orders of the polynomials are assumed to be the same, but this needs not be the case
Chapter 2. Polynomial Nonlinear Systems: Overview
32
in general.
2.3.3
Wiener Model
The Wiener system has the same two type of blocks as the Hammerstein system, but in
the reverse order, as shown in Fig. 2.8.
g(x) Taylor series expansion
x(n)
w(n)
H(q -1)
LTI filter
y(n)
Static nonlinearity
Figure 2.8: The Wiener model.
In the Wiener structure, the input signal x(n) is transformed by the linear dynamic
block to get w(n), and then w(n) goes through the static nonlinear block to produce y(n).
Mathematically, x(n) and y(n) can be expressed as
w(n) = H(q −1 )x(n)
(2.48)
y(n) = g (w(n)) .
(2.49)
and
If the static nonlinear function g is assumed to be approximated by a finite polynomial
expansion, the Wiener model can be described by the following equation
y(n) =
m
∑
[
]i
γi H(q −1 )x(n) .
(2.50)
i=1
2.4
Summary
Polynomial nonlinear systems were described in this chapter. They are used, in this thesis,
to model different TR-UWB systems. After a brief introduction on the different classes
of nonlinear systems, Volterra series expansions were presented. They constitute the basis
of the theory of polynomial nonlinear systems. Further, the Wiener and Hammerstein
models were revisited. They are part of the most frequently studied classes of polynomial
nonlinear models.
Chapter 3
Channel Characterization
3.1
Introduction
The goal of this chapter is the description and discussion of an appropriate channel model
for mobile radio systems, in general, and UWB radio systems, in particular.
Radio propagation in a mobile radio channel is determined mainly by its multipath
nature. Multiple reflections, and sometimes a LOS component of the transmitted signal
arrive at the receiver via different propagation paths and therefore with different amplitudes
and delay times. As an effect of this, the narrowband received power fluctuates dramatically
when observed as a function of location (or time) and frequency. In the early days of mobile
systems, the communications engineer was mainly interested in the time-variability of
narrowband channels, which were thus studied extensively [31]. By that time, transmission
bandwidths were small, hence flat-fading was a reasonable assumption. As the systems
evolved, demand for higher transmission rates has been increasing, making the channels
time dispersion (which is equivalent to its frequency-selectivity) a major issue.
UWB systems transmit signals with very low power levels. Unavoidably, this constraint
limits the range of UWB wireless links to values typically in few meters, making such
systems suitable to short range indoor applications. This is the reason why most of the
research works on the characterization and modelling of UWB channels have focused on
indoor environments with both LOS and NLOS settings.
The following sections review the propagation mechanisms that have to be characterized
by the channel model. Different UWB channels are introduced with emphasis on the
standardized IEEE 802.15.4a realistic channel model.
3.2
3.2.1
Characterization of the Mobile Radio Channel
Components of a Multipath Channel Model
Three mechanisms should be distinguished in order to describe mathematically a multipath
radio channel, namely:
• path loss
33
Chapter 3. Channel Characterization
34
• shadowing
• multipath interference.
The first two are described by large-scale channel models, which essentially provide
information about the average received power at a certain location. Path loss strictly
describes the dependency of this average power on the distance between transmitter and
receiver, while shadowing accounts for the fluctuations observed at a fixed distance, due to
geometric features of the propagation environment. These fluctuations occur for instance
because of the blocking of relevant propagation paths, e.g., the LOS component, as the
mobile moves around. For mobile radio applications, the description of the effects of
multipath interference is required, since the air-interface has to cope with them. These
effects are often referred to as small-scale fading. Small-scale models are valid within
(small) local areas, where the signal fluctuations due to shadowing and path loss can
be neglected. The dimension of such a local area is therefore limited to approximately
5 · · · 40λ, where λ is the wavelength of the RF carrier [32].
The channel model considered in this section is limited to the description of smallscale effects. A set of average parameters specifies the channels behavior within a local
area. These parameters are the normalized received power, P0 , the Ricean K-factor,
K, and the RMS delay spread, τrms . P0 is the ratio of the received power Prx and the
transmitted power Ptx . It is important noting that each realization obtained from the
model has varying instantaneous parameters denoted {P̂0 , K̂, τ̂rms }, since the model is a
stochastic one. The amount of variation of these parameters from the local-area parameters
depends in particular on the observed bandwidth. When the bandwidth is much greater
than the coherence bandwidth, then the multipath is completely resolved and the channel
parameters vary little, since the individual multipath amplitudes do not change rapidly
within a local area. However, if the system is narrowband, then the multipath is not
resolved, and the path amplitudes at each resolvable delay time-bin (being spaced by
the reciprocal of the bandwidth) vary due to multipath interference. This leads to the
fluctuation of the instantaneous channel parameters within the local area [33].
3.2.2
Channel Impulse Response
In complex low-pass equivalent notation, the channel impulse response (CIR) is given as
h(t) =
∞
∑
αn e−jθn δ(t − τn ),
(3.1)
n=0
where {αn }, {θn } and {τn } are the the propagation paths’ amplitudes, phases, and delays,
respectively. Normally, the delay of the first (shortest) ray is defined as τ0 = 0. Since
τn > 0 for i > 0, the channel impulse response is causal. Note that in a real environment,
the parameters {αn }, {θn } and {τn } are time-variant. For the sake of simplicity, this time
dependency was omitted in (3.1).
Chapter 3. Channel Characterization
3.2.3
35
Channel Parameters
The channel parameters P0 ,K and τrms are defined from the static power delay profile
(PDP), which is a function derived from the CIR. The PDP specifies the ray-power versus
time delay, and is given as
ph (t) =
∞
∑
αn2 δ(t − τn ).
(3.2)
n=0
As the ray phases are dropped in this equation, the channel parameters must be constant within the local area, provided that the propagation paths are fully resolvable. The
normalized received power is defined as the sum of the ray powers
P0 =
∞
∑
αn2 .
(3.3)
n=0
The Ricean K-factor is the ratio of the dominant path’s power to the power in the
scattered paths, and it is defined as
K=
2
αn,max
,
2
P0 − αn,max
(3.4)
2
where αn,max
= max{αn2 }. The K-factor specifies the depth of the fades within a local area.
n
Larger K-factors relate to shallower fades, and smaller K-factors relate to deeper fades.
In the presence of a LOS, the first ray is the dominant one, implying that αn,max = α0 at
τ0 = 0.
The RMS delay spread, defined as the second central moment of the PDP, is written
as
√
(3.5)
τrms = τ¯2 − τ̄ 2 ,
∞
∑
where τ¯m =
τnm αi2 /P0 , for m = {1, 2}. τrms is an important parameter for specifying
n=0
the time extent of the dispersive channel. It also characterizes the frequency-selectivity,
since τrms is related to the average number of fades per bandwidth, and to the average
bandwidth of the fades [32].
3.2.4
Channel Description
The channel is characterized by its delay power spectrum (or average power delay Profile
∞
∑
- APDP), Ph (t) = E{h2 (t)} = E{
αn2 δ(t − τn )}. The expected power of ray at delay τn
n=0
given by
E
{
αn2 |τn
}
{
=
ρ2
Ph (τn )
λ(τn )
n=0
,
n>0
(3.6)
Chapter 3. Channel Characterization
36
where λ(τn ) [rays/s] is the density of arriving rays, being a function of the excess delay
time t. ρ2 is the power of the dominant path, given by
K
.
(3.7)
K +1
In agreement with measurements reported in [34], the shape of the APDP is defined as
shown in Fig. 3.1.
2
ρ2 = αn,max
= P0
Ph(t)
r
2
P
e-g (t-tc)
0
tc
Excess delay [s]
Figure 3.1: Model of the average power delay profile
It is specified by four parameters:
• ρ2 - the normalized power of the direct ray
• Π [1/s] - the normalized power density of the constant-level part
• tc [s] - the duration of the constant level part
• γ [1/s] - the decay exponent of the exponentially decaying part.
Mathematically, the APDP [W/s] can be written as

0
t<0



ρ2 δ(t)
t=0
Ph (t) =
.
Π
0 < t ≤ tc


 Πe−γ(t−tc )
t>t
(3.8)
c
The exponentially decaying APDP is a good approximation for most practical channels,
which is implemented by letting tc = 0. The existence of a LOS ray at t = 0 implies that
the fading envelope distribution is Ricean. Rayleigh fading channels have ρ = 0.
The relationship between channel parameters defined in subsection 3.2.3 and the parameters {ρ, Π, γ, tc } is presented below.
P0 relates to the APDP as
Chapter 3. Channel Characterization
∫∞
37
[
]
1
Ph (t)dt = ρ + Π tc +
.
γ
2
P0 =
(3.9)
0
For the special case of the exponentially decaying APDP (tc = 0), we have
∫∞
Ph (t)dt = ρ2 +
P0 =
Π
.
γ
(3.10)
0
The K-factor characterizes the amplitude distribution of Ricean channels, relating the
power of the direct path to the power of the scattered paths.
K=
ρ2
ρ2
[
].
=
P 0 − ρ2
Π tc + γ1
(3.11)
The RMS delay spread τrms was defined in (3.5) as a function of τ̄ and τ¯2 . τ̄ is expressed
as
∫∞
τ̄ =
[ 2
]
Ph (t)
tc tc
1
τ
dt = Π
+ + 2 ,
P0
2
γ
γ
(3.12)
0
and τ¯2 can be written as
∫∞
τ¯2 =
τ
2 Ph (t)
P0
[
]
t3c t2c 2tc
2
dt = Π
+ + 2 + 3 .
3
γ
γ
γ
(3.13)
0
From (3.12), (3.13) and (3.11), τrms is expressed as
√
1
1 u3
1
u22
τrms =
−
,
γ K + 1 u1 (K + 1)2 u21
where u1 = tc γ + 1, u2 = t2c γ 2 /2 + tc γ + 1 and u3 = t3c γ 3 /3 + t2c γ 2 + 2tc γ + 2.
If tc = 0, we have
√
1 2K + 1
.
τrms =
γ K +1
(3.14)
(3.15)
Finally, from (3.7) and (3.11), we have
Π=
If tc = 0, we have
P0 γ
.
K + 1 u1
(3.16)
Chapter 3. Channel Characterization
38
P0
γ.
(3.17)
K +1
Let us define the maximum excess delay as the delay time where the exponentially decaying
part has decreased by about 43 dB. Such attenuation is reached if the duration of the
exponentially decaying part is exactly texp = 10/γ, leading to the maximum delay spread
tmax = tc + texp = tc + 10/γ. Expressed in terms of channel parameters, we have
Π=
u1 (K + 1)
,
tmax = trms (u + 10) √
u1 u3 (K + 1) − u22
(3.18)
which simplifies for tc = 0 to
K +1
tmax = 10trms √
.
(3.19)
2K + 1
It can be seen that tmax and trms are related by a factor which is function of K, tc and
γ. A typical realization of the CIR (ℜe{h(t)}) and its associated PDP are illustrated in
Fig. 3.2 (a) and (b) respectively, with channel parameters shown in Tab. 3.1.
0.25
0.025
0.2
0.02
0.1
Normalized ray powers
Multipath ray amplitudes
0.15
0.05
0
−0.05
−0.1
−0.15
0.015
0.01
0.005
−0.2
−0.25
0
20
40
60
80
Multipath channel delays [ns]
100
120
0
0
20
40
60
80
Multipath channel delays [ns]
(a)
100
120
(b)
Figure 3.2: A typical channel realization (a) Channel impulse response; (b) Associated power
delay profile.
tc
[ns]
0
P0
1
K
NLOS
0
λ
[rays/ns]
5
τrms
[ns]
10
ρ2
0
γ
[ns−1 ]
0.1
Π
0.1
Table 3.1: Simulation values of the channel parameters.
Chapter 3. Channel Characterization
3.3
3.3.1
39
Standardized Channel Models for UWB Communications
Overview
In UWB systems, the intended radiation can cover a bandwidth of almost 10 GHz. This
large bandwidth can give rise to new effects. For example, only few multipath components
overlap within each resolvable delay bin, so the central limit theorem is no more applicable,
and the amplitude fading statistics are no longer Rayleigh. Also, there can be delay bins
into which no multipath components fall, and thus are empty. It is then necessary to
characterize the likelihood that this happens, and that an empty bin is followed by a full
one. A channel model taking into account those characteristics is the well-known SalehValenzuela (S-V) indoor channel model [35]. It is based on measurements utilizing low
power ultra-short pulses (of width 10 ns and center frequency 1.5 GHz) in a medium-size,
two-storey office building. In the S-V model, multipath components arrive at the receiver in
groups (clusters). Cluster arrivals are Poisson distributed with rate Λ. Within each cluster,
subsequent arrivals are also Poisson distributed with rate λ > Λ. With αl,k denoting the
gain of the kth multipath component of the lth cluster, having phase θl,k , the CIR can be
expressed as
h(t) =
L−1 K−1
∑
∑
αl,k ejθl,k δ(t − Tl − τl,k ).
(3.20)
l=0 k=0
The gains αl,k of the kth component in the lth cluster are independent Rayleigh random
2
2
variables with power E{αl,k
}=E{α0,0
}e−Tl /Γ e−τl,k /γ , where Γ is the cluster decay factor, γ is
the ray decay factor. Tl is the delay of the lth cluster, τl,k is the delay of the kth multipath
component relative to the lth cluster arrival time Tl . The phases θl,k are uniformly distributed, i.e., for a bandpass system, the phase is taken as a uniformly distributed random
variable from the range [0,2π). K is the number of multipath components within a cluster,
while L is the number of clusters.
To come up with a statistical model, channel realizations are identified either in the
frequency domain by frequency sweeping or in the time domain using impulsive signals. In
November 2002, the channel modelling subcommittee of the IEEE 802.15.3a Task Group
recommended a channel which captures the aforementioned works, as well as recent refinements [36]. Because the clustering phenomenon has been experimentally confirmed, the
standardized channel model is basically a modified version of the S-V model. To reach
an analytically manageable channel model, the total number of paths is defined as the
number of multipath arrivals with expected power within 10 dB from that of the strongest
arrival path. The Rayleigh distribution in the S-V channel model is replaced by the lognormal distribution. The phases θl,k are also constrained to take values 0 or π with equal
probability to account for signal inversion due to reflection, yielding a real-valued channel
model. With path gains normalized to have unit energy, a log-normal random variable is
introduced to account for shadowing (large-scale fading).
The Task Group 802.15.4a has the mandate to develop an alternative physical layer
Chapter 3. Channel Characterization
40
for sensor networks and similar devices, working with the IEEE 802.15.4 MAC layer. The
main goals for this new standard are energy-efficient data communications with data rates
between 1kbit/s and several Mbit/s; additionally, the capability for geolocation plays an
important role. The channel modelling subgroup started its activities at the meeting in
September 2003 (Singapore), and submitted its final report in September 2004 (Berlin).
The proposed IEEE 802.15.4a channel model was aimed at modelling attenuation and
delay dispersion [37]. The former includes both shadowing and average path loss, while
the latter describes the PDP and the small-scale fading statistics, with parameters such as
RMS delay spread, number of multipath components carrying x% of the energy, etc... This
channel model can be used for UWB systems spanning frequency range 100-1000 MHz and
2-10 GHz.
3.3.2
The IEEE 802.15.4a Channel Model
The key features of the 802.15.4a Channel Model are
• d−n law for the path loss
• frequency dependence of the path loss
• modified S-V model:
– arrival of paths in clusters
– Poisson distribution for ray arrival times
– possible delay dependence of cluster decay times
– some NLOS environments have first increase, then decrease of power delay profile
• Nakagami-distribution of small-scale fading, with different m-factors for different
components
• block fading: channel stays constant over data burst duration.
Path Loss Model
By definition, the attenuation undergone by an electromagnetic wave in transit between a
transmitter and receiver in a communication system is called path loss or path attenuation.
Path loss may be due to many effects, such as: free space loss, refraction, reflection,
diffraction, clutter, aperture-medium coupling loss and absorption. The path loss in a
narrowband system is conventionally defined as
E{Prx (d, fc )}
,
(3.21)
Ptx
where d is the distance between transmitter and receiver, fc is the center frequency, and
the expectation E{·} is taken over an area that is large enough to allow averaging out of
P L(d) =
Chapter 3. Channel Characterization
41
the shadowing as well as the small-scale fading. A frequency-dependent path loss related
to wideband path loss, as suggested in [38] is defined as
f +∆f
∫ /2
P L(f, d) = E{
(
)2
˜
H f , d df˜},
(3.22)
f −∆f /2
where H(f, d) is the transfer function from transmitter antenna connector to receiver antenna connector, and ∆f is chosen small enough so that diffraction coefficients, dielectric
constants, etc., can be considered constant within that bandwidth. The total path loss is
obtained by integrating over the whole bandwidth of interest. To simplify computations,
it is assumed that the path loss as a function of the distance and frequency can be written
as a product of the terms
P L(f, d) = P L(f )P L(d).
The frequency dependence of the path loss is given as [39]
√
P L(f ) ∝ f −κ .
The distance dependence of the path loss in dB is described by
( )
d
P L(d) = P L0 + 10n log10
,
d0
(3.23)
(3.24)
(3.25)
where the reference distance d0 is set to 1 m, and P L0 is the path loss at the reference
distance. n is the path loss exponent. The path loss exponent also depends on the environment, and on whether a LOS connection exists between the transmitter and receiver
or not. LOS path loss exponents in indoor environments range from 1.0 in a corridor to
about 2 in an office environment. NLOS exponents typically range from 3 to 7. The above
model includes the effects of the transmit and the receive antenna, as it defines the path
loss as the ratio of the received power at the receiver antenna connector, divided by the
transmit power (as seen at the transmitter antenna connector).
Fading Statistics
1. Shadowing
Shadowing, or large-scale fading, is defined as the variation of the local mean around
the path loss. Also this process is fairly similar to the narrowband fading. The path loss
in dB can be written as
( )
d
+ S,
(3.26)
P L(d) = P L0 + 10n log10
d0
where S is a Gaussian-distributed random variable with zero mean and standard deviation
σS .
Chapter 3. Channel Characterization
42
2. Power Delay Profile
The discrete-time impulse response, in complex baseband, of the S-V model was expressed in (3.20). The number of clusters L is an important parameter of the model, and
it is assumed to be Poisson-distributed
(L̄)L e−L̄
pL (L) =
.
L!
The distributions of the cluster arrival times are given by a Poisson process
p(Tl /Tl−1 ) = Λe[−Λ(Tl −Tl−1 )] , l > 0
(3.27)
(3.28)
where Λ is the cluster arrival rate, assumed to be independent of l. The classical S-V
model also uses a Poisson process for the ray arrival times. Due to the divergence in the
fitting for the indoor residential, and indoor and outdoor office environments, ray arrival
times are modelled with mixtures of two Poisson processes as follows
p (τl,k |τl,k−1 ) = βλ1 e[−λ1 (τl,k −τl,k−1 )] + (β − 1)λ2 e[−λ2 (τl,k −τl,k−1 )] , k > 0
(3.29)
where p (τl,k |τl,k−1 ) is the conditional probability density function conditioning consecutive
ray arrival times within a cluster. By definition τl,0 = 0. β is the mixture probability,
while λ1 and λ2 are the ray arrival rates. For some environments, like industrial areas, a
“dense” arrival of multipath components is observed, i.e., each resolvable delay bin contains
significant energy. In that case, the concept of ray arrival rates loses its meaning, and a
realization of the CIR based on a tapped delay line model with regular tap spacings is to
be used.
Next, the cluster powers and cluster shapes are determined. The average power delay
profile (APDP), which is the mean power of the different paths, is exponential within each
cluster
2
E{|αl,k
|} = El
1
e−τl,k /γl ,
γl [(1 − β)λ1 + βλ2 + 1]
(3.30)
where El is the integrated energy of the lth cluster, and γl is the intra-cluster decay time
constant. The cluster decay rates are found to depend linearly on the arrival time of the
cluster
γl ∝ kγ Tl + γ0 ,
(3.31)
where kγ describes the increase of the decay constant with delay and γ0 is a constant. The
mean energy (normalized to γl ), of the lth cluster follows in general an exponential decay
)
(
10 log(El ) = 10 log e−Tl /Γ + Xcl ,
(3.32)
where Xcl is a normally distributed variable with standard deviation σcl . Γ is the cluster
decay factor, and is typically around 10-30 ns, while widely differing values (between 1 and
60 ns) have been reported for the intra-cluster constant γl .
Chapter 3. Channel Characterization
43
For the NLOS case of some environments (office and industrial), the shape of the PDP can
be different, namely
2
E{|α1,k
|} = (1 − χe−τl,k /γr )e(−τl,k /γ1 )
E1
γ1 + γr
.
γ1 γ1 + γr (1 − χ)
(3.33)
Here, the parameter χ describes the attenuation of the first component, the parameter
γr determines how fast the APDP increases to its local maximum, and γ1 determines the
decay at late times.
3. Small-scale fading
The distribution density of the small-scale amplitudes is Nakagami
2 ( m )m 2m−1 − mx2
x
e Ω ,
px (x) =
Γ(m) Ω
(3.34)
where m ≥ 1/2 is the is the Nakagami m-factor, Γ(m) is the gamma function, and Ω is the
mean-square value of the amplitude. A conversion to a Rice distribution is approximately
possible with the conversion equations [40]
(K + 1)2
m=
2K + 1
(3.35)
and
√
K=
(m2 − m)
√
,
m − (m2 − m)
(3.36)
where K and m are the Ricean factor and Nakagami-m factor, respectively. The parameter
Ω corresponds to the mean power, and its delay dependence is thus given by the APDP
above. The m-parameter is modelled as a lognormally distributed random variable, whose
logarithm has a mean µm (τ ) and standard deviation σm . Both of these can have a delay
dependence
µm (τ ) = m0 − km τ
σm (τ ) = m̂0 − k̂m τ.
(3.37)
For the first component of each cluster, the Nakagami factor is modelled differently. It is
assumed to be deterministic and independent of delay
m = m̃0 .
(3.38)
A realization of the CIR of the IEEE 802.15.4a Channel Model for indoor office environment with NLOS scenario is illustrated in Fig. 3.3 (a). It can be observed a certain
delay before receiving the first path with significant energy. Fig. 3.3 (b) depicts 100 CIR
realizations with random first path arrival delays. The channel parameters are shown in
Tab. 3.2.
The APDP, averaged over 100 channel realizations, is shown in Fig. 3.4.
Chapter 3. Channel Characterization
44
1.5
0.6
0.5
1
Multipath ray amplitudes
Multipath ray amplitudes
0.4
0.3
0.2
0.1
0
0.5
0
−0.1
−0.5
−0.2
−0.3
−0.4
0
20
40
60
80
Multipath channel delays [ns]
100
120
140
−1
0
20
40
60
80
100
Multipath channel delays [ns]
120
140
160
(b)
(a)
Figure 3.3: Channel impulse response from the IEEE 802.15.4a Channel Model (a) a single
realization; (b) multiple realizations.
Path loss
n
σS
P L0
Power delay profile
L̄
Λ [ns−1 ]
λ1 , λ2 [rays/ns], β
Γ[ns]
kγ , γ0 [ns]
σcl [dB]
Small-scale fading
m0 [dB], km
m̂0 , k̂m
χ
γr , γ1
Office
LOS
NLOS
1.63
1.9
36.6
3.07
3.9
51.4
5.4
0.016
0.19, 2.97, 0.0184
14.6
0, 6.4
3
3.1
0.19
0.11, 2.09, 0.0096
19.8
0, 11.2
3
0.42, 0
0.31, 0
0.5, 0
0.25, 0
0.78
15.21, 11.84
Table 3.2: Channel parameters for indoor office environment.
The different channel realizations are studied independently in terms of their RMS
delay spreads and the number of significant paths they contain with more than 85% of
total energy, as shown in Fig. 3.5 and Fig. 3.6, respectively. The average RMS delay
Chapter 3. Channel Characterization
45
0
−10
Average power (dB)
−20
−30
−40
−50
−60
0
20
40
60
80
100
Multipath channel delay [ns]
120
140
160
Figure 3.4: Average power delay profile.
spread is τ̄rms = 12 ns. The average number of significant paths capturing more than 85%
of total energy is 40.5.
25
Average RMS delay
RMS delays
RMS delay [ns]
20
15
10
5
0
10
20
30
40
50
60
Channel realizations
70
80
90
100
Figure 3.5: RMS delay spreads for different CIR realizations.
Chapter 3. Channel Characterization
46
80
Average number of significant paths
Significant paths per realization
Number of significant paths capturing > 85% energy
70
60
50
40
30
20
10
0
0
10
20
30
40
50
60
Channel realizations
70
80
90
100
Figure 3.6: Number of significant paths.
3.4
Summary
This chapter presented first a generic mobile radio channel, identifying the components
influencing its description. The CIR and the APDP were expressed analytically. The
channel parameters characterizing this channel model are:
• P0 - the normalized received power
• K - the Ricean K-factor
• τrms - the RMS delay spread
• ρ2 - the normalized power of the direct ray
• Π - the normalized power density of the constant-level part
• tc - the duration of the constant level part
• γ - the decay exponent of the exponentially decaying part.
Further, standardized channels models are introduced, especially the IEEE 802.15.4a channel model. The statistical modelling of path loss, path inter-arrival times, large-scale and
small-scale fadings was presented, as well as the determination of the cluster powers and
cluster shapes. The considered parameters characterizing the IEEE 802.15.4a channel
model are:
• P L0 - the path loss at 1m distance
Chapter 3. Channel Characterization
47
• n - the path loss exponent
• σS - the shadowing standard deviation
• Aant - the normalized power of the direct ray
• L̄ - the mean number of clusters
• Λ - the inter-cluster arrival rate
• λ1 , λ2 , β - the ray arrival rates (mixed Poisson model parameters)
• Γ - the inter-cluster decay constant
• kγ , γ0 - the intra-cluster decay time constant parameters
• σcl - the cluster shadowing variance
• m0 , km - the Nakagami-m factor mean
• m̂0 , k̂m - the Nakagami-m factor variance
• m̃0 - the Nakagami-m factor for strong components
• γ - the decay exponent of the exponentially
• γr , γ1 , χ - parameters for alternative PDP shape
The standardized IEEE 802.15.4a UWB channel model, as it is claimed, match better
the measurements, making it a more realistic channel model. However, the log-normal
distribution and the shadowing factor render this model less tractable for theoretical performance analysis and quantification of the channel-induced diversity. For this reason
the channel model presented in section 3.2 will be used whenever performance analysis is
concerned. For most other simulations in this thesis the standardized model will be used.
Chapter 4
Transmitted Reference Systems
4.1
Introduction
The key motivations for using UWB communications systems are the ability of UWB
signals to finely resolve multipath components, as well as the availability of technology
to implement and generate UWB signals with relatively low complexity. Non-coherent
receivers with no channel estimation have been proposed in [41, 42] to make the UWB
technology attractive for applications where low cost and low power consumption are playing an important role, as in the case of WSNs. Compared with coherent receivers that are
facing great design challenges, non-coherent autocorrelation receivers (AcR) represent an
interesting solution for realizing low complexity UWB systems. They can also detect data
with synchronization inaccuracy as large as one frame duration [43].
In AcR front-ends for UWB communications systems, the received signal consisting of
a train of pulses is delayed and correlated with itself. Hence, each information-conveying
pulse is coupled with an unmodulated pulse. Generally, the first pulse is known as the
“reference pulse” and the second pulse is the “data pulse”. The reference pulse acts as
a template for the AcR. Data can be applied, for instance, by differentially modulating
the polarity of the data pulse with respect to the reference pulse. The delay line in an
AcR front-end, also known as “pulse-pair ” correlator, is matched to the lag between the
reference and the data pulses. The overall system is called a Transmitted-Reference (TR)
AcR [41], [43] - [49].
Generally, in AcRs the reference pulse is corrupted by noise and interference, which
is an inherent disadvantage. If the data rate of such systems is increased, interference
among multiple pulses becomes one of the most fundamental deteriorating effects, due to
multipath propagation. This is known as Inter-Frame Interference (IFI) if interference
between multiple pulses of one data symbol is referred to. It is termed Inter-Symbol
Interference (ISI) if interference among consecutive data symbols is considered.
The TR-UWB systems considered, in this thesis, are the Differential TransmittedReference(DTR) UWB system [50], the Low-Duty-Cycle (LDC) DTR-UWB system [51]
and the Dual-Pulse (DP) TR-UWB system [52]. Discrete-time equivalent system models for these TR-UWB systems are derived in [15], [51] and [52]. In this chapter, their
delay-dependence is studied. The delay-dependent system models accurately relate the
48
Chapter 4. Transmitted Reference Systems
49
transmitted data bits di and the timing offset τ to the test statistics at the decision device
ẑ[i]. The data model can be written as
ẑ[i] = zτ [i] + zτ ν [i].
(4.1)
The decision variable ẑ[i] comprises a data and delay dependent term zτ [i] and a noise
term zτ ν [i].
Using the data model and the concept of minimum distance receiver, the BER of lowcomplexity TR-UWB systems was derived, where ISI is minimal or absent.
4.2
4.2.1
TR-UWB Signal and Systems Models
TR-UWB Transmission Schemes
Narrowband signals (i.e. sinusoidal and quasi-sinusoidal signals) have the unique property
of keeping their sinusoidal shape during forms of signal conversions such as addition, subtraction, differentiation and integration. Shape is defined here as the law of change of
a signal in time. The waveforms of sinusoidal and quasi-sinusoidal signals keep a shape
identical to that of the original function and may differ only in their amplitude and time
shift, or phase. On the contrary, the UWB signal has a nonsinusoidal waveform that can
change shape while processing the above specified transformations.
When a UWB signal is generated and transmitted to the antenna in a form of a current
pulse, the first change of the UWB signal shape occurs during pulse radiation, since the
intensity of the radiated electromagnetic field varies proportionally with the derivative of
the antenna current. The second change occurs during signal propagation through the
multipath channel. The third change of the shape occurs during signal reception, for the
same reason as for signal radiation. Thus, when a Gaussian pulse is transmitted, the second
derivative of the Gaussian pulse is expected at the UWB receiver front-end. Hence, the
standard pulse-shape assumed in theoretical work is the second derivative of a Gaussian
pulse (Gaussian doublet), given by
[
]
2
ω(t) = 1 − 4π(t/tm )2 e−2π(t/tm ) .
(4.2)
This pulse is confined in time to an interval [0,Tω ).
A DTR-UWB system sends data as a stream of very narrow pulses. Let us assume each
data symbol is transmitted via Nf consecutive pulses/frames, where i is the symbol index.
A known random sequence bj ∈ {−1, +1} is differentially modulated on the time-hopped
pulses, where j ∈ {0, 1, ..., Nf − 1} is the pulse index within a symbol. The differentially
modulated pulse-polarities are obtained as ai,j+1 = ai,j bj di and ai+1,0 = ai,Nf −1 bNf −1 di ,
where di is the ith data symbol, di ∈ {−1, +1}. The transmitted signal is written as
s(t) =
f −1
∞ N
∑
∑
i=0 j=0
ai,j ω̃(t − ti,j ),
(4.3)
Chapter 4. Transmitted Reference Systems
50
where ω̃(t) is the transmitted pulse shape and ti,j = (j + iNf )Tf + cj = Cj + iTs . Tf is the
average spacing between two pulses, i.e., the average frame duration, cj is the known TH
sequence and Cj = cj + jTf . {Cj } are the relative pulse timings within a symbol. Ts is the
symbol duration. The time shifts between consecutive pulses are given as Dj = ti,j+1 − ti,j .
Since d2i = 1, every second pulse of a symbol is not modulated by data. Each pulse is reused as a reference and data pulse to increase the power efficiency. Indeed, the differential
TR scheme transmits half the number of pulses that would transmit a conventional TR
scheme. Let si (t) be the ith transmitted symbol waveform containing the Nf pulses, i.e.,
Nf −1
si (t) =
∑
ai,j ω̃(t − iTs − jTf − cj ).
(4.4)
j=0
The transmitted signal is visualized in Fig. 4.1(a).
symbol i+1
symbol i
s(t):
timing:
D0
D1
j=0
...
D2
2
1
3
DNf-1
4
N f-1
D0
0
D1
1
...
D2
2
3
4
t
modulation
dib1
dib0
dib2
dibNf-1
...
di+1b0 di+1b1 di+1b2 ...
(a)
r(t):
l=0
1
2
3
...
4
t
r(t+D2 ):
l'=0'
1'
2'
3'
...
4'
t
TI
Integration interval of correlator 3
(b)
Figure 4.1: Diagram of the transmitted and received signals in DTR-UWB.
The burst-oriented LDC DTR-UWB system is fundamentally similar to a conventional
DTR-UWB system. The difference is that the random sequence bj is of length Nf − 1.
This is because the polarity of the first pulse within a symbol is no more dictated by
the polarity of the last pulse of the previous symbol. Hence, the differentially modulated
pulse-polarities are obtained as ai,j+1 = ai,j bj di , j ∈ {0, 1, ..., Nf −2} and ai+1,0 ∈ {−1, +1}.
The LDC DTR-UWB system, as conceived in [51], was designed for efficient narrowband
Chapter 4. Transmitted Reference Systems
51
interference (NBI) mitigation and robustness in multiuser scenarios. Pulses are transmitted
in short bursts, separated by comparably longer idle periods, making up a “low-duty-cycle”
as shown in Fig. 4.2.
symbol i, duty cycle
timing:
D0
D1
j=0
modulation
dib0
...
D2
2
1
dib1
symbol i+1
symbol i, idle
D0
Nf
3
di+1 b0
...
D2
1
0
...
dib2
D1
2
di+1 b1
3
di+1 b2
t
...
Figure 4.2: Diagram of transmitted signals for the burst oriented LDC DTR-UWB systems.
For a DP TR-UWB system, the unmodulated data pulse is followed at a delay D by
the data-modulated pulse, as illustrated in Fig. 4.3. The timing of pulse j = 0 is c0 = 0
and for j = 1 it is c1 = D.
1
2
0
D
r(t)
1'
Tmax
t
2'
r(t-D)
t
TI
Figure 4.3: Received signal for a DP TR-UWB system.
4.2.2
TR-UWB Receiver Front-End
∑
Let g̃(t) = h(t) ∗ ω̃(t) = ∞
n=0 αn ω̃(t − τn ) be the overall channel capturing both pulse
shaping by linear system components and multipath effects. h(t) is a realization of a
simulated CIR, generated according to the channel models presented in sections 3.2 and
3.3. It can also be a measured channel impulse response. The received noisy signal is
expressed as
Chapter 4. Transmitted Reference Systems
52
r(t)
Filter
frx(t)
y0[i]
xb0
yj[i]
xbj
yNf-1[i]
xbNf-1
I&D
TI
Delay
D0
Delay
c0+D0
I&D
z[i]
TI
Delay
Dj
Delay
jTf+cj+Dj
I&D
TI
Delay
DSP:
MUD,
sync.,
etc.
DNf-1
Delay
(Nf-1)Tf+cNf-1+DNf-1
Symb. clk.
1/Ts
Figure 4.4: Receiver front-end for a DTR-UWB system.
r(t) =
f −1
∞ N
∑
∑
ai,j g̃(t − iTs − jTf − cj − τ ) + n(t),
(4.5)
i=0 j=0
where Ts is the symbol period, τ ∈ [0, Ts ) is the timing delay and n(t) represents noise.
A conventional DTR-UWB receiver front-end is shown in Fig. 4.4. It has at its input
a bank of pulse-pair correlators, whose lags are matched to the time shifts Dj . While a
conventional DTR-UWB receiver has Ncr = Nf pulse-pair correlators, a LDC DTR-UWB
receiver has Ncr = Nf − 1 pulse-pair correlators. In a DP TR-UWB receiver, a single
correlator (Ncr = 1) is used for the detection of the pulse-pairs. Integration is performed
over a time interval TI , which can be selected to optimize the signal-to-noise ratio (SNR)
of the AcR outputs [49, 53, 54].
The outputs of the Integrate-and-Dump (I&D) blocks, after being sampled at the symbol rate, are coherently combined by removing the chip-level modulation bj . We obtain
Chapter 4. Transmitted Reference Systems
53
Nf −1
z[i] =
∑
yj [i] bj ,
(4.6)
j=0
where
∫
ti,j +TI
yj [i] =
r̂(t)r̂(t + Dj )dt,
(4.7)
ti,j
and
r̂(t) = r(t) ∗ frx (t) = rτ (t) + ν(t),
(4.8)
where rτ (t) is the delay-dependent noise-free received signal. frx (t) denotes a front-end
analog filter in the receiver. The filtered noise process ν(t) = n(t) ∗ frx (t) is characterized
by its autocorrelation function
N0
(4.9)
frx (κ) ∗ frx (−κ),
2
where N0 /2 is the double-sided power spectral density of the receiver front-end noise.
Rν (κ) = E{ν(t)ν(t + κ)} =
Using (4.7) and (4.8), we obtain
∫ t +T
∫ t +T
yj [i] = ti,ji,j I rτ (t)rτ (t + Dj )dt+ ti,ji,j I ν(t)rτ (t + Dj )dt
∫ t +T
∫ t +T
+ ti,ji,j I rτ (t)ν(t + Dj )dt + ti,ji,j I ν(t)ν(t + Dj )dt
= yτ j [i] + ντ 1j [i] + ντ 2j [i] + ν3j [i].
4.3
(4.10)
Equivalent TR-UWB Systems Models
Among all the TR-UWB systems presented in section 4.2, only the DTR-UWB system is
prone to ISI in the case the average frame duration Tf is less than the maximum delay
spread of the multipath channel Tm . In this thesis, the LDC DTR-UWB and DP TR-UWB
are chosen in such a way that they mitigate ISI effects.
4.3.1
System with ISI: Volterra Equivalent Model
In [15] a discrete-time Volterra equivalent system model is derived for DTR-UWB systems
in a multipath channel, where both IFI and ISI are present. In this subsection, a review
of the system modelling is presented, introducing the delay parameter.
Equivalent System Model without Noise
The noise-free delay-dependent received signal can be written as
Chapter 4. Transmitted Reference Systems
p −1
∞ N
∑
∑
rτ (t) =
ai,l g(t − ti,l − τ ),
54
(4.11)
i=−∞ l=0
where g(t) = g̃(t) ∗ frx (t) is the filtered overall channel. yτ j [i] is expressed as
N∑
f −1
∞
∑
yτ j [i] =
ai+n,l g(t − ti+n,l − τ )
n=−∞ l=0
N∑
f −1
∞
∑
·
=
ai+n′ ,l′ g(t
n′ =−∞ l′ =0
N∑
f −1 ∑
f −1
1 N∑
1
∑
n=−η l=0 n′ =−η l′ =0
− ti+n′ ,l′ − τ )dt
(4.12)
ai+n,l ai+n′ ,l′
·Ig (ti,j − ti+n,l − τ, ti,j − ti+n,l − τ + TI ; Dj + ti+n,l + cl − cl′ ),
where Ig (t1 , t2 ; ζ) =
∫t2
g(t)g(t + ζ)dt. Equation (4.12) accounts for the interference from
t1
all pulses of η previous and one consecutive symbol. η = ⌈Tm /Ts ⌉ is the memory length,
where ⌈·⌉ represents the ceiling function defined by ⌈x⌉ = min {n ∈ Z|x ≤ n}. This IFI
was depicted in Fig. 4.1(b). Equation (4.12) can be written as
yτ j [i] = aT [i]Yτ j a[i],
(4.13)
where a[i] = [ai,0 , ai,1 , . . . , ai,Nf −1 , ai,Nf ]T represents the polarities of all pulses. The elements of the Np (η + 2) × Np (η + 2) matrix Yτ j are given by
[Yτ j ](n+η)Np +l+1,(n′ +η)Np +l′ +1
= Ig (cj − nTs − cl − τ, cj − nTs − cl − τ + TI ; Dj + (n − n′ )Ts + cl − cl′ )
(4.14)
∀ n,n′ ∈ {−η, −η + 1, . . . , 1} and l,l′ ∈ {0, 1, . . . , Np − 1}.
N∑
f −1
The noise-free signal term zτ [i] =
yτ j [i]bj can be modelled as
j=0
Nf −1
zτ [i] =
∑
bj aT [i]Yτ j a[i] =aT [i]Zτ a[i],
(4.15)
j=0
where Zτ =
N∑
f −1
bj Yτ j . It is demonstrated in [15] that (4.15) constitutes a discrete-time
j=0
Volterra model of order two, having the transmitted symbols {di } at its input. The data
and code dependent vector a[i] is expressed as
a[i] = ai−η,0 B(p + Pd[i]),
(4.16)
Chapter 4. Transmitted Reference Systems
55
where d[i] = [di−η , di−η+1 , . . . , di , di+1 ]T , B = diag[1, b0 , b0 b1 , . . .], [B]k,k =
k−2
∏
bµ mod Nf ,
µ=0
k = 1, 2, . . . , Nf . P = Iη+2 ⊗ s is a block-diagonal matrix of dimension Nf (η + 2) × (η + 2),
with the length Nf selection vectors s = [0, 1, 0, 1, . . . , 1]T at its main diagonal. ⊗ denotes
the Kronecker product. The length Nf (η + 2) × (η + 2) vector p = iη+2 ⊗ (iNf − s) contain
ones at the elements of a[i] that are not data dependent. IN and iN denote an identity
matrix of size N × N and an all-ones vector of length N , respectively. Using (4.15) and
(4.16), a second-order Volterra model is obtained
zτ [i] = (p + Pd[i])T BZτ B(p + Pd[i])
(4.17)
= hτ 0 +
hTτ1 d[i]
T
+ d [i]Hτ 2 d[i],
where hτ 0 = pT BZτ Bp, hτ 1 = PT B(Zτ + ZTτ )Bp and Hτ 2 = PT BZτ BP. The bias term
hτ 0 is due to the interference among the (channel’s response to) fixed reference pulses
plus interference between equally modulated data pulses. The linear model coefficient hτ 1
relates to the interference between unmodulated reference and modulated data pulses. The
nonlinear product terms Hτ 2 are attributed to interference between differently modulated
pulses of any two different data symbols. Thus, nonlinear product terms appear only if
significant ISI is present.
Multiuser Equivalent System Model
In a multiuser (MU) equivalent system model, nonlinear terms also exist if data symbols
of various users overlap [15]. An equivalent system model can be derived for the MU case
by including the interfering users’ pulse polarities in the a[i]i -vectors and extending Yj matrices with all channels’ auto- and cross-correlation values for any pulse-pairs. The MU
Volterra model comprises d[i]-vectors extended by the other users’ data and correspondingly extended coefficient vectors and matrices.
Equivalent Noise Model
The three noise terms in (4.10) are considered for the noise analysis. We encounter two
“linear ” terms ντ 1j [i] and ντ 2j [i] which depend on the received signal, and a “product”
noise term ν3j [i]. Following [15] and [55], all three noise components at the output of the
correlators are assumed to be zero-mean Gaussian noise variables. This requires that the
noise process ν(t) is zero mean, that Rν (Dj ) ≈ 0, ∀j, and that the integration-time by
noise-bandwidth product TI W ≫ 1.
The noise component of the decision variable z[i] is defined by zτ ν [i] = ẑ[i]−zτ [i]. Using
(4.6) and (4.10), we obtain
Chapter 4. Transmitted Reference Systems
E{zτ ν [i]zτ ν [i + ι]} =
N∑
f −1 N∑
f −1
j=0
56
bj bj ′ [E{ντ 1j [i]ντ 1j ′ [i + ι]}
j=0
+E{ντ 1j [i]ντ 2j ′ [i + ι]} + E{ντ 2j [i]ντ 1j ′ [i + ι]}
(4.18)
+E{ντ 2j [i]ντ 2j ′ [i + ι]} + E{ν3j [i]ν3j ′ [i + ι]}],
since ν3j [i] is uncorrelated from ντ 1j ′ [i] and ντ 2j ′ [i], ∀ j, j ′ .
The delay and data-dependent covariance of the linear noise terms is expressed by
quadratic forms
N0 T
(νm νm′ )
a[i],
a [i]Vτ ι,j,j
(4.19)
E{ντ mj [i]ντ m′ j ′ [i + ι]} =
′
2
for m, m′ ∈ {1, 2}. It is zero for |ι| > 1 [15]. We derived the expression of the matrix
(νm νm′ )
Vτ ι,j,j
as
′
(ν ν
)
m m′
[Vτ ι,j,j
](n+η)Nf +l+1,(n′ +η)Nf +l′ +1
′
=
·
2
N0
ti,j∫+TI
g(t + (−1)m+1 Dj − ti+n,l − τ )
(4.20)
ti,j
ti+ι,j ′ −t
∫
ti+ι,j ′ −t
′
g(t + κ + (−1)m +1 Dj ′ − ti+n′ ,l′ − τ )Rn (κ)dκdt.
These expressions can be evaluated numerically if the channel response g(t) is known. For
the noise product we obtain
E{ν3j [i]ν3j ′ [i + ι]} ≈ δ[ι]δ[j − j ′ ]TI
≈ δ[ι]δ[j −
∫∞
−∞
N2
j ′ ]TI 40
Rn2 (κ)dκ
∫∞
−∞
(4.21)
|Frx (f )|4 df,
where Frx (f ) is the Fourier transform of the receiver front-end.
Finally, the noise model for the decision variable can be expressed by
E{zτ ν [i]zτ ν [i + ι]} =
where Wτ ι =
N2 = Nf TI
∫∞
−∞
∑ ∑
m,m′
j,j ′
(z z
)
N0 T
N2
a [i]Wτ ι a[i] + δ[ι] 0 N2 ,
2
4
(4.22)
m m′
bj bj ′ Vτ ι,j,j
, ι = 0 (noise variance) or ι = 1 (noise covariance), and
′
|Frx (f )|4 df .
Chapter 4. Transmitted Reference Systems
57
Model Illustration
The equivalent system model is depicted in Fig. 4.5, where the upper branch with the
“Volterra System I” represents the data model, while the lower branch introduces the data
dependency of the noise variance, described by the “Volterra System II”.
delay
data d[i]
Volterra
System I
ẑ [i]
z [i]
z [i]
Channel h(t)
System parameters
Volterra
System II
sqrt
N0 / 2
(N0 )2N2 / 4
Unit variance
Gaussian noise
Noise density N0
Figure 4.5: 2nd order Volterra equivalent system model of the DTR-UWB system.
Model Validation
We select the pulse ω(t) as the second derivative of the Gaussian pulse with unit energy
and duration Tω = 0.7 ns. The pulse parameter τm = 0.2877. The pulse is sampled using
the sampling period Tsamp = 0.05 ns. The precise position of the pulses in each channel
impulse response has been oversampled in order to represent better the unquantized ray
arrival times. The pulse position resolution is Tres = 0.005 ns. The DTR-UWB system’s
key parameters are Nf = 10, Tf = 10 ns, Ts = 100 ns, η = 1 and TI = 20 ns. Random time
hopping (TH) codes were used. All the system parameters are summarized in Tab. 4.1
DTR-UWB
system parameters
Tω
[ns]
0.7
τm
[ns]
0.2877
Tsamp
[ns]
0.05
Tf
[ns]
10
TI
[ns]
20
Ts
[ns]
100
Nf
Ncr
10
10
Table 4.1: DTR-UWB system and simulation parameters
A NLOS channel has been simulated with an exponentially decaying delay profile at
RMS delay spread of τrms = 10 ns using the channel model presented in section 3.2.
The channel simulator generates an unquantized vector of ray arrival times according to
a Poisson process with a mean ray arrival rate of λ = 5 ns−1 and a vector of Rayleigh
distributed ray magnitudes with random polarity. The SNR is defined as the ratio between
the symbol energy and the noise variance, i.e., SNR= Es /σ 2 = Nf Eb /σ 2 , where Eb =
+∞
∫ 2
g (t)dt and σ 2 is the noise variance. Es is the symbol energy.
−∞
Chapter 4. Transmitted Reference Systems
58
The validation of the delay-dependent equivalent system and noise models is shown in
Fig. 4.6 and Fig. 4.7. The data dependence of the models is first studied, followed by the
delay dependence.
(a)
15
10
zτ[i]
5
0
−5
−10
−15
10
Volterra model
conv. sim. w/o noise
data sequence
15
20
25
30
35
40
(b)
1.2
E{zτ ν[i]zτ ν[i+ι]}
1
0.8
noise var.; Volterra m.
0.6
noise var.; conv. sim.
0.4
noise cov.; Volterra m.
0.2
0
−0.2
10
15
20
25
Symbol index i
30
35
40
Figure 4.6: Simulated receiver output compared with the output of the second-order Volterra
equivalent system. (a) Noise-free case (τ = 0). (b) Data dependence of the noise (co)-variance at
SNR= 30 dB (τ = 0).
We compare the noise-free receiver output, simulated by processing the received signal
according to the elements of the receiver front-end shown in Fig. 4.4 (conventional simulations) with the output of the second-order Volterra equivalent system. In Fig. 4.6(a) the
sampled output of the conventional simulation is marked by “◦”. Plus characters “+” show
the output of the equivalent system model at these sampling instants. Perfect agreement
between these two demonstrates the suitability of the proposed equivalent system model.
Fig. 4.6(b) compares the variance of the additive noise var{zτ ν [i]} at the same set of observed decision variables, for a conventional simulation “◦” and for the noise model in
(4.22) “+”. In the conventional simulation runs, one of 1000 independent noise processes
has been added to the receiver front-end at a time, while using a constant data sequence
and channel realization. The simulation results show a significant data-dependence of the
noise variance, whereas the noise covariance, illustrated by asteriks “∗” is insignificant.
After the data dependence, the delay-dependence is illustrated. Fig. 4.7 shows the
simulated receiver output compared with the output of the second-order Volterra equivalent
system model zτ [i], with and without delay. It can be seen that zτ [i] decreases with increase
in time delay, as expected. In fact, zτ [i] decays exponentially with respect to τ as is shown in
Fig. 4.8, where E{zτ [i]/Es } is obtained by averaging zτ [i]/Es over 1000 channel realizations,
for di = 1 and a constant data sequence. On the other hand, under the same simulation
Chapter 4. Transmitted Reference Systems
59
(a)
τ = 0 ns
10
zτ[i]
5
0
−5
−10
10
15
20
25
τ = 15 ns
30
35
(b)
10
Conv. sim. w/o noise
Full Volterra model
5
zτ[i]
40
0
−5
−10
10
15
20
25
Symbol index [i]
30
35
40
Figure 4.7: Simulated receiver output compared with the output of the second-order Volterra
equivalent system. (a) Noise-free case without delay. (b) Noise-free case with delay τ = 15 ns.
conditions, it can be observed that E{zτ ν [i]} and E{zτ2ν [i]} do not exhibit such significant
delay-dependence, as shown in Fig. 4.9.
0.9
0.8
0.7
E{zτ[i]/Es}
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
delay [ns]
30
35
40
45
50
Figure 4.8: Normalized average delay-dependent noise-free decision variable
Chapter 4. Transmitted Reference Systems
60
(a)
1
E{zτ ν[i]}
0.5
0
−0.5
−1
0
2
4
6
8
12
14
16
18
20
12
14
16
18
20
(b)
1.5
E{zτ2 ν[i] }
10
delay [ns]
1
0.5
0
0
2
4
6
8
10
delay [ns]
Figure 4.9: (a) Average output noise versus delay (SNR= 30 dB). (b) Variance of output noise
versus delay (SNR= 30 dB).
4.3.2
Systems without ISI
In this subsection, low complexity TR-UWB systems without ISI are addressed.
DTR-UWB Systems
For DTR-UWB systems with no ISI the memory length η = 0. The noise-free delay
dependent equivalent system model zτ [i] and the variance of the noise variable zτ ν [i] for
ISI-free DTR-UWB systems are expressed as
and
zτ [i] = hτ 0 + hτ 1 di ,
(4.23)
N0
N02
var{zτ ν [i]} =
(hτ ν,0 + hτ ν,1 di ) +
Nf TI W,
2
4
(4.24)
where
hτ 0 = pT BZτ Bp + trace(qT BZτ Bq)
(4.25)
T
hτ 1 = p B(Zτ +
ZT
τ )Bq.
Let b be a vector containing the code sequence {bj }. In the diagonal matrix B, the spreading code b is represented in the form of an accumulated product, B=diag[1 b0 b0 b1 b0 b1 b2 . . .].
q is a vector of length Nf , which in this case is identical to the selection vector s =
∑ cr −1
bj Yτ j ,
[0, 1, 0, 1, . . . , 1]T . The length Nf vector p is defined as p = iNf − s. Zτ = N
j=0
Chapter 4. Transmitted Reference Systems
61
where Yτ j is a Ncr × Ncr matrix containing the autocorrelation integrals Ig (t1 , t2 ; ζ). Yτ j
is expressed as
[Yτ j ]l+1,l′ +1 = Ig (ti,j − ti,l − τ, ti,j − ti,l + TI − τ ; Dj + ti,l − ti,l′ ),
(4.26)
∫
∞
∀ l, l′ ∈ {0, 1, . . . , Ncr − 1}. W = −∞ |Frx (f )|4 df is the equivalent input bandwidth of the
receiver, where Frx (f ) is the Fourier transform of the front-end filter frx (t).
The coefficients of the noise model, hτ ν,0 and hτ ν,1 (4.24), are obtained in a similar way
like hτ 0 and hτ 1 in (4.25), replacing the matrix Zτ by a matrix
Wτ =
∑∑
(m,m′ )
bj bj ′ Vτ,j,j ′ ,
(4.27)
m,m′ j,j ′
where
(m,m)
[Vτ,j,j ′ ]l+1,l′ +1 =
2
N0
·
ti,j∫+TI
g(t + (−1)(m+1) Dj − ti,l − τ )
ti,j
ti+,j ′ −t
∫
(4.28)
(m+1)
g(t + κ + (−1)
ti,j ′ −t
Dj ′ − ti,l′ − τ )Rν (κ)dκdt,
for m, m′ ∈ {1, 2}.
DP TR-UWB Sytems
The equivalent system and noise models for DP TR-UWB systems have been derived
in [52]. The expressions of the noise-free delay dependent equivalent system model zτ [i]
and variance of the noise variable zτ ν [i] for DP TR-UWB systems are the same as in (4.23)
and (4.24), respectively. For these systems the coefficients hτ 0 , hτ 1 , are given as
hτ 0 = Ig (−τ, TI − τ ; D) + Ig (−D − τ, TI − D − τ ; D)
hτ 1 = Ig (−τ, TI − τ ; 0) + Ig (−D − τ, TI − D − τ ; 2D).
(4.29)
For D ≥ TI , hτ ν,0 and hτ ν,1 are expressed as
hτ ν,0 ≈ 2Ig (−τ, TI − τ ; 0) = 2hτ 1
hτ ν,1 ≈ 2Ig (−τ, TI − τ ; D) = 2hτ 0 .
4.4
(4.30)
Bit Error Rate Performance Analysis
In this section, the receiver statistics are analyzed from two perspectives; both perspectives
allow for BER performance analysis to be performed. The first (widely accepted) perspective is to use a Gaussian approximation on the noise terms in the receiver statistics and
determine their respective variances. The second approach is to use the equivalent system
models of the receiver. The average BER is verified through simulation for different SNR
Chapter 4. Transmitted Reference Systems
r(t)
aij
aij+1
aij+2
62
aij+3
Dj
...
...
aij
r(t-D j )
aij+1
aij+2
aij+3
...
...
aij+1 a ij
r(t) r(t-D j )
...
TI
Figure 4.10: Diagram of the received signals in DTR-UWB (without IFI).
values. It is also compared to the average BER derived using the equivalent system model.
Since the derivation of an analytic expression for the average BER would need to solve a
complicated integral, we adopt another approach for computing the average BER, namely
using the concept of minimum distance receiver [56].
4.4.1
Gaussian Approximations approach
As we presented in [57], the goal is to derive the receiver decision statistic in the absence
of any kind of interference. Fig. 4.1(b) shows the correlator output for a duration of TI in
the presence of IFI and Fig. 4.10 shows the correlator output in the absence of noise and
IFI. Equation (4.7) can be rewritten as
∫
ti,j +TI
yj [i] =
[ai,j g(t − ti,j ) + ν(t)][ai,j+1 g(t − ti,j+1 ) + ν(t + Dj )]dt.
(4.31)
ti,j
Since ti,j+1 = Dj + ti,j , (4.31) can be expressed as
∫
ti,j +TI
yj [i] =
[ai,j g(t − ti,j ) + ν(t)][ai,j+1 g(t − ti,j − Dj ) + ν(t + Dj )]dt.
(4.32)
ti,j
Making the appropriate change of variable in (4.32) we finally obtain
yj [i] =
=
∫ TI
0
∫ TI
0
∫ TI
0
[ai,j g(t) + ν(t + ti,j )][ai,j+1 g(t − Dj ) + ν(t + ti,j + Dj )]dt
ai,j+1 ai,j g(t)g(t − Dj )dt+
∫ TI
0
ai,j+1 g(t − Dj )ν(t + ti,j )dt+
= y1j [i] + y2j [i] + y3j [i] + y4j [i].
ai,j g(t)ν(t + ti,j + Dj )dt+
∫ TI
0
(4.33)
ν(t + ti,j )ν(t + ti,j + Dj )dt.
Chapter 4. Transmitted Reference Systems
63
Equation (4.6) can be rewritten as
z[i] =
N∑
f −1
bj y1j +
j=0
N∑
f −1
+
N∑
f −1
bj y2j +
N∑
f −1
j=0
bj y3j
j=0
(4.34)
bj y4j
j=0
= z1 [i] + z2 [i] + z3 [i] + z4 [i].
In (4.34) z1 [i] is the signal term, while z2 [i], z3 [i] and z4 [i] are all noise terms. Let Tm be the
channel maximum excess delay. Since there is no IFI, Dj > Tm ≫ Tω , then {ν(t + ti,j } and
{ν(t + ti,j + Dj } can be regarded as uncorrelated and thus independent zero mean Gaussian random variables. The validity of this assumption is illustrated by Fig. 4.11, where
the noise autocorrelation function is simulated for time delays in the interval [−Tm , Tm ].
Hence z4 [i] can be approximated as a zero mean random variable. g(t) is zero mean for
t > Tω [50], thus z2 [i] and z3 [i] are independent zero mean Gaussian random variables.
Finally, all the three noise terms in (4.34) can be approximated as independent zero-mean
random variables whose variances, given by V2 , V3 and V4 , are equal to the second moments
of z2 [i], z3 [i] and z4 [i], respectively.
Normalized noise autocorrelation function
1
0.8
0.6
0.4
0.2
0
−0.2
−100
−50
0
Time delay [ns]
50
100
Figure 4.11: Simulated noise autocorrelation function.
Analysis of z1 [i]
In subsection 4.2.1 the differentially modulated pulse-polarities were expressed as ai,j+1 =
ai,j bj di , which implies that ai,j+1 ai,j = bj di and
Chapter 4. Transmitted Reference Systems
z1 [i] = di
N∑
f −1 ∫
j=0
TI
0
g(t − Dj )g(t)dt = di z1 .
64
(4.35)
z1 is actually a random variable, considering several realizations of g(t). Since the channel
is quasi-static, g(t−Dj ) is the exact replica of g(t) delayed by Dj . Thus, z1 can be expressed
as
∫ TI
z1 = Nf
g 2 (t)dt.
(4.36)
0
Analysis of z2 [i]
z2 [i] =
N∑
f −1 ∫
j=0
Let z2 [i] =
N∑
f −1
I2j , then V2 =
j=0
∫
2
E(I2j
)
N∑
f −1
ai,j bj g(t)ν(t + ti,j + Dj )dt.
(4.37)
2
E(I2j
), where
j=0
∫
TI
TI
0
TI
E{ai,j bj g(t)ν(t + ti,j + Dj )ai,j bj g(ϱ)ν(ϱ + ti,j + Dj )}dtdϱ.
=
(4.38)
0
0
Considering that g(t) and ν(t + ti,j + Dj ) are uncorrelated, we can split up the expectation term in two separate terms.
∫
2
E(I2j
)
TI
∫
TI
=
E{g(t)g(ϱ)}E{ν(t + ti,j + Dj )ν(ϱ + ti,j + Dj )}dtdϱ.
0
(4.39)
0
Equation (4.39) can be rewritten as
∫
2
E(I2j
)
TI
∫
TI −t
=
0
−t
E{g(t)g(t + κ)}E{ν(t + ti,j + Dj )ν(t + κ + ti,j + Dj )}dtdκ,
(4.40)
where ϱ = t + κ. E{g(t)g(t + κ)} ≈ ρ2 ω(t)ω(t + κ) + P̃h (t)ϕω (κ) [50], with P̃h (t) = Πe−γt
∫ Tω
and ϕω (κ) = 0 ω(µ)ω(µ + κ)dµ is the autocorrelation of the received prototype pulse
ω(t). We define ϕω (0) = 1, i.e., the prototype pulse is assumed to have unit energy.
E{ν(t + ti,j + Dj )ν(t + κ + ti,j + Dj )} = Rν (κ) is the noise autocorrelation function. If
we assume that the received signal is passed through an ideal bandpass filter (BPF) with
one-side bandwidth W and center frequency f0 , where W equals the bandwidth of the
received signal, then
Rν (κ) = W N0
sin(πW κ)
cos(2πf0 κ),
πW κ
(4.41)
Chapter 4. Transmitted Reference Systems
65
where N0 is the noise power. The normalized prototype pulse and noise autocorrelation
functions are shown in Fig. 4.12 and Fig. 4.13, respectively.
Normalized autocorrelation function φω
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−0.8
−0.6
−0.4
−0.2
0
0.2
Time delay [ns]
0.4
0.6
0.8
Figure 4.12: Autocorrelation function of the received prototype pulse.
Normalized noise autocorrelation function Rν
1
0.8
0.6
0.4
0.2
0
−0.2
−30
−20
−10
0
Time delay [ns]
10
20
30
Figure 4.13: Noise autocorrelation function Rν (κ).
Since ρ2 = 0 (NLOS channel),
∫
2
E(I2j ) =
0
TI
−γt
Πe
∫
Tω
dt
−Tω
ϕω (κ)Rν (κ)dκ.
(4.42)
Chapter 4. Transmitted Reference Systems
66
Finally,
V2 =
=
N∑
f −1
Π
γ
(
1 − e−γTI
j=0
ΠNf (1−e−γTI )
γ
)∫ Tω
−Tω
∫ Tω
−Tω
ϕω (κ)Rν (κ)dκ
(4.43)
ϕω (κ)Rν (κ)dκ.
Analysis of z3 [i]
Since bj ai,j+1 = ai,j di ,
z3 [i] = di
N∑
f −1 ∫
TI
0
j=0
N∑
f −1
Let z3 [i] = di
I3j then V3 =
j=0
N∑
f −1
ai,j g(t − Dj )ν(t + ti,j )dt.
(4.44)
2
2
E(I3j
), where E(I3j
) is determined the same way
j=0
2
as E(I2j
).
E
(
2
I3j
)
∫
TI
∫
TI
=
E{g(t − Dj )g(ν − Dj )}E{n(t + ti,j )n(ν + ti,j )}dtdν.
(4.45)
0
0
2
Using the same arguments as the ones used to derive E(I2j
), we have E{g(t−Dj )g(ϱ−Dj )} =
E{g(t)g(ϱ)}, and finally
V3 =
=
Π
γ
N∑
f −1 (
1 − e−γTI
j=0
ΠNf (1−e−γTI )
γ
∫ Tω
−Tω
) ∫ Tω
−Tω
ϕω (κ)Rν (κ)dκ
(4.46)
ϕω (κ)Rν (κ)dκ.
Analysis of z4 [i]
V4 is derived as follows:
V4 =
N∑
f −1 ∫
TI
0
∫ TI
0
E{ν(t + ti,j )n(ϱ + ti,j )}E{ν(t + ti,j + Dj )ν(ϱ + ti,j + Dj )}dtdϱ
∫ T ∫ T −t
= Nf 0 I −tI Rν2 (κ)dtdκ
∫T
= Nf TI −TI I Rν2 (κ)dκ.
j=0
(4.47)
Bit Error Probability
The conditional bit error probability is expressed as
Chapter 4. Transmitted Reference Systems
67

√
z12
P {e|z1 }DT R = Q 
V2 + V3 + V4

(4.48)
√ ∫∞ 2
where Q(x) = 1/ 2π x e−t /2 dt . P {e|z1 }DT R is conditioned by the channel realization.
4.4.2
System Modelling Approach
Using the equivalent system and noise models derived in subsections 4.3.2, the conditional
BER can also be determined as
)
(
(zτ [i]−hτ 0 )di
Pz {e|zτ [i]} = Q √ 2
E{zn [i]}
(
= Q
(4.49)
)
√ hτ 12
E{zn [i]}
,
where hτ 0 is the bias term and hτ 1 is the linear model coefficient.
4.4.3
Derivation of Average BER
The BER of our TR-UWB systems can be obtained by averaging over several conditional
BER values. This average BER can be expressed as
∫∞
P {e|x}fz1 (x)dx,
P̄e,T R =
(4.50)
0
where fz1 (·) is the probability density function (pdf) of z1 . The derivation of an analytic
expression for P̄e,T R by solving the integral in (4.50) is complex. Instead, we propose to
determine the average BER P̄e for low complexity TR-UWB systems, with minimal ISI
(η = 1) or no ISI (η = 0), using the concept of minimum distance receiver.
Probability Density Function of the Equivalent System Model zτ [i]
The equivalent system model zτ [i] is a random process as it depends on the channel realization and data. It is assumed, in this thesis, that zτ [i] is a Gaussian random process. This assumption is justified through simulation as shown in Fig. 4.14, which is obtained after 1000
channel simulation runs, for positive data. The histogram depicts the probability of occurrence of the different values of zτ [i], whereas the solid line is obtained by plotting a Gaussian
probability density function with mean E{zτ [i]} and variance E{zτ2 [i]} − (E{zτ [i]})2 .
Chapter 4. Transmitted Reference Systems
Probability of occurence
Probability of occurence
0.25
68
0.5
p(zτ[i])
p(zτ[i])
0.45
τ =τ12
= 0nsns
τ = 0 ns
0.2
0.4
0.35
0.15
P(zτ[i]=a)
P(zτ[i]=a)
0.3
0.1
0.25
0.2
0.15
0.05
0.1
0.05
0
−10
0
10
a
20
30
0
−10
0
10
a
20
30
Figure 4.14: Probability density function of zτ [i].
Formulation as a Hypothesis Testing Problem
Let us assume a binary detector with the following hypothesis testing problem for detecting
zτ [i]
H−1 : ẑ[i] = s−1τ [i] + zτ ν [i]
(4.51)
H1 : ẑ[i] = s1τ [i] + zτ ν [i],
where s−1τ [i] and s1τ are respectively the equivalent system model for negative and positive
data transmitted at time instant i. s−1τ [i] and s1τ are expressed as
s1τ [i] = hτ 0 + hTτ1 d1 [i] + d1 [i]Hτ 2 d1 [i]
s−1τ [i] = hτ 0 + hTτ1 d−1 [i] + d−1 [i]Hτ 2 d−1 [i]
d1 [i] = [di−η , di−η+1 , . . . , di−1 , 1, di+1 ]T
d−1 [i] = [di−η , di−η+1 , . . . , di−1 , −1, di+1 ]T .
(4.52)
Low complexity TR-UWB systems with minimal ISI (η = 1) are first considered. Since ∀i,
di ∈ {−1, 1}, s1τ and s−1τ can each have 2η+1 possible values skjτ [i], where k ∈ {−1, 1}
and j = 1, 2, . . . , 2η+1 . Thus, at the symbol rate, one of the 2η+2 variables skjτ [i] is
transmitted. This is a classical signal classification problem, which motivates the use of a
minimum distance receiver.
Since zτ [i] is a Gaussian random process, the skjτ [i] are also Gaussian. Moreover they
2
are independent of each other. At time i let we decide Hkj for which Dkj
= (ẑ[i] − skjτ [i])2
is minimum.
Because ẑ 2 [i] is the same ∀(i, j), we decide Hjk for which Tkj (ẑ[i]) = −2ẑ[i]skjτ [i]+s2kj [i]
is minimum, i.e., Tkj (ẑ[i]) = ẑ[i]skjτ [i] − 21 s2kjτ [i] is maximum. Since Hk is decided when
Chapter 4. Transmitted Reference Systems
69
one of the Hkj is decided, and Hkj are independent elements of Hk ,
p(ẑ[i]|Hk ) =
η+1
2∑
p(ẑ[i]|Hkj ).
(4.53)
j=1
This motivates us to decide Hk for which
Tk (ẑ[i]) =
η+1
2∑
1
(ẑ[i]skjτ [i] − s2kjτ [i])
2
j=1
(4.54)
is maximum. Thus, H1 is decided if T1 (ẑ[i]) > T−1 (ẑ[i]).
Let T (ẑ[i]) = T1 (ẑ[i]) − T−1 (ẑ[i]),
T (ẑ[i]) =
η+1
2∑
(
j=1
)
1
ẑ[i](s1jτ [i] − s−1jτ [i]) − (s21jτ [i] − s2−1jτ [i]) .
2
(4.55)
Following the condition that T (ẑ[i]) > 0, we have
T ′ (ẑ[i]) =
η+1
2∑
ẑ[i](s1jτ [i] − s−1jτ [i])
j=1
>
− 21
η+1
2∑
(4.56)
(s21jτ [i]
−
j=1
s2−1jτ [i])
= γτ .
Equation (4.56) shows that the assumed detector consists of a test statistics T ′ (ẑ[i]) and a
threshold γτ .
For the case of ISI-free TR-UWB systems, we have
s1τ [i] = hτ 0 + hTτ1
s−1τ [i] = hτ 0 − hTτ1 .
(4.57)
Th (ẑ[i]) = ẑ[i](s1τ [i] − s−1τ [i])
> − 21 (s21τ [i] − s2−1τ [i]) = ζτ .
(4.58)
H1 is decided if
After simplification we obtain
Th (ẑ[i]) = ẑ[i]
ζτ = hτ 0 .
(4.59)
Analysis of the Test Statistics
T ′ (ẑ[i]) and Th (ẑ[i]) are Gaussian random variables conditioned on either hypothesis. We
already considered zτ ν in subsection 4.3.1 to be zero-mean Gaussian. Using this approxi-
Chapter 4. Transmitted Reference Systems
70
mation we express analytically the moments of T ′ (ẑ[i]) and Th (ẑ[i]).
E {T ′ |H−1 } = E{
= E{
η+1
2∑
(s−1jτ [i] + zτ ν [i])(s1jτ [i] − s−1jτ [i])}
j=1
η+1
2∑
s−1jτ [i](s1jτ [i]
j=1
η+1
2∑
− s−1jτ [i])}
(4.60)
zτ ν [i](s1jτ [i] − s−1jτ [i])}
+E{
j=1
= E{s−1τ (s1τ − s−1τ )T } = ε−1τ ,
where skτ = [sk1τ [i], sk2τ [i], . . . , skN τ [i]] and N = 2η+1 .
E {Th |H−1 } = E{s−1τ }.
var(T ′ |H−1 ) = var(s−1τ (s1τ − s−1τ )T ) + var(zτ ν [i]
(4.61)
η+1
2∑
(s1jτ [i] − s−1jτ [i]))
j=1
=
2
σ−1τ
+
(4.62)
στ2sν ,
where
η+1
2∑
(s1jτ [i] − s−1jτ [i]))2 } × E{zτ2ν }.
(4.63)
var(Th |H−1 ) = var(zτ ) + E{zτ2ν } = στ2 + στ2ν .
(4.64)
στ2sν
= E{(
j=1
Following the same steps as above, we derive
{
}
E {T ′ |H1 } = E s1τ (s1τ − s−1τ )T = ε1τ .
(4.65)
E {Th |H1 } = E{s1τ }.
(4.66)
2
var(T ′ |H1 ) = σ1τ
+ στ2sν ,
(4.67)
2
= var(s1τ (s1τ − s−1τ )T ).
where σ1τ
var(Th |H1 ) = στ2 + στ2ν .
(4.68)
Bit Error Probability
The probability of error P̄e is defined as
P̄e = P (H−1 |H1 )P (H1 ) + P (H1 |H−1 )P (H−1 ),
(4.69)
Chapter 4. Transmitted Reference Systems
71
where P (H1 |H−1 ) is the conditional probability of deciding H1 given that H−1 is true.
Assuming that the prior probabilities are equal, i.e., P (H1 ) = P (H−1 ) = 21 we have
P̄e =
=
1
[P (H−1 |H1 )
2
1
[Pr (T ′ (ẑ[i])
2
P̄e =
=
1
[1
2
+ P (H1 |H−1 )]
< γτ |H1 ) + Pr (T ′ (ẑ[i]) > γτ |H−1 )]
(4.70)
− Q( √γτ2−ε1τ2 ) + Q( √γτ2−ε−1τ2 )]
σ−1τ +στ sν
σ1τ +στ sν
1
[Q( √ε1τ2 −γτ2
2
σ +σ
τ sν
1τ
) + Q( √γτ2−ε−1τ2 )],
(4.71)
σ−1τ +στ sν
√ ∫∞ 2
2
2
where Q(x) = 1/ 2π x e−t /2 dt and Q(−x) = 1 − Q(x). Since σ1τ
≈ σ−1τ
, Pe can be
expressed as
[ (
)]
ε1τ − ε−1τ
1
P̄e =
Q √ 2
.
(4.72)
2
σkτ + στ2sν
For the non-ISI case, we obtain
1
P̄e = [Pr (Th (ẑ[i]) < ζτ |H1 ) + Pr (Th (ẑ[i]) > ζτ |H−1 )],
2
(4.73)
and finally
P̄e =
=
4.4.4
1
2
[ (
)]
E{s1τ −s−1τ }
√ 2 2
Q
στ +στ ν
[ (
)]
2E{hτ 1 }
1
Q √ 2 2
.
2
(4.74)
στ +στ ν
Simulation Results
The channel model is the same as in subsection 4.3.1. The different TR-UWB systems
considered are shown in Table 4.2 with their simulation parameters.
The bit error performance of the different considered TR-UWB systems, in the presence
and absence of timing delay, is demonstrated in Fig. 4.15-Fig. 4.18. The BER curves are
presented as functions of SNR and delay (τ = {0 ns,20 ns}). The solid lines denote the
simulated average BER obtained using Monte Carlo simulation, the dash-dot lines denote
the average BER obtained from (4.72) or (4.74), and the dotted line in Fig. 4.15 denotes
the average BER computed numerically by averaging P {e|z1 } in (4.48). There is a perfect
matching between both BER curves at τ = 0 ns, and good agreement at higher delay
values for the ISI-free DTR-UWB system (η = 0)).
Next, the bit error performance of DTR-UWB system with minimal ISI is illustrated in
Fig. 4.16, in the presence and absence of timing delay (τ = 20 ns and τ = 0 ns, respectively).
It can be observed, for both values of τ , that the solid lines (denoting the simulated average
Chapter 4. Transmitted Reference Systems
TR-UWB systems ⇒
Parameters
⇓
Tω [ns]
τm [ns]
Tsamp [ns]
Tf [ns]
TI [ns]
Ts [ns]
Nf
Ncr
Data rates [Mb/s]
DTR
UWB
η = 1/η = 0
0.7
0.2877
0.05
10/100
20
100/1000
10
10
10/1
72
LDC
DTR-UWB
DP
TR-UWB
0.7
0.2877
0.05
2
20
100
9
8
10
0.7
0.2877
0.05
20
20
200
2
1
5
Table 4.2: TR-UWB systems and simulation parameters.
DTR−UWB no IFI (η = 0), τ = 0 ns
0
10
Avg P{e|z1}
Avg Pe
sim. Avg. BER
−1
Average BER
10
−2
10
−3
10
−4
10
0
5
10
15
SNR [dB]
20
25
30
Figure 4.15: BER performance of the DTR-UWB system (η =0) versus SNR.
BER obtained using Monte Carlo simulation) and the dash-dot lines (denoting the average
BER obtained from (4.72)) are in good agreement.
Finally, the ISI-free systems LDC DTR-UWB and DP TR-UWB are considered. Fig. 4.17
and Fig. 4.18 show the bit error performance of both systems, in the presence and absence
of timing delay. It can again be observed, for both values of τ , that the solid lines (denoting
the simulated average BER obtained using Monte Carlo simulation) and the dash-dot lines
(denoting the average BER obtained from (4.74)) are in good agreement.
Chapter 4. Transmitted Reference Systems
73
DTR−UWB (η = 1)
0
10
−1
Average BER
10
−2
10
sim. Avg. BER (τ = 0 ns)
−3
10
sim. Avg. Pe (τ = 0 ns)
sim. Avg. BER (τ = 20 ns)
sim. Avg. Pe (τ = 20 ns)
−4
10
0
5
10
15
SNR [dB]
20
25
30
Figure 4.16: BER performance of the DTR-UWB system (η =1) versus SNR.
LDC DTR−UWB
0
10
−1
Average BER
10
−2
10
sim. Avg. BER ( τ = 0 ns)
sim. Avg. Pe (τ = 0 ns)
−3
10
sim. Avg. Pe (τ = 20 ns)
sim. Avg. BER ( τ = 20 ns)
0
5
10
15
SNR [dB]
20
25
30
Figure 4.17: BER performance of the LDC DTR-UWB system versus SNR.
4.5
Optimization of the Integration Time Interval TI
From (4.36) it is clear that by increasing TI more energy is collected. Simultaneously, more
noise is also collected as the sum of the noise variances V2 + V3 + V4 is increased.
Chapter 4. Transmitted Reference Systems
74
DP TR−UWB
0
10
−1
Average BER
10
−2
10
sim. Avg. Pe (τ = 0 ns)
sim. Avg. BER ( τ = 0 ns)
−3
10
sim. Avg. Pe (τ = 20 ns)
sim. Avg. BER ( τ = 20 ns)
0
5
10
15
SNR [dB]
20
25
30
Figure 4.18: BER performance of the DP TR-UWB system versus SNR.
90
80
70
Average f(T I)
60
50
40
30
20
10
0
10
20
30
40
50
TI [ns]
60
70
80
Figure 4.19: Optimal choice for TI .
90
100
Chapter 4. Transmitted Reference Systems
75
Equation (4.48) can be rewritten as
P {e|z1 }DT R
v
u
u
= Q t 2ΠN
= Q

)2
(
∫T
Nf 0 I g 2 (t)dt
−γTI
f 1−e
γ
(
) ∫ Tω
−Tω
ϕω (κ)Rn (κ)dκ+Nf TI
∫ TI
−TI

2 (κ)dκ
Rn
(4.75)
(√
)
f (TI ) .
The optimal choice of TI for optimal BER can be found by optimizing the function
TI 7→ f (TI ), which is out of scope of this thesis. Instead, the optimal choice for TI is
made graphically by plotting f (TI ) versus TI , as shown in Fig. 4.19, where the function
TI 7→ f (TI ) is averaged over 1000 channel realizations. This motivates the choice for
TI = 20 ns in all the simulations.
4.6
Summary
Signal and systems models for different TR-UWB systems were presented in this chapter.
The different TR-UWB transmission schemes were introduced, followed by the presentation of the receiver front-end. Delay-dependent discrete time equivalent system and noise
models for different TR-UWB systems were also reviewed in this chapter for TR-UWB
systems with and without ISI. Further, the different receiver statistics are analyzed from
two perspectives:
• Use of Gaussian approximation on the noise terms in the receiver statistics.
• Use of the equivalent system models of the receiver.
Both perspectives allow for BER performance analysis to be performed. The average BER
is verified through simulation for different SNR values. It is also compared to the average
BER derived using the equivalent system models. The derivation of an analytic expression
for the average BER was provided using the concept of minimum distance receiver. Finally,
the optimization of the integration time interval TI was dealt with.
Chapter 5
Synchronization of Transmitted
Reference Systems
5.1
Introduction
The large bandwidth of UWB technology besides providing the possibility to transmit
at a very high data rate also provides very accurate temporal and spatial information
that can be used for precise timing offset estimation. This high accuracy also allows new
perspectives in terms of synchronization, positioning and tracking [9, 58].
Synchronization is a particulary acute problem faced by UWB systems due to the fact
that they employ low-power ultra-short (sub-nanoseconds) pulses. Thus, timing requirements are stringent because minor misalignments may result in a lack of energy capture
which renders symbol detection impossible [59]. Synchronization is typically performed in
two stages: the first stage achieves coarse synchronization to within a reasonable amount of
accuracy, and the second stage is responsible for achieving fine synchronization (timing offset estimation). Synchronization schemes for UWB systems in the literature can be roughly
classified into detection-based schemes [60] - [65] and estimation-based schemes [66] - [70].
The synchronization methods which employ a detection-based approach evaluate a
candidate delay by first obtaining a measure of correlation between the received signal and a
locally generated template signal offset by the candidate delay. This measure of correlation
is then compared to a threshold in order to make a decision. The candidates could be
evaluated in a serial, parallel or hybrid manner. For threshold based synchronization
schemes in multipath fading channels, it was shown in [64] that, no matter how large the
SNR is or how the threshold is chosen, it may not be possible to make the probabilities of
detection and false alarm arbitrarily large and small. Moreover, the large search space in
UWB signal acquisition poses significant challenges in the design and implementation of
practical systems for high data rate applications.
In the estimation-based methods, an estimate of the delay is obtained through maximizing a statistic over a set of candidate delays. This statistic is usually obtained from the
correlation of the received signal with a template. These schemes do not involve a threshold comparison. They rather exploit the cyclostationarity inherent in UWB signaling
due to pulse repetition. Most of these schemes are based on the Nyquist or higher sampling
76
Chapter 5. Synchronization of Transmitted Reference Systems
77
rate of the signal and may not be feasible for low cost UWB devices. To avoid the high
sampling rate and computational complexity associated with channel estimation and time
synchronization, there has been a renewed interest in the TR systems for impulse radio
due to their simple receiver structure.
In this thesis, a novel data-aided timing acquisition technique for frame-level synchronization of DTR-UWB systems is suggested [71]. It is based on incorporating parallel I&D
circuits within pulse-pair correlator branches to improve considerably the energy capture
in the presence of timing offset. Since this technique performs perfectly well when the
noise variance is much less than the signal energy, symbol-long segments of the received
signal are averaged before timing acquisition. A training sequence is chosen to ensure that
pulse polarities within consecutive symbol-long segments of the received signal do not get
cancelled through the averaging operation. The good performance of the proposed timing
acquisition technique is confirmed through simulation results.
Furthermore, a simple timing offset estimator for low complexity TR-UWB systems,
assuming no ISI, is considered. Both the classical and bayesian approaches are used for
estimating the timing offset. It is shown, in this chapter, that both approaches yield the
same timing offset estimate. The proposed algorithm uses energy collected at the symbol
rate, thus reducing considerably the implementation complexity. If coarse synchronization
is achieved (e.g. [71]) and hardware clock timing error is known, timing offset estimation
problems are analogical to ranging problems, and the proposed algorithm can readily be
used for distance measurement. The theoretical performance limit of such a timing offset
estimation is illustrated through the derived Cramér-Rao Lower Bound (CRLB) for the
variance of the timing offset estimates.
5.2
Data-Aided Timing Acquisition
The DTR-UWB receiver illustrated in Fig. 4.4 has at its input a bank of pulse-pair correlators, whose lags are matched to the time shifts Dj . The I&D blocks are triggered at the
arrival-times of the respective pulses, which requires time-synchronization and knowledge
of the TH-sequence.
5.2.1
Timing Acquisition Technique
In this section, we perform a parallel search of the timing offset by incorporating parallel
I&D circuits within pulse-pair correlator branches and using the outputs of these I&D
blocks to determine the frame index that matches the timing offset. It is worth noting
that, in practice, the data obtained at the output of the I&D circuits are actually acquired
serially since the parallel I&D circuits within each pulse-pair correlator branch are actually
fractional parts of a single I&D circuit (fractional sampling). The parallel structure shown
in Fig. 5.1 is used to help understand our timing acquisition technique. This novel timing
acquisition technique relies on judiciously choosing the different triggering times of the
I&D circuits and the time interval TI over which the integration operations are performed,
as shown in Fig. 5.1. Each pulse-pair correlator branch with correlator lag Dj is associated
with Nf parallel I&D blocks which are triggered at different times than the respective
Chapter 5. Synchronization of Transmitted Reference Systems
Integrate-and-Dump blocks
( triggering times)
I&D
( t 0,0 +n 0 )
r ' (t )
I&D
( t 0,0 +n 1 )
X
...
D0
I&D
( t 0,0 +n Nf-1 )
I&D
( t 0,1 +n 0 )
X
I&D
( t 0,Nf-1 +n 1 )
I&D
(t 0,Nf-1 +n Nf-1 )
( )²
y0,Nf-1
( )²
y1,0
( )²
Y0
+
y1,1
Y1
+
( )²
Timing offset
estimate
y1,Nf-1
+
( )²
YN f-1
yNf-1,0
( )²
yNf-1,1
( )²
. . .
DNf-1
...
X
y0,1
. . .
. . .
I&D
( t 0,Nf-1 +n 0 )
( )²
. . .
I&D
( t 0,1 +n Nf-1 )
y0,0
...
...
D1
I&D
( t 0,1 +n 1 )
78
yNf-1,Nf-1
( )²
Figure 5.1: Illustration of the timing acquisition technique
pulses arrival times. Without loss of generality, the synchronization to symbol i = 0 is
considered. Thus, the different pulse arrival times are given by
t′j,k = t0,j + nk ,
(5.1)
where nk = kTf for k ∈ {0, 1, ..., Nf −1 }.
Our timing acquisition technique, through the parallel I&D operations, aims to achieve
efficient multipath energy collection even in presence of timing offset and noise. The
accuracy of timing acquisition is enhanced by reducing the noise level, i.e., by averaging
over different symbol-long segments of the received signal r(t). The shortcomings of such
averaging stem from the fact that the differentially modulated pulse polarities (±1) are
different from one symbol to another. Thus, averaging could result on the cancellation of
some pulses within a symbol waveform and therefore altering the energy collection within
some frames. This is the reason why we use a training sequence, given by d′p = (−1)p ,
where p ∈ {0, 1, ..., P − 1} and P is the length of the training sequence. This sequence
ensures that pulse polarities within the P/2 first odd numbered symbol-long segments of
r(t) are identical, as well as pulse polarities within the P/2 first even numbered symbol-
Chapter 5. Synchronization of Transmitted Reference Systems
79
long segments of r(t). Thus, consecutive segments of r(t) of duration 2Ts are summed
together before we apply our timing acquisition technique.
Indeed,( let us assume )the mth segment of the received signal r(t) of duration 2Ts is
rm (t) = r t + 2(m − 1)Ts for t ∈ [0, 2Ts ), and rm (t) = 0 elsewhere. Then the waveform
used for timing acquisition is
r′ (t) =
2
P
P/2
∑
(5.2)
rm (t).
m=1
The outputs of the Nf × Nf I&D blocks are given by
∫ t0,j +nk +TI ′
yj,k = t0,j
r (t + Dj )r′ (t)dt.
+nk
Let Yk =
N∑
f −1
j=0
2
yj,k
. Fig. 5.2 shows the magnitude of the Yk for different timing offsets.
5
τ = 10 ns
0
Y0
5
0
Y0
5
0
Y0
Y1
Y2
Y3
Y4
Y5
Y6
Y7
Y8
5
Y9
τ = 20 ns
Y1
Y2
Y3
Y4
Y5
Y6
Y7
Y8
Y9
τ = 30 ns
Y1
Y2
Y3
Y4
Y5
Y6
Y7
5
Y8
Y9
τ = 40 ns
0
Y0
5
Y1
0
Y0
Y1
(5.3)
Y2
Y3
Y4
Y5
Y6
Y7
Y8
Y3
Y4
Y5
Y6
Y7
Y8
0
Y0
5
Y9
Y9
τ = 60 ns
Y1
0
Y0
5
Y2
Y3
Y4
Y5
Y6
Y7
Y8
Y9
Y2
Y3
Y4
Y5
Y6
Y7
Y8
Y9
Y2
Y3
Y4
Y5
Y6
Y7
Y8
Y9
Y2
Y3
Y4
Y5
Y6
Y7
Y8
Y9
Y2
Y3
Y4
Y5
Y6
Y7
Y8
Y9
τ = 70 ns
Y1
τ = 80 ns
0
Y0
5
τ = 50 ns
Y2
0
Y0
5
Y1
τ = 90 ns
Y1
τ = 0 ns
0
Y0
Y1
Figure 5.2: Illustration of the outputs Yk for different timing offsets
The first step towards estimating the timing offset is achieved by finding the index n
given by:
n = arg max {Yk } .
k
(5.4)
Finally, the coarse estimate of τ is given as
τ̃ = nTf ,
(5.5)
where n ∈ {0, 1, . . . , Nf −1 }.
5.2.2
Simulation Results
In this subsection, the simulation results are presented. The timing acquisition performance
is discussed by using the following performance metrics: the normalized acquisition esti-
Chapter 5. Synchronization of Transmitted Reference Systems
80
{
}
mation mean square error (MSE) given by E (τ̃ − τ )2 /Ts2 , the probability of detecting
correctly the timing offset PCD and the average bit-error-rate (BER).
Fig. 5.3 depicts the MSE of our timing acquisition technique plotted versus SNR for
values of M = P/2 = 16, 32, 64, 128. As M increases, the normalized MSE decreases
monotonically. The same trend can be observed for increasing SNR. For all the M values,
the MSE curves flatten at high SNR towards a certain error floor. This is due to the
frame resolution limit imposed by the coarse estimation. This frame resolution limit can
be removed through tracking.
0
10
M=128
M=64
M=32
M=16
−1
Normalized MSE
10
−2
10
−3
10
−4
10
0
5
10
15
SNR [dB]
20
25
30
Figure 5.3: Normalized timing acquisition MSE vs. SNR for M = {16, 32, 64, 128}.
This error floor is illustrated in Fig. 5.4 for different timing offsets τ = {41 ns, 42 ns,
45 ns, 48 ns, 49 ns} where the expected estimated timing offsets are given respectively as
τ̃ = {40 ns, 40 ns, 40 ns, 50 ns, 50 ns}.
From the frame resolution limit it is clear that the best performance of our timing
acquisition technique is achieved when the timing offset to be estimated is an integer
multiple of the frame duration. Thus, we have tested the performance of our timing
acquisition technique in terms of its probability of detecting correctly the timing offset
PCD , when the timing offset is an integer multiple of the frame duration. This is shown
in Fig. 5.5 for values of M = 16, 32, 64, 128. It can be observed that the PCD increases
monotonically as M increases. The same trend is observed for increasing SNR. With M =
128, we achieve PCD = 1 from SNR = 10 dB.
Finally, the BER performance of our timing acquisition technique is tested for values of
M = 16, 128. This BER performance is compared to the cases without timing acquisition,
and with perfect timing. It can be observed from Fig. 5.6 that, as M increases, the BER
performance improves monotonically. It is also observed that there is no significant performance loss when we compare the BER performance of our timing acquisition technique to
Chapter 5. Synchronization of Transmitted Reference Systems
M = 64
0
10
τ=41 ns
τ=42 ns
τ=45 ns
τ=48 ns
τ=49 ns
−1
10
Normalized MSE
81
−2
10
−3
10
−4
10
0
5
10
15
SNR [dB]
20
25
30
Figure 5.4: Normalized timing acquisition MSE vs. SNR for τ = {41 ns, 42 ns, 45 ns, 48 ns, 49
ns}.
τ = 40 ns
Probability of detection (PCD)
1
0.8
M=128
M=64
M=32
0.6
M=16
0.4
0.2
0
−5
0
5
10
SNR [dB]
15
20
25
30
Figure 5.5: Probability of detection vs. SNR for M = {16, 32, 64, 128}.
the case where there is perfect timing. Indeed, the performance loss is less than 1 dB for
M = 128 and around 1.2 dB when M = 16. This is due to a combination of the robustness
Chapter 5. Synchronization of Transmitted Reference Systems
82
of the DTR-UWB systems against timing offset and the good performance of our timing
acquisition technique.
0
10
−1
Average BER
10
−2
10
−3
10
Perfect timing
With timing acquisition (M=128)
With timing acquisition (M=16)
Without timing acquisition
−4
10
0
5
10
15
SNR[dB]
20
25
30
Figure 5.6: Average BER vs. SNR for M = {16, 168 }.
5.3
Timing Offset Estimation
The received signal with time delay τ , expressed in (4.5), will start within a window of
duration Td that represent the uncertainty on the arrival time of the received signal. This
time window is allocated in the time line by a previous coarse acquisition process, presented
in the previous section. In this section, two approaches to timing offset estimation are
presented, using the statistical analysis of the delay-dependent equivalent system and noise
models presented in section 4.3 for low complexity TR-UWB systems, and the analysis in
subsection 4.4.3.
5.3.1
Normalization of the decision variable
Let zs [i] = ẑ[i]/Es be the normalized decision variable. This normalization is necessary to
minimize the channel dependence of the decision variable. We obtain
zs [i] = zeτ [i] + zeτ ν [i].
(5.6)
where zeτ [i] = zτ [i]/Es and zeτ ν [i] = zτ ν [i]/Es . At a time instant i, zeτ [i] is a Gaussian ran2
. The data-dependence of zeτ [i] impose
dom variable with mean E{zeτ [i]} and variance σze
a prior knowledge of the data sequence for efficient characterization of zeτ . This implies the
Chapter 5. Synchronization of Transmitted Reference Systems
83
need of a training sequence in order to obtain an efficient timing offset estimation. When
fitted with an exponential function, E{zeτ } can be expressed as
E{zeτ } = A1 exp(Bτ ), di = 1
E{zeτ } = A2 exp(Bτ ), di = −1.
(5.7)
Generally, A2 is slightly different from −A1 , due to the bias term hτ 0 . Hence a time
instant i the realizations of zeτ [i]|di are asymmetric. Nevertheless, for most of the systems
considered in this work, hτ 0 ≈ 0, since the interference among fixed reference pulses plus
interference between equally modulated data pulses is reduced by allowing larger average
pulse separations. Hence, we can write
E{zeτ [i]} = Adi exp(Bτ ).
(5.8)
B takes approximately the same value for all TR-UWB systems considered in this work.
The value of A is proportional to the ratio Ncr /Nf .
5.3.2
Classical Approach to Timing Offset Estimation
Let us consider zeτ to be a deterministic but unknown function of τ . We assume that
the normalized output energy of the TR-UWB systems, collected at the symbol rate, is
composed of the equivalent system model zeτ with zero mean gaussian modelling uncer2
tainty plus system noise zeτ ν . The variance of the modelling error is given by σze
and
2
var(zeτ ν ) = σνe . In [15] it is indicated that the noise zτ ν can be considered as a zero-mean
Gaussian process. Thus, we have
]
[
P
−1
∑
1
1
p(zs ; τ ) =
exp −
(zs [i] − Adi exp(Bτ ))2 ,
2 + σ 2 )]P/2
2 + σ2 )
[2π(σze
2(σ
ze
νe
νe i=0
(5.9)
where P indicate the number of data symbols in the training sequence with the same
polarity. Taking the logarithm of (5.9) and finding its derivative w.r.t τ , we obtain
)
(P −1
∑
ABdi exp(Bτ )
zs [i]−P Adi exp(Bτ )
∂ ln p(z; τ )
i=0
=
.
(5.10)
2 + σ2 )
∂τ
(σze
νe
p(z;τ )
Solving ∂ ln∂τ
= 0 for τ , we obtain the maximum likelihood (ML) estimate of the timing
offset as
(P −1
/
)
∑
1
zs [i] (P Adi )
τ̂ = B ln
(5.11)
( i=0)
= B1 ln |z̄As | ,
Chapter 5. Synchronization of Transmitted Reference Systems
where z̄s = (1/P )
P∑
−1
84
zs [i], and |z̄s | = z̄s /di = zs di .
i=0
The Cramér-Rao Lower Bound (CRLB) for the variance of the timing offset estimates is
derived by finding the derivative of (5.10) w.r.t τ [72].
P AB 2 exp(Bτ ) [|z̄s | − 2A exp(Bτ )]
∂ 2 ln p(z; τ )
.
=
2 + σ2 )
∂τ 2
(σze
νe
{ 2
}
)
Finally, CRLB(τ̂ ) = −E ∂ ln∂τp(z;τ
is expressed as
2
2
2
(σze
+ σνe
)
.
2
P AB exp(Bτ ) [2A exp(Bτ ) − |z̄s |]
CRLB(τ̂ ) =
5.3.3
(5.12)
(5.13)
Bayesian Approach to Timing Offset Estimation
zeτ [i] is now considered to be a stochastic process with pdf
]
[
1
1
2
p(zeτ [i]) = √
exp − 2 (zeτ − Adi exp(Bτ )) .
2
2σze
2πσze
(5.14)
The pdf of zs conditioned on zeτ is given by
]
[
P −1
1
1 ∑
p(zs |zeτ ) =
(zs [i] − zeτ )2 .
exp − 2
2 )P/2
(2πσνe
2σνe i=0
(5.15)
In the Appendix III, we have proven that zeτ |zs is a Gaussian random variable with variance
σz2eτ |zs =
and mean
(
mzeτ |zs =
1
P/ 2 + 1/ 2
σνe
σze
P z̄s Adi exp(Bτ )
+
2
2
σνe
σze
)(
,
(5.16)
P
1
+
2
2
σνe
σze
)−1
.
(5.17)
Equation (5.17) can be written as
mzeτ |zs = α + β exp(Bτ ),
(5.18)
where
(
α=
β=
(
P z̄s
2
σνe
Adi
2
σze
)(
)(
P
2
σνe
+
1
2
σze
P
2
σνe
+
1
2
σze
)−1
)−1
(5.19)
.
The timing offset estimate τ̂ is derived from (5.18) through functional mapping, by miniP∑
−1
mizing the least square (LS) error criterion J(τ ) =
[zs [i] − (α + β exp(Bτ ))]2 . Differeni=0
Chapter 5. Synchronization of Transmitted Reference Systems
85
tiating J(τ ) w.r.t. τ , we obtain
∂J(τ )
= 2P Bβ exp(Bτ )[α + β exp(Bτ ) − z̄s ].
∂τ
Setting
∂J(τ )
∂τ
= 0 we obtain
1
τ̂ = ln
B
Since z̄s − α =
(5.20)
2
z̄s σνe
2 +σ 2 )
(P σze
νe
and β =
(
)
z̄s − α
.
β
(5.21)
2
Adi σνe
2 +σ 2 ) ,
(P σze
νe
(5.21) can be expressed as
(P −1
/
)
∑
1
τ̂ = B ln
zs [i] (P Adi )
( i=0)
= B1 ln |z̄As | ,
(5.22)
which is equal to the ML estimate of τ in (5.11).
5.3.4
Determination of Es , A and B
Estimating the symbol energy
An estimation of the symbol energy at the receiver front-end can be obtained by integrating
the received signal over a reasonably long time interval as follows
KN
s −1
∑
Ês =
rk2 − σ 2 KNs
k=0
,
(5.23)
K
where K is the number of symbols within the time interval used for symbol energy estimation and Ns = Ts /Tsamp is the number of samples per symbol. rk = r(kTsamp ) is the k th
sample of r(t).
Determining A and B
The mathematical procedure for finding the best-fitting curve to a given set of points by
minimizing the sum of the squares of the offsets (“the residuals”) of the points from the
curve is known as least squares fitting [73] [74]. A and B can be determined offline through
least squares exponential fitting of the average of the normalized equivalent system model
zeτ over several channel realizations, as is shown in Appendix IV.
For distance measurement applications, A and B are tuning parameters that can be determined during the calibration phase for making accurate measurements.
5.3.5
Simulation Results
In this subsection, the simulation results are presented. The performance of our dataaided timing offset estimator is demonstrated for different TR-UWB systems by using the
Chapter 5. Synchronization of Transmitted Reference Systems
86
following performance metrics: the normalized estimation mean
√square error (MSE) given
by mse= E{(τ̂ − τ )2 }/Ts2 , the root mean square error rmse= E{(τ̂ − τ )2 } and the bias
(estimation error). We define the bias of our estimator to be Bias(τ̂ ) = E{τ̂ −τ } = E{τ̂ }−τ .
The channel model used, is the IEEE 802.15.4a CM3 model for indoor office environment
[37], with RMS delay spread τrms = 12 ns. The different systems used and their simulation
parameters are shown in Table 5.1.
TR-UWB systems ⇒
Parameters ⇓
Tω [ns]
τm [ns]
Tsamp [ns]
Tf [ns]
TI [ns]
Ts [ns]
Nf
Ncr
DTR
UWB
0.7
0.2877
0.05
100
20
1000
10
10
LDC
DTR-UWB
0.7
0.2877
0.05
16
20
200
5
4
BLDC
DTR-UWB
0.7
0.2877
0.05
2
20
100
9
8
DP
TR-UWB
0.7
0.2877
0.05
20
20
200
2
1
Table 5.1: TR-UWB systems and simulation parameters
Fig. 5.7 - Fig. 5.9 depict the MSE of our timing offset estimator plotted versus SNR for
values of P = 64, 256 and for different TR-UWB systems. Solid lines show the performance
with known Es and dash-dot lines show the performance with estimated Es . For all the
TR-UWB systems considered, it can be observed that, as P increases, the normalized MSE
decreases monotonically. The same trend can be observed for increasing SNR. At SNRs
above 20 dB the MSE of our timing offset estimator with estimated Es get closer to the
MSE of our timing offset estimator with known Es . This confirms the good accuracy of our
energy estimator at SNR>20 dB. For all the TR-UWB systems, the MSE curves flatten
at high SNR towards a certain error floor. This error floor is related to P (amount of
averaging) and Tf (average pulse separation).
IFI plays an important role in the observed error floor as shown in Fig. 5.10, where
the rmse of our timing offset estimation technique is plotted versus SNR for different TRUWB systems. BLDC DTR-UWB system is the burst-oriented type of the low duty cycle
(LDC) DTR-UWB, with pulse-burst of 16 ns compared to the pulse presence of 64 ns for
LDC DTR-UWB system. It can be observed that the BLDC DTR-UWB system, which
has severe IFI, has the highest level of error floor. For other TR-UWB systems, as Tf is
increased, the error floor decreases monotonically.
Fig. 5.11 confirms (5.13) since the performance of our timing offset estimator is related
to the delay. This is due to the exponential fitting, where zeτ values get closer to each
other as delay increases, making it difficult to resolve them.
Chapter 5. Synchronization of Transmitted Reference Systems
87
τ = 5 ns
−3
10
P = 256, Es known
P = 256, Es estimated
P = 64, Es known
P = 64, Es estimated
−4
Normalized MSE
10
−5
10
−6
10
0
5
10
15
SNR [dB]
20
25
30
Figure 5.7: Normalized timing offset estimation MSE vs. SNR for DTR-UWB systems, P =
{64, 256}.
τ = 5 ns
P = 256, Es known
P = 256, Es estimated
P = 64, Es known
P = 64, Es estimated
−2
Normalized MSE
10
−3
10
−4
10
0
5
10
15
SNR [dB]
20
25
30
Figure 5.8: Normalized timing offset estimation MSE vs. SNR for LDC DTR-UWB systems,
P = {64, 256}.
Chapter 5. Synchronization of Transmitted Reference Systems
88
τ = 5 ns
P = 256, Es known
P = 256, Es estimated
P = 64, Es known
−2
P = 64, Es estimated
Normalized MSE
10
−3
10
−4
10
0
5
10
15
SNR [dB]
20
25
30
Figure 5.9: Normalized timing offset estimation MSE vs. SNR for DP TR-UWB systems, P =
{64, 256}.
τ = 5 ns P = 256
DTR−UWB
LDC DTR−UWB
BLDC DTR−UWB
DP TR−UWB
1
rmse [ns]
10
0
10
10
12
14
16
18
20
SNR [dB]
22
24
26
28
30
Figure 5.10: Timing offset estimation rmse vs. SNR for different TR-UWB systems.
Chapter 5. Synchronization of Transmitted Reference Systems
89
P = 128 SNR = 30 dB
2
10
DTR−UWB
LDC DTR−UWB
DP DTR−UWB
1
rmse [ns]
10
0
10
−1
10
0
5
10
15
20
25
Delay [ns]
30
35
40
45
50
Figure 5.11: Timing offset estimation rmse vs. delay for different TR-UWB systems.
(a)
10
(b)
55
Mean est. delays
50
Unbiased line
45
P=128
SNR = 30 dB
5
Mean estimated delays
Bias of delay estimation
40
0
35
30
25
20
15
10
P=256, Es estimated
5
P=64, Es estimated
−5
0
10
SNR [dB]
20
30
0
0
10
20
30
Delay [ns]
40
50
Figure 5.12: Biasedness of timing offset estimator for DTR-UWB systems, (a) vs. SNR (b) vs.
delay.
Chapter 5. Synchronization of Transmitted Reference Systems
(a)
10
(b)
55
P=256, Es estimated
P=64, Es unknown
Mean est. delays
50
Unbiased line
45
P=128
SNR = 30 dB
Mean estimated delays
Bias of delay estimation
40
5
0
90
35
30
25
20
15
10
5
−5
0
10
SNR [dB]
20
0
30
0
10
20
30
Delay [ns]
40
50
Figure 5.13: Biasedness of timing offset estimator for LDC DTR-UWB systems, (a) vs. SNR
(b) vs. delay.
(a)
10
(b)
55
P=256, Es estimated
P=64, Es estimated
Mean est. delays
50
Unbiased line
45
P=128
SNR=30 dB
5
Mean estimated delays [ns]
Bias of delay estimation
40
0
35
30
25
20
15
10
5
−5
0
10
SNR [dB]
20
30
0
0
10
20
30
Delay [ns]
40
50
Figure 5.14: Biasedness of timing offset estimator for DP TR-UWB systems, (a) vs. SNR (b)
vs. delay.
Chapter 5. Synchronization of Transmitted Reference Systems
91
Fig. 5.12 - Fig. 5.14 depict the biasedness of our timing offset estimator plotted versus
SNR and delay for values of P = 64, 256 and for different TR-UWB systems. This
knowledge contributes in deciding the operating SNR and range (looking also at Fig. 5.11)
of our timing offset estimator. For conventional DTR-UWB and dual pulse (DP) TR-UWB
systems the biasedness of our estimator is negligible from 20 dB onwards, which is not the
case for LDC DTR-UWB system which exhibits some small biasedness.
5.4
Summary
In this chapter, a novel data-aided timing acquisition technique for DTR-UWB systems
was presented. The main signal processing techniques used were:
• Averaging of the received signal
• Fractional sampling
• Maximum selection criterion
The technique proposed is limited by the frame resolution which can be overcome through
timing offset estimation, which was also presented, in this chapter, for low complexity TRUWB systems. The proposed data-aided timing offset estimator uses energy collected at
the symbol rate, thus reducing considerably the implementation complexity.
Simulation results provided a thorough performance analysis of the timing acquisition
technique in terms of timing acquisition MSE, probability of correct detection and average
BER. For the timing offset estimator, the performance metrics used were the normalized
estimation MSE and estimation error (biasedness). The CRLB for such timing offset
estimation approach was also derived.
Chapter 6
Localization with Transmitted
Reference Systems
6.1
Introduction
Emergence of location based services requires a technical solution for accurate indoor localization of persons and objects. As satellite based positioning systems are not readily
applicable to indoor localization, alternative solutions are needed.
Indoor wireless systems have to cope with severe multipath scenarios. Due to their
extremely large bandwidth, UWB signals offer a good multipath resolution and enable
accurate localization. UWB localization approaches can be divided into Time of Arrival
(TOA), Time-difference of Arrival (TDOA), Angle of Arrival (AOA) and Received Signal
Strength (RSS) based systems.
There are various existing and potential applications of high precision localization using
UWB technology. An UWB precision location system [75] can be used to identify and locate
valuable assets, for example, in hospitals, industrial fields and government offices. In [12] a
low cost and small size UWB ASIC (application specific integrated circuit) is proposed for
outdoor sport and recreational activities. The architecture and implementation status of a
low-cost experimental UWB localization system is reported in [76]. It is designed to enable
practical evaluations of UWB based positioning concepts. Another potential application is
in WSN. Awareness of sensor positions may effectively improve network performance. For
instance, location-aware routing protocols can reduce routing overhead and save energy
by avoiding route search [77]. In [78] a novel packet routing scheme for UWB ad-hoc
network is proposed and evaluated. This routing scheme employs UWB’s features and
controls spreading factor according to the distance between the nodes. When controlling
the maximum transmission distance and adopting multi-hop transmission, the throughput
performance can be improved as compared to single-hop transmission. Another position
based routing strategy which exploits the high precision ranging capabilities offered by
UWB is presented in [79]. It is shown that if a position-aware routing protocol and a
power-aware routing metric based on ranging measurements are adopted, emitted power
levels as well as multi-user interference are significantly reduced.
The system proposed in this chapter deals with UWB localization based on the timing
92
Chapter 6. Localization with Transmitted Reference Systems
93
offset estimation algorithm presented in section 5.3. This algorithm can readily be used
for distance measurement since c = d/τ , where c is the speed of light and d is the distance
travelled by light or other electromagnetic waves during a time span τ . Our algorithm
provides a new aspect in distance measurement and is similar to the RSS technique. The
difference is that the RSS technique relies on the knowledge of the path loss model for
accurate and reliable distance estimation, whereas our algorithm uses the statistical analysis of the energy collected at the output of the receiver. Generally, the RSS metric is
relatively simple to detect and no high synchronization effort and protocols are needed like
in the TOA based solutions. Moreover, TOA based systems are sensitive to the available
bandwidth, and also to the occurrence of UDP channel conditions, whereas RSS based systems are less sensitive to the available bandwidth and more resilient to UDP conditions.
Nevertheless, it is not expected that RSS methods can match the precision of time-based
methods. The achievable accuracy in RSS localization decreases with the distance, which
has to been taken into consideration in the system design.
6.2
Localization Systems
Locating a node in a wireless system involves the collection of location information from
radio signals traveling between the target node and a number of beacon nodes. Different
measurement techniques for localization are discussed in this section from the viewpoint
of UWB systems. The general system architecture of localization systems will be briefly
discussed.
6.2.1
Ranging-based Systems
The most popular measurement technique for localization is ranging. There are two methods to obtain range measurements: timing and signal strength.
Timing
Time-based techniques rely on measurements of travel times of signals between nodes. If
two nodes have a common clock, the node receiving the signal can determine the TOA of
the incoming signal that is time-stamped by the reference node. The measured time can
provide a circle of radius representing the distance between the node to be localized and
the receiving node, centered at the latter. In two-dimensional localization, at least three
circles are required as shown in Fig. 6.1.
This technique, also known as trilateration, is easy to implement; however, it requires
knowledge of the transmission time of the emitted signal as well as synchronization of the
transmitter and receivers clocks. Otherwise, considerable localization errors can occur. For
instance, a clock inaccuracy of just 1 µs will lead to a position error of 300 m. Furthermore,
this technique can suffer from multipath propagation effects.
In the TDOA technique the difference in time at which the signal from the node to be localized arrives at two different receivers is measured. Each time difference is then converted
into a hyperbola with a constant distance between the two receivers. In two-dimensional
Chapter 6. Localization with Transmitted Reference Systems
94
Location
Receiver 1
d1 = ct1
d2 = ct2
d3 = ct3
Receiver 2
Receiver 3
Figure 6.1: TOA localization principle.
localization, at least two pairs of receivers are required; the location is the intersection of
the hyperbolas. This technique, known as multilateration or hyperbolic positioning, requires
synchronization of the receivers’ clocks; however, knowledge of the absolute transmission
time of the emitted signal is no longer required. Here again, multipath propagation effects
can influence the accuracy of the location of the receivers. Fig. 6.2 shows an example of
TDOA localization. For both TOA and TDOA techniques, the wider the bandwidth of the
signal the lower the measurement error becomes. This property allows extremely accurate
location estimates using time-based techniques via UWB radios.
Location
Receiver 1
Hyperbola d1,2
Receiver 2
Receiver 3
Hyperbola d1,3
Hyperbola d2,3
Figure 6.2: TDOA localization principle.
Signal Strength
With this technique the signal strength or energy of the node to be localized is measured
at several stationary receivers. Ideally, each measurement will provide a circle of radius
representing the distance between the node to be localized and the receiver that made this
Chapter 6. Localization with Transmitted Reference Systems
95
measurement, centered at the corresponding receiver. The location of the node is then
given by the intersection of these circles. In two-dimensional localization, and assuming
no measurement error occurs, at least three circles are required in order to resolve the
ambiguities arising from multiple crossings of the circles. The accuracy of the estimated
location can be improved by increasing the number of measurements and then averaging
the results. However, this approach requires an exact knowledge of the path loss in order
to get an accurate estimation of the signal strength at the results.
Our Approach
We proposed a RSS-like technique that uses the statistical analysis of the distance-dependent
discrete-time TR-UWB system models, instead of the path loss model, to estimate the distance d between the transmitter and the receiver. Indeed, considering c = d/τ , where c is
the speed of light in vacuum, (5.6) can be written as
zs [i] = czeτ [i] + czeτ ν [i] = zed [i] + zedν [i].
(6.1)
At a time instant i, zed [i] is a Gaussian random variable with mean E{zed [i]} = cE{zeτ [i]}
2
and variance c2 σze
. Hence, (5.8) becomes
E{zed [i]} = Adi exp(Bd/c).
(6.2)
0.9
0.8
0.7
E{zed}
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
distance [m]
10
15
Figure 6.3: Normalized average distance-dependent noise-free decision variable
The distance dependence of E{zed [i]} is shown in Fig. 6.3, and is obtained by averaging
zed over 1000 channel realizations. The maximum likelihood (ML) estimate of the distance
between the transmitter and the receiver can be expressed as
Chapter 6. Localization with Transmitted Reference Systems
c
dˆ = ln
B
(
|z̄s |
A
96
)
.
(6.3)
The CRLB for the variance of the distance estimates is derived from (5.13) and is given by
ˆ =
CRLB(d)
6.2.2
2
2
c2 (σze
+ σνe
)
.
2
P AB exp(Bd/c) [2A exp(Bd/c) − |z̄s |]
(6.4)
Directionality-based Systems
Another way of estimating location is to measure the signal direction or angle of arrival
(DOA/AOA) of the target node seen by reference nodes by steering the main lobe of a
directional antenna or an adaptive antenna array. To determine the location of a node
in a two-dimensional plane, it is sufficient to measure the angles of the straight lines that
connect the node and two reference nodes, as shown in Fig. 6.4.
Location
y
Receiver 2
Receiver 1
x
Figure 6.4: AOA localization principle.
This method is also called triangulation and it has the advantage of not requiring
synchronization of the receivers nor an accurate timing reference. On the other hand,
receivers require regular calibration in order to compensate for antenna mismatches.
The AOA approach is not suited to UWB localization for the following reasons:
• The use of antenna arrays increases the system cost, annulling the main advantage
of a UWB radio equipped with low-cost transceivers.
• Due to the large bandwidth of a UWB signal, the number of paths may be very large,
especially in indoor environments. Therefore, accurate angle estimation becomes very
challenging due to scattering from objects in the environment.
• Time-based approaches can provide very precise location estimates, and therefore
they are better motivated for UWB over the most costly AOA-based techniques.
Chapter 6. Localization with Transmitted Reference Systems
97
Ranging and directionality-based systems can be termed as Network-based systems.
6.2.3
System Architecture
Localization systems using similar measurement techniques can differ considerably in their
system architecture. We basically have centralized, hierarchical or decentralized architectures. The design of a localization system can also be largely influenced by application
requirements. Certain applications require tightly coupled systems using beacon nodes that
are wired to a centralized controller and placed at fixed positions, whereas for other applications loosely coupled systems are more appropriate. They use beacon nodes that are
wireless and coordinate in a completely decentralized manner without central control.
Tightly coupled Systems
Several of the traditional localization technologies have a tightly coupled system architecture, motivated by application requirements. These applications generally require high
accuracy and real-time tracking. Problems of time synchronization and coordination among
beacon nodes are easily resolved because these systems are wired and have a centralized
controller. These systems achieve high accuracy, but their drawback is that the centralized
location estimation limits the number of devices they can simultaneously track. Secondly,
wiring significantly impedes deployment. A key research challenge in these systems is
achieving similar performance with outdoor applications, where deployment cannot be
controlled and wiring may be infeasible.
Loosely coupled Systems
Motivated by deployment concerns, recently proposed localization systems are decentralized and completely wireless [80] - [84]. From all these systems the three approaches that
meet the basic requirements for self-organization, robustness, and energy-efficiency are:
• Ad-hoc positioning [82]
• N-hop multilateration [84]
• Robust positioning [83].
These three algorithms are fully distributed and use local broadcast for communication with
immediate neighbors. This last feature allows them to be executed before any multihop
routing is in place, hence, they can support efficient location-based routing schemes. They
use a three-phase approach for determining the individual node positions:
1. Determination of the distances between nodes and anchor nodes.
2. Derivation for each sensor node of its location from its beacon distances.
3. Refinement of the node locations using information about the range to, and locations
of neighboring nodes.
The refinement phase is optional and may be included to obtain more accurate locations.
Chapter 6. Localization with Transmitted Reference Systems
6.3
98
Location Estimation
In this section, the standard Least Squares (LS) [81] [82] and spherical-interpolation (SI)
[85] methods for location estimation are described. For convenience, we assume that the
nodes with known positions, (xk , yk , zk ), k = 1, 2, . . . , N , are beacons while the nodes with
unknown positions, (x, y, z) are sensors.
6.3.1
The LS algorithm
Let ti be the TOA of the signal at anchor k, t0 be the transmit time at the sensor and
dk = c(tk − t0 ) be the range between the sensor and the k th beacon node. Then, we obtain
d2k = (xk − x)2 + (yk − y)2 + (zk − z)2 .
(6.5)
Subtracting (6.5) for k = 2, 3, . . . , N by (6.5) for k = 1 produces
xk,1 x + yk,1 y + zk,1 z = bk,1 , k = 2, . . . , N ,
(6.6)
where xk,1 = xk − x1 , yk,1 = yk − y1 , zk,1 = zk − z1 and bk,1 = 0.5[(x2k + yk2 + zk2 ) − (x21 +
y12 + z12 ) + d21 − d2k ]. Equation (6.6) can be written in a compact form as
Ap=b,
(6.7)
where



 
x2,1 y2,1 z2,1
b2,1
x
 x3,1 y3,1 z3,1 
 b3,1
A=
, p =  y , b = 
..
.. 
 ...
 ...

.
.
z
xN,1 yN,1 zN,1
bN,1


.

(6.8)
The standard LS solution to (6.7) is derived, as shown in Appendix V, as
(
)−1 T
p̂LS = AT A
A b,
(6.9)
where the matrix inverse is supposed to exist and dk are replaced by their corresponding
estimates.
6.3.2
The SI Method
When the transmit time t0 is unknown, TDOA measurements can be employed as for
centralized networks. In this case, the SI method would be a suitable candidate. First, the
spatial origin is mapped
receiver√1, as shown in Fig. 6.5.
√to one of the beacon nodes, say
T
2
2
2
By defining Rk = xk + yk + zk , pk = [xk , yk , zk ] , R = x2 + y 2 + z 2 and dk,l =
dk − dl , dk can be expressed using the Pythagorean theorem as
d2k = R2 + Rk2 − 2pTk p.
(6.10)
Chapter 6. Localization with Transmitted Reference Systems
99
Location
z
d1
=R
di
Receiver 1
( x1 , y1 , z1 ) =(0,0,0)
x
R
i
y
Receiver i
( x i , yi , z i )
Figure 6.5: Illustration of the spherical interpolation approach.
where p is the location of the sensor node to be estimated. Since d1 = R, R − d1 + dk =
dk ⇒ dk = R + dk1 . (6.10) can be rewritten as
(R + dk1 )2 = R2 + Rk2 − 2pTk p, k = 2, 3, . . . , N.
(6.11)
In the presence of measurement errors, (6.11) becomes
(R + dk1 )2 + ek = R2 + Rk2 − 2pTk p
⇒ ek = (Rk2 − dk1 )2 − 2Rdk1 − 2pTk p,
(6.12)
where ek is the equation error. (6.12) can be written in compact form as
e = rd − 2Rd − 2Ap.
(6.13)
2
− d2N 1 ]T
A is given by (6.8), e is the equation error vector, rd = [R22 − d221 , R32 − d231 , . . . , RN
T
and d = [d21 , d31 , . . . , dN 1 ] .
The standard LS solution for p, given R, is
1 ( T )−1 T
A A
A (rd − 2Rd).
(6.14)
2
A closed-form solution can be derived by substituting (6.14) in (6.13) and minimizing the
equation again with respect to R.
Rewriting the equation error (6.13) to eliminate p by substituting the value from (6.14),
we get
p̂ =
Chapter 6. Localization with Transmitted Reference Systems
(
)−1 T
ẽ = rd − 2Rd − A AT A
A (rd − 2Rd) = (IN −1 − As )(rd − 2Rd),
100
(6.15)
(
)−1 T
where As = A AT A
A .
The error criterion can be expressed as
J(R) = ẽT ẽ = (rd − 2Rd)T B(rd − 2Rd),
(6.16)
where B = (IN−1 −As )T (IN−1 −As ) and IN−1 denotes an identity matrix of size N −1×N −1
. Minimizing J with respect to R is a form of weighted LS in which the weighting matrix B
is idempotent of rank N − 4 [85]. For this reason more than four beacon nodes are needed
for the SI method.
∂J
Differentiating (6.16) with respect to R and setting ∂R
= 0 we obtain
1 dT Brd
.
(6.17)
2 dT Bd
Substituting this solution back into (6.14) yields the sensor node location estimate
R̃ =
p̂SI
6.3.3
1 ( T )−1 T
dT Brd d
=
A A
A (rd − T
).
2
d Bd
(6.18)
Simulation results
The simulation results are presented in this subsection. The performance of the proposed
distance estimator is demonstrated
√ for different TR-UWB systems by using the following
ˆ = E{(dˆ − d)2 } and the distance estimation variance which
performance metrics: rmse(d)
ˆ E{(dˆ − d)2 } − [Bias(d)]
ˆ 2.
can be defined from the MSE and the bias of d as Var(d)=
The IEEE 802.15.4a CM3 channel model for indoor office environment, with RMS delay
spread τrms = 12 ns, is used once again. The different systems used and their simulation
parameters are shown in Table 5.1.
Using a training sequence of length P = 128 and operating the TR-UWB receivers at
a SNR of 30 dB, the rmse of the distance estimator in (6.3) is shown in Fig. 6.6.
It can be observed that, for all TR-UWB systems considered, the rmse increase monotonically with distance. At distance within 15 m the DTR-UWB system has a superior
distance measuring performance. The TR-UWB system with relatively worst distance
measuring performance is the low duty cycle (LDC) DTR-UWB system. This is because
it has the lowest average frame duration, which means it is more affected by IFI. It should
also be noted that the DTR-UWB system is able to provide centimeter accuracy for a
transmitter-receiver separation up to 10 m. The variance of the distance estimator for the
three TR-UWB systems considered is shown in Fig. 6.7.
Next, the localization capability of our three TR-UWB systems is analyzed. For this
purpose the proposed simulation setup is shown in Fig. 6.8. A typical laboratory room or
field size of 10 m × 6 m is illustrated. A local 2D reference system is used.
Chapter 6. Localization with Transmitted Reference Systems
P = 128
5
101
SNR = 30 dB
DTR−UWB
LDC DTR−UWB
DP DTR−UWB
4.5
4
3.5
rmse [m]
3
2.5
2
1.5
1
0.5
2
4
6
8
distance [m]
10
12
14
Figure 6.6: Distance estimation rmse vs. distance transmitter-receiver for different TR-UWB
systems.
P = 128
11
SNR = 30 dB
DTR−UWB
10
LDC DTR−UWB
DP DTR−UWB
9
8
Variance [m2]
7
6
5
4
3
2
1
2
4
6
8
distance [m]
10
12
14
Figure 6.7: Distance estimation variance vs. distance transmitter-receiver for different TRUWB systems.
Chapter 6. Localization with Transmitted Reference Systems
102
(10,6)
(3,5)
3.6
6.4
0m
6m
1m
5m
(5,2)
(0,2)
3m
4.12 m
m
(0,0)
2.2
5.38
(9,1)
(4,0)
10 m
(xk,yk)
(x,y)
kth beacon node
sensor node
Figure 6.8: 2D localization scenario.
Both location estimation methods described in section 6.3 are used to localize the sensor
node. The different beacon nodes are positioned in such a way that the matrix A is nonsingular, hence invertible. This is ensured by not allowing more than two beacon nodes to
have the same X or Y coordinates. For simulation purpose, the sensor node is localized
at the location (x = 5, y = 2). Using both the LS and SI localization methods, different
location estimates for different TR-UWB systems are shown in Fig. 6.9 - Fig. 6.11.
The dots “·” represent the location estimates of the sensor node using 100 Monte Carlo
runs, whereas the star “⋆” indicates the true sensor node location. It can be observed, for
all TR-UWB systems considered, that the SI method exhibits less dispersion of the location
estimates than the LS method. Indeed, with the LS method, more outliers are observed.
Thus, the SI method is more precise than the LS method. Moreover, the SI method is more
accurate than the LS method as can be seen in Tab. 6.1 where the localization accuracy is
illustrated. This is because the SI method is analogous to the weighted LS method. It can
also be observed that the DTR-UWB and dual pulse (DP) TR-UWB systems exhibit less
outliers than the LDC DTR-UWB system, and they
√ are more suitable for localization. In
Tab. 6.1 the localization error is defined by ϵ = (x̂ − x)2 + (ŷ − y)2 , where x̂ and ŷ are
the estimates of the X and Y coordinates of the sensor node, respectively.
Chapter 6. Localization with Transmitted Reference Systems
DTR−UWB P = 128 SNR = 30 dB
6
Y coordinate [m]
Sensor true location
Location estimates
LS Method
5
4
3
2
1
0
0
1
2
3
4
5
6
X coordinate [m]
7
6
Y coordinate [m]
9
10
Sensor true location
Location estimates
SI Method
5
8
4
3
2
1
0
0
1
2
3
4
5
6
X coordinate [m]
7
8
9
10
Figure 6.9: Location estimation with DTR-UWB systems.
LDC DTR−UWB
6
SNR = 30 dB
Sensor true location
Location estimates
LS Method
5
Y coordinate [m]
P = 128
4
3
2
1
0
0
1
2
3
4
5
6
X coordinate [m]
7
8
9
10
6
Y coordinate [m]
5
Sensor true location
SI Method
Location estimates
4
3
2
1
0
0
1
2
3
4
5
6
X coordinate [m]
7
8
9
10
Figure 6.10: Location estimation with LDC DTR-UWB systems.
103
Chapter 6. Localization with Transmitted Reference Systems
DP TR−UWB
6
Sensor true location
Location estimates
LS Method
5
Y coordinate [m]
P = 128 SNR = 30 dB
4
3
2
1
0
0
1
2
3
4
5
6
X coordinate [m]
7
6
Y coordinates [m]
9
10
Sensor true location
Location estimates
SI Method
5
8
4
3
2
1
0
0
1
2
3
4
5
X coordinates
6
7
8
9
10
Figure 6.11: Location estimation with DP TR-UWB systems.
TR-UWB systems ⇒
Performance
⇓
Average p̂LS
Average p̂SI
Average ϵLS [cm]
Average ϵSI [cm]
DTR
UWB
η=0
(5.041,1.944)
(4.982,1.999)
48.43
28.57
LDC
DTR-UWB
DP
TR-UWB
(4.886,1.341)
(4.822,1.802)
127.6
58.702
(5.035,1.948)
(4.976,2.015)
59.95
33.60
Table 6.1: Localization accuracy of different TR-UWB systems.
104
Chapter 6. Localization with Transmitted Reference Systems
6.4
105
Summary
Localization with different TR-UWB systems was dealt with in this chapter. Different
distance measurement techniques were presented, namely the TOA, TDOA, AOA and
RSS methods. A new approach to distance measurement, similar to the RSS technique
was introduced. It uses the statistical analysis of the energy collected at the output of the
receiver rather than relying on the knowledge of the path loss model in RSS technique. The
performance of the proposed distance estimator was demonstrated for different TR-UWB
systems. Further, the localization capability of the considered TR-UWB systems were
analyzed using the LS and SI localization methods. Simulation results have shown the
superiority of the SI method over the LS method. It was also shown that the DTR-UWB
and DP TR-UWB are more suitable for localization than LDC DTR-UWB system.
Chapter 7
Conclusions and Outlook
7.1
Conclusions
The main objective of this doctoral thesis is to present the conceptual design and analysis of a transmitted reference (TR)-UWB impulse response (IR) receiver that has dual
functionality in providing robust, low power data communication and accurate location
information in wireless sensor networks (WSNs). TR signaling in combination with an autocorrelation receiver (AcR) is especially suitable in this context because it allows a lower
design complexity compared to coherent receivers. It allows higher accuracy in distance
measurement as well.
In this thesis, the theory of polynomial nonlinear systems was revisited. Volterra series
expansions were also presented. Further, the Wiener and Hammerstein models, which are
the simplified forms of the Volterra model, were discussed.
The efficient design of UWB communications systems requires an appropriate channel
model that accurately describes the UWB propagation. The description and discussion of
appropriate channel models for mobile radio systems and UWB radio systems, in particular,
were provided in this thesis. Emphasis was made on the IEEE 802.15.4a standardized
channel model for UWB Communications.
The polynomial nonlinear models and the channel models are important in the simulation and analysis of low complexity TR-UWB systems. The TR-UWB systems considered, in this thesis, are the differential transmitted reference (DTR)-UWB system, the
low duty cycle (LDC) DTR-UWB system and the dual pulse (DP) TR-UWB system.
Discrete-time equivalent system models for these TR-UWB systems were derived and their
delay-dependence studied. TR-UWB receiver statistics were analyzed from two perspectives: Gaussian approximation on the noise terms in the receiver and the analysis using
the equivalent system and noise models of the receiver. Both perspectives led to the BER
performance analysis.
Synchronization is a particularly acute problem faced by UWB systems due to the fact
that they employ low-power ultra-short pulses. Coarse synchronization and fine synchronization algorithms were developed and evaluated in this thesis, utilizing a known training
sequence. A novel data-aided timing acquisition technique for frame-level synchronization
of DTR-UWB systems was suggested. Furthermore, a simple timing offset estimator for
106
Chapter 7. Conclusions and Outlook
107
low complexity TR-UWB systems, assuming no ISI, was considered. Both the classical
and bayesian approaches were used for estimating the timing offset. It was shown that
both approaches yield the same timing offset estimate. The theoretical performance limit
of such timing offset estimation was illustrated through the derived Cramer-Rao Lower
Bound (CRLB) for the variance of the timing offset estimates.
Assuming that coarse synchronization has been achieved and hardware clock timing
error is known, the proposed timing offset algorithm was used as a new approach for
distance measurement. It is a RSS-like technique that uses the statistical analysis of the
distance-dependent discrete-time TR-UWB system models, instead of the path loss model,
to estimate the distance between the transmitter and the receiver. Different measurement
techniques for localization were discussed in this thesis from the viewpoint of UWB systems.
The general system architectures of localization systems were presented. Further, the
performance of the proposed distance estimator was demonstrated for different TR-UWB
systems. Finally, the localization capability of the considered TR-UWB systems were
analyzed using the LS and SI localization methods.
7.2
Summary of Contributions
The main contributions of this thesis are summarized below


Systems modelling {







Gaussian approximation


Low complexity TR systems Performance analysis


using derived models




Average
BER
derivation



 Coarse synchronization




Fine synchronization
Synchronization
and
Localization


 Distance and location estimation

7.3
Discussion and Recommendations
The equivalent system models for TR-UWB systems presented in this thesis constitute
an important step towards developing an emulator for TR-UWB systems. Such emulator
would be very useful for developing optimal and suboptimal detection algorithms for differential UWB communications systems under ISI or without ISI. It would also be a powerful
tool for developing signal processing algorithms for equalization and synchronization.
The performance of the different TR-UWB systems studied is summarized in Tab. 7.1.
DTR-UWB systems are recommended for short-range radio location-aware WSN applications with lower data rate requirement but higher localization accuracy requirement. DP
TR-UWB systems have lower complexity, but provide higher data rate, when compared to
DTR-UWB systems, at the expense of loosing some localization accuracy. Thus, they are
recommended for short-range radio location-aware WSN applications with stringent low
cost requirements.
TR-UWB systems ⇒
Performance
⇓
Symbol rate [Mb/s]
Timing offset rmse [ns]
(τ = 5 ns, P = 256, SNR = 30 dB)
Mean localization error [cm] (LS method)
Mean localization error [cm] (SI method)
DTR
UWB
η=0
1
1.65
LDC
DTR-UWB
DP
TR-UWB
5
1.9
5
1.7
48.43
28.57
127.6
58.702
59.95
33.60
Table 7.1: Performance of different TR-UWB systems.
Sensor network localization has been an active area of research for the last few years. For
WSNs, the ability for sensor nodes to determine their position through automatic means
is recognized as an essential capability. The research community has made great strides
in ranging technologies, systems for infrastructure-based localization, and algorithms and
techniques for ad-hoc localization. UWB systems are being investigated for ad-hoc and
wireless sensor networks, where it is not necessary to have fixed base stations like with
conventional radio systems. Hence, each node need to have a routing function. A multihop transmission will also be necessary if the area cover is larger than the permitted
distance of communication between nodes.
As already discussed in this thesis, the very large bandwidth of UWB technology, besides providing the possibility to transmit at very high data rate, also provides very accurate
temporal and spatial information. This high accuracy allows for new prospectives in terms
of synchronization, positioning and tracking. In the case of tracking a mobile sensor node
in a wireless system, two recursive tracking algorithms, namely the extended Kalman filter
(EKF) and the unscented Kalman filter (UKF), can be employed as smoothing algorithms
on the location estimates. The Kalman filter is one of the best optimal recursive Bayesian
estimator for linear tracking problems.
The design of an UWB indoor network as a local augmentation to the global navigation
satellite system (GNSS) positioning, can be envisaged. It would enable a seamless absolute
positioning and ensure a smooth outdoor-indoor transition. It would also help overcoming
signal attenuation and extending coverage, by pseudo-ranging from low-level ground based
UWB beacon nodes.
Further in this context is the development of a joint UWB-IR/inertial measurement
unit (IMU) system for joint communication and positioning/location in industry and logistics scenarios. Such system be based on peer-to-peer mechanisms with network routing
functionality. The system elements might be either mobile system elements or fixed system elements. The major difference is that fixed nodes are furnished with their absolute or
relative position, whereas mobile nodes use fixed nodes to obtain their absolute or relative.
Whether mobile or fixed, systems elements would comprise two loosely coupled parts: the
UWB subsystem and the IMU subsystem. Both would provide data into a data fusion
module computing enhanced position co-ordinates.
108
Appendix I
Gaussian Integral
The Gaussian integral, or probability integral, is an integral of the Gaussian function e−x .
It is expressed analytically as
2
∫∞
e−x dx =
2
√
π.
(1)
−∞
2
bx+c
The integral of any Gaussian function ae−( f ) is reducible in terms of the Gaussian
integral.
Let make the change of variable y = bx + c ⇒ dx = dy/b,
∫∞
ae−(
) dx = a
b
bx+c 2
f
−∞
∫+∞
y 2
e−( f ) dy.
(2)
−∞
Substituting y with f z, and using (1) gives
∫∞
ae−(
bx+c 2
f
) dx =
−∞
=
af
b
∫∞
e−z dz
2
(3)
−∞
af √
π.
b
The integrals of the exponential functions x2 e−bx and xn e−bx are definite and are
expressed analytically as
2
∫∞
x2 e−bx dx =
−∞
∫∞
xn e
−∞
−bx2
dx =
2



1
2
(2k−1)!! √ π
b
2k+1 bk
k!
2bk+1
√π
b3
2
b > 0,
(4)
b > 0, n = 2k, k ∈ Z
,
(5)
n = (2k + 1)
where Z is the field of integers, n!! denotes the double factorial of n and is defined recursively by
109
{
n!! =
1
ifn ∈ {−1, 0, 1}
.
n(n − 2)!! ifn ≥ 2
(6)
For example, 8!!= 2 · 4 · 6 · 8 = 384 and 9!!= 1 · 3 · 5 · 7 · 9 = 945. Some identities involving
double factorials are
(2n)!! = 2n n!
.
(2n + 1)!! = (2n+1)!
2n n!
(7)
Following the same change of variables as in (2) and (3), in this thesis, more general
analytical expressions to (2) and (3) are given as
∫∞
)
(
bx+c2
2 −
f
ax e
−∞
∫∞
−∞
axn e−(
bx+c 2
f
) dx =
dx =
a
2
( f )3 √
b
π b > 0,
(8)

( )2k+1 √
(2k−1)!!

π n = 2k
 a 2k+1 bk fb


a k!2
110
( f )2k+2
b
.
n = 2k + 1
(9)
Appendix II
Energy of Gaussian pulses
∫∞
∫∞
−∞
K12 e−4π(t/tm ) dt.
2
ωg2 (t)dt =
Eg =
(10)
−∞
√
Using (3) with a = K12 , b = 2 π and f = tm , we have
Eg =
∫∞
∫∞
ωg21 (t)dt
Eg1 =
Using (8) with a =
K12
=
−∞
−∞
16π 2 K12
t4m
K12 tm
.
2
(11)
16π 2 −4π(t/tm )2
e
dt.
t4m
(12)
√
, b = 2 π and f = tm , we have
K12 π
.
tm
∫∞ 2
∫∞ 16π2 K12 (
1−
ωg2 (t)dt =
t4
(13)
Eg1 =
Eg2 =
=
−∞
∫∞
−∞
+
=
m
−∞
16π 2 K12
t4m
∫∞
2
e−4π(t/tm ) dt −
−∞
16π 2
Eg
t4m
−
8π
E
t2m g1
−
8π
E
t2m g1
2
e−4π(t/tm ) dt
128π 3 K12 2 −4π(t/tm )2
te
dt
t6m
(14)
+ Eg21 .
162 π 4 K12
,
t8m
Eg21 =
16π 2
Eg
t4m
−∞
)2
162 π 4 K12 4 −4π(t/tm )2
te
dt
t8m
Eg21 is computed using (9) with a =
Since
∫∞
4πt2
t2m
√
b = 2 π, f = tm , n = 4 and k = 2.
6π 2 K12
.
t5m
= 0, we finally have
111
(15)
Eg2 =
6π 2 K12
.
t5m
112
(16)
Appendix III
Posterior Probability Density
Function
Using the Bayes’ rule, we can express p(zeτ |zs ) as
p(zs |zeτ )p(zeτ )
p(zs )
p(zs |zeτ )p(zeτ )
.
∞
∫
p(zs |zeτ )p(zeτ )dzeτ
p(zeτ |zs ) =
=
(17)
−∞
Using (5.14) and (5.15), (17) can be expressed as
[
]
]
[
P∑
−1
1
2
(zs [i] − zeτ ) exp − 2σ12 (zeτ − θ(τ ))2
exp − 2σ2
νe
ze
i=0
[
]
p(zeτ |zs ) = ∞
,
]
[
P
−1
∫
∑
1
1
2
2
exp − 2σ2
(zs [i] − zeτ ) exp − 2σ2 (zeτ − θ(τ )) dzeτ
νe
∞
(18)
ze
i=0
where θ(τ ) = Adi exp(Bτ ). After simplification, we obtain
p(zeτ |zs ) =
[ (
)]
1
2 −2P z z̄ )+ 1 (z −θ(τ ))2
exp −1
(P
z
eτ
s
eτ
eτ
2
2
σ2
σze
)]
[ ( νe
∞
∫
−1
1
1
2 −2P z z̄ )+
2
exp 2
(P
z
(z
−θ(τ
))
dzeτ
eτ
s
eτ
eτ
2
2
σνe
∞
=
(
P
2
σνe
1
2
σze
νe
ze
f (zeτ ) =
+
(
Let v = σP2 + σ12
)
2
zeτ
)−1
−2
(
∞
∫
∞
P z̄s
2
σνe
exp[ −1
f (zeτ )]
2
exp[ −1
f (zeτ )]dzeτ
2
+
(
and m =
f (zeτ ) =
=
=
θ(τ )
2
σze
P z̄s
2
σνe
)
σze
(19)
.
2
)
zeτ + θ(τ
2 . This function is quadratic in zeτ .
σze
)
)
+ θ(τ
v, then we have
2
σ
ze
2 )
1 2
z − 2 mv zeτ + θσ(τ
2
v eτ
ze
2
1 2
2
(z − 2mzeτ + m2 ) − mv
v eτ
2 )
2
1
(z − m)2 − mv + θσ(τ
2 .
v eτ
ze
Putting the expression of (20) back into (17), we have
113
+
θ2 (τ )
2
σze
(20)
p(zeτ |zs ) =
[
(
)]
2
θ 2 (τ )
1
exp[− 2v
(zeτ −m)2 ] exp − 21
− mv
2
σze
[
(
)]
.
∞
∫
1 θ 2 (τ )
m2
1
2
−
dzeτ
exp[− 2v (zeτ −m) ] exp − 2
2
v
(21)
σze
∞
Following the requirement that p(zeτ |z) integrate to 1 [72], we finally have
[
]
1
1
2
p(zeτ |zs ) = √
exp − (zeτ − m) .
(22)
2v
2πv
This shows
that )z(eτ conditioned
on zs is a Gaussian( random )variable with mean
(
)
mzeτ |zs =
P z̄s
2
σνe
+
θ(τ )
2
σze
P
2
σνe
+
1
2
σze
−1
and variance σz2eτ |zs =
114
P
2
σνe
+
1
2
σze
−1
.
Appendix IV
Least Squares Exponential Fitting
Let E{zeτ } = z¯e (n) = A exp(Bn), where n ∈ {0, 1, . . . , N − 1} are the different delay values
in ns, as shown in Fig. 4.8 for N = 51.
Since ln(z¯e (n)) = ln(A) + Bn = ε + γn, to determine A and B we start by minimizing the
function
χ2 (ε, γ) =
N
−1
∑
z̄e (n)[ln(z̄e (n)) − ε − γn]2 .
(23)
n=0
Applying least squares fitting gives
 N −1
N
N∑
−1
∑
∑


z̄e (n) ln(z̄e (n))
nz̄e (n) =
z̄e (n) + γ
 ε
n=1
n=0
n=0
(24)
N
N∑
−1
N∑
−1
∑


 ε
nz̄e (n) ln(z̄e (n))
n2 z̄e (n) =
nz̄e (n) + γ
n=1
n=0
n=0
The system of equation in (24) can be rewritten in matrix format

 N −1
 N −1
N∑
−1
∑
∑
[
]
z̄e (n) ln (z̄e (n))
z̄
(n)
nz̄
(n)
e
e
 ε


n=0

 n=0
 n=0
=
−1
−1
N∑
−1
 γ
 N∑
 N∑
nz̄e (n)
n2 z̄e (n)
nz̄e (n) ln (z̄e (n))
n=0




(25)
n=0
n=0
or
[
M
ε and γ are obtained by solving (26).
[
ε
γ
ε
γ
]
= m.
(26)
]
= M−1 m.
(27)
Finally, A and B are expressed as
A = exp(ε)
B = γ.
115
(28)
Appendix V
Linear Least Squares
Let us consider a data model expressed as
x[n] = s[n] + w[n], n = 0, 1, · · · , N − 1.
(29)
w[n] is a zero mean noise process and s[n] is the signal model. (29) can be expressed in
compact form as
x = s + w.
(30)
Let us consider a unknown vector parameter θ= [θ1 , θ2 , . . . , θp ] of dimension p × 1 to
be estimated. For the signal s=[s[0], s[1], . . . , s[N − 1]]T to be linear in the unknown
parameters, it is assumed, using matrix notation, that
s = Hθ,
(31)
where H is a known N × p matrix (N > p) of rank p. The matrix H is known as the
observation matrix. The least square error (LSE) is found by minimizing
J(θ) =
N∑
−1
(x[n] − s[n])2
n=0
= (x − Hθ)T (x − Hθ)
= xT x − xT Hθ − θ T HT x + θ T HT Hθ
= xT x − xT − 2xT Hθ + θ T HT Hθ.
(32)
It is important noting that xT Hθ is a scalar and J is a quadratic function of θ. The
gradient ∂J(θ)/∂θ is expressed as
∂J(θ)
= −2HT x + 2HT Hθ.
∂θ
Setting the gradient equal to zero yields the LSE
θ̂ = (HT H)−1 HT x.
116
(33)
(34)
Appendix VI
Design of ZESS wireless sensor nodes
Design of sensor nodes
1. Hardware Selection
In our completed experimental implementation of a sensor node we feed the digital
input data from a digital source such as a personal computer (PC) via a USB interface to an 8051-compatible microcontroller. The sensor module is represented by a
GPS module.
The signal processing and network protocol functions are handled by the Cypress
Semiconductor’s EZ-USB which is an enhanced 8051 core with fast execution time
and added features. It uses internal RAM for program and data storage, making it
a soft solution. The EZ-USB chip operates at 3.3V.
The radio module used is a single chip 2.4 GHz Gaussian Frequency Shift Keying
(GFSK) transceiver based on the HomeRF technology and capable of delivering data
rate up to 1 Mbps.
An illustration of the experimental design of a sensor node with wireless communication capabilities is shown in Fig. 1.
2. Firmware Development
Generally, a firmware is software (programs or data) that is embedded in a hardware
device, allowing reading and execution of the software, but does not allow modification, as for example, writing or deleting data by an end user. A great feature of the
firmware we developed is that it can be modified. This is because of the soft (RAMbased) feature of the EZ-USB microcontroller that allows unlimited configuration
and upgrades.
Integration of GPS data
The main objective of this work is to provide our sensor nodes with positioning information for localization purposes in WSN. This work consists of three tasks:
1. Serial Interface of a Handheld GPS Receiver
Based on the strictly low energy consumption requirements of sensor nodes, we have
R
decided to select the Lassen⃝
IQ GPS module whose main features are: ultra low
power consumption - economical price - Aided GPS (A-GPS) capability through TSIP
117
Clock signal
8051 simulation tools
C51 compiler
EZ-USB control panel
Matlab DLL
Data
Sensor Node
GPS Sensor
USB 2.0
dio
Ra
ne
an
Ch
Programming
Configuration
Cypress EZ-USB FX2 µC
Control
Packetization
l
Home RF
UWB
Zigbee
Verifying the clock and data timing
diagrams with an oscilloscope
USB 2.0
Figure 1: Experimental design of wireless sensor nodes
(Trimble Standard Interface Protocol) for faster acquisition - possibility to switch the
module automatically to higher sensitivity when satellite signals are week - 12 channel simultaneous operation for complete position, velocity and time (PVT) solutions
- supports standards NMEA (National Marine Electronics Association ) 0183, TSIP,
TAIP (Trimble ASCII Interface Protocol) and Differential GPS (DGPS).
R
As an embedded design, the Lassen⃝
IQ GPS module provides direct CMOS compatible TTL level serial I/O. The RX and TX signals are driven directly by the dual
R
universal asynchronous receiver/transmitter (UART) on the Lassen⃝
IQ GPS receiver. Interfacing these signals directly to the UART of our 8051-Microcontroller
provides direct serial communication without the complication of RS-232 or RS-422
line drivers.
2. The NMEA Protocol
The hardware interface for GPS units is designed to meet the NMEA requirements.
R
The Lassen⃝
IQ GPS receiver normally operates using one of the three protocols
- TSIP, TAIP or NMEA 0183. Our program to dissect the desired GPS message
string and extract the required information is based on the NMEA 0183 which is an
industry standard protocol common to marine applications. The signal characteristics of NMEA 0183 serial data transmission are: Baud rate=4800, Data bits=8, Stop
bits=1, Parity bits=none.
The entire protocol encompasses over 50 messages, but only a subset of these messages apply to a particular GPS receiver. Indeed, at 4800 baud we can only send 480
characteres per seconds. Since NMEA 0183 messages can have up to 79 characters,
118
we are limited to 6 different messages.
3. Dissection of the desired GPS message
The NMEA protocol is designed to run as a process in the background spitting out
messages which are then captured as needed by a program. Thus our program, written in C51, will sample the data stream, use that data for screen display, and then
sample the data again, based on our information requirements. Depending on the
time needed to use the data there can be easily a lag of 4 seconds in the responsiveness to changed data.
As an example, a dissected NMEA message structure, based on some GPS measurements (June 11th 2005) at ZESS is shown and explained below:
$ GPGGA,215354.463,5054.2113,N,00801.7561, E,1,07,1.1,370.4,M,,,1.1,
0000*2A
The “$” sign indicates the start of a message.
The GP talker identification signifies a GPS source (source of navigation information.
The GGA message identification describe the message content and the number and
order of the data fields. GGA describes the Global Positioning System fixed data
The “,” serve as delimiters for the data fields.
215354.463 is a specific data field in the GPGGA message. In fact it describes the
UTC Time given as 21h53mn54.463s. The other following fields describe, respectively,
the latitude, the North/South indicator, the longitude, the East/West indicator, the
position fix indicator, the satellites used (Range 0 to 12), the Horizontal Dilution of
Precision (HDOP), the Mean Sea Level (MSL) altitude, the unit used (M=meters).
0000 is the differential referential station ID.
The “*” sign serves as a checksum delimiter and 2A are the two ASCII characters
which indicates the hexadecimal value of the checksum.
Not seen in the message above are the carriage return [CR] and line feed [LF] whose
combination terminate the message.
4. Extraction of the required information
Upon dissection of the required GPS message, our C51 program extract the desired
fields of interest (UTC Time, Position etc...) and store the data in dedicated buffers
for further use including data display, localization solutions, Aided GPS experiments,
emulation of position-based routing algorithms, etc...
The algorithm flowchart for extracting the required position information is shown in
Fig. 2.
Communication Radio Module - Personal Computer
A dynamic-link library (DLL) named USB HRF DLL and its corresponding source-code
for the microcontroller were developed with the objective of enabling a relatively simple
communication capability between the Home RF hardware and a Personal Computer (PC)
via the USB interface. Since the USB Framework in the microcontroller exhibits a relatively complex interrupt processing system, it was advantageous to make a change on the
hardware. This concerns the Interrupt signal INT0 for signaling that Home RF has received data. This was set to INT5 (Trigger in). This change is absolutely necessary for
the signaling of a receive packet to occur.
119
Start
No
Check for rx GPS
msg string
Yes
Read GPS msg string
and store in buffer
Is the stored string
a <$GPGGA>?
Refresh
buffer
No
Yes
Extract latitude longitude
and UTC msgs
Display
Figure 2: Algorithm flowchart
A special driver for the USB system must be installed. It can be found in the folder
Treiber USB CAM of the provided CD. Loading the DLL and corresponding microcontroller
programs, will allow the so-called renumeration to take place. This causes the PC to detect
the microcontroller as a new sensor interface. It is important noting that the hardware
and the DLL can be utilized fully only after the driver is correctly installed.
Two types of DLL can be exported, one for use under Matlab (Macro “Matlab” in
USB HRF DLL.h) and the other one under C++. Using the Matlab environment, the program HomeRF Matlab.m was written as a test-program which depicts the DLL principal
functionality.
Home RF Functionalities
The DLL and the programs which run on the microcontroller are closely interrelated
to each other. Each exported DLL function calls certain functions in the microcontroller
via the USB Protocol. Thus it is possible for the source code of the microcontroller to
adapt itself to new tasks without knowledge of the USB protocols or the entire USB
Frameworks. The basic idea in implementing the functions in the microcontroller is to try
to separate the USB specific functions from the Home RF functions. In this way, for any
user of the DLL and the frameworks in the microcontroller, only two files are important:
HRF.c and HRF.h. Both can be found in the directory . . . /Config USB HRF/Control8051
(see provided CD), including also all relevant functions addressed by the HomeRF radio
module. The block diagram in Fig. 3 shows the communication between the Home RF
device, the microcontroller, the DLL and the software application.
For the communication between the DLL and the Home RF hardware via the microcontroller four DLL functions with different tasks are provided:
120
I2C EEPROM
Hardware
PC with application software
Application
USB_HRF_DLL
USB framework
Home RF
functionality
Home RF
Microcontroller
USB
connection
ICs
Figure 3: Interaction between the application and the microcontroller functions used by the
radio module.
• USB Dll HRF Set Config: this function allows for the configuration settings to be
sent to the Home RF device (length of data payload, address width, destination
address for received data, enable/disable cyclic redundancy check (CRC), communication mode (Direct or ShockBurstTM ), RF data rate, crystal frequency, RF output
power, frequency channel and receive/transmit operation ).
• USB Dll HRF Transfer Data: this function allows the Home RF radio module to operate in Active Receive/Transmit modes (Direct or ShockBurstTM ). The ShockBurst
technology uses on-chip FIFO (First In, First Out) to clock in data at a low data
rate and transmit at a high rate (1 Mbps) thus enabling extremely power reduction.
In direct mode the radio module functions like a traditional RF device. Data must
be sent at 1Mbps or 250kbps at low data rate setting, for the receiver to detect the
signals.
• USB Dll HRF Receive Init: this function prepares the microcontroller programs to
accept data from the Home RF chip, at the rising edge of the DR signal, and to store
it internally. No data are passed on to the DLL and thus to the PC.
• USB Dll HRF Receive Data: this function allows to read in the received data via the
DLL.
These four DLL functions are related to the following three functions:
HomeRF Set Config, HomeRF Transmit Data and HomeRF Receive Data in the microcontroller, as shown in Fig. 4. When the exported functions of the DLL are called (through the
framework of the microcontroller), the necessary Home RF functions belonging to them
are activated.
121
HomeRF_Set_Config
USB_Dll_HRF_Set_Config
HomeRF_Transmit_Data
USB_Dll_HRF_Transfer_Data
USB_Dll_Receive_Data
HomeRF_Receive_Data
USB_Dll_Receive_Init
DLL
Microcontroller
Figure 4: Relation between the DLL functions and the HomeRF functions in the microcontroller
122
List of Publications
Journal Papers
1. G.F. Tchere, P. Ubolkosold, S. Knedlik, O. Loffeld., “Timing Offset Estimation for
Transmitted-Reference Impulse Radio UWB Systems”, IEEE Journal of Communications and Networks, submitted October 2008.
Conference Papers
1. P. Ubolkosold, G. F. Tchere, S. Knedlik, O. Loffeld, “A New Closed-Form Frequency
Estimator in the Presence of Fading-Induced Multiplicative Noise”, 67th IEEE Vehicular Technology Conference, Singapore, May 2008.
2. P. Ubolkosold, G. F. Tchere, S. Knedlik, O. Loffeld, “An Improved Frequency
Offset Estimator for SIMO-OFDM with Maximum Estimation Range”, 67th IEEE
Vehicular Technology Conference, Singapore, May 2008.
3. P. Ubolkosold, G.F. Tchere, S. Knedlik, O. Loffeld, “Simple Carrier Frequency Offset Estimators in Frequency Flat-fading Channels”, IEEE International Conference
on Communications (ICC 2007), Glasgow, Scotland, June 2007.
4. G.F. Tchere, P. Ubolkosold, S. Knedlik, O. Loffeld, “Bit Error Performance of
Differential Impulse Radio UWB systems”, 65th IEEE Vehicular Technology Conference, VTC Spring 2007, Dublin, Ireland, April 2007.
5. G.F. Tchere, P. Ubolkosold, S. Knedlik, O. Loffeld, “Communication and Localization in Wireless Sensor Networks using Differential Impulse Radio UWB systems”,
4th Workshop on Positioning, Navigation and Communication (WPNC 2007), Hannover, Germany, March 2007.
6. G.F. Tchere, P. Ubolkosold, S. Knedlik, O. Loffeld, “Bit Error Performance of
UWB Differential Transmitted Reference systems”, IEEE International Symposium
on Communications an Information Technologies (ISCIT), Bangkok, Thailand, October 2006.
7. G.F. Tchere, P. Ubolkosold, S. Knedlik, O. Loffeld, K. Witrisal, “Data-Aided Timing Acquisition in UWB Differential Transmitted Reference systems”, IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC),
Helsinki, Finland, September 2006.
123
8. P. Ubolkosold, G.F. Tchere, S. Knedlik, O. Loffeld, “Nonlinear Least-Squares Frequency Offset Estimation and its Simplified Versions for Flat fading Channels ”,
IEEE International Symposium on Communications an Information Technologies
(ISCIT), Bangkok, Thailand, October 2006.
9. P. Ubolkosold, G.F. Tchere S. Knedlik , O. Loffeld, “Data-Aided Joint estimation
of Phase and Doppler Parameters using an Extended Kalman Filter ”, IEEE International Symposium on Control, Communications and Signal Processing (ISCCSP
2006), Marrakech, Morocco, March 2006.
10. P. Ubolkosold, O. Loffeld, S. Knedlik, G.F. Tchere, “A Robust Data-Aided Frequency Offset Estimation for Single Frequency Signals ”, IEEE Sarnoff Symposium,
Princeton, New Jersey, USA, 2006
124
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