Table Of ContentMOBILE
FADING
CHANNELS
MOBILE
FADING
CHANNELS
Matthias Pätzold
Professor of Mobile Communications
Agder University College, Grimstad, Norway
Originally published in the German language by Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, D-65189
Wiesbaden, Germany, under the title “Matthias Pätzold: Mobilfunkkanäle. 1. Auflage (1st Edition)”. Copyright ©
Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, 1999.
Copyright © 2002 by John Wiley & Sons, Ltd
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V
Preface
This book results from my teaching and research activities at the Technical Uni-
versity of Hamburg-Harburg (TUHH), Germany. It is based on my German book
“Mobilfunkkan¨ale — Modellierung, Analyse und Simulation” published by Vieweg &
Sohn, Braunschweig/Wiesbaden, Germany, in 1999. The German version served as a
text for the lecture Modern Methods for Modelling of Networks, which I gave at the
TUHH from 1996 to 2000 for students in electrical engineering at masters level.
The book mainly is addressed to engineers, computer scientists, and physicists, who
workintheindustryorinresearchinstitutesinthewirelesscommunicationsfieldand
therefore have a professional interest in subjects dealing with mobile fading channels.
In addition to that, it is also suitable for scientists working on present problems of
stochastic and deterministic channel modelling. Last, but not least, this book also is
addressed to master students of electrical engineering who are specialising in mobile
radio communications.
In order to be able to study this book, basic knowledge of probability theory and
system theory is required, with which students at masters level are in general
familiar. In order to simplify comprehension, the fundamental mathematical tools,
which are relevant for the objectives of this book, are recapitulated at the beginning.
Starting from this basic knowledge, nearly all statements made in this book are
derived in detail, so that a high grade of mathematical unity is achieved. Thanks
to sufficient advice and help, it is guaranteed that the interested reader can verify
the results with reasonable effort. Longer derivations interrupting the flow of the
content are found in the Appendices. There, the reader can also find a selection
of MATLAB-programs, which should give practical help in the application of the
methods described in the book. To illustrate the results, a large number of figures
have been included, whose meanings are explained in the text. Use of abbreviations
has generally been avoided, which in my experience simplifies the readability consid-
erably.Furthermore,alargenumberofreferencesisprovided,sothatthereaderisled
tofurthersourcesofthealmostinexhaustibletopicofmobilefadingchannelmodelling.
My aim was to introduce the reader to the fundamentals of modelling, analysis,
and simulation of mobile fading channels. One of the main focuses of this book is
the treatment of deterministic processes. They form the basis for the development
of efficient channel simulators. For the design of deterministic processes with given
correlation properties, nearly all the methods known in the literature up to now
are introduced, analysed, and assessed on their performance in this book. Further
focus is put on the derivation and analysis of stochastic channel models as well
as on the development of highly precise channel simulators for various classes of
frequency-selective and frequency-nonselective mobile radio channels. Moreover, a
primarytopicisthefittingofthestatisticalpropertiesofthedesignedchannelmodels
VI
to the statistics of real-world channels.
At this point, I would like to thank those people, without whose help this book
would never have been published in its present form. First, I would like to express
my warmest thanks to Stephan Kraus and Can Karadogan, who assisted me with
the English translation considerably. I would especially like to thank Frank Laue for
performingthecomputerexperimentsinthebookandformakingthegraphicalplots,
whichdecisivelyimprovedthevividnessandsimplifiedthecomprehensionofthetext.
Sincerely,IwouldliketothankAlbertoD´ıazGuerreroandQiYaoforreviewingmost
parts of the manuscript and for giving me numerous suggestions that have helped me
toshapethebookintoitspresentform.Finally,IamalsogratefultoMarkHammond
and Sarah Hinton my editors at John Wiley & Sons, Ltd.
Matthias P¨atzold
Grimstad
January 2002
VII
Contents
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 THE EVOLUTION OF MOBILE RADIO SYSTEMS . . . . . . . . . 1
1.2 BASIC KNOWLEDGE OF MOBILE RADIO CHANNELS . . . . . . 3
1.3 STRUCTURE OF THIS BOOK . . . . . . . . . . . . . . . . . . . . . 7
2 RANDOM VARIABLES, STOCHASTIC PROCESSES, AND
DETERMINISTIC SIGNALS . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 RANDOM VARIABLES . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Important Probability Density Functions . . . . . . . . . . . . 15
2.1.2 Functions of Random Variables . . . . . . . . . . . . . . . . . . 19
2.2 STOCHASTIC PROCESSES . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Stationary Processes . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 Ergodic Processes . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3 Level-Crossing Rate and Average Duration of Fades . . . . . . 25
2.3 DETERMINISTIC CONTINUOUS-TIME SIGNALS . . . . . . . . . . 27
2.4 DETERMINISTIC DISCRETE-TIME SIGNALS . . . . . . . . . . . . 29
3 RAYLEIGH AND RICE PROCESSES AS REFERENCE MODELS 33
3.1 GENERAL DESCRIPTION OF RICE AND RAYLEIGH PROCESSES 34
3.2 ELEMENTARY PROPERTIES OF RICE AND RAYLEIGH PRO-
CESSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 STATISTICAL PROPERTIES OF RICE AND RAYLEIGH PRO-
CESSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.1 Probability Density Function of the Amplitude and the Phase . 39
3.3.2 Level-Crossing Rate and Average Duration of Fades . . . . . . 41
3.3.3 The Statistics of the Fading Intervals of Rayleigh Processes . . 46
4 INTRODUCTION TO THE THEORY OF DETERMINISTIC
PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1 PRINCIPLE OF DETERMINISTIC CHANNEL MODELLING . . . . 56
4.2 ELEMENTARY PROPERTIES OF DETERMINISTIC PROCESSES 59
4.3 STATISTICAL PROPERTIES OF DETERMINISTIC PROCESSES . 63
4.3.1 Probability Density Function of the Amplitude and the Phase . 64
4.3.2 Level-Crossing Rate and Average Duration of Fades . . . . . . 72
4.3.3 Statistics of the Fading Intervals at Low Levels . . . . . . . . . 77
VIII Contents
4.3.4 Ergodicity and Criteria for the Performance Evaluation . . . . 78
5 METHODS FOR THE COMPUTATION OF THE MODEL
PARAMETERS OF DETERMINISTIC PROCESSES . . . . . . . . 81
5.1 METHODS FOR THE COMPUTATION OF THE DISCRETE
DOPPLER FREQUENCIES AND DOPPLER COEFFICIENTS . . . 83
5.1.1 Method of Equal Distances (MED) . . . . . . . . . . . . . . . . 83
5.1.2 Mean-Square-Error Method (MSEM). . . . . . . . . . . . . . . 90
5.1.3 Method of Equal Areas (MEA) . . . . . . . . . . . . . . . . . . 95
5.1.4 Monte Carlo Method (MCM) . . . . . . . . . . . . . . . . . . . 104
5.1.5 L -Norm Method (LPNM) . . . . . . . . . . . . . . . . . . . . 113
p
5.1.6 Method of Exact Doppler Spread (MEDS) . . . . . . . . . . . . 128
5.1.7 Jakes Method (JM) . . . . . . . . . . . . . . . . . . . . . . . . 133
5.2 METHODS FOR THE COMPUTATION OF THE DOPPLER PHASES143
5.3 FADINGINTERVALSOFDETERMINISTICRAYLEIGHPROCESSES145
6 FREQUENCY-NONSELECTIVE STOCHASTIC AND DETER-
MINISTIC CHANNEL MODELS . . . . . . . . . . . . . . . . . . . . . 155
6.1 THE EXTENDED SUZUKI PROCESS OF TYPE I . . . . . . . . . . 157
6.1.1 Modelling and Analysis of the Short-Term Fading . . . . . . . 157
6.1.1.1 ProbabilityDensityFunctionoftheAmplitudeandthe
Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.1.1.2 Level-Crossing Rate and Average Duration of Fades . 166
6.1.2 Modelling and Analysis of the Long-Term Fading . . . . . . . . 169
6.1.3 The Stochastic Extended Suzuki Process of Type I . . . . . . . 172
6.1.4 The Deterministic Extended Suzuki Process of Type I . . . . . 176
6.1.5 Applications and Simulation Results . . . . . . . . . . . . . . . 181
6.2 THE EXTENDED SUZUKI PROCESS OF TYPE II . . . . . . . . . 185
6.2.1 Modelling and Analysis of the Short-Term Fading . . . . . . . 186
6.2.1.1 ProbabilityDensityFunctionoftheAmplitudeandthe
Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
6.2.1.2 Level-Crossing Rate and Average Duration of Fades . 193
6.2.2 The Stochastic Extended Suzuki Process of Type II . . . . . . 196
6.2.3 The Deterministic Extended Suzuki Process of Type II. . . . . 200
6.2.4 Applications and Simulation Results . . . . . . . . . . . . . . . 205
6.3 THE GENERALIZED RICE PROCESS . . . . . . . . . . . . . . . . . 208
6.3.1 The Stochastic Generalized Rice Process . . . . . . . . . . . . . 209
6.3.2 The Deterministic Generalized Rice Process . . . . . . . . . . . 213
6.3.3 Applications and Simulation Results . . . . . . . . . . . . . . . 217
6.4 THE MODIFIED LOO MODEL . . . . . . . . . . . . . . . . . . . . . 218
6.4.1 The Stochastic Modified Loo Model . . . . . . . . . . . . . . . 218
6.4.1.1 AutocorrelationFunctionandDopplerPowerSpectral
Density . . . . . . . . . . . . . . . . . . . . . . . . . . 222
6.4.1.2 ProbabilityDensityFunctionoftheAmplitudeandthe
Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
6.4.1.3 Level-Crossing Rate and Average Duration of Fades . 228
6.4.2 The Deterministic Modified Loo Model . . . . . . . . . . . . . 230
Contents IX
6.4.3 Applications and Simulation Results . . . . . . . . . . . . . . . 236
7 FREQUENCY-SELECTIVE STOCHASTIC AND DETERMIN-
ISTIC CHANNEL MODELS . . . . . . . . . . . . . . . . . . . . . . . . 241
7.1 THE ELLIPSES MODEL OF PARSONS AND BAJWA . . . . . . . . 244
7.2 SYSTEM THEORETICAL DESCRIPTION OF FREQUENCY-
SELECTIVE CHANNELS . . . . . . . . . . . . . . . . . . . . . . . . . 245
7.3 FREQUENCY-SELECTIVE STOCHASTIC CHANNEL MODELS. . 250
7.3.1 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . 250
7.3.2 The WSSUS Model According to Bello. . . . . . . . . . . . . . 251
7.3.2.1 WSS Models . . . . . . . . . . . . . . . . . . . . . . . 251
7.3.2.2 US Models . . . . . . . . . . . . . . . . . . . . . . . . 253
7.3.2.3 WSSUS Models . . . . . . . . . . . . . . . . . . . . . 253
7.3.3 The Channel Models According to COST 207 . . . . . . . . . . 259
7.4 FREQUENCY-SELECTIVE DETERMINISTIC CHANNEL MODELS 267
7.4.1 SystemFunctionsofFrequency-SelectiveDeterministicChannel
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
7.4.2 Correlation Functions and Power Spectral Densities of DGUS
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
7.4.3 Delay Power Spectral Density, Doppler Power Spectral Density,
and Characteristic Quantities of DGUS Models . . . . . . . . . 276
7.4.4 Determination of the Model Parameters of DGUS Models . . . 281
7.4.4.1 Determination of the discrete propagation delays and
delay coefficients . . . . . . . . . . . . . . . . . . . . . 281
7.4.4.2 DeterminationofthediscreteDopplerfrequenciesand
Doppler coefficients . . . . . . . . . . . . . . . . . . . 283
7.4.4.3 Determination of the Doppler phases . . . . . . . . . 284
7.4.5 Deterministic Simulation Models for the Channel Models
According to COST 207 . . . . . . . . . . . . . . . . . . . . . . 284
8 FAST CHANNEL SIMULATORS . . . . . . . . . . . . . . . . . . . . . 289
8.1 DISCRETE DETERMINISTIC PROCESSES . . . . . . . . . . . . . . 290
8.2 REALIZATION OF DISCRETE DETERMINISTIC PROCESSES . . 292
8.2.1 Tables System . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
8.2.2 Matrix System . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
8.2.3 Shift Register System . . . . . . . . . . . . . . . . . . . . . . . 297
8.3 PROPERTIES OF DISCRETE DETERMINISTIC PROCESSES . . . 297
8.3.1 Elementary Properties of Discrete Deterministic Processes . . . 298
8.3.2 Statistical Properties of Discrete Deterministic Processes . . . 305
8.3.2.1 Probability Density Function and Cumulative Distri-
bution Function of the Amplitude and the Phase . . . 306
8.3.2.2 Level-Crossing Rate and Average Duration of Fades . 313
8.4 REALIZATION EXPENDITURE AND SIMULATION SPEED. . . . 315
8.5 COMPARISON WITH THE FILTER METHOD . . . . . . . . . . . . 317
