Linear time variant

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Dissertation

A TIME-FREQUENCY CALCULUS

FOR TIME-VARYING SYSTEMS

AND NONSTATIONARY PROCESSES

WITH APPLICATIONS

Gerald Matz([email protected])

Institute of Communicationsand Radio-Frequency Engineering

Vienna University of Technology

This dissertation is available online at

http://www.nt.tuwien.ac.at/dspgroup/tfgroup/doc/psfiles/GM-phd.ps.gz

NACHRICHTENTECHNIKINSTITUT FR

UND HOCHFREQUENZTECHNIK


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DISSERTATION

A TIME-FREQUENCY CALCULUS

FOR TIME-VARYING SYSTEMS

AND NONSTATIONARY PROCESSES

WITH APPLICATIONS

ausgefuhrt zum Zwecke der Erlangung des akademischen Grades eines

Doktors der technischen Wissenschaften

unter der Leitung von

Ao. Univ.-Prof. Dipl.-Ing. Dr. Franz HlawatschInstitut fur Nachrichtentechnik und Hochfrequenztechnik

eingereicht an der Technischen Universitat Wien

Fakultat fur Elektrotechnik

vonGerald Matz

Servitengasse 13/9

1090 Wien

Wien, im November 2000


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Die Begutachtung dieser Arbeit erfolgte durch:

1. Ao. Univ.-Prof. Dipl.-Ing. Dr. F. Hlawatsch

Institut fur Nachrichtentechnik und Hochfrequenztechnik

Technische Universitat Wien

2. O. Univ.-Prof. Dipl.-Ing. Dr. W. Mecklenbrauker

Institut fur Nachrichtentechnik und Hochfrequenztechnik

Technische Universitat Wien


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To Petra


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Abstract

This thesis introduces an approximate time-frequency calculus for underspread linear time-varyingsystems (i.e., time-varying systems that effect only small time-frequency shifts of the input signal)

and underspread nonstationary random processes (i.e., nonstationary processes that feature only small

time-frequency correlations).

After briefly describing the major difficulties encountered with time-varying systems and non-

stationary processes, we introduce an extended definition of underspread systems. Our extended

underspread concept is based on weighted integrals and moments of the systems generalized spread-

ing function. Subsequently, numerous approximations are presented which show that in the case of

underspread systems the generalized Weyl symbol constitutes an approximate time-frequency trans-

fer function. As a mathematical underpinning of our transfer function approximations, we provide

bounds on the associated approximation errors that involve the previously defined weighted integrals

and moments of the generalized spreading function.

We then consider nonstationary random processes and provide an extended definition of under-

spread processes. This extended underspread concept is based on weighted integrals and moments of

the generalized expected ambiguity function of the process. Subsequently, two fundamental classes of

time-varying power spectra are introduced and analyzed: type I spectra that extend the generalized

Wigner-Ville spectrum and type II spectra that extend the generalized evolutionary spectrum. We

show that in the case of underspread processes, the various members of these two classes of spectra

are approximately equivalent to each other and (at least) approximately satisfy several desirable prop-

erties. Our approximations are again supported by bounds on the associated approximation errors.

These bounds are formulated in terms of the previously defined weighted integrals and moments of

the generalized expected ambiguity function. The definition and analysis of time-frequency coherence

functions concludes our discussion of time-varying power spectra.

Finally, we illustrate the practical relevance of our theoretical findings by considering several ap-

plications in the areas of statistical signal processing and wireless communications. These applications

include nonstationary signal estimation and detection, the sounding of mobile radio channels, multi-

carrier communications over time-varying channels, and the analysis of car engine signals.

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Kurzfassung

Diese Dissertation beschreibt einen Zeit-Frequenz-Kalkul fur lineare zeitvariante Systeme mitgeringen Zeit-Frequenz-Verschiebungen (underspread-Systeme) und fur instationare stochastische

Prozesse mit schwachen Zeit-Frequenz-Korrelationen (underspread-Prozesse).

Nach einer kurzen Darstellung der Probleme, welche bei zeitvarianten Systemen und instation aren

Prozessen auftreten, stellen wir ein erweitertes Konzept von underspread-Systemen vor. Dieses

beruht auf gewichteten Integralen und Momenten der verallgemeinerten Spreading-Funktion des Sys-

tems. Im Weiteren werden zahlreiche Approximationen formuliert, welche zeigen, dass fur die Klasse

der underspread-Systeme das verallgemeinerte Weyl-Symbol naherungsweise als Zeit-Frequenz-

Ubertragungsfunktion interpretiert und verwendet werden kann. Die angesprochenen Approxima-

tionen werden durch obere Schranken fur die zugehorigen Approximationsfehler mathematisch un-

termauert, wobei diese Schranken mit den zuvor definierten gewichteten Integralen und Momenten

formuliert werden.

Danach betrachten wir instationare stochastische Prozesse und geben eine erweiterte Definition

von underspread-Prozessen. Diese beruht auf einer globalen Charakterisierung der Zeit-Frequenz-

Korrelation des Prozesses mittels gewichteter Integrale und Momente des Erwartungswertes der ver-

allgemeinerten Ambiguitatsfunktion. Schlielich werden zwei Klassen von zeitvarianten Spektren

vorgestellt und analysiert: Spektren vom Typ I (eine Erweiterung des verallgemeinerten Wigner-

Ville-Spektrums) und Spektren vom Typ II (eine Erweiterung des verallgemeinerten evolutionaren

Spektrums). Wir zeigen, dass im Fall von underspread-Prozessen die verschiedenen Spektren bei-

der Klassen naherungsweise aquivalent sind und gewisse wunschenswerte Eigenschaften (zumindest)

naherungsweise erfullen. Wieder werden alle Naherungen durch obere Schranken fur die entsprechen-

den Approximationsfehler untermauert. Die Diskussion zeitvarianter Spektren wird mit der Definition

und Analyse von Zeit-Frequenz-Koharenzfunktionen beendet.

Abschlieend illustrieren wir die Anwendung unserer theoretischen Ergebnisse auf gewisse Be-

reiche der statistischen Signalverarbeitung und der Mobilkommunikation. Diese Bereiche umfassen

die instationare Signalschatzung und -detektion, die Messung von Mobilfunkkanalen, Mehrtrager-

Ubertragungsverfahren fur zeitvariante Kanale und die Analyse instationarer Motorsignale.

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Acknowledgements

It is my pleasure to express my sincere thanks to several people who have contributed to this thesis

in various ways.

I am particularly indebted to F. Hlawatsch who continually furthered my personal and pro-

fessional development. His constant advice and thorough proofreading resulted in countless useful

suggestions that helped a lot to improve this thesis with regard to both technical content and presen-

tation.

W. Mecklenbrauker kindly agreed to act as a referee and pointed me to generalized Chebyshev

inequalities. His support and continual interest in the progress of this work are gratefully acknowl-

edged.

W. Kozek pioneered the theory of underspread systems and processes. His influence on this

thesis and my research in general is sincerely appreciated.

I am grateful to J. F. Bohme, S. Carstens-Behrens, and M. Wagner for introducing me to the

problems of car engine diagnosis and providing me with the car engine data used in Chapter 4.

I am indebted to A. Molisch for several enlightening discussions in which he generously shared

with me his expertise on mobile radio.

Finally and most importantly, I have been permanently backed up by my wife Petra. Her

sympathy, encouragement, love, and support have been vital for the completion of this thesis. I owe

her more than anyone else.

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Contents

1 Introduction 11.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Review of Time-Invariant/Stationary Theory . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Transfer Functions of Time-Invariant and Frequeny-Invariant Linear Sytems . . . . 3

1.2.2 Power Densities of Stationary and White Processes . . . . . . . . . . . . . . . . . 4

1.3 Time-Varying Systems and Nonstationary Random Processes . . . . . . . . . . . . . . . . 5

1.3.1 Time-Varying Systems and the Generalized Weyl Symbol . . . . . . . . . . . . . . 6

1.3.2 Nonstationary Processes and Time-Varying Power Spectra . . . . . . . . . . . . . 6

1.4 The Importance of Being Underspread . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Underspread Linear Time-Varying Systems . . . . . . . . . . . . . . . . . . . . . . 8

1.4.2 Underspread Nonstationary Processes . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Signal Processing Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 Overview of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Underspread Systems 152.1 Operators with Compactly Supported Spreading Function . . . . . . . . . . . . . . . . . . 16

2.1.1 General Support Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.2 Definition of Displacement-limited Underspread Operators . . . . . . . . . . . . . 17

2.1.3 Unitary Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.4 Operator Sums, Adjoints, Products, and Inverses . . . . . . . . . . . . . . . . . . 20

2.2 Operators with Rapidly Decaying Spreading Function . . . . . . . . . . . . . . . . . . . . 21

2.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.2 Weighted Integrals and Moments of the Generalized Spreading Function . . . . . . 22

2.2.3 Unitary Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.4 Underspread Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.5 Operator Sums, Adjoints, Products, and Inverses . . . . . . . . . . . . . . . . . . 32

2.2.6 Non-Band-Limited Parts of Operators with Rapidly Decaying Spreading Function . 35

2.3 Underspread Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3.1 Approximate Uniqueness of the Generalized Weyl Symbol . . . . . . . . . . . . . . 37

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2.3.2 The Generalized Weyl Symbol of Operator Adjoints . . . . . . . . . . . . . . . . . 40

2.3.3 Approximate Real-Valuedness of the Generalized Weyl Symbol . . . . . . . . . . . 41

2.3.4 Composition of Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3.5 Composition ofH with H+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.3.6 Operator Inversion Based on the Generalized Weyl SymbolPart I . . . . . . . . . 502.3.7 Operator Inversion Based on the Generalized Weyl SymbolPart II . . . . . . . . 57

2.3.8 Approximate Eigenvalues and Eigenfunctions . . . . . . . . . . . . . . . . . . . . 61

2.3.9 Approximate Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.3.10 Input-Output Relation for Deterministic Signals Based on the Generalized Weyl

Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.3.11 (Multi-Window) STFT Filter Approximation of Time-Varying Systems . . . . . . . 73

2.3.12 Infimum and Supremum of the Weyl Symbol . . . . . . . . . . . . . . . . . . . . 77

2.3.13 Approximate Non-Negativity of the Generalized Weyl Symbol of Positive Semi-

Definite Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

2.3.14 Boundedness of the Generalized Weyl Symbol of Self-Adjoint Operators . . . . . . 842.3.15 Maximum System Gain (Operator Norm) . . . . . . . . . . . . . . . . . . . . . . 87

2.3.16 Approximate Commutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

2.3.17 Approximate Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

2.3.18 Sampling of the Generalized Weyl Symbol of Underspread Operators . . . . . . . . 92

3 Underspread Processes 973.1 Time-Frequency Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.1.2 Time-Frequency Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . 983.1.3 The Expected Ambiguity Function . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.1.4 Extended Concept of Underspread Processes . . . . . . . . . . . . . . . . . . . . . 103

3.1.5 Innovations System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.1.6 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.2 Elementary Time-Varying Power Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.2.1 Generalized Wigner-Ville Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.2.2 Generalized Evolutionary Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.3 Type I Time-Varying Power Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

3.3.1 Definition and Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

3.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293.3.3 Approximate Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

3.3.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

3.4 Type II Time-Varying Power Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

3.4.1 Definition and Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

3.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

3.4.3 Approximate Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

3.4.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

3.5 Equivalence of Time-Varying Power Spectra . . . . . . . . . . . . . . . . . . . . . . . . . 146

3.5.1 Equivalence of Generalized Wigner-Ville Spectrum and Generalized Evolutionary

Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147


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3.5.2 Equivalence of Type I and Type II Spectra . . . . . . . . . . . . . . . . . . . . . . 149

3.6 Input-Output Relations for Nonstationary Random Processes . . . . . . . . . . . . . . . . 151

3.6.1 Input-Output Relation Based on the Generalized Wigner-Ville Spectrum . . . . . . 151

3.6.2 Input-Output Relation Based on the Generalized Evolutionary Spectrum . . . . . . 154

3.7 Approximate Karhunen-Loeve Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 1573.8 Time-Frequency Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

3.8.1 Spectral Coherence and Coherence Operator . . . . . . . . . . . . . . . . . . . . . 159

3.8.2 Time-Frequency Formulation of the Coherence Operator . . . . . . . . . . . . . . 160

3.8.3 The Generalized Time-Frequency Coherence Function . . . . . . . . . . . . . . . . 164

4 Applications 1694.1 Nonstationary Signal Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

4.1.1 Time-Varying Wiener Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

4.1.2 Time-Frequency Formulation of the Time-Varying Wiener Filter . . . . . . . . . . 170

4.1.3 Time-Frequency Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

4.1.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

4.2 Nonstationary Signal Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.2.1 Optimal Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

4.2.2 Time-Frequency Formulation of Optimal Detectors . . . . . . . . . . . . . . . . . 176

4.2.3 Time-Frequency Detector Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 178


4.3 Sounding of Mobile Radio Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

4.3.1 Channel Sounder Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

4.3.2 Analysis of Measurement Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . 1834.3.3 Optimization of PN Sequence Length . . . . . . . . . . . . . . . . . . . . . . . . 185


4.4 Multicarrier Communication Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

4.4.1 Pulse-Shaping OFDM and BFDM Systems . . . . . . . . . . . . . . . . . . . . . 188

4.4.2 Approximate Input-Output Relation for OFDM/BFDM Systems . . . . . . . . . . 189


4.5 Analysis of Car Engine Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

4.5.1 Time-Varying Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

4.5.2 TF Coherence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

4.5.3 Subspace Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

5 Conclusions 1975.1 Summary of Novel Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

5.2 Open Problems for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

A Linear Operator Theory 207A.1 Basic Facts about Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

A.2 Kernel Representation of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

A.3 Eigenvalue Decomposition and Singular Value Decomposition . . . . . . . . . . . . . . . 210


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A.4 Special Types of Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

B Time-Frequency Analysis Tools 213B.1 Time-Frequency Representations of Linear, Time-Varying Systems . . . . . . . . . . . . . 214

B.1.1 Generalized Spreading Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

B.1.2 Generalized Weyl Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

B.1.3 Generalized Transfer Wigner Distribution and Generalized Input and Output Wigner

Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

B.2 Time-Frequency Signal Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

B.2.1 Short-Time Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

B.2.2 Generalized Wigner Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

B.2.3 Spectrogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

B.2.4 Generalized Ambiguity Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

B.3 Time-Frequency Representations of Random Processes . . . . . . . . . . . . . . . . . . . 224

B.3.1 Generalized Wigner-Ville Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 225

B.3.2 Generalized Evolutionary Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 226

B.3.3 Physical Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

B.3.4 Generalized Expected Ambiguity Function . . . . . . . . . . . . . . . . . . . . . . 227

C The Symplectic Group and Metaplectic Operators 229C.1 The Symplectic Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

C.2 Metaplectic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

C.3 Effects on Time-Frequency Representations . . . . . . . . . . . . . . . . . . . . . . . . . 233

Bibliography 237

List of Abbreviations 249


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1Introduction

[. . .] there is no doubt that linear systems will continue to be an objectof study for as long as one can foresee. Thomas Kailath

L

INEAR systems and random processes are the foundations of numerous signal modelling, analysis,

and processing schemes. In a formal sense, the characterization of linear systems and the second-order description of random processes can be based on the same mathematics: linear operator theory.

This viewpoint will be emphasized throughout this thesis.

A fundamental distinction must be made between time-invariant systems and stationary processes

on the one hand and time-varying systems and nonstationary processes on the other. The majority of

books and research papers restrict to time-invariant systems and stationary processes, both of which

can be treated in an efficient and intuitively appealing manner using Fourier analysis. In contrast,

time-varying systems and nonstationary processes have received much less attention.

In this introductory chapter, we review some basic facts of the theories of time-invariant linear

systems and stationary processes (Section 1.2). Further, in Section 1.3 we discuss the fundamentalproblems encountered when dealing with time-varying/nonstationary scenarios. Section 1.4 outlines

the central results of this thesis, which consist in the (approximate) solution of these basic problems

by means of an approximate time-frequency calculus of time-varying transfer functions and time-

varying power spectra. This time-frequency calculus is valid for the practically important classes of

underspread systems and underspread processes and it has the advantage of being conceptually simple,

computationally attractive, and physically intuitive. The chapter continues with an outline of signal

processing applications of our results in Section 1.5, a brief account of related work in Section 1.6, and

a summary of the major contributions of this thesis in Section 1.7.

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2 Chapter 1. Introduction

1.1 General Remarks

Linear systems and random processes are of fundamental importance in many engineering applica-

tions. In particular, linear systems provide useful models for communication channels and speech

production, are vital in transmitter/receiver design, and are used as filters in processing schemes

for signal separation, enhancement, and detection. Similarly, random processes are used to model

phenomena as diverse as speech and audio, communication signals, visual data, biological signals, or

signals arising in machine monitoring. Furthermore, undesired disturbances (noise and interference)

are usually modelled as being random, too. Hence, linear systems and random processes lie at the

heart of numerous (statistical) signal processing methods. The above short list of their applications

is far from being complete and, indeed, falls short of illustrating their ubiquity.

In a formal sense, linear systems and (second-order statistics of) random processes allow a unified

mathematical treatment via linear operator theory1 (a brief review of certain elements of linear oper-

ator theory is given in Appendix A; far more comprehensive treatments can be found in [69,158]). In

particular, a linear system can be associated to a linear operator H whose kernel equals the systems

impulse response h(t, t) relating the input signal x(t) and the output signal y(t) as2

y(t) = (Hx)(t) =

t

h(t, t) x(t) dt. (1.1)

Throughout this thesis, when talking about linear systems we will restrict our attention to Hilbert-

Schmidt (HS) operators (see Appendix A). The only exceptions to this general rule are i) linear

time-invariant (LTI) systems and linear frequency-invariant (LFI) systems (which are never HS); and

ii) unitary systems (see Subsection 2.1.3). In a similar way, a correlation operator Rx can be used

as second-order description of a random process x(t) in the sense that the kernel of Rx equals the

correlation function rx(t, t) = E{x(t) x(t)} of x(t) (here, E denotes expectation). We note that

the set of all correlation operators equals the subclass of positive semi-definite linear operators (see

Appendix A). Except in the case of stationary or white processes, we will implicitly assume the

processes involved to have finite mean energy, Ex E{x22} < . This implies that the correspondingcorrelation operator is trace-class (or nuclear, see Appendix A).

LTI systems and stationary processes can be efficiently dealt with using convolution and Fourier

transform techniques. The corresponding theories are well developed and allow to gain useful insights

(as discussed in Section 1.2). Unfortunately, Fourier transform techniques lose much of their appeal

and usefulness in the case of linear time-varying (LTV) systems and nonstationary processes. This

explains why LTV systems and nonstationary processes, while providing a very general framework

(much more general than LTI systems and stationary processes), are considerably more difficult to

treat (see Section 1.3). This problem motivated much of the work in this thesis, which is concerned

1We note that while the theoretical treatment of linear systems and random processes can be cast in the same

mathematical framework, the interpretation of the corresponding ob jects is quite different. In particular, in the case of

a linear system we are mainly interested in characterizing its effects on various input signals. In contrast, in the case of

a random process we are concerned with a description of its correlative properties and its power distribution.2

Throughout this thesis, integrals are from to unless stated otherwise.


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1.2 Review of Time-Invariant/Stationary Theory 3

with the development of a time-frequency (TF) calculus for LTV systems and nonstationary processes.

This TF calculus is valid for underspread systems and underspread processes which will be discussed

in Section 1.4.

1.2 Review of Time-Invariant/Stationary Theory

LTI systems and their dual, LFI systems, as well as stationary processes and their dual, white processes,

are quite restrictive models (see Section A.4). However, their treatment using convolution and Fourier-

type methods is comparatively simple. Thus, we subsequently outline several well-known results

for LTI (LFI) systems and stationary (white) random processes (details can be found in standard

textbooks like [160, 163]). Our emphasis will be placed on those properties whose extension to time-

varying systems and nonstationary processes will be developed in Chapters 2 and 3, respectively.

1.2.1 Transfer Functions of Time-Invariant and Frequeny-Invariant Linear Sytems

LTI systems are characterized by an impulse response of the form h(t, t) = g(t t), so that thegeneral input-output relation (1.1) specializes to a convolution (denoted by an asterisk),

y(t) = (g x)(t) =

tg(t t) x(t) dt.

LFI systems are characterized by an impulse response of the form h(t, t) = m(t) (t t); the input-output relation (1.1) here reduces to a time-domain multiplication,

y(t) =

t

m(t) (t t) x(t) dt = m(t) x(t).

LTI (LFI) systems are always normal and any two LTI systems (LFI systems) commute with each

other. The spectral transfer function (frequency response) of LTI systems is given by the Fourier

transform of g(),

G(f)

g() ej2f d , (1.2)

and the temporal transfer function of LFI systems is given by the multiplier function m(t). Thesespectral and temporal transfer functions are extremely simple and efficient system descriptions. This

is due to the following properties:

The complex sinusoids {ej2f t} (parametrized in a physically meaningful manner by frequency f)are the generalized eigenfunctions [68] of any LTI system, with G(f) the associated generalized

eigenvalues, i.e. (Hx)(t) = G(f) ej2f t for x(t) = ej2f t. Similarly, the Dirac impulses (t t)(parametrized in a physically meaningful manner by time t) are the generalized eigenfunctions

of LFI systems, with m(t) the associated generalized eigenvalues: (Hx)(t) = m(t) (t t) forx(t) = (t t

).


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As a consequence of the previous property, for LTI systems the Fourier transform of (Hx)(t)equals the Fourier transform of x(t) multiplied by G(f), and for LFI systems the output signal

(Hx)(t) equals the input signal x(t) multiplied by m(t). Hence, for LTI and LFI systems, the

input-output relation simplifies to the multiplication of two functions in the frequency domain

and in the time domain, respectively.

The spectral (temporal) transfer function of the series connection (composition) of two LTI(LFI) systems with transfer functions G1(f) (m1(t)) and G2(f) (m2(t)) equals G1(f) G2(f)

(m1(t) m2(t)).

The adjoint H+ of an LTI (LFI) system has impulse response g() (m(t) (t t)), and henceits spectral (temporal) transfer function is simply the complex conjugate of G(f) (m(t)).

The inverse H1 of LTI (LFI) systemif it existscorresponds to the reciprocal of the spectral

(temporal) transfer function.

For LTI (LFI) systems the maximum system gain (i.e., operator norm, see Section A.1) is equalto the supremum of |G(f)| (|m(t)|).

1.2.2 Power Densities of Stationary and White Processes

The correlation function rx(t1, t2) = E{x(t1) x(t2)} of (wide-sense) stationary processes depends onlyon the difference t1 t2, i.e., rx(t1, t2) = rx(t1 t2). Hence, the 1-D correlation function rx() is a

complete second-order description. The power spectral density (PSD) [163] of the stationary processx(t) is defined as the Fourier transform of the correlation function (Wiener-Khintchine relation), i.e.,

Px(f)

rx() ej2f d . (1.3)

Similarly, the correlation function of a white process is of the form rx(t1, t2) = qx(t1) (t1 t2) andthe temporal power density is given by the mean instantaneous intensity [163] qx(t). The PSD and

mean instantaneous intensity are very simple, physically intuitive, and useful second-order statisti-

cal descriptions of the process. The following properties and interpretations of the PSD and mean

instantaneous intensity are of fundamental importance:

The complex exponentials ej2f t are the generalized eigenfunctions of the (convolution type)correlation operator Rx of stationary processes. This implies that the Fourier transform diago-

nalizes the correlation operator and thus provides a decorrelation of stationary processes,

E

X(f) X(f)

= Px(f) (f f) .

Here, X(f) =

t x(t) ej2f t dt is the process Fourier transform. For white processes, the corre-

lation operator is diagonalized by the Dirac impulses (t t) and decorrelation is obtained inthe time domain, E {x(t) x

(t)} = qx(t) (t t

).


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1.3 Time-Varying Systems and Nonstationary Random Processes 5

Let h() denote the impulse response of a (time-invariant) innovations system of x(t) [38,163],i.e., x(t) =

h() n(t ) d, with n(t) normalized stationary white noise. Then it is known

that

Px(f) =

|H(f)

|2, (1.4)

where H(f) is the transfer function of the innovations system as defined in (1.2). Similarly, if

h(t, t) = m(t) (t t) is the (frequency-invariant) innovations system of a white process, i.e.,x(t) = m(t) n(t), then qx(t) = |m(t)|2.

The PSD (mean instantaneous intensity) is a complete second-order statistics of stationary(white) processes since the correlation function rx(t1, t2) can be completely recovered from Px(f)

(qx(t)).

The PSD and the mean instantaneous intensity are nonnegative quantities that integrate to themean temporal and spectral power, respectively:

Px(f) 0 ,

fPx(f) df = E

|x(t)|2 ,qx(t) 0 ,

t

qx(t) dt = E|X(f)|2 .

These properties are essential for an interpretation as average power densities.

If a stationary process x(t) is passed through an LTI system with impulse response k(), theoutput process y(t) = (k

x)(t) is again stationary with PSD

Py(f) = |K(f)|2 Px(f) . (1.5)

Furthermore, the cross PSD of y(t) and x(t) is given by

Py,x(f)

ry,x() ej2f d = K(f) Px(f) , (1.6)

where ry,x() = E{y(t) x(t )} is the cross correlation function of y(t) and x(t). Similarly,passing a white process x(t) through an LFI system with temporal transfer function m(t) again

yields a white process y(t) = m(t) x(t) with mean instantaneous intensity qy(t) =

|m(t)

|2 qx(t)

and mean instantaneous cross intensity qy,x(t) = m(t) qx(t).

1.3 Time-Varying Systems and Nonstationary Random Processes

LTV systems and nonstationary random processes provide very general and thus powerful models

for a large variety of engineering applications. Unfortunately, as discussed below, this generality is

paid for with an increased difficulty in describing LTV systems and nonstationary processes. The

situation gets more comforting if one considers the classes of underspread systems and underspread

processes [117120, 126, 127, 129, 144, 145] which can be viewed as natural extensions of LTI (LFI)


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systems and stationary (white) processes, respectively. The basic concepts related to underspread

systems and underspread processes as well as their importance for an approximate TF calculus will

be outlined in Section 1.4, while a detailed discussion of the approximate TF calculus will be provided

in Chapters 2 and 3.

1.3.1 Time-Varying Systems and the Generalized Weyl Symbol

For LTV systems, the generalized Weyl symbol (GWS) is defined as [114,118, 144]

L()H

(t, f)

h()(t, ) ej2f d , (1.7)

with h()(t, ) = h(t + (1/2 ), t (1/2 + )). Important properties and relations of the GWS aresummarized in Section B.1.2. The GWS has been recognized as a potential candidate for a TF (or,

time-varying) transfer function, i.e., as a generalization of the spectral (temporal) transfer function.Unfortunately, in contrast to LTI (LFI) systems, LTV systems and the GWS are generally much more

cumbersome to work with. This is due to the following reasons :

Contrary to LTI and LFI systems, LTV systems are not always normal. In the non-normalcase, one has to deal with (numerically more expensive) singular value decompositions instead

of eigenvalue decompositions [69,158] (see also Appendix A).

In general, the eigenfunctions or singular functions of distinct LTV system are different andpossess no simple specific structure.

Since the singular value decomposition (eigenvalue decomposition) of LTV systems has no specificstructure, it can only be interpreted using signal space concepts, thus lacking a specific physical

interpretation. In particular, the parameter k in (A.4) and (A.5) has in general no physical

meaning. Furthermore, there exists no fast implementation (like the FFT for LTI systems) of

the transform associated to these unstructured singular functions (eigenfunctions).

None of the practically convenient properties of the spectral (temporal) transfer function of LTI(LFI) systems (see the list in Subsection 1.2.1) is valid any longer for the GWS of LTV systems.

1.3.2 Nonstationary Processes and Time-Varying Power Spectra

In the context of random processes, the eigenvalue decomposition of the correlation operator, referred

to as Karhunen-Loeve (KL) decomposition [136], has a specific interpretation that is important for

statistical signal processing schemes. We shall thus discuss very briefly the KL expansion of non-

stationary processes. Let us consider a finite-energy, zero-mean, nonstationary process x(t). The

associated correlation operator Rx is trace-class (cf. Appendix A) and has (orthonormal) eigenfunc-

tions uk(t) and absolutely summable nonnegative eigenvalues k. The KL theorem [136] states that

x(t) can be expanded into the eigenfunctions uk(t) (in the mean-square sense) and that the expansion


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1.3 Time-Varying Systems and Nonstationary Random Processes 7

coefficients are uncorrelated with mean power equal to the eigenvalues k, i.e.,3

x(t) =

k=1

x, uk uk(t) , E {x, ukx, ul} = k kl . (1.8)

The double orthogonality of the KL expansion (orthogonality of the basis functions uk(t) and orthog-onality of the random coefficients x, uk) is the reason for the (theoretical) optimality and usefulnessof the KL transform (i.e., the transform mapping the random process x(t) to the coefficients x, uk)in various applications like transform coding or signal detection.

In the case of stationary processes, the KL transform reduces to the Fourier transform. With X(f)

denoting the Fourier transform of x(t), (1.8) reads

x(t) =

f

X(f) ej2f t df , E

X(f) X(f)

= Px(f) (f f) ,

i.e., one formally has uk

(t)

ej2f t and k

Px

(f), with the continuous parameter f (frequency)

replacing the discrete parameter k. For white processes there is a similar integral representation where

formally uk(t) (t t) and k qx(t), with the continuous parameter t (time) replacing thediscrete parameter k.

Hence, in a certain sense, the KL eigenvalues provide a generalization of the PSD and the mean

instantaneous intensity to general nonstationary processes. Unfortunately, the KL transform in general

suffers from several drawbacks:

The KL basis functions uk(t) are not known a priori and in general are different for differentprocesses.

In general, the basis {uk(t)} lacks a simple structure; this is the reason why typically no efficientimplementation of the KL transform exists.

In general, the parameter k of the KL transform has no physical meaning like frequency f ortime t.

These drawbacks are well recognized and have been a major driving force for research into approxi-

mations of the KL transform [50, 60,61, 118, 120, 137, 138]: depending on the specific application, the

goal has been either to develop approximate KL transforms using highly structured and efficiently

implementable bases or to provide physically more relevant definitions of spectra for nonstationaryprocesses. Potential candidates for the latter are the generalized Wigner-Ville spectrum (GWVS),

defined as [60,61,63,140,145]

W()

x (t, f)

r()x (t, ) ej2f d ,

with r()x (t, ) = rx(t + (1/2 ), t (1/2 + )), and the generalized evolutionary spectrum (GES),

defined as [145, 148] (cf. also [49, 118, 170, 171])

G()x (t, f)

L

()H

(t, f)

2

,3

The inner product is defined as usual, x, y = Rt x(t) y(t) dt.


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with H an innovations system of x(t), i.e., a system satisfying HH+ = Rx. Further details about the

GWVS and the GES can be found in Sections B.3.1 and B.3.2, respectively. Unfortunately, none of

the practically important and convenient properties of the PSD or the mean instantaneous intensity

as detailed in Subsection 1.2.2 are satisfied by the GWVS or GES of general nonstationary processes.

1.4 The Importance of Being Underspread

In the foregoing subsections, we saw that in general the GWS of an LTV system and the GWVS or

GES of a nonstationary process fail to provide tools that are as efficient and meaningful as the transfer

function of LTI (LFI) systems and the power densities of stationary (white) processes, respectively.

However, for underspread systems (see Chapter 2) the GWS features similar properties as the transfer

function (at least in an approximate sense) and for underspread processes (see Chapter 3) the GWVS

and GES (approximately) satisfy similar properties as the PSD. Making this statement precise andproving this claim is the main theme of this thesis. In the next two subsections, we outline the main

results of the theory developed in Chapters 2 and 3.

1.4.1 Underspread Linear Time-Varying Systems

For our subsequent discussion, it is necessary to consider the TF shifts introduced by an LTV system

H. These can be characterized by the generalized spreading function (GSF) [114,118]

S()H

(, ) t h()(t, ) ej2 t dt ,with h()(t, ) as before. Important properties and relations of the GSF are summarized in Section

B.1.1. In particular,

y(t) = (Hx)(t) =

S()H

(, ) (S(),x)(t) d d , (1.9)

with S(), denoting the generalized TF shift operator,

(S(),x)(t) = x(t ) ej2t ej2 (1/2) .

It is seen from (1.9) that the spread, or extension, of the GSF about the origin of the ( , )-planeprovides a global characterization of the TF shifts introduced by the system. The GSF extension in

the direction characterizes the systems length of memory whereas the extension in the direction

determines the fastness of the systems time-variations or fluctuations. Conceptually, an LTV system

is called underspread if its GSF is concentrated in a small region about the origin of the ( , ) plane,4

which indicates that the system introduces only small TF shifts , or, in other words, that the

systems memory is short and/or its time-variations are slow (see Subsections 2.1.2 and 2.2.4). In

contrast, systems introducing large TF shifts , are referred to as overspread.

4In fact, in most cases it suffices that the GSF is concentrated around some point (0, 0) since the corresponding

offset TF shift can be split off from the system.


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1.4 The Importance of Being Underspread 9

The underspread concept was first used for random LTV systems in the context of fading multipath

channels [11,109,172,203] and for doubly spread radar targets [72,74,203]. Adopting the terminology

of the random case, Kozek [118120] introduced underspread deterministic LTV systems by requiring

that their GSF be exactly zero outside a small rectangular support region about the origin of the (, )

plane. Systems being underspread in Kozeks sense, i.e., having small compact GSF support, will be

treated in Section 2.1 and we will refer to them as displacement-limited (DL) operators.

In practice, the condition of small compact GSF support is often not satisfied exactly but only

effectively. This raises the question of how to choose the effective support region and how the modeling

error resulting from a specific choice of this effective support region affects the validity of the results

obtained using the compact support model. Furthermore, the small compact support requirement is

often unnecessarily restrictive since several important results hold for much wider classes of systems,

including systems with a GSF that satisfies certain support constraints but still has infinite support.

Thus, in this thesis we introduce and use an extended underspread concept based on operators withrapidly decaying GSF (see Section 2.2). As a foundation for this extension, we use weighted integrals

and moments of the GSF as measures of the global TF shifts of a system, without requiring the GSF

to have compact support.

For LTV systems of the underspread type (be they DL operators or operators with rapidly decaying

GSF), all of the difficulties connected with LTV systems and the GWS as listed in Subsection 1.3.1 are

(at least approximately) removed. In fact, one can establish a GWS-based approximate TF transfer

function calculus that yields the following useful results (the details and bounds on the associated

approximation errors will be presented in Section 2.3):

Underspread operators are approximately normal (see Subsection 2.3.17) and have approximateeigenfunctions which, being time and frequency translates of a prototype signal, are highly

structured (see Subsection 2.3.8). Hence, these approximate eigenfunctions also allow an intuitive

physical interpretation.

The GWS approximately reflects the systems maximum gain (see Subsection 2.3.15).

The GWS of the adjoint operator H+ is approximately equal to the complex conjugate of theGWS of H (see Subsection 2.3.2).

The GWS of the product (composition) of two (jointly) underspread operators (systems) isapproximately given by the product of the respective individual GWS (see Subsection 2.3.4).

Jointly underspread LTV systems are approximately commuting (see Subsection 2.3.16).

In a certain sense to be discussed in Subsection 2.3.6, the GWS of the inverse of H approximatelycorresponds to 1/L

()H

(t, f).

All these approximations can be summarized by stating that the GWS of (jointly) underspread systems

can be interpreted as a TF transfer function which can be used in exactly the same manner as the

spectral (temporal) transfer function of LTI (LFI) systems.


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1.4.2 Underspread Nonstationary Processes

TF correlations play a similar role for random processes as TF shifts do for linear systems. They can

be compactly characterized by the generalized expected ambiguity function (GEAF) [117, 118, 126]

A()x (, ) S()Rx

(, ) = t

r()x (t, ) ej2 t dt ,

with r()x (t, ) as before (further details can be found in Sections 3.1 and B.3.4). In particular, the

GEAF can be interpreted as a global measure of the TF correlation of process components separated

by in time and by in frequency. Hence, the extension of the GEAF about the origin of the (, )

plane provides a global characterization of the TF correlations of the process. The GEAF extension

in the direction describes the temporal correlation width whereas the extension in the direction

characterizes the spectral correlation width of the process. Conceptually, we will refer to a random

process as underspread if its GEAF is sufficiently concentrated about the origin of the (, ) plane.

This is equivalent to the requirement that the process features only small TF correlations. Since

the GEAF is the GSF of the correlation operator Rx, it follows that the correlation operator of an

underspread process is underspread in the sense of Subsection 1.4.1. In contrast, processes with large

TF correlations are referred to as overspread.

Underspread processes were first introduced by Kozek [117, 118, 126] in analogy to underspread

LTV systems. Kozeks definition of underspread processes was similarly based on the requirement

that their GEAF is exactly zero outside a small rectangular support region about the origin of the

(, ) plane. We will refer to processes that are underspread according to this original definition as

correlationlimited (CL) processes.

Similar to the case of LTV systems, the condition of small compact GEAF support will often be

satisfied only effectively. Hence, for the same reasons as mentioned in the foregoing subsection, we

shall introduce and use in this thesis an extended underspread concept that is based on rapid decay

of the GEAF. The GEAF decay (and thus the global TF correlations) will be measured via weighted

integrals and moments. For this extended version of the underspread property, no compact GEAF

support is required.

The important fact about underspread processes is that the difficulties connected with nonstation-

ary random processes and the GWVS/GES as listed in Subsection 1.3.2 are alleviated. In particular,

one can show that the GWVS and GES (as well as other time-varying spectrasee Chapter 3) ofunderspread processes (approximately) satisfy the following useful properties (the details and bounds

on the associated approximation errors will be presented in Chapter 3):

Time and frequency translates of a reasonable prototype signal are approximate KL eigen-functions of underspread processes. These approximate KL eigenfunctions allow a physically

meaningful interpretation and, due to their underlying structure, the associated transform can

be efficiently implemented.

The GWVS and GES are (approximately) positive and describe the mean TF energy distributionof the process in a meaningful way.


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1.5 Signal Processing Applications 11

For both the GWVS and GES, simple and intuitive approximate input-output relations similarto (1.5) and (1.6) can be derived.

Almost all definitions of time-varying power spectra proposed up to now in the literature (inparticular, the GWVS and GES) are approximately equivalent.

In an approximate sense, most of the time-varying power spectra are complete second-ordercharacterizations of the process (for the GWVS, this holds exactly, see Subsection B.3.1).

In essence, the above approximations imply that the GWVS and GES of underspread processes are

reasonable definitions of time-varying power spectra which extend the PSD (mean instantaneous in-

tensity) of stationary (white) processes in a meaningful way to the nonstationary case.

1.5 Signal Processing ApplicationsAs explained above, Chapters 2 and 3 establish that the GWS is a meaningful TF transfer function for

underspread LTV systems and that the GWVS and GES are meaningful mean TF energy distributions

of underspread nonstationary random processes. These findings have important implications for the

following applications that will be discussed in more detail in Chapter 4.

Nonstationary Signal Estimation and Enhancement. The Wiener filter is known to be

the optimal linear signal estimator with respect to a mean square error criterion [106,187, 197, 202].

Unfortunately, in the case of nonstationary processes, the design of the Wiener filter is based on

the solution of an operator equation and requires a computationally costly and potentially unstableoperator inversion. Applying the results of Subsection 2.3.7 allows to use a computationally efficient,

numerically stable, and physically intuitive approximate TF design of time-varying Wiener filters

[92,111] (see Section 4.1).

Nonstationary Signal Detection. In the case of nonstationary Gaussian processes, the design

of the likelihood ratio detector [108,168,187,202] and the deflection-optimal detector [7,168] involves the

solution of an operator equation that requires expensive and potentially unstable operator inversions.

Similar to the signal estimation problem, we show how the results of Subsection 2.3.6 can be used

to obtain a computationally less costly, stable, and intuitive approximate TF design of time-varying

signal detectors [141143,146] (see Section 4.2).

Sounding of Mobile Radio Channels. Accurate wideband measurements of mobile radio

channels by means of correlative channel sounders are the cornerstone of any design or simulation

of mobile radio systems with high data rate [40, 52, 164]. While most channel sounders assume the

channel to be time-invariant, practical mobile radio channels are time-varying. For this reason, the

measurements are typically affected by systematic errors [149,150,156]. Using the results of Subsections

2.3.1, 2.3.4, and 2.3.18, these errors can be quantified and bounded (see Section 4.3).

Multicarrier Communication Systems. Multicarrier communication systems like orthogonal

frequency division multiplex (OFDM) and discrete multi-tone (DMT) [2830, 133, 182, 209, 215] are


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closely related to TF signal expansions. Recent work showed that pulses other than the usual rectan-

gular one as well as an extension to biorthogonal transmit and receive filters can be advantageous in

the case of fast time-varying channels [18,19,128]. In Section 4.4, we will point out a close relation of

these recent results with approximate eigenfunctions of LTV systems as discussed in Subsection 2.3.8.

Analysis of Car Engine Signals. Pressure and vibration signals measured in combustion

engines can be modelled successfully as nonstationary random processes. They are important for

knock detection and other car engine diagnosis tasks [17, 26,27, 113, 143,146, 181,207]. Application of

the time-varying spectra and TF coherence function considered in Chapter 3 allows to extract useful

time-varying and nonstationary features of this type of real data (see Section 4.5).

1.6 Related Work

As mentioned above, underspread LTV systems with a GSF satisfying a rectangular support constraint

were first considered in the pioneering work of W. Kozek [118120, 127]. Several transfer function

approximations have been derived within this framework [118120, 127]. Furthermore, results in a

similar spirit have been obtained for the symbol calculi in the context of quantum-mechanical quan-

tization [64, 167, 206] and pseudo-differential operators [64, 73, 94, 112], with the difference that these

theories define specific symbol classes directly in phase space whereas in our approach we formulate

growth conditions in the (dual) spreading function domain. Also, whereas in quantization theory

and pseudo-differential operator theory one studies the operators corresponding to a given symbol,

we consider a given LTV system (operator) and investigate how close its GWS comes to the intu-

itive engineering notion of a TF transfer function. We will further comment on these differences and

similarities in Sections 2.2 and 2.3.

Several results for underspread random processes with rectangularly supported GEAF that are

related to the approximations in Chapter 3 were also derived by W. Kozek in [115, 118, 120, 126].

Furthermore, results close in spirit to our considerations have been presented in [191193]. There, the

analysis and processing of random processes with slowly time-varying statistics based on innovation

systems and the evolutionary spectrum is discussed. In [60,61,63], observations regarding the Wigner-

Ville spectrum and the evolutionary spectrum were made which are closely related to our discussion of

time-varying power spectra. Finally, [32] considers LTV systems and nonstationary random processesthat are regionally underspread, i.e., require the underspread condition to hold only for a portion

of the system (process) that is localized in a specific TF region.

1.7 Overview of Contributions

We conclude this introductory chapter with an overview of the major contributions of this thesis (in

order of appearance).

LTV systems with compactly supported GSF (Section 2.1): We introduce novel parameters


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1.7 Overview of Contributions 13

characterizing the TF shifts of LTV systems with compact GSF support, and we extend the

previous definitions [118120,127] of (jointly) underspread systems to accommodate oblique ori-

entations of GSF support regions. Furthermore, the spread parameters of unitarily transformed

systems as well as of sums, products, and adjoints of underspread LTV systems are analyzed.

LTV systems with rapidly decaying GSF (Section 2.2): We present an extension of the un-

derspread concept to LTV systems whose GSF does not necessarily have compact support but

features rapid decay. This extension uses weigthed integrals and moments of the GSF that

provide novel measures of the TF shifts introduced by a system. We derive various relations

for weighted GSF integrals and moments of unitarily transformed LTV systems and of sums,

products, and adjoints of LTV systems. Furthermore, we present Chebyshev-like inequalities

inequalities that are useful for assessing the errors made by approximating an arbitrary LTV

system by a system with compactly supported GSF.

TF transfer function approximations (Section 2.3): This large section contains numerous re-

sults that establish a GWSbased approximate TF transfer function calculus for underspread

LTV systems. All TF transfer function approximations are underpinned by bounds on the as-

sociated approximation errors which are formulated in terms of weighted GSF integrals and

moments. While numerous TF transfer function approximations are completely new, others are

extensions of existing result for the special case of LTV systems with compactly supported GSF

in [118120,127].

TF correlation analysis (Section 3.1): We present methods for the analysis of the TF correlationsof a random process. Furthermore, we provide a novel concept of underspread processes that is

based on weighted integrals and moments of the GEAF and that extends previous definitions

of underspread processes that were based on the assumption of a compactly supported GEAF

[117,118,126].

Elementary time-varying spectra (Section 3.2): For underspread processes, the GWVS and

GES are shown to be smooth, effectively real-valued, and positive quantities. We furthermore

discuss the occurrence of statistical cross-terms in the case of overspread processes and we

present uncertainty relations for the GWVS and GES.

Type I and type II spectra (Sections 3.33.5): Type I time-varying power spectra (previously

considered in [3,60,61,63]) are introduced in an axiomatic fashion and shown to be extensions of

the GWVS. Similarly, we extend the GES by introducing the novel class of type II time-varying

power spectra. We show that these spectra satisfy (at least approximately) several desirable

properties. Furthermore, we prove that for underspread processes the members of these two

classes of spectra are approximately equivalent (for the special case of GWVS and GES and

processes with compactly supported GEAF, part of these results has been shown previously

in [118,126]).


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Input-output relations (Section 3.6): We present novel approximations that relate the GWVS

(GES) of the nonstationary output process of an LTV system to the GWVS (GES) of the

nonstationary input process.

Approximate KL expansion (Section 3.7): We present approximate KL expansions for under-spread nonstationary random processes. This extends existing results derived for processes with

compactly supported GEAF [115, 116, 118, 120] to more general scenarios and is furthermore

related to results obtained in [137,138] for locally stationary processes.

TF coherence function (Section 3.8): We discuss the concept ofcoherence for nonstationary ran-

dom processes and introduce a novel class of TF coherence functions. While a TF coherence

has been defined in [210] in an ad hoc fashion, we prove that our TF coherence functions can be

viewed as approximate TF formulations of a coherence operator and we provide several approx-

imations that justify their interpretation as a coherence function.

Applications (Chapter 4): Chapter 4 applies the theoretical results developed in Chapters 2 and

3 to the problems of signal estimation, signal detection, channel sounding, multicarrier commu-

nications, and car engine diagnosis (see Section 1.5 for an outline of these applications).


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2Underspread Systems

My work always tried to unite the truth with the beautiful, but when I had tochoose one or the other, I usually chose the beautiful. Hermann Weyl

ALTHOUGH there exist several useful time-frequency tools for characterizing linear time-varyingsystems (linear operators), such as the generalized Weyl symbol and the generalized spreadingfunction, these time-frequency representations are in general difficult to work with. For example,series connections and inverses of linear time-varying systems in general cannot be expressed via the

generalized Weyl symbol in as simple a manner as it is possible with the transfer function of linear

time-invariant systems. Hence, there remain several problems as to how these representations are to

be interpreted and how they can be used in specific signal processing applications.

A key concept that allows to answer these questions is that of underspread systems. Such systems

introduce only limited time-frequency shifts and are characterized by a spreading function concentrated

around the origin of the (, )-plane. Underspread systems essentially come in two flavors: The first

type, proposed in this thesis and discussed in detail in Section 2.2, builds on the requirement of rapid

decay of the spreading function such that specific weighted integrals and moments are sufficiently

small. The second type (which can be viewed as a special case of the first), is based on a strict

support constraint of the spreading function, similar to strictly bandlimited signals. This latter type

of underspread systems, reviewed in Section 2.1, has been introduced and extensively analyzed in the

pioneering work of W. Kozek.

In Section 2.3 of this chapter, numerous approximations based on specific underspread assumptions

are proved that establish an approximate time-frequency transfer function calculus. We note that some

parts of our analysis are parallel to Kozeks work, although our (more general) definition of underspread

systems using weighted integrals and moments and the approximations based on it are original.

15


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16 Chapter 2. Underspread Systems

2.1 Operators with Compactly Supported Spreading Function

In this section we consider operators with compactly supported GSF and review Kozeks original def-

inition of underspread operators. We furthermore discuss the effect of unitary metaplectic transforms

on the operators GSF support and study the GSF supports of the sum, product, and inverses of

operators.

2.1.1 General Support Constraints

The definition of underspread (deterministic) LTV systems by requiring the support of their GSF to

be confined to a rectangular region about the origin has first been proposed and extensively studied

in the pioneering work of W. Kozek [117120, 123, 126]. This support constraint on the GSF implies

that the system introduces only limited time displacements and frequency displacements and hence

we refer to such systems as displacement-limited (DL) systems (operators). Since GSF and GWS are2-D Fourier transform pairs, it further follows that the GWS of a DL system is a 2-D band-limited

function. The existing extensive theory of band-limited signals [162] thus serves as an additional

motivation for this definition of underspread systems. In particular, a strict band-limitation of the

GWS allows the GWS to be sampled on a 2-D sampling grid without information loss.

The following discussion is intended as a review of some of the concepts introduced in [117119,

123,126], with some slight modifications and extensions comprising more general support constraints

than rectangular ones (e.g., oblique regions in the (, )-plane). This will also yield bounds on the

errors incurred by the transfer function approximations in Section 2.3 that are slightly more tight

and/or valid under more general conditions than previous bounds in [117119,123,126].

Consider an LTV system (operator) H with GSF S()H

(, ). We require that the support of the

GSF is confined to a compact region GH, i.e., |SH(, )| = 0 for (, ) GH. For such systems, we canwrite1

GH =

(, ) : |SH(, )| > 0

.

Thus, with the indicator function IGH(, ) of GH, defined as

IGH(, ) = 1 , (, ) GH ,0 , (, ) GH ,

the GSF satisfies the condition

S()H

(, ) IGH(, ) = S()H

(, ). (2.1)

While in general there exist infinitely many indicator functions which satisfy (2.1) for a given S()H

(, ),

our definition corresponds to the indicator function with minimal support, i.e., where the associated

region GH has minimal area. The multiplicative relation (2.1) in the (, )-domain corresponds to a1

Note that the GSF magnitude is independent of , |S()H (, )| = |SH(, )| (see Subsection B.1.1).


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2.1 Operators with Compactly Supported Spreading Function 17

convolution relation in the TF domain, i.e.,t

f

L()H

(t, f) L()T

(t t, f f) dt df = L()H

(t, f). (2.2)

Here, L()T (t, f) is the GWS of an operator T defined via its GSF as

S()T

(, ) = IGH(, ). (2.3)

Note (2.2) holds for any operator T with GSF satisfying S()eT

(, ) = 1 for (, ) GH (while beingarbitrary for (, ) GH). However, due to (B.9), the operator T defined via (2.3) has minimum normwithin the class of all operators satisfying (2.2).

To each (, )-region G or indicator function IG(, ), there corresponds a linear subspace SG thatconsists of all linear operators H satisfying the corresponding support constraint (2.1),

SG = H : S()H (, )IG(, ) = S()H (, ) .The orthogonal complement of SG corresponds to the complement G = R2\G of G whose indicatorfunction is IG(, ) = 1 IG(, ), such that for any operator Hc in the complement space ScG = SGthere is S

()Hc

(, ) IG(, ) = 0 or equivalently S()Hc

(, ) IG(, ) = S()Hc

(, ) [1 IG(, )] = S()Hc (, ).Obviously, we also have

H SG1 and G1 G2 = H SG2 .Furthermore, given a region G or indicator function IG(, ), any operator H can be split into twoorthogonal parts lying respectively inside and outside the associated operator space SG ,

H = HG + HG. (2.4)

The orthogonal compononents HG and HG are found by projecting H onto the respective subspace.

This projection can easily be accomplished in the GSF domain according to

S()

HG(, ) = S

()H

(, )IG(, ), S()

HG(, ) = S

()H

(, ) [1 IG(, )] . (2.5)

We call HG the DL part of H corresponding to the region G. It is easily checked using (B.8) thatthese two systems are indeed orthogonal, i.e.

HG, HG

= Tr

HG(HG)+

= 0. Note that this does

not in general imply HGHG = 0 or HGHG = 0. These concepts will be important in Sections 2.3.6

and 2.3.18 when discussing operator inverses and TF-sampling of the GWS, respectively.

2.1.2 Definition of Displacement-limited Underspread Operators

In the following, we recall from [118] the definition of underspread operators in the constrained support

sense:

Definition 2.1. LetH be a DL system with GSF supportGH and let(max)H and(max)H be the maximumtime and frequency shift, respectively, introduced by the system,

(max)H

max(,)GH |

|

, (max)H

max(,)GH |

|

. (2.6)


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(max)H

(max)H

(max)

H(max)

H

GH

GH

(a) (b)

Figure 2.1: Illustration of (a) a rectangular and (b) a hyperbolic compact support constraint.

Then,H = 4

(max)H

(max)H

is called the (strict-sense) displacement spread of the DL system H, and H is called (strict-sense) DL

underspread if

H = 4(max)H

(max)H

1 . (2.7)

Condition (2.7) was first introduced in [118120]. We note that 4(max)H

(max)H

is the area of a

rectangle with sides of length 2(max)H

and 2(max)H

, respectively, which contains the GSF support

region

GH. Hence, H measures the support of the GSF of H via a circumscribed rectangle (cf. Fig.

2.1(a)). Sometimes, it will suffice to require the less restrictive condition

H max(,)GH

| | 1 . (2.8)

Since H measures only the maximum product | | for points within GH, we have

H H/4 , (2.9)

with equality if and only if one of the four corner points (max, max) GH. Condition (2.8) issomewhat different in spirit from (2.7) since it does not require the area of the support of the GSF tobe small or even finite; however, it still implies a support constraint since the GSF has to lie within

the hyperbolae defined by | | = H (see also Fig. 2.1(b)).It is important to note that DL operators [118120] are a special case of operators with rapidly

decaying GSF to be defined in Section 2.2. This will further be discussed in Subsection 2.2.2.

2.1.3 Unitary Transformations

Let us now consider the influence of some specific unitary transformations of operators on the GSF

support, i.e., we consider the transformed operator H = UHU+ with some unitary operator U.


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2.1 Operators with Compactly Supported Spreading Function 19

We first study the effects of TF shifts, i.e., H = St,fHS+t,f where S()t,f denotes the joint TF shiftoperator (see Appendix B). Since TF shifts leave the GSF magnitude unchanged, the quantities H

and H remain unchanged too, i.e.,

eH = H, eH = H.

Hence, any class of operators defined by some type of GSF support constraint also comprises all TF

shifted versions of each member of this class,

H SG = St,fHS+t,f SG.

In particular, all TF shifted versions of an underspread DL system are also DL underspread.

Next let us consider the class M of metaplectic transformations U = (A) that correspond to thearea-preserving linear (symplectic) TF coordinate transforms (see Appendix C and [46,64,162,208]).

For any metaplectic operator U = (A), the GSF with = 0 of H = UHU+ is given by (cf. (C.4))S

(0)eH

(, ) = S(0)H

(a + b,c+ d).

Since |S()H

(, )| is independent of (cf. Section B.1.1), there is also2S()eH

(, ) = S()

H(a + b,c + d)

for all . (2.10)Specific metaplectic operators which depend only on a single parameter are the TF scaling operator

(a = 1/d, b = c = 0), the Fourier transform operator (a = d = 0, b = 1/c = T2), the chirpmultiplication operator (a = d = 1, b = 0), the chirp convolution operator (a = d = 1, c = 0), and the

fractional Fourier transform operator (a = d = cos , b = T2 sin , c = (sin )/T2).It is straightforward to show that only for TF scalings and Fourier transforms there is eH = H

and eH = H. In all other cases, H and H must be expected to change. In some cases, it is

undesirable that systems whose GSFs are equal up to area-preserving linear TF coordinate transforms

are assigned different displacement spreads. Hence, we extend the definition of the DL underspread

property in (2.7) such that the displacement spread of all operators whose GSFs are obtained from

each other by symplectic group TF coordinate transforms is equal:

Definition 2.2. An operator H is called wide-sense DL underspread if there exists a metaplectic

operator U M such that the displacement spread of H = UHU+ satisfies eH 1. We callminH inf

UMeH

the wide-sense displacement spread of H.

Note that all systems which are unitarily equivalent via some U M are thus assigned thesame wide-sense displacement spread. This is of specific importance when discussing transfer function

approximations for the case = 0 where shearings and rotations of the GSF can be of particular

interest (see, e.g., Subsections 2.3.4 and 2.3.15).2

Note that for = 0 this equality is valid only for the magnitude of the GSF and not for the GSF itself.


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2.1.4 Operator Sums, Adjoints, Products, and Inverses

We shall next consider the displacement spreads of sums, adjoints, products, and inverses of operators.

Sum. With regard to the sum of two operators H1 and H2, we obviously have

|SH1+H2 (, )| =S()H1

(, ) + S()H2

(, ) |SH1 (, )| + |SH2 (, )| .

Without making any additional assumptions, we have to deal with the worst case, i.e., GSFs SH1 (, ),

SH2 (, ) which do not cancel anywhere in the sum S()H1

(, ) + S()H2

(, ). Here, the region of sup-

port of SH1+H2 (, ) is given by GH1,H2 GH1 GH2 and the corresponding indicator function isIGH1,H2 (, ) = IGH1 (, ) + IGH2 (, ) IGH1 (, )IGH2 (, ). For the displacement spreads we thenobtain

H1+H2

H1,H2 4max(max)H1

, (max)H2 max

(max)H1

, (max)H2 , (2.11)

H1+H2 H1,H2 max {H1 , H2} .

It follows that H1,H2 max {H1 , H2} and hence, in general, H1+H2 must be expected to be largerthan both H1 and H2 (except if the support of one GSF is totally contained within that of the other

GSF, or the GSFs add to zero in specific peripheral regions of the ( , )-plane). This shows that the

sum of two DL underspread operators is DL but not necessarily DL underspread. On the other hand,

H1,H2 is never larger than the maximum of H1 and H2 . Hence, the class of operators defined by

H A is closed under addition.

In [118] two systems are called jointly underspread if the area of the union of their GSF supportscan be circumscribed by an axis-parallel rectangle of area less than one. However, based on the above

observations and using the definition

minH1,H2 infUM

UH1U+,UH2U+

we here introduce the following definition of jointly DL operators:

Definition 2.3. Two systems H1 and H2 are said to be jointly strict-sense (wide-sense) DL under-

spread if H1,H2 1 (minH1,H1 1).

Hence, jointly DL underspread systems are individually DL underspread and also their sum H1+H2

is DL underspread. We will call H1,H2 (minH1,H1

) the strict-sense (wide-sense) joint displacement spread

of H1 and H2. In essence, jointly DL underspread operators have GSFs satisfying similar support

constraints.

Adjoint. SinceSH+ (, ) = SH(, ) according to (B.6), the displacement spreads of a

system and its adjoint are equal, H+ = H, and furthermore H+ = H.

Product. The following inequalities can be easily deduced from the bound (B.13):

(max)H2H1

(max)H1 +

(max)H2 ,

(max)H2H1

(max)H1 +

(max)H2 .


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2.2 Operators with Rapidly Decaying Spreading Function 21

In particular, the above inequalities imply that H2 can be as large as 4H but not larger,

H2 = 4(max)H2

(max)H2

4(2(max)H

)(2(max)H

) = 4H.

This bound can be shown to hold also for the composition of a system and its adjoint, i.e., HH+

4H

and H+H 4H.Inverse. The GSF of the inverse of an operator H is difficult to analyze. From the Neumann

series [158]

H1 =

k=0

(I H)k ,

it is seen that due to the higher powers of (I H) the support of the GSF of H1 may grow adinfinitum. Thus, in general the inverse of a DL underspread operator need not be even DL. Yet, this

does not imply that the GSF of H1 may not be concentrated around the origin. This will further be

discussed in Subsection 2.3.6.

2.2 Operators with Rapidly Decaying Spreading Function

After this discussion of DL and DL underspread operators, we now turn to a novel extended concept

of underspread operators that replaces the compact support constraint on the GSF by generalized

decay constraints.

2.2.1 Motivation

In many cases, the assumption that the support of the GSF of an operator is exactly confined to some

small area around the origin (as used in [118, 119, 127] for the definition of underspread operators)

appears to be too restrictive. Often, the GSF is merely concentrated around the origin and has rapid

decay away from the origin. A useful measure of the decay of the GSF is in terms of weighted GSF

integrals and moments which describe the effective, rather than exact, support of the GSF. Hence,

a reasonable and practically relevant extension of the underspread concept can be based on such

measures of effective GSF support. This is the point of view we adopt in this section.

Based on our extended concept of underspread operators with rapidly decaying GSF, we will

prove the validity of several underspread approximations in Section 2.3. This approach has several

advantages as compared to the results obtained for DL operators:

In many practical situations, a theory based on weighted GSF integrals and moments is closerto physical reality than a theory based on exact support constraints.

The results can be used to judge how well the results obtained for DL operators apply to operatorswith rapidly decaying GSF (see Subsection 2.2.6).

In general, an operator and its inverse cannot be simultaneously underspread in the DL sense,whereas it is possible that they both have fast decaying GSF.


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(a) (b) (c) (d) (e)

Figure 2.2: Gray-scale plots (darker shades correspond to larger values) of several specific weighting

functions: (a) (, ) = ||k, (b) (, ) = ||k, (c) (, ) = | |k, (d) (, ) = |1 A(0)s (, )|, (e)(, ) =

1 1A

(0)s (, )

with A(0)s (, ) the ambiguity function (see Section B.2.4) of a normalizedGaussian function.

On the other hand, a drawback of our extended theory of underspread operators is that it does notallow for an exact TF sampling of the GWS as discussed in [118] for the case of DL underspread

operators. We will further comment on the sampling problem in Subsection 2.3.18.

2.2.2 Weighted Integrals and Moments of the Generalized Spreading Function

To circumvent the problems and limitations associated the DL underspread concept as mentioned

above, we here propose to globally characterize the TF shifts of a system H by means of the weighted

GSF integrals

m()H

(, ) |SH(, )| dd

|SH(, )| dd

=1

SH1

(, ) |SH(, )| d d , (2.12-a)

M()H

2(, ) |SH(, )|2 dd

|SH(, )|2 dd

1/2

=1

H2

2(, ) |SH(, )|2 d d1/2

, (2.12-b)

which are normalized by the L1 or L2 norm of S()H (, ) (recall that SH2 = H2). We note thatdue to (1.9), the implicit assumption that the GSF has finite L1 norm, SH1 < , is a sufficientcondition for the bounded input bounded output stability of the system H. Typically, (, ) in (2.12-a)

and (2.12-b) is a nonnegative weighting function which satisfies (, ) (0, 0) = 0 and penalizesGSF contributions lying away from the origin. We note that m

()H

and M()H

do not depend on the

GWS parameter . Since the GSF magnitude of the adjoint system H+ is given by |SH+ (, )| =|SH(, )|, we have m()H+ = m

()H

and M()H+

= M()H

whenever the weighting function is even-

symmetric, i.e., (, ) = (, ). Fig. 2.2 shows some specific weighting functions which will beused in later subsections.


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2.2 Operators with Rapidly Decaying Spreading Function 23

Absolute Moments of the GSF. As a special case of the above weighted integrals using the

weighting functions (, ) = ||k||l, we also introduce the normalized moments

m(k,l)H

1

SH

1 ||k ||l |SH(, )| d d , (2.13-a)

M(k,l)H

1

H2

2k 2l |SH(, )|2 d d1/2

, (2.13-b)

with integers3 k, l N0. If one views |SH(, )|/SH(, )1 and |SH(, )|2/H22 as probabilitydensity functions (they are positive and integrate to one), the above moments are analogous to the

(absolute) moments of random variables [163]; thus, we call m(k,l)H

and M(k,l)H

the (absolute) moments

of order (k, l) of H. Note that m(0,0)H

= M(0,0)H

= 1 for any system H. In almost all cases we will use

those (absolute) moments where either k = 0 or l = 0 or k = l.

Moments with k = 0 or l = 0 p enalize mainly GSF contributions located away from the axis oraway from the axis, respectively. Thus, systems with GSF concentrated along the axis (i.e.,

quasi-LTI systems) have small m(0,l)H

and small M(0,l)H

, whereas systems with GSF concentrated

along the axis (i.e., quasi-LFI systems) have small m(k,0)H

and small M(k,0)H

(cf. Figs. B.1 and

2.2).

Moments with k = l penalize mainly GSF contributions located away from the and axes,i.e., lying in oblique directions in the (, ) plane. This is due to the fact that the corresponding

weighting function is constant along the hyperbolae | | = c (cf. Fig. 2.2). In particular, asuperposition (i.e., parallel connection) of a (quasi-) LTI system and a (quasi-) LFI system has

a GSF concentrated along the and axes and will thus have small m(k,k)H

and M(k,k)H

.

Note that M(k,l)H

in (2.13-b) is well-defined only for Hilbert-Schmidt (HS) operators, i.e., for systems

with finite HS norm (cf. Appendix A). As such, this definition is not directly applicable to LTI and

LFI systems. However, for LTI and LFI systems appropriately modified moment definitions can be

given in terms of the impulse response g() or the Fourier transform of the modulation function,

M() = (Fm)(), respectively:

M(k,0)HLTI

1

g

2 2k |g()|2 d

1/2

, M(0,l)HLFI

1

M

2

2l |M()|2 d

1/2

. (2.14)

Note that these specific moments characterize the time displacements and frequency displacements of

LTI and LFI systems, respectively; since LTI systems do not cause frequency displacements and LFI

systems do not cause time displacements, it does not make sense to ask about moments characterizing

the frequency displacements of LTI systems or the time displacements of LFI systems. Furthermore,

for systems with GSF perfectly concentrated along the and axes, i.e., for any superposition of LTI

and LFI systems with impulse response h(t, t) = g(t t) + m(t) (t t), one has m(k,l)H

= 0 for any

k, l both > 0.

3Note that theoretically k, l could be any positive real-valued numbers. However, in the subsequent sections only

integer values are required.


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Due to Schwarz inequality, the moments satisfy the following inequalities:

m(k,l)H

m(2k,0)H

m(0,2l)H

, M(k,l)H

M(2k,0)H

M(0,2l)H

. (2.15)

Since m

(2k,0)

H and m

(0,2l)

H and similarly M

(2k,0)

H and M

(2k,0)

H measure the average extension of the G

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