Pseudo Random Signal Processing

Pseudo Random Signal Processing

Theory and Application

Hans-Jiirgen Zepernick Blekinge Institute of Technology, Sweden

Adolf Finger Dresden University of Technology, Germany

John Wiley &. Sons, Ltd

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Contents

Preface ix

List of abbreviations xi

List of common symbols xvii

1 Introduction 1 1.1 Prologue 1 1.2 Elements of pseudo random signal processing 2 1.3 Outline of the book 5

2 Characterization of signals and sequences 7 2.1 Classification of signals and sequences 7

2.1.1 Morphological classification 8 2.1.2 Phenomenological classification 9 2.1.3 Energy classification 12 2.1.4 Spectral classification 12

2.2 Transformations of signals and sequences 13 2.2.1 Basic transformations 13

2.3 Correlation measures 16 2.3.1 Autocorrelation and crosscorrelation functions 17 2.3.2 Discrete periodic correlation functions 19 2.3.3 Aperiodic correlation functions 23 2.3.4 Other properties and relationships 26 2.3.5 Correlation of binary sequences 34 2.3.6 Orthogonality 37

2.4 Power spectral density 39 2.4.1 Power spectral density of analog signals 39 2.4.2 Power spectral density of periodic signals 41 2.4.3 Power spectral density of periodic pulse trains 44

2.5 Pseudo random signals and sequences 45 2.5.1 Pseudo randomness criteria 45 2.5.2 Pseudo randomness and power spectral density 48 2.5.3 Pseudo randomness and polyphase sequences 49

vi CONTENTS

3 Mathematical foundations 3.1 Algebraic structures

3.1.1 Binary algebra, semigroup, and monoid 3.1.2 Groups, rings, and fields

3.2 Polynomials over finite fields 3.2.1 Polynomials and polynomial rings 3.2.2 Euclidean algorithm for polynomials 3.2.3 Irreducible polynomials 3.2.4 Cyclotomic cosets and minimal polynomials 3.2.5 Primitive polynomials

4 Binary pseudo random sequences 4.1 Classification 4.2 Maximal-length sequences

4.2.1 Linear recurring sequences 4.2.2 Maximal-length sequences 4.2.3 Properties of maximal-length sequences 4.2.4 Autocorrelation functions of maximal-length sequences

4.3 Binary sequences with good autocorrelation 4.3.1 Difference sets 4.3.2 De Bruijn sequences 4.3.3 Quadratic residue sequences 4.3.4 Other difference set sequences 4.3.5 Barker sequences and Williard sequences

4.4 Binary sequences with special crosscorrelation 4.4.1 Transorthogonal and orthogonal sequences 4.4.2 Gold sequences 4.4.3 Gold-like sequences 4.4.4 Kasarni sequences

5 Nonbinary pseudo random sequences 5.1 Classification 5.2 Interference-free window sequences

5.2.1 Large-area synchronous codes 5.3 Complex-valued sequences

5.3.1 Complex maximal-length sequences 5.3.2 Polyphase sequences 5.3.3 Quadriphase sequences

5.4 Polyphase sequences with special correlations 5.4.1 Equivalent odd and even correlation sequences 5.4.2 Oppermann sequences

6 Generating pseudo random signals 6.1 Linear autonomous automata

6.1.1 Mathematical description 6.1.2 Canonical forms 6.1.3 State cycles

51 51 51 53 61 61 62 67 73 78

87 87 89 90 92 95

102 106 106 108 109 110 112 114 114 118 122 123

127 127 131 131 147 147 149 154 159 159 163

173 173 173 175 179

CONTENTS vii

6.2 Generating maximal-length sequences 182 6.2.1 Standard circuits for binary maximal-length sequences 182 6.2.2 Special cases of modulo 2 arithmetic 184 6.2.3 High-speed sequence generation 189 6.2.4 Nonbinary sequence generation with binary encoding 193

6.3 Transformations of maximal-length sequences 195 6.3.1 Transversal filtering 195 6.3.2 Histogram transformation through mapping 201 6.3.3 Generation of phase-shifted maximal-length sequences 204

6.4 Combinations of maximal-length sequences 209 6.4.1 Modifications of binary maximal-length sequences 209 6.4.2 Product sequences 210 6.4.3 Combination sequences 210 6.4.4 Concatenated sequences 211

6.5 Pseudo random signal processing with microprocessors and memory circuits 212 6.5.1 Realizations with microprocessors 212 6.5.2 Realizations with memory circuits 213 6.5.3 Realizations with programmable logic devices 215 6.5.4 WIND-FLEX 218 6.5.5 Pseudo random signal generators 220

7 Applications of pseudo random signal processing 225 7.1 Spread spectrum communications 226

7.1.1 Basic concepts 227 7.1.2 Basic spread spectrum systems 228 7.1.3 Spread spectrum communication systems 236 7.1.4 Universal mobile telecommunications system 242 7.1.5 Bluetooth 263

7.2 Ranging and navigation systems 269 7.2.1 Ranging principles 269 7.2.2 Correlation receivers 272 7.2.3 Synchronization 275 7.2.4 Global positioning system 288 7.2.5 Galileo 301 7.2.6 Other ranging and navigation systems 310

7.3 Scrambling 314 7.3.1 Scrambling functions 315 7.3.2 Scrambling techniques 316 7.3.3 Scramblers for wireline systems 325 7.3.4 Scramblers for wireless systems 331

7.4 Automatic testing and system verification 340 7.4.1 Signature analysis 341 7.4.2 Built-in self-test schemes 349 7.4.3 Bit error analysis 352

7.5 Cryptology 359 7.5.1 Cryptosystems 360 7.5.2 Generators for stream ciphers 361 7.5.3 Feedback carry shift registers 366

viii CONTENTS

7.5.4 Content scrambling system for digital versatile discs 372 7.5.5 Encryption in radio and television systems 373 7.5.6 Security encryption algorithm AS of the global system for mobile communication 376

7.6 Other applications 379 7.6.1 Correlation analysis of linear systems 379 7.6.2 Optical fiber systems 381 7.6.3 Angular sensor systems 384 7.6.4 Add-on data transmission in analog television 386

Bibliography 391

Index 403

Preface

Pseudo random signal processing has emerged from space and military applications with a history of research and development in these areas spanning a period of more than 40 years. The main focus in these applications was on signal formats and processing techniques that can ensure signal integrity, especially immunity against jamming attacks. The evolution of digital mobile radio systems and the increasing demand for positioning systems along with the advances in integrated circuit complexity have resulted in frequent use of pseudo random signal processing as a viable technique for many civilian and commercial applications. Especially with the introduction of cellular mobile radio systems, pseudo random signal processing has received increased attention during the course of the late 1980s. The numerous features of these processing techniques that are important for cellular radio include the ability to eliminate or alleviate multipath propagation, the resistance to interference, and the potential of sharing allocated bandwidth with other users or even sharing it as an overlay with other commu.nication systems. More recently, successful applications in commercial satellite navigation systems and third-generation mobile communication systems have proven the concepts of pseudo random signal processing as being an important and critical enabler of modern communication and information systems. In addition, the methodology of pseudo random signal processing has evolved into fields such as acoustics, biomedicine, and sensor systems, to mention just a few of the specialized application areas. It can be expected that an in-depth knowledge of pseudo random signal processing will provide the basis for the development of many new applications in communication, information, and computer technologies, in both the short and long term.

The objective of this book is to provide an important transition from covering the mathematical foundations to conveying the powerful engineering concepts of pseudo random signal processing. In particular, the far-reaching signal processing principles will serve as the connecting link between theory and practice. The book is intended to provide comprehensive coverage of the theoretical foundations of pseudo random signal processing, which makes it timeless and independent of the actual state of the art in circuit and system technology. The reader will also gain insights into the increasingly sophisticated applications of the described techniques in modem communication and information technologies such as mobile radio systems, navigation systems, scrambling, circuit testing, cryptology, and a number of selected specialized applications. The combination of theory and practice makes the book attractive as a practically oriented introduction to researchers and it provides essential reading for practicing engineers.

The book is based on our experience in the area of pseudo random signal processing obtained from numerous research projects, teaching senior graduate courses at universities, and delivering short courses to industry. The book is based on a course entitled "Digital Signal Structures," which has been taught by Adolf Finger. It also draws upon material from the senior graduate courses "Advanced Communications" and "Error Control Coding," which have been taught by Hans-Jiirgen Zepemick.

x PREFACE

These courses are aimed at students in their final year of studies to provide insights into the areas of advanced communications theory and how it can be utilized in practice.

The book is aimed at academics and students in the areas of electrical, electronic, and computer engineering as well as scientists and practicing engineers in research and development. It is suitable for a wide audience working in the fields of telecommunications, information technology, and computer science. The book is accessible to readers with at least an undergraduate electrical engineering or computer science background in signals and systems, communications, and electronics. The book is written at an advanced level and will enable the reader to access the more specialized technical articles and textbooks.

We are grateful to the many researchers whose original contributions form the foundations of pseudo random signal processing and who basically have made this book possible. We would also like to thank the anonymous reviewers who provided constructive suggestions and valuable comments that guided us in the early stages to shape the content of the book into its present form.

We would like to extend warm thanks to our students from Europe, the United States, Asia, and Australia, whose questions have helped us to refine the presentation. Special thanks go to our doctoral students and post-doctoral research fellows for the intellectual stimulation they have provided over the years. We would like to thank our friends and colleagues for many helpful discussions and support during the course of writing this book. We also wish to express our deep gratitude to Dr. Manora Caldera and Dr. Helmut Wiehl for proof-reading various parts of the manuscript and their valuable comments and suggestions which have helped us to improve the book.

Finally, we wish to thank the editorial and publishing team of John Wiley & Sons for their enormous assistance in the preparation of this book. In particular, we are very grateful to Birgit Gruber, Sarah Corney, Kathryn Sharples, Claire Twine, Simone Taylor, Emily Bone, and Wendy Hunter for guiding us safely through all the phases of the book project and for their professional work.

Hans-Jorgen Zepemick Adolf Finger

List of abbreviations

3GPP AACF ACCF ACF ACL ADSL AMPS AO ASIC ATM AU AWGN BCD BCR BER B-ISDN BIST BOC BOT BPF BPSK BS CA CCD CCD-PNMF CCF CD CDMA CIW CMOS CO CQI CS CSMA CSMA-CD

Third-generation partnership project Aperiodic autocorrelation function Aperiodic crosscorrelation function Autocorrelation function Asynchronous connectionless Asymmetric digital subscriber line Advanced mobile phone service Auto-optimal Application-specific integrated circuit Asynchronous transfer mode Astronomical unit Additive white Gaussian noise Binary-coded decimal Bose-Chaudhuri-Rocquenghem Bit error rate Broadband integrated services digital network Built-in self-test Binary offset carrier Broadcast online television Bandpass filter Binary phase shift keying Base station Conditional access Charge-coupled device CCD pseudo noise matched filter Crosscorrelation function Collision detection Code-division multiple-access Container identification word Complementary metal oxide semiconductor Cross-optimal Channel-quality indication Commercial service Carrier sense multiple-access CSMA with collision detection

xii ABBREVIATIONS

CSS CUT CVBS CW DAB DAC DC DECT DEMUX DES DGPS DH DK DLL DMT DPCCH DPDCH DS DS-CDMA DSL DSP DSR DSS DSSS DVB DVD ECL EEG EGNOS EK EOE EPROM ESA ETSI EU EUVE FBG FBI FCC FCSR FDD FDMA FFH FH FH-CDMA FHS FHSS FLL

Content scrambling system Circuit -under-test Composite video blanking and synchronization Control word Digital audio broadcast Digital-to-analog converter Direct current Digital enhanced cordless telecommunications Demultiplexer Data encryption standard Differential GPS Data high Distribution key Delay-lock loop Discrete multi-tone Dedicated physical control channel Dedicated physical data channel Direct-sequence Direct-sequence code-division multiple-access Digital subscriber line Digital signal processing Digital satellite radio Distributed sample scrambling Direct-sequence spread spectrum Digital video broadcast Digital versatile disc Emitter coupled logic Electroencephalograph European geostationary navigation overlay service Entitlement key Equivalent odd and even Erasable programmable read-only memory European Space Agency European Telecommunications Standards Institute European Union Extreme Ultraviolet Explorer Fiber Bragg grating Feedback information Federal Communications Commission Feedback with carry shift register Frequency-division duplex Frequency-division multiple-access Fast frequency hopping Frequency hopping Frequency hopping CDMA Frequency hop synchronization Frequency hopping spread spectrum Frequency-lock loop

FM FPGA FSK FSS FZC Gbps GLONASS GMW GPS GRO GSM GSRx HARQ HARQ-ACK HBI HDSL HEC HPA HPSK HS-DPCCH HS-PDSCH IEEE IF IFW 1R IS-95 ISDN lSI ISM ITU LA LAB LAP LAS LC LED LEO LFSR LORAN LPF LS LSB LSE LTI LUT MAC MAl MASER

Frequency modulation Field programmable logic gate array Frequency shift keying Frame synchronous scrambling Frank-Zadoff-Chu Gigabits per second Global navigation satellite system Gordon-Mills-Welch Global positioning system Gamma Ray Observatory Global system for mobile communication Ground station reference receiver Hybrid automatic repeat request HARQ acknowledgment Horizontal blanking interval High-bit-rate digital subscriber line Header error control High-power amplifier Hybrid PSK

ABBREVIATIONS xiii

High-speed dedicated physical control channel High-speed dedicated physical downlink shared channel Institute of Electrical and Electronics Engineers Intermediate frequency Interference-free window Infrared Interim Standard 95 Integrated services digital network Intersymbol interference Industrial, scientific, and medical International Telecommunications Union Large-area Logic array block Lower address part Large-area synchronous Linear complexity Light-emitting diode Low earth orbit Linear feedback shift register Long-range navigation Lowpass filter Loosely synchronous codes Least significant bit Least sidelobe energy Linear time-invariant Look-up table Medium access control Multiple access interference Microwave amplification by stimulated emission of radiation

xiv ABBREVIA nONS

MC MCC MC-CDMA MEO MF MFSK MLSSA MPDU MPEG MS MSAC MSAS MSB MSCC MSPD MSPR MT-CDMA MUI MUX NASA NAVSTAR NCO NLES OCQPSK OFDM OOK OQPSK OS OSI OVSF PACF PAL PAR PCCF PCM PCPCH PCS PD PDF PHY PLA PLCP PLD PLL PN PPDU PRACH PRBS

Multi-carrier Master control center Multi-carrier CDMA Medium earth orbit Merit factor M-ary FSK Maximal-length sequence system analyzer MAC sublayer protocol data unit Moving Pictures Experts Group Mobile station Mean-square out-of-phase aperiodic autocorrelation MT sat-based augmentation system Most significant bit Mean-square aperiodic crosscorrelation Maximum peak-to-side-peak distance Maximum peak-to-side-peak ratio Multi-tone CDMA Multi-user interference Multiplexer National Aeronautics and Space Administration Navigation system time and ranging Number-controlled oscillator Navigation land earth station Orthogonal complex quadrature phase shift keying Orthogonal frequency-division multiplexing On-off keying Offset quadrature phase shift keying Open services Open system interconnection Orthogonal variable spreading factor Periodic autocorrelation function Phase alternate line Peak-to-average ratio Periodic crosscorrelation function Pulse code modulation Physical common packet channel Personal communications system Photo diode Probability density function Physical layer Programmable logic array Physical layer convergence protocol Programmable logic device Phase-lock loop Pseudo noise PLCP protocol data unit Physical random access channel Pseudo random bit sequence

PRC PRN PRPG PROM PRS PSC P-SCH PSD PSDU PSK QAM QoS QPSK RAM RARASE RASE RF RIMS RNSS ROM RSA SA SAR SAW SCH SDH SDU SFD SFH SIG SK SNR SOH SOL SIP SS S-SCH SSS STM TD-CDMA TDD TDL TDMA TDRSS TFC TFCI THSS TIA

Pseudo random code Pseudo random noise Pseudo random pattern generator Programmable read-only memory Public regulated services Primary synchronization code Primary SCH Power spectral density PLCP service data unit Phase shift keying Quadrature amplitude modulation Quality-of-service Quadrature phase shift keying Random access memory Recursion-aided RASE Rapid acquisition by sequential estimation Radio frequency Ranging and monitoring station Radio navigation satellite system Read-only memory Rivest-Shamir-Adleman Selective availability Search and rescue Surface acoustic wave Synchronization channel Synchronous digital hierarchy Service data unit Start frame delimiter Slow frequency hopping Special interest group Service key Signal-to-noise ratio Section overhead Safety-of-life Serial-to-parallel Spread spectrum Secondary SCH Self-synchronous scrambling Synchronous transport module Time-division CDMA Time-division duplex Tau-dither loop Time-division multiple-access Tracking and data relay satellite system Transport format combination Transport format combination indicator Time hopping spread spectrum Telecommunications Industry Association

ABBREVIATIONS xv

xvi ABBREVIATIONS

TPC TPG TTL TV UAP UHF UMTS U-NII UTRA VBI VCO VDSL VEP VHDL WAAS WCDMA WLAN WPAN WSS XNR ZC ZCZ

Transmit power control Test pattern generator Transistor-transistor logic Television Upper address part Ultra high frequency Universal mobile telecommunications system Unlicensed national information infrastructure UMTS terrestrial radio access Vertical blanking interval Voltage controlled oscillator Very high-bit -rate digital subscriber line Visual evoked potential Very high-speed integrated circuit hardware description language Wide area augmentation system Wideband CDMA Wireless local area network Wireless personal area network Wide-sense stationary Exclusive-NOR Zadoff-Chu Zero correlation zone

List of common symbols

a; 4)

(I) a;

{a;} {a;k)} {a;l)} a a(k)

a(l)

a(x) a'(x) a(x,x- I )

a(D) arg(c) As b; b(k)

b(l) I

(bJ (bjk) }

{bi/)} b b(k)

b(l)

b(x) b*(x) b(x, X-I)

beD) 'B; ci

(k) ci

eil)

Element i of sequence {a;} Element i of kth sequence (a?)} Element i of [th sequence (ai/)} Unipolar binary, multilevel, or q-ary sequence

eh unipolar binary, multilevel, or q-ary sequence

[th unipolar binary, multilevel, or q-ary sequence Unipolar binary, multilevel, or q-ary sequence of length or period N e h unipolar binary, multilevel, or q-ary sequence of length or period N [th unipolar binary, multilevel, or q-ary sequence of length or period N Polynomial Reciprocal polynomial of a(x) Laurent polynomial Power series representation of sequence {ai} Argument of a complex number c = C I + jC2

Agreements for discrete-time shift s Element i of sequence {bJ Element i of ell sequence {b?)} Element i of [th sequence (bi/)} Unipolar binary, multilevel, or q-ary sequence

k'h unipolar binary, multilevel, or q-ary sequence

Ilh unipolar binary, multilevel, or q-ary sequence Unipolar binary, multilevel, or q-ary sequence of length or period N k'h unipolar binary, multilevel, or q-ary sequence of length or period N [th unipolar binary, multilevel, or q-ary sequence of length or period N Polynomial Reciprocal polynomial of b(x) Laurent polynomial Power series representation of sequence {bJ Conjugacy class of f3i with respect to a given Galois field Element i of sequence {cd Element i of k'h sequence (c?)} Element i of [th sequence (cil)}

xviii

{cJ {c?J} { cj/}} c C(k)

C(l)

C(X)

Cam

Cern C max

Ca,a(S) Ca,b(S) Cx,As) CX,y(s) Cu,u(S) Cu,v(S) ck,k(s) Ck,/(S) ei

D

Ds detO deg[a(x)] expO Ex EO f fe f(x) .'TO .'T-) 0 g(x) gcd(a, b) GF GF(p) GF(q) GF(p) [x] HO i lim Icm(a, b) mx m(x) max{A} min{A} mod n

SYMBOLS

Unipolar binary, multilevel, or q-ary sequence

e h unipolar binary, multilevel, or q-ary sequence

It" unipolar binary, multilevel, or q-ary sequence Unipolar binary, multilevel, or q-ary sequence of length or period N kth unipolar binary, multilevel, or q-ary sequence of length or period N Ith unipolar binary, multilevel, or q-ary sequence of length or period N Characteristic polynomial Maximum out-of-phase aperiodic autocorrelation magnitude Maximum aperiodic crosscorrelation magnitude Maximum nontrivial aperiodic correlation value Aperiodic autocorrelation function of a Aperiodic crosscorrelation function of a and b Aperiodic autocorrelation function of x Aperiodic crosscorrelation function of x and y Aperiodic autocorrelation function of u Aperiodic crosscorrelation function of u and v Aperiodic autocorrelation function of k t

" sequence O(kJ

Aperiodic crosscorrelation function of klh sequence O(k) and II" sequence 0(1)

Cyclotomic coset Shift operator Disagreements for discrete-time shift s Determinant of a matrix Degree of polynomial a(x) Exponential function Energy of x(t) Expectation Frequency Chip rate General polynomial Fourier transform Inverse Fourier transform Generator polynomial Greatest common divisor of a and b Galois field Galois field or prime field of order p Galois field or extension field of order q = pm Set of polynomials f(x) of arbitrary degree over GF(p) Histogram Discrete-time variable Limit Least common multiple of a and b Mean of a random process X(t) Minimal polynomial Largest element of a set A = {a), a2' ... ,aA} Smallest element of a set A {a), a2,· .. ,aA} Modulo operation Polynomial degree

N N/n) ord(f3) p(x) Pe Px

PG

rxAr) rx:y( r) rectO Rac Ram Ram Rec Rem Rem Rmax

Ra.a(s) Ra b(S) Rx:xCs) Rx.y(s) Ru.u(s) Ru u(s) Ru:v(s) Ru.v<s) Rx,xCr) Rx.y(r) Ru,u( r) R U • ll ( r) Rx.x(r) Rx.x(t;, t) Ru(t;. tj) Rd(n) :Rr

ffi{ c} s sgnO sinc(') step(-)

Sx.Af) Su,u(/) Sx . .(k) Sx.y(k) t

T

Tc U i

SYMBOLS xix

Length or period of a sequence Number of monic irreducible polynomials p(x) in GF(p)[x] of degree n Order of an element f3 of a finite field Irreducible or primitive polynomial Bit error rate Power of x(t) Processing gain Normalized autocorrelation coefficient of x(t) Normalized crosscorrelation coefficient of x(t) and yet) Rectangular pulse Average mean-square out-of-phase aperiodic autocorrelation value Maximum out-of-phase periodic autocorrelation magnitude Maximum out-of-phase periodic odd autocorrelation magnitude Average mean-square aperiodic crosscorrelation value Maximum periodic crosscorrelation magnitude Maximum periodic odd crosscorrelation magnitude Maximum nontrivial periodic correlation value Periodic autocorrelation function of a Periodic crosscorrelation function of a and b Periodic autocorrelation function of x Periodic crosscorrelation function of x and y Periodic autocorrelation function of u Odd periodic autocorrelation function of u Periodic crosscorrelation function of u and v Odd periodic crosscorrelation function of u and v Autocorrelation function of x(t) Crosscorrelation function of x(t) and yet) Autocorrelation function of u(t) Crosscorrelation function of u(t) and vet) Autocorrelation function of a wide-sense stationary random process X(t) Autocorrelation function of a random process X(t) Crosscorrelation function of two random processes X(t) and yet) Residue or remainder Residue class Real part of a complex number c C1 + jC2

Discrete-time shift variable Signum function Sinc function Step function Power spectral density of x(t) Power spectral density of u(t) Autocorrelation spectrum of x Crosscorrelation spectrum of x and y Continuous-time variable Period of continuous-time signal Chip duration Element i of sequence {u i }

xx

(k) U;

(I) U;

{U;} (U?)} (ui!)} u U(k)

U(!)

U(t) V;

(k) Vi viI) {v;} {V;kl} ( v;!)} v V(k)

V(l)

vet) wt(a) Xi

X?) xiI) {X;} (xik

)}

(x;!)} x X(k)

x(1)

x(t) X X(t) X(f)

Yi y;k)

y;!) (y;) (y?)} (l)} y y(k)

y(l)

yet) Y yet) Y(f)

SYMBOLS

Element i of eh sequence (uik)}

Element i of lth sequence (u;l)} Complex-valued or polyphase sequence

k'h complex-valued or polyphase sequence

[fit complex-valued or polyphase sequence Complex-valued or polyphase sequence of length or period N klh complex-valued or polyphase sequence of length or period N lIlt complex-valued or polyphase sequence of length or period N Complex-valued signal Element i of sequence {Vi}

Element i of eh sequence {v;kl} Element i of lIlt sequence (viI)} Complex-valued or polyphase sequence

k'lt complex-valued or polyphase sequence

lIlt complex-valued or polyphase sequence Complex-valued or polyphase sequence of length or period N k,lt complex-valued or polyphase sequence of length or period N lth complex-valued or polyphase sequence of length or period N Complex-valued signal Weight of a Element i of sequence {x;} Element i of eh sequence (x;k)} Element i of ltlt sequence (xi!)} Bipolar binary or real-valued sequence

kth bipolar binary or real-valued sequence

/'It bipolar binary or real-valued sequence Bipolar binary or real-valued sequence of length or period N kIlt bipolar binary or real-valued sequence of length or period N It" bipolar binary or real-valued sequence of length or period N Real-valued signal Random variable Random process Fourier transform of x(t) Element i of sequence {y;} Element i of ktlt sequence {y;k)} Element i of lth sequence (y;')} Bipolar binary or real-valued sequence

ktlt bipolar binary or real-valued sequence

lth bipolar binary or real-valued sequence Bipolar binary or real-valued sequence of length or period N eh bipolar binary or real-valued sequence of length or period N lth bipolar binary or real-valued sequence of length or period N Real-valued signal Random variable Random process Fourier transform of yet)

SYMBOLS xxi

Zi Element i of sequence {ztl Z~k) Element i of klh sequence {z?)} z;i) Element i of Ilh sequence {z}l)} {Z;} Bipolar binary or real-valued sequence (Z)k)} eh bipolar binary or real-valued sequence

{z)l)} Ilh bipolar binary or real-valued sequence z Bipolar binary or real-valued sequence of length or period N Z(k) klh bipolar binary or real-valued sequence of length or period N z(l) Ilh bipolar binary or real-valued sequence of length or period N z(t) Real-valued signal a Primitive element of a Galois field f3 Element of a Galois field 'Y Element of a Galois field 80 Dirac impulse 7J Energy efficiency 110 Mobius function l1ij Covariance coefficient 4>0 Euler's totient function t/lj(x) Cyclotomic polynomial Pij Correlation coefficient T Continuous-time shift variable I1t Chip duration A (-) Triangular pulse C Set of complex numbers IF Finite field N Set of natural numbers N+ Set of nonzero natural numbers lR Set of real numbers Z Set of integer numbers I . I Absolute value or magnitude II . II Cardinality or size of a set (·f Transposition of a vector or matrix 0- 1 Inversion of a matrix O' Complex conjugation Lx J Greatest integer less than or equal to x r x 1 Smallest integer greater than or equal to x (f) Modulo addition ® Kronecker product 1\ Conjunction v Disjunction V For all E Is an element of U Union C Subset

Negation _ Equivalence CHt Fourier transform

xxii SYMBOLS

a! alb (:) (~) aCt) * bet) aCt) * bet) A\13 00

Factorial function a divides b Binomial coefficient Legendre symbol Convolution Correlation Difference of set A and set 13 Infinity

1 Introduction

1.1 PROLOGUE

The perfonnance of modem communication and infonnation systems is influenced by the potential of the available integrated circuits and by the efficiency of the algorithms chosen for the actual signal processing. The tremendous advances in integrated circuit technology facilitate the implementation of increasingly sophisticated signal processing algorithms. This has resulted in strong interactions between theoretical concepts and technological developments, which in tum have produced a wide range of practical applications.

This general trend applies in particular to the fields of pseudo random signals and sequences, which constitute an important element of efficient signal processing in almost every modem communication and infonnation system. These types of signals and sequences are strictly detenninistic in nature but offer similar characteristics as random signals. The strong mathematical structure associated with pseudo random signals not only provides a solid foundation for systematic signal set design but also guides the development of extremely powerful signal processing techniques. The distinct benefits of pseudo random signal processing compared to standard processing techniques include a very robust immunity to hostile jamming and a superior operation against several fonns of unintentional interfer­ence. These advantages have been exploited first for military applications, where a secure, reliable, and robust communication link is of major concern. Later, with the advent of multi-user communications along with rapid advances in technology, more intricate pseudo random signal processing techniques were introduced to the civilian and commercial fields. In the meantime, implementation costs for these techniques have been largely reduced and this allows the many attractive features of pseudo random signal processing to be used extensively in practice.

The wide range of applications includes areas such as signal analysis, correlation analysis, bit error measurements, scramblers, positioning, ranging and navigation systems, spread spectrum techniques, code-division multiple-access systems, and cryptology. This indicates the importance of pseudo random signal processing for communication, infonnation, and computer technologies. From a conceptual point of view, we think it is meaningful and justified to introduce the generic tenn of pseudo random signal processing.

Pseudo Random Signal Processing Theory and Application Hans-Jtirgen Zepemick and Adolf Finger © 2005 John Wiley & Sons. Ltd

2 INTRODUCTION

1.2 ELEMENTS OF PSEUDO RANDOM SIGNAL PROCESSING

The field of pseudo random signal processing centers around the terms "signal" and "structure." Essentially, it is concerned with the analysis and synthesis of random-like signals as well as their generation, realization, and processing. The related pseudo random signal processing algorithms are often associated with a particular application.

A signal in the technological and scientific sense is usually considered as a carrier of information. Signals are often associated with technical, physical, or biological processes and as such may be described by mathematical functions that evolve over time. The fundamental properties of a signal can be relatively complex and significantly influence the mathematical framework to be used for their description. In fact, it is often found that the most important characteristic of a signal serves as an attribute when referring to a particular signal type. Some examples may illustrate the variety of signal terminologies used to indicate signal properties:

• Shape of mathematical function. Sinusoidal signal, pulse signal, rectangular signal, triangular signal.

• Purpose of use. Test signal, measurement signal, control signal, training signal, timing signal, synchronization signal, clock signal.

• Particulars in relation to transmission. Message signal, noise signal, interfering signal, carrier signal, modulation signal, transmit signal, receive signal.

• Time and amplitude characteristics. Periodic signal, aperiodic signal, continuous signal, discrete signal, digital signal, binary signal, nonbinary signal.

• Predictability and structure. Deterministic signal, random signal, cyclostationary signal, pseudo random signal.

• Origin and application field. Data signal, speech signal, audio signal, video signal, satellite signal, radio signal, radar signal.

As with the numerous signal terminologies, there exists a large variety on how the term "structure" is used. Some of the more frequently applied notions include word combinations such as system structure, circuit structure, network structure, code structure, algebraic structure, fine structure, and coarse structure. In technical, physical, and biological processes, the term "signal structure" typically corresponds to the level of uncertainty with respect to the signal amplitude over time. The uncertainty can be quantified by characterizing the level of similarity between signals and their shifted versions using correlation functions. In partiCUlar, the autocorrelation measures the dependence of one signal on itself and the crosscorrelation measures the dependence of one Signal on another. In the latter case of dealing with more than one signal, the considered sets of signals are frequently referred to as codes.

Figure 1.1 shows a classification of structured signals with reference to the example of communi­cation systems. In the case when the amplitudes of a signal are given over discrete time, the signal is referred to as a sequence. Accordingly, pseudo random signals have their origins in correlation codes and in signals with special correlation functions. Due to the close relationship between these two roots of pseudo random signals, the terms pseudo random code and pseudo random signal are both used synonymously in the literature and without differentiating whether a set of sequences or an individual signal is considered. Also, it should be mentioned that pseudo random signals are called pseudo noise signals in some applications.

I Source codes

I

Coded Signals

Channel codes

I Compression Error

control

ELEMENTS OF PSEUDO RANDOM SIGNAL PROCESSING 3

Structured

Sequences Modulated Signals

Line codes

Spectral shaping

Correlation codes

Signals with special Baseband correlation functions structure

I

Synchronization codes Orthogonal Comma-free codes _........ signals

Spreading/Addressing Ranging/Radar/Sonar

Figure 1.1 Classification of structured signals.

Multi-phase modulation

Chirp signals

Signals with special ambiguity function

The design and generation of pseudo random signals are closely related to the particular require­ments given by the specifications of the considered application. Characteristics such as those derived from the different autocorrelation and crosscorrelation functions are commonly used to pose the design objectives for a given application. Some applications may require a pseudo random sequence with special autocorrelation properties. A different application may need pseudo random codes with certain crosscorrelation characteristics while another application may rely on some specific structural prop­erties. Table 1.1 gives an overview of the correlation requirements of some prominent applications together with examples of the related practical systems. It is not intended to be exhaustive with this

Requirement

Good autocorrelation

Good crosscorrelation

Others such as linear complexity

Table 1.1 Examples of application areas of pseudo random signals.

Application area

Ranging and navigation Spread spectrum

communications Scrambling

System test and analysis Spread spectrum

communications Navigation Multi-input system test

and analysis Cryptology

Practical system example

GPS, Galileo, TDRSS, etc. IS-95, cdma2000, UMTS, IEEE 802.11,

Bluetooth, etc. DECT, DSR, DAB, DVB, PCM in wireline

systems, ATM, Bluetooth, IEEE 802.11, etc. MLSSA 2000 etc. IS-95, cdma2000, UMTS, IEEE 802.11,

Bluetooth, etc. GPS, Galileo, etc. Test-per-clock BIST, test-per-scan BlST, etc.

DVD, Pay-TV, GSM, etc.

4 INfRODUCTION

overview but to illustrate the different requirements of contemporary applications of pseudo random signals. The same applies to the selected practical system examples of which many will be described in more detail in succeeding chapters. The first group of applications requires good autocorrelation properties, which translates to an impulse·like function with the corresponding frequency spectrum being uniformly distributed over a wide bandwidth. These properties facilitate tasks such as precise range measurements, recovery of synchronization, and spectral shaping, but may also serve for hiding a signal below the noise level to protect it against interference. The next group of applications asks for good crosscorrelation characteristics implying that sets of pseudo random sequences are used. As a number of different pseudo random signals are usually to be distinguished with these types of systems, similarities among the sequences of a given set must be kept to a minimum. Therefore, low crosscorrelations between pairs of sequences are considered as favorable in the related application areas. In multi-user communication scenarios, this property allows different users simultaneous access to a common transmission medium. As each user has been assigned its own pseudo random signal, identification of the communicating partners is easily obtained. Finally, a third group of applications may draw upon a wide range of other specific structural signal properties. Here, the linear complexity serves as an example of such a specific structural requirement. In brief, the linear complexity specifies the length of the shortest shift register that is capable of generating a given pseudo random sequence. This measure enables quantification of the effort needed to recover the entire structure contained in a particular pseudo random sequence given that only a snapshot of the sequence was available. As such, large linear complexities are preferred in applications associated with cryptology to ensure that secret information cannot be recovered by an eavesdropper. Table 1.1 also indicates that some applica­tions and the related practical systems may rely on both favorable autocorrelation and crosscorrelation properties or may even have additional specific requirements. In these cases, a trade-off between the sometimes conflicting objectives needs to be considered in the pseudo random signal design.

It is evident from the above brief discussions that the rich area of pseudo random signal processing is influenced by many different mathematical and engineering areas. Some of the major relationships of pseudo random signal processing to other fields are shown in the overview given in Table 1.2.

Table 1.2 Relationships of pseudo random signal processing to other fields.

Mathematics

Abstract algebra Polynomial algebra Number theory Recursive sequences Random numbers Combinatorics

Transmission techniques

Modulation Noise processes Noise reduction Synchronization Signal theory

Pseudo random signal processing

System theory

System analysis Linear systems Binary systems Digital systems Discrete systems Stochastic systems

Automata theory

Shift registers Stochastic automata Linear automata Nonlinear automata Autonomous automata

Communications theory

Optimal receivers Coding theory Correlation electronics Stochastic processes Information theory

Micro electronics

Logic circuits Memory circuits Filters Microprocessors Converters Circuit testing

OUTLINE OF THE BOOK 5

Several of these different connections between pseudo random signal processing and other fields will be revealed in more detail throughout the succeeding chapters of the book.

Over the past four decades, the theoretical understanding of pseudo random signals and sequences has very much matured. There exists a considerable amount of mathematical contributions reporting on many discrete random-like sequences, their properties, and the various means of sequence design [53, 65, 96, 111, 161]. On the other hand, it can be observed that engineering-oriented publications mainly concentrate on an in-depth coverage of a particular application. For example, the number of texts entirely devoted to the subject of spread spectrum systems continues to grow and the interested reader may be directed to [54, 67, 143, 222, 248, 272]. The same applies to the coverage of the even more specific spread spectrum-based mobile radio systems [33, 57, 117, 143, 175, 181, 227] and wireless networks [28, 180], for which a tremendous body of literature exists. Modem navigation systems can be found in textbooks such as [100, 114,242], scrambling techniques and their applications are the focus of [139, 142], and in-depth treatments of cryptology can be found in [216, 232] and many other excellent texts. However, an effective attempt that brings together the underlying theoretical concepts, the powerful signal processing techniques, the general practical aspects, and the numerous application fields of pseudo random signal processing seems to be missing.

1.3 OUTLINE OF THE BOOK

This book will provide a transition from covering the engineering and mathematical foundations to conveying the powerful signal processing principles, which serve as the connecting link between theory and application. Here, the practical applications not only are drawn from the fields of communications but also span the boundaries of several technical disciplines including examples from test, information, and computer systems. The four key segments of the book can be broadly described as foundations, designs, realizations, and applications of pseudo random signal processing.

Foundations The engineering foundations cover the basic signal processing concepts associated with pseudo random signals and sequences. Special attention is given to describing and discussing important signal charac­teristics such as correlation measures and power spectral density. The offered insights will enable the reader not only to identify design objectives but also to guide the assessment of processing techniques in view of their ability to cope with constraints given by practical applications. The mathematical foundations are mainly concerned with the relevant topics of abstract algebra such as algebraic struc­tures, finite fields, and the corresponding arithmetic. This introduces the feature of strong mathematical structure to the area of pseudo random signal processing as required for the systematic design of signal and sequence sets and for the efficient realization of advanced processing techniques.

Designs Equipped with solid insights into the theoretical foundations of pseudo random signal processing, the reader is then introduced to a discussion of prominent designs of conventional binary sequences as well as modem classes of nonbinary and complex-valued sequences. This segment provides a sound knowledge about design objectives, design methodologies, and properties of the respective sequence designs. It inherently points to means of sequence generation and to potential areas of their application. It also indicates the general trend from simple to more advanced approaches. The presented designs give a thorough overview of sequences and related processing techniques that are used in the traditional and contemporary applications as well as the directions the technology may support in the future.

6 INTRODUCTION

However, it is not intended to provide an exhaustive survey of the vast number of sequence designs which have been proposed over the previous decades.

Realizations This segment of the book is dedicated to the technical realizations of pseudo random signal processing. The standard circuits that are used in practice for implementing the respective processing algorithms are presented together with case studies of typical building blocks and realizations. In this context, some important properties of the underlying logic are emphasized. In addition to these hard-wired realizations, the more flexible system architectures based on microprocessors, digital signal processors, memory units, and software suits are presented.

Applications The fourth segment of the book provides a survey and a discussion of important applications of pseudo random signal processing in modem communication, information, and computer technologies as well as several applications in other specialized fields. The background to each of the different applications is provided first and to a degree required for understanding the relationship to pseudo random signal processing. This is then followed by a more detailed coverage of the representative practical realizations, standards, and systems in the considered areas. In this way, interconnections and potential synergies among the different tlelds are revealed. In particular, the considered applications are drawn from the fields of spread spectrum systems, ranging and navigation systems, scrambling, automatic test and system verification, cryptology, and other applications.

These four segments are accommodated within the succeeding six chapters of the book. Chapters 2 and 3 contain the engineering and mathematical foundations of pseudo random signal processing, respectively. Binary sequence designs and nonbinary sequence designs are presented in Chapters 4 and 5, respectively. In Chapter 6, the realizations together with implementation issues of pseudo random signals and generators are covered. Chapter 7 is dedicated to the enormous range of applications and as such also details several signal processing techniques.