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Signal Design for Good CorrelationFor Wireless Communication, Cryptography, and Radar

S O L O M O N W . G O L O M BUniversity of Southern California

G U A N G G O N GUniversity ofWaterloo, On tario

K l CAMBRIDGE^ P UNIVE RS IT Y PRE SS

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Contents

Preface page xiAcknowledgments xiiiHistorical Introduction xv

1 Gen eral Properties of Correlation 11.1 W hat is correlation? 11.2 Continuous correlation 21.3 Binary correlation 21.4 Com plex correlation 31.5 M utual orthogonality 31.6 The simplex bound on mutual negative correlation 41.7 Au tocorrelation 61.8 Crosscorrelation 72 Applications of Correlation to the Com mu nication

of Information 102.1 The maxim um likelihood detector 102.2 Coherent versus incoherent detection 122.3 Orthogonal, biorthogonal, and simplex codes 142.4 Hadam ard matrices and code construction 152.5 Cyclic Hadam ard matrices 183 Finite Fields 223.1 Algebraic structures 223.2 Construction of GF(pn) 313.3 The basic theory of finite fields 343.4 M inimal polynomials 413.5 Trace functions 523.6 Powers of trace functions 583.7 The num bers of irreducible polynom ials and coset leaders 69

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viii Contents

App endix A: A M aple program for step 3 in Algorithm 3.1 72Appendix B: Primitive polynom ials 72Appendix C: M inimal polynom ials 77Exercises for Chapter 3 79

4 Feedback Shift Register Sequ ences 814.1 Feedback shift registers 824.2 Definition of LFSR sequences in terms of polynom ial rings 904.3 Minimal polynomials and periods 944.4 Decom position of LFSR sequences 1034.5 The matrix representation 1064.6 Trace representation of LFSR s 1084.7 Generating functions of LFSRs 112

Exercises for C hapter 4 1145 Random ness M easurements and m -Sequences 1175.1 Golom b's random ness postulates and random ness criteria 1175.2 Rand om ness properties of m-sequences 1275.3 Interleaved structu reof m-sequences 1355.4 Trinom ial property 1455.5 Constant-on-cosets property 1485.6 Two-tuple balance property 1525.7 Classification of binary sequences of period 2" 1 155

Exercises for Chapter 5 1596 Transforms of Sequences and Fun ctions 1626.1 The (discrete) Fourier transform 1626.2 Trace representation 1666.3 Linear spans and spectral sequences 1746.4 One-to-one correspondence between sequences and functions 1776.5 Hadam ard transform and convolution transform 1856.6 Correlation of functions 1906.7 Laws of the Hadam ard transform and convolution transform 1936.8 The matrix representation of the DF T and the

Hadamard transform 197Exercises for Chapter 6 199

7 Cyclic Difference Sets and Binary Sequen ces withTwo-Level Autocorrelation 2027.1 Cyclic difference sets and their relationship to binary sequences

with two-level autocorrelation 2027.2 More results about C 2077.3 Fourier spectral constraints 211

Exercises for Chapter 7 218

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Contents ix

8 Cyclic Hadam ard Sequences, Part 1 2198.1 Constructions with subfield decom position 2208.2 GM W constructions 2348.3 Statistical properties of GM W sequences of all types 2488.4 Linear spans of GM W sequences of all types 2508.5 Shift-distinct sequences from GM W cons tructions 2558.6 Implem entation aspects of GM W constructions 257

Exercises for Chapter 8 2649 Cyclic Hadam ard Sequences, Part 2 2679.1 M ultiple trace term sequences 2679.2 Hyperoval constructions 2829.3 Kasami power function construction 2949.4 The iterative decimation Hadam ard transform 303

Appendix: Known 2-level autocorrelation sequences of periods1023, 2047 , and 4095 and their trace representations 318

Exercises for Chap ter 9 32110 Signal Sets with Low Crosscorrelation 32310.1 Crosscorre lation, signal sets, and boolean functions 32310.2 Odd case: Go ld-pair signal sets and their generalization 33610.3 Even case: Kasam i (small) signal sets and their generalization 34410.4 Even case: Bent function signal sets 35310.5 Interleaved construction of signal sets 36310.6 Z 4 signal sets 371Exercises for Chapter 10 38011 Correlation of Boolean Fun ctions 38211.1 Invariants, resiliency, and nonlinearity 38311.2 Dual functions and resiliency 39211.3 Dual functions, additive autoco rrelation, and the

propagation property 395Exercises for Chapter 11 401

12 App lications to Radar, Sonar, Synch ronization, and CDM A 40212.1 Overview 40212.2 Types of Signals and correlations 40312.3 Barker sequences 40412.4 Generalized Barker sequences 40512.5 Huffman's impulse-equivalent pulse trains 40612.6 Pulse patterns and optimal rulers 40812.7 Perfect circular rulers from cyclic projective planes 41212.8 Tw o-dimensional pulse patterns 41512.9 Periodic modulation 417

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