Performance Analysis and Modeling of Digital Transmission Systems

450 Pages·2004·11.542 MB·English

by  William Turin (auth.)

PERFORMANCE ANALYSIS
AND MODELING OF DIGITAL
TRANSMISSION SYSTEMS
Information Technology: Transmission, Processing, and Storage
Series Editor: Jack Keil Wolf
University of California at San Diego
La Jolla, California
Editorial Board: Robert J. McEliece
California Institute of Technology
Pasadena, California
John Proakis
Northeastern University
Boston, Massachusetts
William H. Tranter
Virginia Polytechnic Institute and State University
Blacksburg, Virginia
Coded Modulation Systems
John B. Anderson and Arne Svensson
Communication System Design Using DSP Algorithms: With
Laboratory Experiments for the TMS320C6701 and TMS320C6711
Steven A. Tretter
A First Course in Information Theory
Raymond W. Yeung
Interference Avoidance Methods for Wireless Systems
Dimitrie C. Popescu and Christopher Rose
Nonuniform Sampling: Theory and Practice
Edited by Farokh Marvasti
Performance Analysis and Modeling of Digital Transmission Systems
William Turin
Stochastic Image Processing
Chee Sun Won and Robert M. Gray
A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume
immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact
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PERFORMANCE ANALYSIS
AND MODELING OF DIGITAL
TRANSMISSION SYSTEMS
William Thrin
AT&T Labs-Research
Florham Park, New Jersey
Springer Science+Business Media, LLC
Library of Congress Cataloging-in-Publication Data
Turin, Williarn.
Performance analysis and modeling of digital transmission systems/Williarn Turin.
p. cm. -(Information technology: transmission, processing, and storage)
Rev. ed. of: Digital transmission systems, ci999.
Includes bibliographical references and index.
Additional material to this book can be downloaded from http://extra.springer.com.
ISBN 978-1-4613-4781-1 ISBN 978-1-4419-9070-9 (eBook)
DOI 10.1007/978-1-4419-9070-9
1. Digital communications-Evaluation. 2. Digital communications-Mathematical
models. I. Turin, Williarn. Digital transmission systems. 11. Title. Ill. Series.
TK5103.7.T8675 2004
621.382-dc22
200404I4I7
All files on this CD-ROM are covered by the following statement:
Copyright © 2004 by Williarn Turin
The computer code and or data in this file is provided for demonstration purposes only with no
guarantee or warranty of any kind that it is correct or produces correct results. By using the code
and or data in this file the user agrees to accept all risks and liabilities associated with the code
and or data. The computer code and or data in this file may be copied (and used) for
noncommercial academic or research purposes only, provided that this notice is included. This file
or any portion of it may not be resold, rented, or distributed without written permission of the
author.
The files are organized into the following directories: Chapter I through Chapter 7, and Appendix
5 through Appendix 7. For details on using the CD-ROM, please refer to the book Introduction.
For further information, please write to the publisher at [email protected] to obtain the
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Additional material to tbis book can be downloaded from http://extras.springer.com.
©2004 Springer Science+ Business Media New York
Originally published by Kluwer Academic/Plenum Publishers, New York in 2004
Softcover reprint of the hardcover 1st edition 2004
http://www.kluweronline.com
ill 9 8 7 6 5 4 3 2
A C.I.P. record for this book is available from the Library of Congress
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or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise,
without written permission from the Publisher, with the exception of any material supplied
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use by the purchaser of the work.
Perrnissions for books published in Europe: [email protected]
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To my wife Galina
CONTENTS
PREFACE .................................................................................................................. xv
NOTATION ............................................................................................................... xix
INTRODUCTION .................................................................................................... .
1. ERROR SOURCE MODELS ......................... .................................................. 9
1.1 DESCRIPTION OF ERROR SOURCES BY
HIDDEN MARKOV MODELS .................................. ................... ............. 9
1.1.1 Finite State Channel................................................................. ............. 9
1.1.2 Hidden Markov Models .................. ................................................ ....... 11
1.1.3 Discrete-Time Finite State Systems ................................ .................. .... 12
1.1.4 Error Source HMM ............ ............. ...................................................... 14
1.1.5 Gilbert's Model ..................................................................................... 15
1.1.6 Equivalent Models.. ................................................ ......... ............. ......... 17
1.1.7 The HMM Model Generality............... ................................................. 18
1.1.8 Satellite Channel Model............ ........... ..................................... ............ 19
1.1.9 Fading Wireless Channels ...................................... ......... .......... ............ 22
1.1.10 Concatenated Channel Error Sources ....... ........................................... 24
1.2 BINARY SYMMETRIC STATIONARY CHANNEL ............................... 25
1.2.1 Model Description .. ................................................. ......... ..................... 25
1.2.2 Diagonalization of the Matrices ............................................................ 28
1.2.3 Models with Two Sets of States ....................................... ..................... 28
1.2.4 Block Matrix Representation ................................ ................................. 30
1.3 ERROR SOURCE DESCRIPTION BY MATRIX PROCESSES .............. 33
1.3.1 Matrix Process Definition ...................................................................... 33
vii
viii Contents
1.3.2 Block Matrix Representation . ................................................................ 35
1.3.3 Matrix Processes and Difference Equations .......................................... 36
1.304 Matrix Processes and Markov Functions .............................................. 38
104 ERROR SOURCE DESCRIPTION BY SEMI-MARKOV PROCESSES .. 39
104.1 Semi-Markov Processes ..................................................... ................... 39
104.2 Semi-Markov Lumping of Markov Chain ............................................. 42
1.5 SOME PARTICULAR ERROR SOURCE MODELS ................................ 49
1.5.1 Single-Error-State Models ..................................................................... 49
1.5.2 Alternating Renewal Process Models................ ... ................................. 51
1.5.3 Alternating State HMM ......................................................................... 54
1.504 Fading Channel Errors ........................................................................... 55
1.6 CONCLUSION ......................... ......................... .............................. ............ 56
References ............ ...................... ..... .................. ...... ...... ... ... ............. ...... ..... ....... 57
2. MATRIX PROBABILITIES ............................................................................. 61
2.1 MATRIX PROBABILITIES AND THEIR PROPERTIES ......................... 61
2.1.1 Basic Definitions ........... ..... .... ....... ....... ............ ...... ..... .............. ............ 61
2.1.2 Properties of Matrix Probabilities ............. ............................................ 63
2.1.3 Random V ariables ................................................................................. 65
2.2 MATRIX TRANSFORMS .......................................................................... 66
2.2.1 Matrix Generating Functions ............................. .............................. ...... 66
2.2.2 Matrix z-Transforms .............................................................................. 69
2.2.3 Matrix Fourier Transform ................................. ......... ..................... ...... 70
2.204 Matrix Discrete Fourier Transform .................. ......... ............................ 71
2.2.5 Matrix Transforms and Difference Equations .. ..................................... 73
2.3 MATRIX DISTRIBUTIONS ....................................................................... 74
2.3.1 Matrix Multinomial Distribution ........................................................... 74
2.3.2 Matrix Binomial Distribution ................................................................ 76
2.3.3 Matrix Pascal Distribution ..................................................................... 80
204 MARKOV FUNCTIONS ............................................................................. 81
204.1 Block Matrix Probabilities ............................. ................... .................... 81
204.2 Matrix Multinomial Distribution .................... ................... .................... 83
204.3 Interval Distributions of Markov Functions ... ....................................... 84
20404 Signal Flow Graph Applications .................... ....................................... 86
2.5 MONTE CARLO METHOD ............................... ................... ..................... 90
2.5.1 Error Source Simulation ........................................................................ 90
2.5.2 Performance Characteristic Calculation ................................................ 92
2.6 COMPUTING SCALAR PROBABILITIES ................ ............................... 93
2.5.1 The Forward and Backward Algorithms ............................................... 93
2.5.2 Scalar Generating Functions ................................................................. 94
2.7 CONCLUSION ............................................................................................ 97
References ....................... ............................ ....... ...... ........................ ........... ... .... 99
Contents ix
3. MODEL PARAMETER ESTIMATION ......................................................... 101
3.1 The EM ALGORITHM ................................................................................ 101
3.1.1 Kullback-Leibler Divergence ................................................................ 101
3.1.2 Minimization Algorithms 102
3.1.3 The EM Algorithm ................................................................................ 103
3.1.4 Maximization of a Function .................................................................. 109
3.1.5 EM Algorithm Acceleration .................................................................. 111
3.1.6 Statistical EM Algorithm .................................................... .................. 111
3.2 BAUM-WELCH ALGORITHM ................................................................. 112
3.2.1 Phase-Type Distribution Approximation .............................................. 113
3.2.2 HMM Approximation ............................................................................ 120
3.2.3 Fitting HMM to Experimental Data ...................................................... 127
3.2.4 Matrix Form of the BWA ...................................... .............. .................. 130
3.2.5 Scaling ................................................................................................... 133
3.3 MARKOV RENEWAL PROCESS ............................................................. 134
3.3.1 Fast Forward and Backward Algorithms ............................................... 135
3.3.2 Fitting MRP ........................................................................................... 136
3.4 MATRIX-GEOMETRIC DISTRIBUTION
PARAMETER ESTIMATION ................................................................... 142
3.4.1 Distribution Simplification .................................................................... 142
3.4.2 ML Parameter Estimation ..................................................................... 144
3.4.3 Sequential Least Mean Square Method ................................................. 145
3.4.4 Piecewise Logarithmic Transformation ................................................ 147
3.4.5 Prony's Method ..................................................................................... 147
3.4.6 Matrix Echelon Parametrization ............................................................ 151
3.4.7 Utilization of the Cumulative Distribution Function ............................ 152
3.4.8 Method of Moments .............................................................................. 154
3.4.9 Transform Domain Approximation ....................................................... 155
3.5 MATRIX PROCESS PARAMETER ESTIMATION ................................. 156
3.5.1 ML Parameter Estimation ..................................................................... 156
3.5.2 Interval Distribution Estimation ............................................................ 157
3.5.3 Utilization of Two-Dimensional Distributions ...................................... 157
3.5.4 Polygeometric Distributions .................................................................. 159
3.5.5 EITor Bursts ........................................................................................... 160
3.6 HMM PARAMETER ESTIMATION ......................................................... 161
3.6.1 Multilayered Error Clusters ................................................................... 161
3.6.2 Nested Markov Chains .......................................................................... 164
3.6.3 Single-EITor-State Models ..................................................................... 166
3.7 MONTE CARLO METHOD OF MODEL BUILDING .............................. 167
3.8 ERROR SOURCE MODEL IN SEVERAL CHANNELS .......................... 168
3.9 CONCLUSION ............................................................................................ 169
x Contents
References ................ ... ........ ...... ........ .... .... ........ ...... ............ ...... .................. ....... 172
4. PERFORMANCE OF FORWARD ERROR-CORRECTION SYSTEMS .. 175
4.1 BASIC CHARACTERISTICS OF ONE-WAY SYSTEMS ....................... 175
4.2 ELEMENTS OF ERROR-CORRECTING CODING ................................. 176
4.2.1 Field Galois Arithmetic ..................................................... .................... 176
4.2.2 Linear Codes .......................................................................................... 179
4.2.3 Cyclic Codes .......................................................................................... 181
4.2.4 Bose-Chaudhuri-Hocquenghem Codes ................................................. 184
4.2.5 Reed-Solomon Codes ............................................................................ 186
4.2.6 Convolutional Codes ............................................................................. 186
4.2.6 Trellis-Code Modulation ....................................................................... 194
4.3 MAXIMUM A POSTERIORI DECODING ............................................... 196
4.3.1 MAP Symbol Estimation ...................................................................... 197
4.3.2 MAP' Sequence Decoding ..................................................................... 204
4.4 BLOCK CODE PERFORMANCE CHARACTERIZATION ..................... 210
4.4.1 The Probability of Undetected Error ..................................................... 211
4.4.2 Performance of Forward Error-Correcting Codes ................................. 213
4.4.3 Symmetrically Dependent Errors .......................................................... 216
4.4.4 Upper and Lower Bounds ...................................................................... 219
4.4.5 Postdecoding Error Probability............................................................. 223
4.4.6 Soft Decision Decoding ......................................................................... 225
4.4.7 Correcting Errors and Erasures ............................................................. 227
4.4.8 Product Codes ........................................................................................ 230
4.4.9 Concatenated Codes .............................................................................. 232
4.5 CONVOLUTIONAL CODE PERFORMANCE ......................................... 233
4.5.1 Viterbi Algorithm Performance .............................................. ............... 233
4.5.2 Syndrome Decoding Performance ......................................................... 236
4.6 COMPUTER SIMULATION .............. .......................... .............. .......... ...... 239
4.7 ZERO-REDUNDANCY CODES ................................................................ 240
4.7.1 Interleaving............ ............ ............ ................. ....................................... 240
4.7.2 Encryptors and Scramblers .................................................................... 241
4.8 CONCLUSION ............................... ............................................................. 242
References .......................................................................................................... 242
5. PERFORMANCE ANALYSIS OF COMMUNICATION PROTOCOLS ... 245
5.1 BASIC CHARACTERISTICS OF TWO-WAY SYSTEMS ....................... 245
5.2 RETURN-CHANNEL MESSAGES ............................................................ 248
5.3 SYNCHRONIZATION ................................................................................ 250
5.4 ARQ PERFORMANCE CHARACTERISTICS ......................................... 253
5.4.1 Stop-and-Wait ARQ .............................................................................. 254
Contents xi
5.4.2 Message Delay Distribution .................................................................. 257
5.4.3 Insertion and Loss Interval Distribution ................................................ 258
50404 Accepted Messages ............................................................................... 258
504.5 Alternative Assumptions ....................................................................... 259
5.4.6 Modified Stop-and-Wait ARQ .............................................................. 260
5.4.7 Go-Back-N ARQ ................................................................................... 263
5.4.8 Multiframe Messages ............................................................................ 270
5.5 DELAY-CONSTAINED SYSTEMS ........................................................... 273
5.5.1 Queue-Length Probability Distribution ................................................. 274
5.5.2 Packet Delay Probability....................................................................... 275
5.6 CONCLUSION ............................................................................................ 276
References .......................................................................................................... 277
6. CONTINUOUS TIME HMM ............................................................................ 279
6.1 CONTINUOUS AND DISCRETE TIME HMM ........................................ 279
6.1.1 Markov Arrival Processes ..................................................................... 279
6.1.2 MAP Discrete Skeleton ....... .................................................................. 282
6.1.3 Matrix Poisson Distribution .................................................................. 283
6.2 FITTING CONTINUOUS TIME HMM ...................................................... 286
6.2.1 Fitting Phase-Type Distribution .......... .................................................. 286
6.2.2 HMM Approximation. ........ ................................................................... 290
6.2.5 CONCLUSION ............... ........ .................................................................. 292
References .......................................................................................................... 292
7. CONTINUOUS STATE HMM ......................................................................... 295
7.1 CONTINUOUS AND DISCRETE STATE HMM ...................................... 295
7.2 OPERATOR PROBABILITy..................................................................... 297
7.3 FILTERING, PREDICTION, AND SMOOTHING .................................... 299
7 A LINEAR SYSTEMS .................................................................................... 302
704.1 Autocovariance Function ...................................................................... 305
7.4.2 Observation Sequence PDF ................................................................... 307
704.3 Kalman Filter PDF ................................................................................ 308
7.404 The Innovation Representation PDF ..................................................... 314
704.5 The Backward Algorithm ...................................................................... 316
7.4.6 The Forward-Backward Algorithm PDF ............................................... 320
704.7 RTS Smoother ....................................................................................... 320
704.8 Viterbi Algorithm .................................................................................. 324
7.5 AUTOREGRESSIVE MOVING AVERAGE PROCESSES ...................... 325
7.5.1 State Space Representation ................................................................... 326
7.5.2 Autocovariance Function ...................................................................... 328
7.5.3 The Forward Algorithm ......................................................................... 329