+DIgital signal processing principles algoristhms and applicatin 3rd by proakis

The author and publisher o f this book have used their best efforts in preparing this book.. 2324.1.2 Power Density Spectrum of Periodic Signals.. 235 4.1.3 The Fourier Transform for Con

It is not to be reexported and it is not for sale in the U S.A , Mexico, or Canada.

© 1996 by Prentice-Hall, Inc.

Simon & Schuster/A Viacom Company

U pper Saddle River, N ew Jersey 07458

A ll rights reserved No part o f this book may be

reproduced, in any form or by any means,

without permission in writing from the publisher.

The author and publisher o f this book have used their best efforts in preparing this book These efforts include the developm ent, research, and testing o f the theories and programs to determine their effectiveness The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out

of the furnishing, performance, or use o f these programs.

Printed in the United States o f America

10 9 8 7 6 5

ISBN 0-13-3TM33fl-cl

Prentice-Hall International (U K ) Limited L o n d o n

Prentice-Hall of Australia Pty Limited, Sydney

Prentice-Hall Canada, Inc., Toronto

Prentice-Hall Hispanoamericana S.A., M exico

Prentice-Hall o f India Private Limited, N ew D elhi

Prentice-Hall o f Japan, Inc., T o kyo

Simon & Schuster Asia Pie, Ltd., Singapore

Editora Prentice-Hall do Brasil, Ltda., R io de Janeiro

S ig n als, S y stem s, and S ig n al P ro c e ssin g 2

1.1.1 Basic Elem ents of a Digital Signal Processing System 4

1.1.2 A dvantages of Digital over Analog Signal Processing, 5

C la ssific a tio n o f S ignals 6

1.2.1 Multichannel and M ultidimensional Signals 7

1.2.2 Continuous-Time Versus D iscrete-Tim e Signals 8

1.2.3 Continuous-Valued Versus D iscrete-V alued Signals 10

1.2.4 Determ inistic Versus Random Signals, 11

T h e C o n c e p t o f F re q u e n c y in C o n tin u o u s -T im e a n d

D isc re te -T im e S ignals 14

1.3.1 Continuous-Tim e Sinusoidal Signals, 14

1.3.2 Discrete-Tim e Sinusoidal Signals 16

1.3.3 Harmonically R elated Complex Exponentials, 19

A n a lo g -to -D ig ita l a n d D ig ita l-to -A n a lo g C o n v e rs io n 21

1.4.1 Sampling of Analog Signals, 23

1.4.2 The Sampling Theorem , 29

1.4.3 Q uantization of Continuous-A m plitude Signals, 33

1.4.4 Q uantization of Sinusoidal Signals, 36

1.4.5 Coding of Quantized Samples, 38

1.4.6 Digital-to-A nalog Conversion, 38

1.4.7 Analysis of Digital Signals and Systems Versus Discrete-Time Signals and Systems, 39

S u m m a ry a n d R e fe re n c e s 39

Problems 40

2 DISCRETE-TIME SIGNALS AND SYSTEMS 43

2.1 D isc re te -T im e S ignals 43

2.1.1 Some Elem entary Discrete-Tim e Signals, 45

2.1.2 Classification of Discrete-Tim e Signals, 47

2.1.3 Simple Manipulations of Discrete-Tim e Signals, 52

2.2 D isc re te -T im e S y stem s 56

2.2.1 In p u t-O u tp u t D escription of Systems, 56

2.2.2 Block Diagram R epresentation of Discrete-Tim e Systems, 59

2.2.3 Classification of Discrete-Tim e Systems, 62

2.2.4 Interconnection of D iscrete-Tim e Systems, 70

2.3 A n aly sis o f D isc re te -T im e L in e a r T im e - I n v a ria n t S y stem s 722.3.1 Techniques for the Analysis of Linear Systems, 72

2.3.2 Resolution of a Discrete-Tim e Signal into Impulses, 74

2.3.3 Response of LTI Systems to A rbitrary Inputs: The Convolution Sum, 75

2.3.4 Properties of Convolution and the Interconnection of LTI

Systems, 82

2.3.5 Causal Linear T im e-Invariant Systems 86

2.3.6 Stability of Linear Tim e-Invariant Systems, 87

2.3.7 Systems with Fim te-D uration and Infinite-D uration Impulse

Response 90

2.4 D isc re te -T im e S y stem s D e s c rib e d by D iffe re n c e E q u a tio n s 912.4.1 Recursive and N onrecursive Discrete-Tim e Systems, 92

2.4.2 Linear Tim e-Invariant Systems C haracterized by

Constant-Coefficient D ifference Equations, 95

2.4.3 Solution of Linear Constant-C oefficient Difference E quations 1002.4.4 The Impulse Response of a Linear T im e-Invariant Recursive System, 108

2.5 Im p le m e n ta tio n o f D isc re te -T im e S y stem s 111

2.5.1 Structures for the Realization of Linear Tim e-Invariant

Systems, 111

2.5.2 Recursive and N onrecursive Realizations of F IR Systems, 1162.6 C o rre la tio n of D isc re te -T im e S ig n als 118

2.6.1 Crosscorrelation and A utocorrelation Sequences, 120

2.6.2 Properties of the A utocorrelation and Crosscorrelation

Sequences, 122

2.6.3 Correlation of Periodic Sequences, 124

2.6.4 Com putation of Correlation Sequences, 130

2.6.5 In p u t-O u tp u t Correlation Sequences, 131

2.7 S u m m a ry a n d R e fe re n c e s 134

3 THE Z-TRANSFORM AND ITS APPLICATION TO THE ANALYSIS

OF LTI SYSTEMS

3.1 T h e r -T ra n s fo rm 151

3.1.1 The D irect ^-Transform 152

3.1.2 The inverse : -Transform, 160

3.2 P ro p e rtie s o f th e ; -T ra n sfo rm 161

3.3 R a tio n a l c -T ra n sfo rm s 172

3.3.1 Poles and Zeros, 172

3.3.2 Pole Location and Tim e-Dom ain Behavior for Causal Signals 1783.3.3 The System Function of a Linear T im e-Invariant System 181

3.4 In v e rs io n o f th e ^ -T ra n sfo rm 184

3.4.1 The Inverse ; -Transform by Contour Integration 184

3.4.2 The Inverse ;-Transform by Pow er Series Expansion 186

3.4.3 The Inverse c-Transform by Partial-Fraction Expansion 188

3.4.4 Decom position of Rational c-Transforms 195

3.5 T h e O n e -sid e d ^ -T ra n sfo rm 197

3.5.1 Definition and Properties, 197

3.5.2 Solution of Difference Equations 201

3.6 A n a ly sis o f L in e a r T im e -In v a ria n t S y stem s in th e D o m a in 2033.6.1 Response of Systems with Rational System Functions 203

3.6.2 Response of P o le-Z ero Systems with N onzero Initial

Conditions 204

3.6.3 T ransient and Steady-State Responses, 206

3.6.4 Causality and Stability 208

3.6.5 P o le-Z ero Cancellations 210

3.6.6 M ultiple-O rder Poles and Stability 211

3.6.7 The S chur-C ohn Stability Test, 213

3.6.8 Stability of Second-O rder Systems 215

3.7 S u m m a ry a n d R e fe re n c e s 219

P ro b le m s 220

4 FREQUENCY ANALYSIS OF SIGNALS AND SYSTEMS

4.1 F re q u e n c y A n a ly sis o f C o n tin u o u s -T im e S ig n als 230

4.1.1 The Fourier Series for Continuous-Tim e Periodic Signals 2324.1.2 Power Density Spectrum of Periodic Signals 235

4.1.3 The Fourier Transform for Continuous-Time A periodic

Signals, 240

4.1.4 Energy Density Spectrum of Aperiodic Signals 243

4.2 F re q u e n c y A n a ly sis o f D isc re te -T im e S ig n als 247

151

230

4.2.2 Power Density Spectrum of Periodic Signals 250

4.2.3 The Fourier Transform of Discrete-Tim e A periodic Signals 2534.2.4 Convergence of the Fourier Transform 256

4.2.5 Energy D ensity Spectrum of A periodic Signals, 260

4.2.6 Relationship of the Fourier Transform to the i-T ransform , 2644.2.7 The Cepstrum, 265

4.2.8 The Fourier Transform of Signals with Poles on the Unit

Circle, 267

4.2.9 The Sampling T heorem Revisited, 269

4.2.10 Frequency-Dom ain Classification of Signals: The Concept of Bandwidth, 279

4.2.11 The Frequency Ranges of Some N atural Signals 282

4.2.12 Physical and M athem atical Dualities 282

4.3 P ro p e rtie s of th e F o u rie r T ra n s fo rm fo r D isc re te -T im e

S ignals 286

4.3.1 Symmetry Properties of the Fourier Transform , 287

4.3.2 Fourier Transform Theorem s and Properties, 294

4.4.3 Steady-State Response to Periodic Input Signals, 315

4.4.4 Response to A periodic Input Signals 316

4.4.5 Relationships Betw een the System Function and the Frequency Response Function 319

4.4.6 Com putation of the Frequency Response Function 321

4.4.7 In p u t-O u tp u t Correlation Functions and Spectra, 325

4.4.8 Correlation Functions and Power Spectra for R andom Input Signals 327

4.5 L in e a r T im e -In v a ria n t S y stem s as F re q u e n c y -S e le c tiv e

F ilte rs 330

4.5.1 Ideal Filter Characteristics, 331

4.5.2 Lowpass, Highpass, and Bandpass Filters, 333

4.6.1 Invertibility of Linear Tim e-Invariant Systems, 356

4.6.2 M inimum-Phase M aximum-Phase, and Mixed-Phase Systems 3594.6.3 System Identification and D econvolution, 363

5.1.2 The Discrete Fourier Transform (D FT) 399

5.1.3 The D FT as a Linear Transform ation 403

5.1.4 Relationship of the D FT to O ther Transform s, 407

5.2 P ro p e rtie s o f th e D F T 409

5.2.1 Periodicity Linearity, and Symmetry Properties, 410

5.2.2 M ultiplication of Two DFTs and Circular Convolution 415

5.2.3 Additional D FT Properties, 421

5.3 L in e a r F ilte rin g M e th o d s B a sed o n th e D F T 425

5.3.1 Use of the DFT in Linear Filtering 426

5.3.2 Filtering of Long D ata Sequences 430

5.4 F re q u e n c y A n a ly sis o f S ignals U sin g th e D F T 433

5.5 S u m m a ry a n d R e fe re n c e s 440

P ro b le m s 440

6 EFFICIENT COMPUTATION OF THE DFT: FAST FOURIER

TRANSFORM ALGORITHMS

6.1 E ffic ie n t C o m p u ta tio n o f th e D F T : F F T A lg o rith m s 448

6.1.1 Direct Com putation of the DFT, 449

6.1.2 D ivide-and-C onquer A pproach to Com putation of the D FT 4506.1.3 Radix-2 FFT Algorithms 456

6.1.4 Radix-4 FFT Algorithms 465

6.1.5 Split-Radix FFT Algorithms, 470

6.1.6 Im plem entation of FFT Algorithms 473

6.2 A p p lic a tio n s o f F F T A lg o rith m s 475

6.2.1 Efficient Com putation of the D FT of Two Real Sequences 4756.2.2 Efficient Com putation of the D FT of a Z N -Point Real

Sequence, 476

6.2.3 Use of the FFT Algorithm in Linear Filtering and Correlation, 4776.3 A L in e a r F ilte rin g A p p ro a c h to C o m p u ta tio n o f th e D F T 4796.3.1 The G oertzel Algorithm, 480

394

448

6.4 Q u a n tiz a tio n E ffe c ts in th e C o m p u ta tio n o f th e D F T 486

6.4.1 Quantization Errors in the D irect Com putation of the DFT 4876.4.2 Quantization Errors in FFT Algorithm s 489

6.5 S u m m ary an d R e fe re n c e s 493

P ro b le m s 494

7 IMPLEMENTATION OF DISCRETE-TIME SYSTEMS

7.1 S tru c tu re s fo r th e R e a liz a tio n o f D isc re te -T im e S y ste m s 500

7.3.5 Lattice and Lattice-Ladder Structures for IIR Systems, 531

S ta te -S p a c e S ystem A n a ly sis a n d S tru c tu re s 539

7.4.1 State-Space D escriptions of Systems Characterized by DifferenceEquations 540

7.4.2 Solution of the State-Space Equations 543

7.4.3 Relationships Between In p u t-O u tp u t and State-Space

Descriptions, 545

7.4.4 State-Space Analysis in the z-Dom ain, 550

7.4.5 Additional State-Space Structures 554

R e p re s e n ta tio n o f N u m b e rs 556

7.5.1 Fixed-Point R epresentation of N um bers 557

7.5.2 Binary Floating-Point R epresentation of Numbers 561

7.5.3 E rrors Resulting from R ounding and Truncation 564

Q u a n tiz a tio n o f F ilte r C o e ffic ie n ts 569

7.6.1 Analysis of Sensitivity to Q uantization of Filter Coefficients 5697.6.2 Q uantization of Coefficients in FIR Filters 578

7.7 R o u n d -O ff E ffe c ts in D ig ita l F ilte rs 582

7.7.1 Limit-Cycle Oscillations in Recursive Systems 583

7.7.2 Scaling to Prevent Overflow, 588

7.7.3 Statistical Characterization of Q uantization Effects in Fixed-Point Realizations of Digital Filters 590

7.8 S u m m a ry a n d R e fe re n c e s 598

500

8 DESIGN OF DIGITAL FILTERS

8.1 G e n e r a l C o n s id e ra tio n s 614

8.1.1 Causality and Its Implications 615

8.1.2 Characteristics of Practical Frequency-Selective Filters 619

8.2 D e s ig n o f F I R F ilte rs 620

8.2.1 Symmetric and A ntisym m eiric F IR Filters, 620

8.2.2 Design of Linear-Phase F IR Filters Using W indows, 623

8.2.3 Design of Linear-Phase F IR Filters by the Frequency-Sampling

M ethod, 630

8.2.4 Design of Optim um Equiripple Linear-Phase F IR Filters, 6378.2.5 Design of F IR D ifferentiators, 652

8.2.6 Design of H ilbert Transform ers, 657

8.2.7 Comparison of Design M ethods for L inear-Phase FIR Filters, 6628.3 D e sig n o f I I R F ilte rs F ro m A n a lo g F iite rs 666

8.3.1 IIR Filter Design by A pproxim ation of Derivatives 667

8.3.2 IIR Filter Design by Impulse Invariance 671

8.3.3 IIR Filter Design by the Bilinear Transform ation, 676

8.3.4 The M atched-; Transform ation, 681

8.3.5 Characteristics of Commonly Used Analog Filters 681

8.3.6 Some Examples of Digital Filter Designs Based on the Bilinear Transform ation 692

8.4 F re q u e n c y T ra n s fo rm a tio n s 692

8.4.1 Frequency T ransform ations in the Analog Dom ain, 693

8.4.2 Frequency T ransform ations in the Digital D om ain 698

8.5 D e sig n o f D ig ita l F ilte rs B a se d o n L e a s t-S q u a re s M e th o d 7018.5.1 Pade A pproxim ation M ethod, 701

8.5.2 Least-Squares Design M ethods, 706

8.5.3 FIR Least-Squares Inverse (W iener) Filters, 711

8.5.4 Design of IIR Filters in the Frequency D om ain, 719

8.6 S u m m a ry a n d R e fe re n c e s 724

P ro b le m s 726

9 SAMPLING AND RECONSTRUCTION OF SIGNALS

9.1 S am p lin g o f B a n d p a ss S ig n als 738

9.1.1 R epresentation of Bandpass Signals, 738

9.1.2 Sampling of Bandpass Signals, 742

9.1.3 Discrete-Tim e Processing of Continuous-Tim e Signals, 746

9.2 A n a lo g -to -D ig ita l C o n v e rs io n 748

9.2.1 Sam ple-and-H old 748

9.2.2 Q uantization and Coding, 750

9.2.3 Analysis of Q uantization Errors, 753

614

738

9.3 D ig ita l-to -A n a lo g C o n v e rs io n 763

9.3.1 Sample and Hold, 765

9.3.2 First-O rder Hold 768

9.3.3 Linear Interpolation with Delay, 771

10.5.1 Direct-Form FIR Filter Structures, 793

10.5.2 Polyphase Filter Structures, 794

10.5.3 Tim e-V ariant Filter Structures 800

10.6 M u ltista g e I m p le m e n ta tio n o f S a m p lin g -R a te C o n v e rs io n 80610.7 S a m p lin g -R a te C o n v e rsio n o f B a n d p a ss S ig n als 810

10.7.1 Decim ation and Interpolation by Frequency Conversion, 812

10.7.2 M odulation-Free M ethod for D ecim ation and Interpolation 81410.8 S a m p lin g -R a te C o n v e rs io n by a n A r b itra r y F a c to r 815

10.8.1 First-O rder A pproxim ation, 816

10.8.2 Second-O rder Approxim ation (Linear Interpolation) 819

10.9 A p p lic a tio n s o f M u ltira te Signal P ro c e ss in g 821

10.9.1 Design of Phase Shifters 821

10.9.2 Interfacing of Digital Systems with D ifferent Sampling Rates, 82310.9.3 Im plem entation of Narrow band Lowpass Filters, 824

10.9.4 Im plem entation of Digital Filter Banks 825

10.9.5 Subband Coding of Speech Signals, 831

10.9.6 Q uadrature M irror Filters 833

10.9.7 Transm ultiplexers 841

10.9.8 Oversampling A/D and D /A Conversion, 843

10.10 S u m m a ry a n d R e fe r e n c e s 844

782

11 LINEAR PREDICTION AND OPTIMUM LINEAR FILTERS

11.1 In n o v a tio n s R e p re s e n ta tio n o f a S ta tio n a ry R a n d o m

P ro c e ss 852

11.1.1 R ational Power Spectra 854

11.1.2 Relationships Between the Filter P aram eters and the

A utocorrelation Sequence, 855

11.2 F o rw a rd a n d B a c k w a rd L in e a r P re d ic tio n 857

11.2.1 Forw ard Linear Prediction, 857

11.2.2 Backward Linear Prediction, 860

11.2.3 The Optim um Reflection Coefficients for the Lattice Forw ard and Backward Predictors, 863

11.2.4 Relationship of an A R Process to Linear Prediction 864

11.3 S o lu tio n o f th e N o rm a l E q u a tio n s 864

11.3.1 The Levinson-D urbin Algorithm 865

11.3.2 The Schiir Algorithm 868

11.4 P ro p e rtie s o f th e L in e a r P re d ic tio n - E rr o r F ilte rs 873

11.5 A R L a ttic e a n d A R M A L a ttic e -L a d d e r F ilte rs 876

11.5.1 A R LaLtice Structure 877

11.5.2 A R M A Processes and Lattice-Ladder Filters 878

11.6 W ie n e r F ilte rs fo r F ilte rin g a n d P re d ic tio n 880

11.6.1 FIR W iener Filter, 881

11.6.2 Orthogonality Principle in Linear M ean-Square Estim ation, 88411.6.3 IIR W iener Filter 885

11.6.4 Noncausal W iener Filter 889

11.7 S u m m a ry an d R e fe re n c e s 890

P ro b le m s 892

12 POWER SPECTRUM ESTIMATION

12.1 E s tim a tio n o f S p e c tra fro m F in ite -D u ra tio n O b s e rv a tio n s o f

Signals 896

12.1.1 Com putation of the Energy Density Spectrum 897

12.1.2 Estim ation of the A utocorrelation and Power Spectrum of

R andom Signals: The Periodogram 902

12.1.3 The Use of the DFT in Power Spectrum E stim ation, 906

12.2 N o n p a r a m e tr ic M e th o d s fo r P o w e r S p e c tru m E s tim a tio n 908

12.2.1 The B artlett Method: Averaging Periodogram s, 910

12.2.2 The Welch Method: Averaging Modified Periodogram s, 911

12.2.3 The Blackman and Tukey Method: Smoothing the

Periodogram , 913

12.2.4 Perform ance Characteristics of N onparam etric Power Spectrum

852

896

12.2.5 Com putational Requirem ents of N onparam etric Pow er Spectrum Estim ates, 919

12.3 P a ra m e tric M e th o d s fo r P o w e r S p e c tru m E stim a tio n 920

12.3.1 Relationships Between the A utocorrelation and the Model

12.3.7 MA Model for Power Spectrum Estim ation, 933

12.3.8 A R M A Model for Pow er Spectrum Estim ation, 934

12.3.9 Some E xperim ental Results, 936

12.4 M in im u m V a ria n c e S p e c tra l E s tim a tio n 942

12.5 E ig e n a n a ly sis A lg o rith m s fo r S p e c tru m E s tim a tio n 946

12.5.1 Pisarenko Harm onic Decom position M ethod, 948

12.5.2 Eigen-decomposition of the A utocorrelation Matrix for Sinusoids

in White Noise, 950

12.5.3 M U SIC Algorithm 952

12.5.4 ESPR IT Algorithm, 953

12.5.5 O rder Selection Criteria 955

12.5.6 Experim ental Results, 956

12.6 S u m m a ry a n d R e fe re n c e s 959

P ro b le m s 960

C TABLES OF TRANSITION COEFFICIENTS FOR THE DESIGN OF

L j _ Preface

T h is b o o k w as d e v e lo p e d b a se d o n o u r te a c h in g o f u n d e r g ra d u a te an d g r a d u ­

a te level c o u rse s in d ig ita l signal p ro c e ssin g o v e r th e p a s t se v e ra l y ears In this

b o o k w e p re s e n t th e fu n d a m e n ta ls o f d isc re te -tim e sig n a ls, sy stem s, an d m o d e rn

d ig ital p ro c e ssin g a lg o rith m s a n d a p p lic a tio n s fo r s tu d e n ts in e le c tric a l e n g in e e r­ing c o m p u te r e n g in e e rin g , a n d c o m p u te r sc ien ce T h e b o o k is s u ita b le fo r e ith e r

a o n e -s e m e s te r o r a tw o -se m e s te r u n d e r g ra d u a te level c o u rse in d isc re te sy stem s

a n d d ig ital sig n al p ro c e ssin g It is also in te n d e d fo r u se in a o n e -s e m e s te r first-y e a r

g r a d u a te -le v e l c o u rse in d ig ital signal p ro c e ssin g

It is a s su m e d th a t th e s tu d e n t in e le c tric a l an d c o m p u te r e n g in e e rin g h as h a d

u n d e r g ra d u a te c o u rse s in a d v a n c e d calcu lu s (in c lu d in g o rd in a ry d iffe re n tia l e q u a ­tio n s) an d lin e a r sy ste m s fo r c o n tin u o u s-tim e signals, in c lu d in g a n in tro d u c tio n

to th e L a p la c e tra n sfo rm A lth o u g h th e F o u rie r se rie s a n d F o u rie r tra n sfo rm s o f

p e rio d ic a n d a p e rio d ic signals a re d e s c rib e d in C h a p te r 4, we e x p e c t th a t m a n y

s tu d e n ts m a y hav e h a d th is m a te ria l in a p r io r c o u rse

A b a la n c e d c o v e ra g e is p r o v id e d o f b o th th e o ry a n d p ra c tic a l a p p lic a tio n s

A larg e n u m b e r o f w ell d e sig n e d p ro b le m s a re p ro v id e d to h e lp th e s tu d e n t in

m a s te rin g th e su b je c t m a tte r A so lu tio n s m a n u a l is a v a ila b le fo r th e b e n e fit o f

th e in s tr u c to r an d can b e o b ta in e d fro m th e p u b lish e r

T h e th ird e d itio n o f th e b o o k c o v ers basically th e sa m e m a te ria l as th e se c ­

o n d e d itio n , b u t is o rg a n iz e d d iffe re n tly T h e m a jo r d iffe re n c e is in th e o r d e r in

w hich th e D F T a n d F F T a lg o rith m s a re c o v e re d B a se d o n su g g e stio n s m a d e by

se v e ra l re v ie w e rs, w e n o w in tro d u c e th e D F T a n d d e s c rib e its effic ie n t c o m p u ta ­tio n im m e d ia te ly fo llo w in g o u r tr e a tm e n t o f F o u rie r an aly sis T h is re o rg a n iz a tio n

h a s also a llo w e d us to e lim in a te r e p e titio n o f so m e to p ic s c o n c e rn in g th e D F T an d its a p p lic a tio n s

In C h a p te r 1 w e d e s c rib e th e o p e ra tio n s in v o lv e d in th e a n a lo g -to -d ig ita l

c o n v e rsio n o f a n a lo g signals T h e p ro c e ss o f sa m p lin g a sin u so id is d e s c rib e d in

so m e d e ta il a n d th e p ro b le m o f aliasin g is e x p la in e d Signal q u a n tiz a tio n an d

d ig ita l-to -a n a lo g c o n v e rsio n a re also d e s c rib e d in g e n e ra l te rm s, b u t th e a n aly sis

is p r e s e n te d in su b s e q u e n t c h a p te rs

C h a p te r 2 is d e v o te d e n tire ly to th e c h a ra c te riz a tio n a n d a n a ly sis o f lin e a r tim e -in v a ria n t (s h ift-in v a ria n t) d isc re te -tim e sy ste m s a n d d isc re te -tim e signals in

th e tim e d o m a in T h e c o n v o lu tio n sum is d e riv e d a n d sy stem s a re c a te g o riz e d

a c c o rd in g to th e d u r a tio n o f th e ir im p u lse r e sp o n s e as a fin ite -d u ra tio n im p u lse

re sp o n s e (F IR ) a n d as a n in fin ite -d u ra tio n im p u lse re sp o n s e ( I I R ) L in e a r tim e -

in v a ria n t sy stem s c h a ra c te riz e d by d iffe re n c e e q u a tio n s a re p r e s e n te d a n d th e so ­

lu tio n o f d iffe re n c e e q u a tio n s w ith in itial c o n d itio n s is o b ta in e d T h e c h a p te r

co n c lu d e s w ith a tr e a tm e n t o f d isc re te -tim e c o rre la tio n

T h e z -tra n s fo rm is in tro d u c e d in C h a p te r 3 B o th th e b ila te r a l an d th e

u n ila te ra l z -tra n s fo rm s a re p re s e n te d , a n d m e th o d s fo r d e te r m in in g th e in v e rse

z -tra n s fo rm a re d e s c rib e d U s e o f th e z -tra n s fo rm in th e a n a ly sis o f lin e a r tim e -

in v a ria n t sy stem s is illu stra te d , a n d im p o rta n t p r o p e rtie s o f sy stem s, su c h as c a u s a l­ity a n d sta b ility , a re r e la te d to z -d o m a in c h a ra c te ristic s

C h a p te r 4 tr e a ts th e an aly sis o f signals an d sy ste m s in th e fre q u e n c y d o m a in

F o u rie r se rie s an d th e F o u rie r tra n sfo rm a re p r e s e n te d fo r b o th c o n tin u o u s-tim e

a n d d isc re te -tim e signals L in e a r tim e -in v a ria n t ( L T I) d isc re te sy ste m s a re c h a r ­

a c te riz e d in th e fre q u e n c y d o m a in by th e ir fre q u e n c y re sp o n se fu n c tio n a n d th e ir

re sp o n s e to p e rio d ic a n d a p e rio d ic sig n a ls is d e te rm in e d A n u m b e r o f im p o rta n t

ty p e s o f d isc re te -tim e sy stem s a re d e s c rib e d , in c lu d in g r e s o n a to r s , n o tc h filters,

co m b filters, all-p ass filters, a n d o sc illa to rs T h e d e sig n o f a n u m b e r o f sim p le

F IR a n d IIR filters is also c o n s id e re d In a d d itio n , th e stu d e n t is in tro d u c e d to

th e c o n c e p ts o f m in im u m -p h a s e , m ix e d -p h a se , a n d m a x im u m -p h a s e sy stem s a n d

to th e p ro b le m o f d e c o n v o lu tio n

T h e D F T its p r o p e rtie s a n d its a p p lic a tio n s, a re th e to p ic s c o v e re d in C h a p ­

te r 5 T w o m e th o d s a re d e s c rib e d fo r u sin g th e D F T to p e rfo rm lin e a r filtering

T h e u se o f th e D F T to p e rfo rm fre q u e n c y a n aly sis o f sig n a ls is a lso d e sc rib e d

C h a p te r 6 c o v e rs th e efficien t c o m p u ta tio n o f th e D F T In c lu d e d in th is c h a p ­

te r a re d e s c rip tio n s o f rad ix -2 , ra d ix -4, a n d sp lit-ra d ix fa st F o u rie r tra n s fo rm (F F T )

a lg o rith m s, a n d a p p lic a tio n s o f th e F F T a lg o rith m s to th e c o m p u ta tio n o f c o n v o ­

lu tio n a n d c o rre la tio n T h e G o e rtz e l a lg o rith m a n d th e c h irp -z tra n sfo rm a re

in tro d u c e d as tw o m e th o d s fo r c o m p u tin g th e D F T u sin g lin e a r filterin g

C h a p te r 7 tr e a ts th e re a liz a tio n o f I I R a n d F IR sy stem s T h is tr e a tm e n t

in c lu d e s d ire c t-fo rm , c a sc a d e , p a ra lle l, la ttic e , a n d la ttic e -la d d e r re a liz a tio n s T h e

c h a p te r in c lu d e s a tr e a tm e n t o f sta te -s p a c e a n aly sis a n d s tru c tu re s fo r d isc re te -tim e

sy stem s, a n d e x a m in e s q u a n tiz a tio n e ffe c ts in a d ig ita l im p le m e n ta tio n o f F IR an d

I I R sy stem s

T e c h n iq u e s fo r d esig n o f d ig ital F IR a n d I I R filte rs a re p r e s e n te d in C h a p ­

te r 8 T h e d esig n te c h n iq u e s in c lu d e b o th d ire c t d esig n m e th o d s in d isc re te tim e

a n d m e th o d s in v o lv in g th e c o n v e rsio n o f a n a lo g filte rs in to d ig ital filte rs by v a rio u s tra n sfo rm a tio n s A lso tr e a te d in this c h a p te r is th e d e sig n o f F I R a n d I I R filters

by le a s t-s q u a re s m e th o d s

C h a p te r 9 fo c u se s o n th e sa m p lin g o f c o n tin u o u s-tim e sig n a ls a n d th e r e ­

c o n s tru c tio n o f su c h sig n a ls fro m th e ir sa m p le s In th is c h a p te r, w e d e riv e th e

sa m p lin g th e o r e m fo r b a n d p a s s c o n tin u o u s-tim e -s ig n a ls an d th e n c o v e r th e A /D

a n d D /A c o n v e rsio n te c h n iq u e s , in clu d in g o v e rsa m p lin g A /D a n d D /A c o n v e rte rs

C h a p te r 10 p ro v id e s a n in d e p th tr e a tm e n t o f s a m p lin g -ra te c o n v e rsio n an d its a p p lic a tio n s to m u ltira le d ig ita l signal p ro c e ssin g In a d d itio n to d e sc rib in g d e c ­

c o n v e rsio n by a n a r b itra ry fa c to r S e v e ra l a p p lic a tio n s to m u ltira te sig n al p ro c e ss­

in g a re p r e s e n te d , in c lu d in g th e im p le m e n ta tio n o f d ig ita l filters, s u b b a n d co d in g

o f sp e e c h sig n a ls, tra n sm u ltip le x in g , a n d o v e rsa m p lin g A /D a n d D /A c o n v e rte rs

L in e a r p re d ic tio n a n d o p tim u m lin e a r (W ie n e r) filte rs a re tr e a te d in C h a p ­

te r 11 A lso in c lu d e d in th is c h a p te r a re d e s c rip tio n s o f th e L e v in s o n - D u rb in

a lg o rith m a n d Schiir a lg o rith m fo r solving th e n o rm a l e q u a tio n s , as w ell as th e

A R la ttic e a n d A R M A la ttic e -la d d e r filters

P o w e r s p e c tru m e s tim a tio n is th e m a in to p ic o f C h a p te r 12 O u r c o v e ra g e

in c lu d e s a d e s c rip tio n o f n o n p a r a m e tric a n d m o d e l-b a se d (p a ra m e tr ic ) m e th o d s

A lso d e s c rib e d a re e ig e n -d e c o m p o sitio n -b a se d m e th o d s, in c lu d in g M U S IC a n d

E S P R IT

A t N o r th e a s te r n U n iv e rsity , w e h a v e u se d th e first six c h a p te rs o f this b o o k

fo r a o n e - s e m e s te r (ju n io r lev el) c o u rse in d isc re te sy ste m s a n d d ig ita l signal p r o ­cessing

A o n e - s e m e s te r se n io r level c o u rse fo r s tu d e n ts w h o h a v e h a d p r io r e x p o s u re

to d isc re te sy ste m s c a n u se th e m a te ria l in C h a p te rs 1 th r o u g h 4 fo r a q u ic k re v ie w

a n d th e n p ro c e e d to c o v e r C h a p te r 5 th ro u g h 8

In a first-v e a r g ra d u a te level c o u rse in d ig ital signal p ro c e ssin g , th e first five

c h a p te rs p ro v id e th e s tu d e n t w ith a g o o d re v ie w o f d isc re te -tim e sy stem s T h e

in s tru c to r c a n m o v e q u ic k ly th ro u g h m o st o f th is m a te ria l a n d th e n c o v e r C h a p te rs

6 th ro u g h 9, fo llo w e d by e ith e r C h a p te rs 10 an d 11 o r by C h a p te rs 11 a n d 12

W e h a v e in c lu d e d m a n y e x a m p le s th ro u g h o u t th e b o o k a n d a p p ro x im a te ly

500 h o m e w o rk p ro b le m s M a n y o f th e h o m e w o rk p ro b le m s c a n b e so lv e d n u m e r ­ically o n a c o m p u te r, u sin g a so ftw a re p a c k a g e such as M A T L A B © T h e se p r o b ­lem s a re id e n tifie d by a n a ste risk A p p e n d ix D c o n ta in s a list o f M A T L A B fu n c ­tio n s th a t th e s tu d e n t c a n u se in so lv in g th e s e p ro b le m s T h e in s tr u c to r m ay also

H L e v -A ri, L M e ra k o s , W M ik h a e l, P M o n tic c io lo , C N ik ias, M S c h e tz e n ,

H T ru sse ll, S W ilso n , a n d M Z o lto w s k i W e a re also in d e b te d to D r R , P ric e fo r

re c o m m e n d in g th e in c lu sio n o f sp lit-ra d ix F F T a lg o rith m s a n d r e la te d su g g e stio n s

F in ally , w e w ish to a c k n o w le d g e th e su g g e stio n s a n d c o m m e n ts o f m a n y f o rm e r

g r a d u a te s tu d e n ts , a n d esp ecially th o s e by A L K o k , J L in a n d S S rin id h i w h o

a ssiste d in th e p r e p a r a tio n o f se v e ra l illu stra tio n s a n d th e so lu tio n s m a n u a l

J o h n G P ro a k is

D im itris G , M a n o la k is

D ig ita l signal p ro c e ssin g is a n a re a o f sc ie n c e a n d e n g in e e rin g th a t h a s d e v e lo p e d

ra p id ly o v e r th e p a st 30 y ears T h is ra p id d e v e lo p m e n t is a re su lt o f th e sig n if­

ic a n t a d v a n c e s in digital c o m p u te r te c h n o lo g y a n d in te g ra te d -c irc u it fa b ric a tio n

T h e digital c o m p u te rs a n d a sso c ia te d d ig ital h a rd w a re o f th r e e d e c a d e s ago w e re

re la tiv e ly larg e an d e x p e n siv e an d , as a c o n s e q u e n c e , th e ir u se w as lim ite d to

g e n e ra l-p u rp o s e n o n -re a l-tim e (o ff-lin e) scientific c o m p u ta tio n s a n d b u sin e ss a p ­

p lic a tio n s T h e ra p id d e v e lo p m e n ts in in te g ra te d -c irc u it te c h n o lo g y , s ta rtin g w ith

m e d iu m -sc a le in te g ra tio n (M S I) a n d p ro g re s sin g to la rg e -sc a le in te g ra tio n (L S I),

a n d now , v e ry -la rg e -sc a le in te g ra tio n (V L S I) o f e le c tro n ic circu its has sp u rre d

th e d e v e lo p m e n t o f p o w e rfu l, sm a lle r, fa ste r, a n d c h e a p e r d ig ital c o m p u te rs a n d

s p e c ia l-p u rp o se d ig ital h a rd w a re T h e se in e x p e n siv e an d re la tiv e ly fa st digital c ir­

c u its h a v e m a d e it p o ssib le to c o n s tru c t highly s o p h is tic a te d d ig ital sy ste m s c a p a b le

o f p e rfo rm in g co m p le x d ig ital signal p ro c e ssin g fu n c tio n s a n d task s, w hich a re u s u ­ally to o difficult a n d /o r to o ex p en siv e to be p e rfo rm e d by a n a lo g c irc u itry o r a n a lo g signal p ro c e ssin g sy stem s H e n c e m a n y of th e signal p ro c e ss in g ta sk s th a t w e re

c o n v e n tio n a lly p e rfo rm e d by a n a lo g m e a n s a re re a liz e d to d a y by less ex p e n siv e

a n d o fte n m o re re lia b le d ig ital h a rd w a re

W e d o n o t w ish to im p ly th a t d ig ital signal p ro c e ssin g is th e p r o p e r so lu ­tio n fo r all sig n al p ro c e ssin g p ro b le m s In d e e d , fo r m a n y sig n a ls w ith e x tre m e ly

w id e b a n d w id th s , re a l-tim e p ro c e ssin g is a r e q u ire m e n t F o r such signals, a n a ­

lo g o r, p e rh a p s, o p tical sig n a l p ro c e ssin g is th e o n ly p o ssib le so lu tio n H o w e v e r,

w h e re d ig ital circu its a re a v a ila b le a n d h a v e su fficien t sp e e d to p e rfo rm th e signal

p ro c e ssin g , th e y a re u su a lly p re fe ra b le

N o t o n ly d o d ig ital c irc u its yield c h e a p e r a n d m o re re lia b le sy stem s fo r signal

p ro c e ssin g , th e y h a v e o th e r a d v a n ta g e s as w ell In p a rtic u la r, d ig ital p ro c e ssin g

h a rd w a re allow s p ro g ra m m a b le o p e ra tio n s T h ro u g h s o ftw a re , o n e can m o re e asily

m o d ify th e sig n a l p ro c e ssin g fu n c tio n s to b e p e rfo rm e d b y th e h a rd w a re T h u s

d ig ital h a rd w a re a n d a s so c ia te d s o ftw a re p ro v id e a g r e a te r d e g re e o f flexibility in

sy ste m d esig n A lso , th e r e is o fte n a h ig h e r o r d e r o f p re c isio n a c h ie v a b le w ith

d ig ital h a rd w a re an d so ftw a re c o m p a re d w ith a n a lo g c irc u its a n d a n a lo g signal

p ro c e ss in g sy stem s F o r all th e s e re a so n s , th e re h a s b e e n an e x p lo siv e g ro w th in

In this b o o k o u r o b je c tiv e is to p re s e n t an in tro d u c tio n o f th e basic an aly sis

to o ls a n d te c h n iq u e s fo r d ig ita l p ro c e ssin g o f sig n als W e b e g in by in tro d u c in g

so m e o f th e n e c e ssa ry te rm in o lo g y a n d by d e s c rib in g th e im p o rta n t o p e ra tio n s

a s so c ia te d w ith th e p ro c e ss o f c o n v e rtin g an a n a lo g sig n al to d ig ita l fo rm su ita b le

fo r d ig ital p ro c e ssin g A s w e shall se e , d ig ital p ro c e ssin g o f a n a lo g sig n a ls has

so m e d ra w b a c k s F irst, a n d fo re m o s t, c o n v e rsio n o f an a n a lo g sig n a l to digital

fo rm , a c c o m p lish e d by sa m p lin g th e signal a n d q u a n tiz in g th e sa m p le s, re su lts in a

d is to rtio n th a t p re v e n ts us fro m re c o n s tru c tin g th e o rig in a l a n a lo g sig n al fro m th e

q u a n tiz e d sa m p le s C o n tro l o f th e a m o u n t o f th is d is to rtio n is a c h ie v e d by p ro p e r

ch o ice o f th e sa m p lin g ra te a n d th e p re c isio n in th e q u a n tiz a tio n p ro c e ss S e c o n d ,

th e re a re finite p re c isio n e ffe c ts th a t m u st be c o n s id e re d in th e d ig ita l p ro c e ssin g

o f th e q u a n tiz e d sa m p le s W h ile th e s e im p o rta n t issu es a re c o n s id e re d in so m e

d e ta il in this b o o k , th e e m p h a sis is o n th e a n a ly sis a n d d e sig n o f d ig ital signal

p ro c e ssin g sy stem s a n d c o m p u ta tio n a l te c h n iq u e s

1.1 SIGNALS, SYSTEMS, AND SIGNAL PROCESSING

A signal is d efin ed as any p hysical q u a n tity th a t v a rie s w ith tim e , sp a c e , o r an y

o th e r in d e p e n d e n t v a ria b le o r v a ria b le s M a th e m a tic a lly , w e d e s c rib e a sig n al as

a fu n c tio n o f o n e o r m o re in d e p e n d e n t v a ria b le s F o r e x a m p le , th e fu n c tio n s

* i( r ) = 5/

(1.1.1)

S2(t) = 20 r

d e s c rib e tw o signals, o n e th a t v a rie s lin e a rly w ith th e in d e p e n d e n t v a ria b le t (tim e )

an d a se c o n d th a t v a rie s q u a d ra tic a lly w ith t A s a n o th e r e x a m p le , c o n s id e r the

fu n c tio n

v) = 3x + 2 x y + 1 0 y 2 ( 1 1 2 )

T h is fu n c tio n d e s c rib e s a sig n al o f tw o in d e p e n d e n t v a ria b le s x a n d y th a t co u ld

r e p re s e n t th e tw o sp a tia l c o o rd in a te s in a p la n e

T h e signals d e s c rib e d by (1.1.1) a n d (1.1.2) b e lo n g to a class o f sig n a ls th a t

a re p re c ise ly d e fin e d by sp e cify in g th e fu n c tio n a l d e p e n d e n c e o n th e in d e p e n d e n t

v a ria b le H o w e v e r, th e r e a re cases w h e re su c h a fu n c tio n a l r e la tio n s h ip is u n k n o w n

o r to o h ighly c o m p lic a te d to b e o f an y p ra c tic a l use

F o r ex a m p le , a sp e e c h sig n al (see Fig 1.1) c a n n o t be d e s c rib e d fu n c tio n a lly

w h e re {/!,(/)}, { F ,(r)j, a n d {t9,(r)} a re th e se ts o f (p o ssib ly tim e -v a ry in g ) a m p litu d e s, fre q u e n c ie s, an d p h a s e s, re sp e c tiv e ly , o f th e sin u so id s In fact, o n e w ay to in te r p re t

Sec 1.1 Signals, Systems, and Signal Processing

' W W W ’ Figure 1.1 Example o f a speech signal.

sp e e c h signal is to m e a s u re th e a m p litu d e s, fre q u e n c ie s, a n d p h a s e s c o n ta in e d in

th e sh o rt tim e s e g m e n t o f th e signal

A n o th e r e x a m p le o f a n a tu r a l signal is an e le c tro c a rd io g ra m (E C G ) Such a signal p ro v id e s a d o c to r w ith in fo rm a tio n a b o u t th e c o n d itio n o f th e p a tie n t's h e a rt

S im ila rly , a n e le c tro e n c e p h a lo g ra m ( E E G ) signal p ro v id e s in fo rm a tio n a b o u t th e

a c tiv ity o f th e b rain

S p e e c h , e le c tro c a rd io g ra m , a n d e le c tro e n c e p h a lo g ra m signals a re e x a m p le s

o f in fo rm a tio n -b e a rin g sig n a ls th a t ev o lv e as fu n c tio n s o f a single in d e p e n d e n t

v a ria b le , n a m e lv , tim e A n e x a m p le o f a signal th a t is a fu n c tio n o f tw o in d e ­

p e n d e n t v a ria b le s is an im ag e signal T h e in d e p e n d e n t v a ria b le s in th is case a re

th e sp a tia l c o o rd in a te s T h e se a re b u t a few e x a m p le s o f th e c o u n tle ss n u m b e r o f

n a tu r a l sig n a ls e n c o u n te r e d in p ra c tic e

A s so c ia te d w ith n a tu r a l signals a re th e m e a n s by w h ich su ch sig n a ls a re g e n ­

e ra te d F o r e x a m p le , sp e e c h sig n a ls a re g e n e ra te d by fo rc in g a ir th ro u g h th e v ocal

co rd s Im a g e s a re o b ta in e d by e x p o sin g a p h o to g ra p h ic film to a sc e n e o r a n o b ­

je c t T h u s sig n al g e n e ra tio n is u su a lly a sso c ia te d w ith a s y s t e m th a t r e sp o n d s to a

stim u lu s o r fo rc e In a sp e e c h sig n a l, th e sy stem co n sists o f th e vocal c o rd s a n d

th e vocal tra c t, also c a lle d th e v o cal cavity T h e stim u lu s in c o m b in a tio n w ith th e

sy ste m is c a lle d a signal source T h u s w e h av e sp e e c h so u rc e s, im a g e s so u rc e s, a n d

v a rio u s o th e r ty p e s o f sig n al so u rc e s

A sy ste m m ay also b e d efin e d as a p h y sic al d ev ice th a t p e rfo rm s a n o p e r a ­

tio n o n a signal F o r e x a m p le , a filte r u se d to re d u c e th e n o ise a n d in te rfe re n c e

c o rru p tin g a d e s ire d in fo rm a tio n -b e a rin g signal is called a sy stem In th is case th e filte r p e rfo rm s so m e o p e ra tio n (s ) o n th e signal, w hich h a s th e effe c t o f re d u c in g (filte rin g ) th e n o ise a n d in te rfe re n c e fro m th e d e s ire d in f o rm a tio n -b e a rin g signal

W h e n w e p ass a sig n a l th ro u g h a sy stem , as in filte rin g , w e say th a t we h a v e

p ro c e ss e d th e sig n al In this case th e p ro c e ssin g o f th e sig n a l in v o lv es filterin g th e

n o ise a n d in te r fe r e n c e fro m th e d e s ire d signal In g e n e ra l, th e sy ste m is c h a ra c ­

te riz e d by th e ty p e o f o p e r a tio n th a t it p e rfo rm s on th e sig n a l F o r ex a m p le , if

th e o p e r a tio n is lin e a r, th e sy stem is called lin ear If th e o p e r a tio n o n th e signal

is n o n lin e a r, th e sy stem is said to b e n o n lin e a r, a n d so fo rth S u ch o p e ra tio n s a re

F o r o u r p u rp o se s , it is c o n v e n ie n t to b r o a d e n th e d e fin itio n o f a sy stem to

in clu d e n o t o n ly ph y sical dev ices, b u t also so ftw a re re a liz a tio n s o f o p e ra tio n s on

a signal In d ig ital p ro c e ssin g o f signals o n a d ig ital c o m p u te r, th e o p e ra tio n s p e r ­

fo rm e d o n a sig n al c o n sist o f a n u m b e r o f m a th e m a tic a l o p e ra tio n s as sp e cified by

a so ftw a re p ro g ra m In th is case, th e p ro g ra m r e p re s e n ts an im p le m e n ta tio n o f th e

sy stem in software T h u s we h av e a sy ste m th a t is re a liz e d on a d ig ita l c o m p u te r

by m e a n s o f a se q u e n c e o f m a th e m a tic a l o p e r a tio n s ; th a t is, w e h a v e a digital signal p ro c e ssin g sy stem re a liz e d in so ftw a re F o r e x a m p le , a d ig ita l c o m p u te r can

b e p r o g ra m m e d to p e rfo rm d ig ital filterin g A lte rn a tiv e ly , th e d ig ita l p ro cessin g

o n th e signal m ay be p e rfo rm e d by d ig ital h a rd w a re (lo g ic c irc u its) c o n fig u re d to

p e rfo rm th e d e s ire d specified o p e ra tio n s In su c h a re a liz a tio n , w e h a v e a physical

d ev ice th a t p e rfo rm s th e sp e cified o p e ra tio n s In a b r o a d e r se n se , a d ig ital sy stem can be im p le m e n te d as a c o m b in a tio n o f d ig ital h a rd w a re an d s o ftw a re , e a c h of

w hich p e rfo rm s its o w n se t o f specified o p e ra tio n s

T h is b o o k d e a ls w ith th e p ro c e ssin g o f signals by d ig ital m e a n s , e ith e r in so ft­

w a re o r in h a rd w a re Since m a n y o f th e sig n a ls e n c o u n te r e d in p ra c tic e a re a n alo g ,

w e will also c o n s id e r th e p ro b le m of c o n v e rtin g an a n a lo g signal in to a d ig ital sig­

n al fo r p ro cessin g T h u s we will be d e a lin g p rim a rily w ith d ig ita l sy stem s T h e

o p e ra tio n s p e rfo rm e d by such a sy stem can u su a lly b e sp e cified m a th e m a tic a lly

T h e m e th o d o r set o f ru le s fo r im p le m e n tin g th e sy s te m by a p ro g ra m th a t p e r ­

fo rm s th e c o rre sp o n d in g m a th e m a tic a l o p e ra tio n s is c a lle d a n alg o ri th m U su ally ,

th e re a re m a n y w ays o r a lg o rith m s by w h ich a sy stem can be im p le m e n te d , e ith e r

in so ftw a re o r in h a rd w a re , to p e rfo rm th e d e s ire d o p e ra tio n s a n d c o m p u ta tio n s

In p ra c tic e , we h av e an in te re s t in d e v isin g a lg o rith m s th a t a re c o m p u ta tio n a lly efficien t, fast, a n d easily im p le m e n te d T h u s a m a jo r to p ic in o u r stu d y o f d ig i­tal signal p ro c e ssin g is th e discu ssio n o f efficien t a lg o rith m s fo r p e rfo rm in g such

o p e ra tio n s as filterin g , c o rre la tio n , a n d s p e c tra l an aly sis

1.1.1 Basic Elements of a Digital Signal Processing

System

M o st o f th e signals e n c o u n te re d in sc ien ce a n d e n g in e e rin g a re a n a lo g in n a tu re

T h a t is th e signals a re fu n c tio n s of a c o n tin u o u s v a ria b le , such a s tim e o r sp a ce,

a n d u su a lly ta k e o n v alu es in a c o n tin u o u s ra n g e S u ch signals m a y b e p ro c e sse d

d ire c tly by a p p r o p ria te a n a lo g sy stem s (su ch as filte rs o r fre q u e n c y a n a ly z e rs) or fre q u e n c y m u ltip lie rs fo r th e p u rp o s e of c h a n g in g th e ir c h a ra c te ristic s o r e x tra c tin g

so m e d e s ire d in fo rm a tio n In su ch a case w e say th a t th e signal h a s b e e n p ro c e sse d

d ire c tly in its a n a lo g fo rm , as illu stra te d in Fig 1.2 B o th th e in p u t signal a n d th e

Analog output signal

input

signal

Analogoutput

signal

Digital input

signal

Digital output signal Figure 1.3 Block diagram o f a digital signal processing system.

D ig ita l signal p ro c e ssin g p ro v id e s an a lte rn a tiv e m e th o d fo r p ro c e ssin g th e

a n a lo g sig n a l, as illu stra te d in Fig 1.3 T o p e r fo r m th e p ro c e ss in g d ig itally , th e r e

is a n e e d fo r an in te rfa c e b e tw e e n th e a n a lo g signal a n d th e d ig ital p ro c e sso r

T h is in te rfa c e is c a lle d an analog-to-digital ( A / D ) con verter T h e o u tp u t of th e

A /D c o n v e rte r is a d ig ita l signal th a t is a p p r o p ria te as a n in p u t to th e d ig ita l

p ro c e ss o r

T h e d ig ital signal p ro c e ss o r m a y be a la rg e p ro g ra m m a b le d ig ital c o m p u te r

o r a sm a ll m ic ro p ro c e s s o r p ro g ra m m e d to p e rfo rm th e d e s ire d o p e ra tio n s on th e

in p u t sig n al It m ay also b e a h a rd w ire d d ig ital p ro c e ss o r co n fig u re d to p e r fo r m

a sp e cified se t o f o p e ra tio n s o n th e in p u t sig n al P ro g ra m m a b le m a c h in e s p r o ­

v id e th e flex ib ility to c h a n g e th e sig n al p ro c e ssin g o p e r a tio n s th ro u g h a c h a n g e

in th e so ftw a re , w h e re a s h a rd w ire d m a c h in e s a re difficult to re c o n fig u re C o n s e ­

q u e n tly , p ro g ra m m a b le signal p ro c e ss o rs a re in v ery c o m m o n u se O n th e o th e r

h a n d , w h e n signal p ro c e ssin g o p e r a tio n s a re w ell d e fin e d , a h a rd w ire d im p le m e n ­

ta tio n o f th e o p e ra tio n s can b e o p tim iz e d , re su ltin g in a c h e a p e r signal p ro c e ss o r

a n d , u su a lly , o n e th a t ru n s fa ste r th a n its p ro g ra m m a b le c o u n te r p a r t In a p p li­

c a tio n s w h e re th e d ig ita l o u tp u t fro m th e d ig ita l signal p ro c e ss o r is to be given

to th e u se r in a n a lo g fo rm , such as in sp e e c h c o m m u n ic a tio n s, w e m u st p r o ­

v id e a n o th e r in te rfa c e fro m th e d ig ital d o m a in to th e a n a lo g d o m a in S u ch an

in te rfa c e is c a lle d a digital-to-analog ( D / A ) converter T h u s th e signal is p r o ­

v id e d to th e u se r in a n a lo g fo rm , a s illu stra te d in th e b lo c k d ia g ra m o f Fig 1.3

H o w e v e r, th e r e a re o th e r p ra c tic a l a p p lic a tio n s in v o lv in g signal an aly sis, w h e re

th e d e s ire d in fo rm a tio n is c o n v e y e d in d ig ital fo rm a n d n o D /A c o n v e rte r is

r e q u ire d F o r e x a m p le , in th e d ig ita l p ro c e ssin g o f r a d a r signals, th e in f o rm a ­tio n e x tra c te d fro m th e r a d a r sig n a l, such as th e p o sitio n o f th e a irc ra ft a n d its

s p e e d , m ay sim p ly b e p rin te d o n p a p e r T h e re is n o n e e d fo r a D /A c o n v e rte r in

th is case

1.1.2 Advantages of Digital over Analog Signal

Processing

T h e re a re m a n y re a s o n s w hy d ig ita l signal p ro c e ssin g o f a n a n a lo g signal m a y be

p r e fe ra b le to p ro c e ssin g th e sig n al d ire c tly in th e a n a lo g d o m a in , as m e n tio n e d briefly e a rlie r F irst, a digital p ro g ra m m a b le sy ste m allo w s flex ib ility in r e c o n ­

R e c o n fig u ra tio n o f an a n a lo g sy stem u su a lly im p lies a re d e s ig n o f th e h a rd w a re

fo llo w ed b y te s tin g a n d v e rific a tio n to se e th a t it o p e r a te s p ro p e rly

A c c u ra c y c o n s id e ra tio n s also p la y an im p o rta n t ro le in d e te rm in in g th e fo rm

o f th e sig n al p ro c e ss o r T o le ra n c e s in a n a lo g c irc u it c o m p o n e n ts m a k e it e x tre m e ly difficult fo r th e sy stem d e s ig n e r to c o n tro l th e ac c u ra c y o f an a n a lo g signal p r o ­cessing sy stem O n th e o th e r h a n d , a digital sy ste m p r o v id e s m u ch b e tte r c o n tro l

o f accu racy r e q u ire m e n ts S uch re q u ire m e n ts , in tu rn , re s u lt in sp e cify in g th e a c ­

c u ra c y r e q u ire m e n ts in th e A /D c o n v e rte r a n d th e d ig ita l sig n a l p ro c e ss o r, in te rm s

o f w o rd le n g th , flo a tin g -p o in t v e rsu s fix e d -p o in t a rith m e tic , a n d sim ila r fa c to rs

D ig ita l sig n a ls a re easily sto re d o n m a g n e tic m e d ia ( ta p e o r disk ) w ith o u t d e ­

te r io ra tio n o r loss o f sig n al fidelity b e y o n d th a t in tro d u c e d in th e A /D co n v e rsio n

A s a c o n s e q u e n c e , th e sig n a ls b e c o m e tra n s p o r ta b le a n d c a n b e p ro c e ss e d off-line

in a re m o te la b o ra to ry T h e d ig ital sig n a l p ro c e ssin g m e th o d also allow s fo r th e im ­

p le m e n ta tio n o f m o re s o p h is tic a te d sig n al p ro c e ssin g a lg o rith m s It is u su a lly very

d ifficu lt to p e rfo rm p re c ise m a th e m a tic a l o p e r a tio n s on sig n a ls in a n a lo g fo rm b u t

th e se sa m e o p e ra tio n s c a n b e ro u tin e ly im p le m e n te d on a d ig ita l c o m p u te r u sin g

so ftw a re

In so m e cases a d ig ita l im p le m e n ta tio n o f th e signal p ro c e ssin g sy stem is

c h e a p e r th a n its a n a lo g c o u n te rp a rt T h e lo w e r co st m ay b e d u e to th e fact th a t

th e d ig ital h a rd w a re is c h e a p e r, o r p e r h a p s it is a re su lt o f th e flexibility fo r m o d ­ific atio n s p ro v id e d by th e d ig ital im p le m e n ta tio n

A s a c o n s e q u e n c e o f th e se a d v a n ta g e s , d ig ita l sig n a l p ro c e ss in g has b e e n

a p p lie d in p ra c tic a l sy ste m s c o v e rin g a b ro a d ra n g e o f d isc ip lin e s W e cite, fo r e x ­

am p le , th e a p p lic a tio n o f d ig ita l sig n al p ro c e ssin g te c h n iq u e s in sp e e c h p ro c e ssin g

an d signal tra n sm iss io n o n te le p h o n e c h a n n e ls, in im ag e p ro c e ss in g a n d tra n sm is­sio n , in se ism o lo g y a n d g eo p h y sics, in oil e x p lo ra tio n , in th e d e te c tio n o f n u c le a r

e x p lo sio n s, in th e p ro c e ssin g o f sig n a ls re c e iv e d fro m o u te r sp a c e , a n d in a vast

v a rie ty o f o th e r a p p lic a tio n s S o m e o f th e s e a p p lic a tio n s a re c ite d in s u b s e q u e n t

c h a p te rs

A s a lre a d y in d ic a te d , h o w e v e r, d ig ital im p le m e n ta tio n h a s its lim ita tio n s

O n e p ra c tic a l lim ita tio n is th e sp e e d o f o p e r a tio n o f A /D c o n v e r te r s a n d digital signal p ro c e sso rs W e sh a ll se e th a t sig n a ls h a v in g e x tre m e ly w id e b a n d w id th s r e ­

q u ire fa s t-sa m p lin g -ra te A /D c o n v e rte rs an d fa st d ig ita l sig n al p ro c e sso rs H e n c e

th e r e a re a n a lo g sig n a ls w ith la rg e b a n d w id th s fo r w hich a d ig ital p ro c e ssin g a p ­

p ro a c h is b e y o n d th e s ta te o f th e a rt o f d ig ital h a rd w a re

1.2 CLASSIFICATION OF SIGNALS

T h e m e th o d s we u se in p ro c e ssin g a sig n al o r in a n a ly z in g th e r e s p o n s e o f a system

to a sig n a l d e p e n d h e a v ily o n th e c h a ra c te ristic a ttr ib u te s o f th e specific signal

T h e re a re te c h n iq u e s th a t a p p ly only to specific fa m ilie s o f sig n a ls C o n s e q u e n tly ,

an y in v e s tig a tio n in sig n al p ro c e ssin g sh o u ld sta rt w ith a cla ssific a tio n o f th e signals

1.2.1 Multichannel and Multidimensional Signals

A s e x p la in e d in S e c tio n 1.1, a sig n al is d e s c rib e d by a fu n c tio n o f o n e o r m o re

in d e p e n d e n t v a ria b le s T h e v alu e o f th e fu n c tio n (i.e., th e d e p e n d e n t v a ria b le ) can

be a re a l-v a lu e d sc a la r q u a n tity , a c o m p le x -v a lu e d q u a n tity , o r p e r h a p s a v e c to r

F o r e x a m p le , th e signal

s i( r ) = A sin37rr

is a re a l-v a lu e d sig n al H o w e v e r, th e signal

s2(f) = A e ji7Tt = A cos 37t t j'A sin 3:r r

is c o m p le x v a lu e d

In so m e a p p lic a tio n s, signals a re g e n e r a te d by m u ltip le so u rc e s o r m u ltip le

se n so rs Such sig n a ls, in tu rn , can be re p re s e n te d in v e c to r fo rm F ig u re 1.4 show s

th e th r e e c o m p o n e n ts o f a v e c to r sig n al th a t r e p re s e n ts th e g ro u n d a c c e le ra tio n

d u e to an e a r th q u a k e T h is a c c e le ra tio n is th e re su lt o f th r e e b asic ty p e s o f e la stic

w av es T h e p rim a ry (P ) w av es a n d th e se c o n d a ry (S) w av es p r o p a g a te w'ithin th e

b o d y o f ro ck a n d a re lo n g itu d in a l a n d tra n sv e rsa l, re sp e c tiv e ly T h e th ird ty p e

o f ela stic w av e is called th e su rfa c e w av e, b e c a u s e it p r o p a g a te s n e a r th e g ro u n d

su rfa c e If $*(/) k = 1 2 3 d e n o te s th e e le c tric a l signal fro m th e £ th se n so r as a

fu n c tio n o f tim e , th e se t o f p = 3 signals can b e re p re s e n te d by a v e c to r S?(f )< w h e re

r si (O '

S;,(r) = S i ( t )

- S l ( t ) J

W e re fe r to su c h a v e c to r o f sig n als as a m u l t i c h a n n e l signal In e le c tr o c a r d io g ra ­

p h y fo r e x a m p le , 3 -le a d a n d 1 2 -lead e le c tro c a rd io g ra m s ( E C G ) a re o fte n u se d in

p ra c tic e , w hich re su lt in 3 -c h a n n e l a n d 1 2 -ch an n el signals

L e t us n o w tu rn o u r a tte n tio n to th e in d e p e n d e n t v a ria b le (s ) If th e signal is

a fu n c tio n o f a single in d e p e n d e n t v a ria b le , th e signal is c a lle d a o n e - d i m e n s i o n a l signal O n th e o th e r h a n d , a signal is c a lle d M -d i m e n s i o n a l if its v a lu e is a fu n c tio n

of M in d e p e n d e n t v a ria b le s.

T h e p ic tu re sh o w n in Fig 1.5 is a n e x a m p le o f a tw o -d im e n sio n a l signal, sin c e

th e in te n sity o r b rig h tn e ss I ( x y) a t e a c h p o in t is a fu n c tio n o f tw o in d e p e n d e n t

v a ria b le s O n th e o th e r h a n d , a b la c k -a n d -w h ite te le v isio n p ic tu re m ay be r e p ­

r e s e n te d as I ( x y t ) sin c e th e b rig h tn e ss is a fu n c tio n o f tim e H e n c e th e T V

p ic tu re m ay b e tr e a te d as a th re e -d im e n s io n a l signal In c o n tra s t, a c o lo r T V p ic ­

tu re m a y b e d e s c rib e d by th r e e in te n sity fu n c tio n s o f th e fo rm Ir (x, y ?), Is (x y t ),

a n d I i , ( x y , t ) , c o rre s p o n d in g to th e b rig h tn e ss o f th e th re e p rin c ip a l c o lo rs (re d

from the epicenter of an earthquake (From Earthquakes, by B A Bold © 1988

by W H Freeman and Company Reprinted with permission of the publisher.)

te rm s th e se sig n a ls a re d e s c rib e d by a fu n c tio n o f a single in d e p e n d e n t v ariab le

A lth o u g h th e in d e p e n d e n t v a ria b le n e e d n o t b e tim e , it is c o m m o n p ra c tic e to use

t as th e in d e p e n d e n t v a riab le In m a n y cases th e signal p ro c e ss in g o p e ra tio n s an d

a lg o rith m s d e v e lo p e d in this te x t fo r o n e -d im e n sio n a l, sin g le -c h a n n e l signals can

b e e x te n d e d to m u ltic h a n n e l a n d m u ltid im e n sio n a l signals

1.2.2 Continuous-Time Versus Discrete-Time Signals

Signals can b e f u rth e r classified in to fo u r d if fe re n t c a te g o rie s d e p e n d in g o n th e

c h a ra c te ristic s o f th e tim e (in d e p e n d e n t) v a ria b le a n d th e v a lu e s th e y ta k e

Sec 1.2 Classification of Signals

Figure 1.5 Example of a two-dimensional signal.

th e y ta k e on v alu es in th e c o n tin u o u s in te rv a l (a b ) w h e re a c a n be —oc a n d b

can be oc M a th e m a tic a lly , th e se sig n als c a n be d e s c rib e d by fu n c tio n s o f a c o n ­

tin u o u s v a ria b le T h e sp e e c h w a v e fo rm in Fig 1.1 a n d th e sig n a ls x i(r) = c o s 7i t ,

x j { t ) = e ^ 1'1, —oc < t < oq a re e x a m p le s o f a n a lo g sig n als D isc rete-time signals

a re d efin e d o n ly a t c e rta in specific v a lu e s o f tim e T h e se tim e in sta n ts n e e d n o t be

e q u id is ta n t, b u t in p ra c tic e th e y a re u su a lly ta k e n a t e q u a lly sp a c e d in te rv a ls fo r

c o m p u ta tio n a l c o n v e n ie n c e a n d m a th e m a tic a l tra c ta b ility T h e sig n al x(t„) =

n = 0, ± 1 , ± 2 , p ro v id e s an e x a m p le o f a d isc re te -tim e signal If we use th e

in d e x n o f th e d isc re te -tim e in sta n ts as th e in d e p e n d e n t v a ria b le , th e signal v a lu e

b e c o m e s a fu n c tio n o f an in te g e r v a ria b le (i.e., a s e q u e n c e o f n u m b e rs) T h u s a

d is c re te -tim e signal can be re p re s e n te d m a th e m a tic a lly by a se q u e n c e o f re a l o r

c o m p le x n u m b e rs T o e m p h a siz e th e d isc re te -tim e n a tu r e o f a sig n al, w e sh a ll

d e n o te su c h a signal as x{ n) in ste a d o f x ( t ) If th e tim e in sta n ts t„ a re e q u a lly

s p a c e d (i.e., t„ = n T ), th e n o ta tio n x ( n T ) is also u se d F o r e x a m p le , th e se q u e n c e

x (n ) if n > 0

o th e rw ise (1.2.1)

is a d isc re te -tim e sig n al, w hich is r e p re s e n te d g ra p h ic a lly as in Fig 1.6

In a p p lic a tio n s, d isc re te -tim e signals m a y a rise in tw o ways:

1 B y se le c tin g v alu es o f an a n a lo g sig n a l a t d isc re te -tim e in sta n ts T h is p ro c e ss

is c a lle d s a m p li n g a n d is d isc u sse d in m o re d e ta il in S e c tio n 1.4 A ll m e a s u r ­

in g in stru m e n ts th a t ta k e m e a s u re m e n ts a t a re g u la r in te rv a l o f tim e p ro v id e

2 By a c c u m u la tin g a v a ria b le o v e r a p e rio d o f tim e F o r e x a m p le , c o u n tin g th e

n u m b e r o f cars u sing a g iv en s tr e e t ev e ry h o u r, o r re c o rd in g th e v a lu e o f gold

ev e ry day, re su lts in d isc re te -tim e signals F ig u re 1.7 sh o w s a g ra p h o f th e

W o lfe r s u n s p o t n u m b e rs E a c h sa m p le o f th is d isc re te -tim e signal p ro v id e s

th e n u m b e r o f s u n s p o ts o b se rv e d d u rin g a n in te rv a l o f 1 y e a r

1.2.3 Continuous-Valued Versus Discrete-Valued Signals

T h e v a lu e s o f a c o n tin u o u s-tim e o r d isc re te -tim e sig n al can be c o n tin u o u s o r d is­cre te If a signal ta k e s o n all p o ssib le v a lu e s o n a finite o r a n in fin ite ra n g e , it

Year

is said to b e c o n tin u o u s-v a lu e d signal A lte rn a tiv e ly , if th e sig n al ta k e s o n v a lu e s fro m a fin ite se t o f p o ssib le v alu es, it is said to be a d isc re te -v a lu e d signal U su a lly ,

th e s e v a lu e s a re e q u id is ta n t a n d h e n c e can be e x p re ss e d as a n in te g e r m u ltip le of

th e d ista n c e b e tw e e n tw o successive v alu es A d isc re te -tim e signal h av in g a set o f

d isc re te v a lu e s is called a digital signal F ig u re 1,8 show s a d ig ita l signal th a t ta k e s

o n o n e o f f o u r p o ssib le v alu es

In o r d e r fo r a sig n a l to b e p ro c e ss e d d ig itally , it m u st be d isc re te in tim e

a n d its v a lu e s m u st b e d isc re te (i.e., it m u st b e a d ig ital sig n a l) If th e signal to

b e p ro c e ss e d is in a n a lo g fo rm , it is c o n v e rte d to a d ig ital sig n al by sa m p lin g th e

a n a lo g sig n al at d isc re te in sta n ts in tim e , o b ta in in g a d isc re te -tim e signal, an d th e n

by q u a n t i z i n g its v a lu e s to a set o f d isc re te v a lu e s, as d e s c rib e d la te r in th e c h a p te r

T h e p ro c e ss o f c o n v e rtin g a c o n tin u o u s-v a lu e d signal in to a d isc re te -v a lu e d sig n al,

called qu a n tiza tio n , is basically an a p p ro x im a tio n p ro c e ss It m ay b e a c c o m p lish e d

sim ply bv ro u n d in g o r tru n c a tio n F o r e x a m p le , if th e a llo w a b le signal v a lu e s

in th e d ig ita l signal a re in te g e rs, say 0 th ro u g h 15, th e c o n tin u o u s-v a lu e signal is

q u a n tiz e d in to th e se in te g e r v alu es T h u s th e signal v alu e 8.58 will be a p p ro x im a te d

by th e v a lu e 8 if th e q u a n tiz a tio n p ro c e ss is p e rfo rm e d by tr u n c a tio n o r by 9 if

th e q u a n tiz a tio n p ro c e ss is p e rfo rm e d by ro u n d in g to th e n e a re s t in te g e r A n

e x p la n a tio n o f th e a n a lo g -to -d ig ita l c o n v e rsio n p ro c e ss is giv en la te r in th e c h a p te r

Figure 1.8 Digital signal with four different amplitude values.

1.2.4 Deterministic Versus Random Signals

T h e m a th e m a tic a l an aly sis a n d p ro c e ssin g o f sig n als r e q u ire s th e a v a ila b ility o f a

m a th e m a tic a l d e s c rip tio n fo r th e signal itself T h is m a th e m a tic a l d e s c rip tio n , o fte n

re fe rr e d to as th e signa l m o d e l , le a d s to a n o th e r im p o rta n t classificatio n of signals

A n y signal th a t can b e u n iq u e ly d e s c rib e d by a n e x p licit m a th e m a tic a l e x p re ssio n ,

a ta b le o f d a ta , o r a w ell-d efin ed ru le is called det erministic T h is te rm is u se d to

e m p h a siz e th e fact th a t all p a st, p r e s e n t, a n d f u tu re v alu es o f th e signal a re k n o w n

p re c ise ly , w ith o u t an y u n c e rta in ty

In m a n y p ra c tic a l a p p lic a tio n s, h o w e v e r, th e r e a re sig n a ls th a t e ith e r c a n n o t

b e d e s c rib e d to an y r e a s o n a b le d e g re e o f a c c u ra c y by ex p licit m a th e m a tic a l f o r ­

o f su c h a re la tio n s h ip im p lie s th a t su c h signals e v o lv e in tim e in a n u n p re d ic ta b le

th e an aly sis a n d d e s c rip tio n o f ra n d o m signals u sin g statistical te c h n iq u e s in ste a d

o f e x p lic it fo rm u la s T h e m a th e m a tic a l fra m e w o rk fo r th e th e o re tic a l an aly sis of

r a n d o m sig n a ls is p ro v id e d by th e th e o ry o f p ro b a b ility a n d sto c h a stic p ro cesses

S o m e b asic e le m e n ts o f this a p p ro a c h , a d a p te d to th e n e e d s o f th is b o o k , a re

p r e s e n te d in A p p e n d ix A

It sh o u ld b e e m p h a s iz e d a t th is p o in t th a t th e classificatio n o f a real-w orld

sig n a l a s d e te rm in is tic o r ra n d o m is n o t alw ays clear S o m e tim e s, b o th a p p ro a c h e s

tim e s, th e w ro n g classificatio n m ay le a d to e r ro n e o u s re su lts , since so m e m a th e ­

m a tic a l to o ls m ay a p p ly o n ly to d e te rm in is tic sig n a ls w hile o th e r s m a y a p p ly o n ly

to ra n d o m signals T h is will b e c o m e c le a re r as w e e x a m in e specific m a th e m a tic a l

to o ls

1.3 THE CONCEPT OF FREQUENCY IN CONTINUOUS-TIME AND

DISCRETE-TIME SIGNALS

T h e c o n c e p t o f fre q u e n c y is fa m ilia r to s tu d e n ts in e n g in e e rin g a n d th e sciences

T h is c o n c e p t is b asic in fo r e x a m p le , th e d esig n o f a ra d io re c e iv e r, a h ig h -fid elity

sy stem , o r a sp e c tra l filte r fo r c o lo r p h o to g ra p h y F ro m p h y sic s w e k n o w th a t fre q u e n c y is clo sely r e la te d to a specific ty p e o f p e rio d ic m o tio n called h a rm o n ic

o sc illa tio n , w hich is d e s c rib e d by sin u so id a l fu n c tio n s T h e c o n c e p t o f fre q u e n c y

is d irectly re la te d to th e c o n c e p t o f tim e A c tu a lly , it h as th e d im e n sio n o f in v erse tim e T h u s w e sh o u ld e x p e c t th a t th e n a tu r e o f tim e (c o n tin u o u s o r d isc re te ) w o u ld affe c t th e n a tu r e o f th e fre q u e n c y a c co rd in g ly

1.3.1 Continuous-Time Sinusoidal Signals

A sim ple h a rm o n ic o sc illa tio n is m a th e m a tic a lly d e s c rib e d by th e follow ing

c o n tin u o u s-tim e sin u so id a l signal:

x a(t) = A cos(Q t + 0) —oc < t < oc (1.3.1)

sh o w n in Fig 1.10 T h e su b s c rip t a u se d w ith x { t ) d e n o te s an a n a lo g signal T h is signal is c o m p le te ly c h a ra c te riz e d by th re e p a ra m e te rs : A is th e a m p l i t u d e o f th e sin u so id ft is th e f r e q u e n c y in r a d ia n s p e r se c o n d (ra d /s), a n d 6 is th e p h a s e in

ra d ia n s In ste a d o f ft, w e o fte n u se th e fre q u e n c y F in cycles p e r se c o n d o r h e rtz

T h e a n a lo g sin u so id a l signal in (1.3.3) is c h a ra c te riz e d by th e fo llo w in g p r o p ­erties:

A L F o r e v e ry fixed v a lu e o f th e fre q u e n c y F, x a(r) is p e rio d ic In d e e d , it can

easily b e sh o w n , u sin g e le m e n ta ry trig o n o m e try , th a t

x a (.t + Tp ) = A„(r)

w h e re Tp = 1 / F is th e fu n d a m e n ta l p e rio d o f th e sin u so id a l signal.

A 2 C o n tin u o u s -tim e sin u s o id a l sig n a ls w ith d istin c t (d iffe re n t) fre q u e n c ie s a re

th e m se lv e s d istin c t

A 3 In c re a s in g th e fre q u e n c y F re su lts in a n in c re a se in th e r a te o f o s c illa tio n

o f th e signal, in th e se n se th a t m o re p e rio d s a re in c lu d e d in a given tim e

in te rv a l

W e o b se rv e th a t fo r F = 0 th e v alu e Tp — oc is c o n siste n t w ith th e f u n ­

d a m e n ta l re la tio n F = 1 / T r D u e to c o n tin u ity o f th e tim e v a ria b le r, w e can

in c re a se th e fre q u e n c y F, w ith o u t lim it, w ith a c o r re sp o n d in g in c re a se in th e ra te

B y d e fin itio n , fre q u e n c y is a n in h e re n tly p o sitiv e p h y sic al q u a n tity T h is

is o b v io u s if w e in te r p r e t fre q u e n c y as th e n u m b e r o f cycles p e r u n it tim e in a

p e rio d ic signal H o w e v e r, in m a n y cases, o n ly fo r m a th e m a tic a l c o n v e n ie n c e , w e

n e e d to in tro d u c e n e g a tiv e fre q u e n c ie s T o se e th is w e recall th a t th e sin u so id a l sig n al (1.3.1) m ay be e x p re ss e d as

x a (t) = A c o s ( ^ r + 6 ) = j eJ(Q,+f>) + ~ e - J(a+9) (1.3.6)

w hich follow s fro m (1.3.5) N o te th a t a sin u so id a l sig n al can b e o b ta in e d by a d d in g

tw o e q u a l-a m p litu d e c o m p le x -c o n ju g a te e x p o n e n tia l sig n a ls, so m e tim e s called p h a -

so rs, illu stra te d in Fig 1.11 A s t i m e p ro g re s se s th e p h a s o rs r o ta te in o p p o site

d ire c tio n s w ith a n g u la r f re q u e n c ie s ±£2 r a d ia n s p e r se c o n d Since a p o sitiv e f r e ­

q u e n c y c o rre s p o n d s to c o u n te rc lo c k w ise u n ifo rm a n g u la r m o tio n , a neg ative f r e ­

q u e n c y sim p ly c o rre s p o n d s to c lo ck w ise a n g u la r m o tio n

F o r m a th e m a tic a l c o n v e n ie n c e , w e u se b o th n e g a tiv e a n d p o sitiv e fre q u e n c ie s

th r o u g h o u t th is b o o k H e n c e th e f re q u e n c y ra n g e fo r a n a lo g sin u so id s is —oo <

16 Introduction Chap 1

Re

Figure 1.11 Representation o f a cosine function by a pair o f complex-conjugate exponentials (phasors).

1.3.2 Discrete-Time Sinusoidal Signals

A d isc re te -tim e sin u so id a l signal m ay be e x p re ss e d as

x ( n ) — A cos (ton + 8), —oo < n < oc (1.3.7)

w h e re n is an in te g e r v a ria b le , c a lle d th e sa m p le n u m b e r A is th e am p l i t u d e o f the sin u so id , co is th e f r e q u e n c y in ra d ia n s p e r sa m p le , an d 8 is th e p h a s e in rad ian s

If in ste a d o f a> w e u se th e fre q u e n c y v a ria b le / d efin e d by

th e re la tio n (1.3.7) b e c o m e s

x ( n ) — A co s(2 n f n + 8) — oc < n < oc (1.3.9)

T h e fre q u e n c y / h a s d im e n sio n s o f cycles p e r sa m p le In S e c tio n 1.4 w h ere

we c o n s id e r th e sa m p lin g o f a n a lo g sin u so id s, w e re la te th e f re q u e n c y v a ria b le

/ o f a d is c re te -tim e sin u so id to th e fre q u e n c y F in cycles p e r se c o n d fo r th e

a n a lo g sin u so id F o r th e m o m e n t w e c o n s id e r th e d isc re te -tim e sin u so id in (1.3.7)

in d e p e n d e n tly of th e c o n tin u o u s-tim e sin u so id g iv en in (1.3.1) F ig u re 1.12 show s

a sin u so id w ith fre q u e n c y co — n /6 ra d ia n s p e r sa m p le ( f ~ ~ cycles p e r sa m p le )

a n d p h a s e 8 — n / 3

x(n) - A cos (urn + 8)

Figure 1.12 Example of a discrete-time

In c o n tr a s t to c o n tin u o u s-tim e sin u so id s, th e d isc re te -tim e sin u so id s a re c h a r ­

a c te riz e d by th e fo llo w in g p ro p e rtie s :

B l A discrete -t ime si n u s o id is p e r i o d ic o n l y i f its f r e q u e n c y f is a ra tional n u m b e r

B y d e fin itio n , a d isc re te -tim e signal x ( n ) is p e rio d ic w ith p e rio d N ( N > 0) if

a n d o n ly if

x ( n + N ) = x ( n ) fo r all n (1.3.10)

T h e sm a lle st v alu e o f N fo r w hich (1.3.10) is tru e is c a lle d th e f u n d a m e n t a l p e r io d

T h e p r o o f o f th e p e rio d ic ity p r o p e rty is sim p le F o r a sin u so id w ith fre q u e n c y /o to b e p e rio d ic , w e sh o u ld hav e

cos[27t /o( A7 + n) + 8} — c o s(2 ,t/o « + 6)

T h is r e la tio n is tru e if a n d only if th e re e x ists an in te g e r k such th a t

2 n f ) N = 2 k n

o r, e q u iv a le n tly

N

A c c o rd in g to (1.3.11) a d isc re te -tim e sin u so id a l signal is p e rio d ic only if its f r e ­

q u e n c y /o can be e x p re ss e d as th e ra tio o f tw o in te g e rs (i.e / () is ra tio n a l)

T o d e te r m in e th e fu n d a m e n ta l p e rio d N o f a p e rio d ic sin u so id , w e e x p re ss its fre q u e n c y /o as in (1.3.11) a n d can ce l co m m o n fa c to rs so th a t k a n d N a re re la tiv e ly

p rim e T h e n th e fu n d a m e n ta l p e rio d o f th e sin u so id is e q u a l to N O b s e rv e th a t a

sm a ll c h a n g e in fre q u e n c y can re su lt in a large c h an g e in th e p e rio d F o r e x a m p le ,

a re u n iq u e A n y se q u e n c e re su ltin g fro m a sin u so id w ith a fre q u e n c y M > n , o r

| / | > j , is id e n tic a l to a s e q u e n c e o b ta in e d fro m a sin u s o id a l sig n a l w ith fre q u e n c y

\co\ < n B e c a u se o f th is sim ilarity , w e call th e sin u so id h a v in g th e fre q u e n c y M >

tt an alias o f a c o rre s p o n d in g sin u so id w ith fre q u e n c y jwj < n T h u s w e re g a rd

fre q u e n c ie s in th e ra n g e — tt < a> < tt , o r — 1 < / < 1 as u n iq u e a n d all fre q u e n c ie s

|o>[ > t t , o r | / | > ~, as aliases T h e r e a d e r sh o u ld n o tic e th e d iffe re n c e b e tw e e n

d isc re te -tim e sin u so id s a n d c o n tin u o u s-tim e sin u so id s, w h e re th e la tte r re s u lt in

d istin c t signals fo r £2 o r F in th e e n tir e ra n g e —o c < £2 < oc o r —o c < F < oc.

B 3 Th e h ighe s t rate o f oscillation in a disc rete-tim e s i n u s o i d is attained w he n

to — 7i (or cu = — tt ) or, eq u iv a le n tly , f — \ (o r f = — \ )

T o illu stra te th is p r o p e rty , le t u s in v e stig a te th e c h a ra c te ristic s o f th e sin u ­

so id a l signal se q u e n c e

x ( n ) = cos c^o n

w h en th e fre q u e n c y v a rie s fro m 0 to tt T o sim p lify th e a r g u m e n t, w e ta k e valu es

o f ( l > o = 0, 7t /8, tt/4 , jt/2 , n c o rre sp o n d in g to / = 0, 5, w hich re su lt in

p e rio d ic se q u e n c e s h a v in g p e rio d s N = oc, 16, 8, 4, 2 as d e p ic te d in Fig 1.13 W e

n o te th a t th e p e rio d o f th e sin u so id d e c re a s e s as th e f re q u e n c y in c re a se s In fact,

we can se e th a t th e ra te o f o sc illa tio n in c re a se s as th e f re q u e n c y in crease s

,, xin)

T o se e w h a t h a p p e n s fo r tt < ioq < 2tt we c o n s id e r th e sin u so id s w ith fre q u e n c ie s a>\ = a>(> a n d 0 J 2 — 2 n — ojq N o te th a t as co\ v aries fro m tt to 2n a>z

v a rie s fro m ir to 0 it can b e easily se e n th a t

= A cos co} n — A cos won

X 2 (n) = A cos uhn — A cos(27r — coo)n (1.3.14)

= A c o s (— coqii ) — x \ (h)

H e n c e ur± is an alias o f w\ If w e h a d u se d a sin e fu n c tio n in ste a d o f a co sin e fu n c ­

tio n , th e re su lt w o u ld basically be th e sa m e , e x c e p t fo r a 180' p h a s e d iffe re n c e

b e tw e e n th e sin u so id s A](«) an d x i ( n ) In an y case, as we in c re a se th e re la tiv e fre q u e n c y coo o f a d isc re te -tim e sin u so id fro m tt to 27r its ra te of o sc illa tio n d e ­

creases F o r coo = 2 tt th e re su lt is a c o n s ta n t signal, as in th e case fo r oju = 0

O b v io u sly , fo r co{) = tt (o r f = k) w e h av e th e h ig h e st ra te o f o sc illatio n

A s fo r th e case o f c o n tin u o u s-tim e sig n als, n e g a tiv e fre q u e n c ie s can b e in ­tro d u c e d as w ell fo r d isc re te -tim e signals F o r this p u rp o s e w e use th e id e n tity

Since d isc re te -tim e sin u so id a l signals w ith f re q u e n c ie s th a t a re se p a ra te d by

an in te g e r m u ltip le o f 27r a re id e n tic a l, it follow s th a t th e fre q u e n c ie s in any in te rv a l

co] < a> < co\ + 2 tt c o n s titu te all the ex istin g d isc re te -tim e sin u so id s o r c o m p le x

e x p o n e n tia ls H e n c e th e fre q u e n c y ra n g e fo r d isc re te -tim e sin u so id s is finite w ith

d u ra tio n 2 n U su ally , we c h o o se th e ra n g e 0 < co < 2 n o r — tt < co < tt ({) < f < 1.

— 1 < / < | ) , w hich we call th e f u n d a m e n t a l range.

1.3.3 Harmonically Related Complex Exponentials

S in u so id a l sig n a ls a n d c o m p lex e x p o n e n tia ls p lay a m a jo r ro le in th e an aly sis o f

sig n als an d sy stem s In so m e cases w e d e a l w ith sets o f h a rm o n ic a lly related c o m ­

p lex e x p o n e n tia ls (o r sin u so id s) T h e s e a re se ts o f p e rio d ic co m p le x e x p o n e n tia ls

w ith fu n d a m e n ta l fre q u e n c ie s th a t a re m u ltip le s o f a single p o sitiv e fre q u e n c y

A lth o u g h we co n fin e o u r d isc u ssio n to c o m p lex e x p o n e n tia ls , th e sa m e p r o p e r ­ties c learly h o ld fo r sin u so id a l sig n als W e c o n s id e r h a rm o n ic a lly re la te d c o m p le x

e x p o n e n tia ls in b o th c o n tin u o u s tim e an d d isc re te tim e

Continuous-time exponentials T h e basic sig n a ls fo r c o n tin u o u s-tim e ,

h a rm o n ic a lly re la te d e x p o n e n tia ls are

sk (t) = ejkno' = e ll7TkFn' jt = 0 ± l ± 2 (1.3.16)

W e n o te th a t fo r e a c h value o f k, s^U) is p e rio d ic w ith fu n d a m e n ta l p e rio d

1 /( k Fo ) = Tp / k o r fu n d a m e n ta l fre q u e n c y kFo Since a sig n al th a t is p e rio d ic

w ith p e rio d Tp / k is also p e rio d ic w ith p e rio d k ( T p / k ) = Tp fo r an y p o sitiv e in te g e r

to S ectio n 1.3.1, Fo is a llo w e d to ta k e any v a lu e a n d all m e m b e rs o f th e set are

w h e re ck, k = 0, ± 1 , ± 2 a re a r b itra ry c o m p le x c o n s ta n ts T h e signal x a(t)

is p e rio d ic w ith f u n d a m e n ta l p e rio d Tp = l / f o , a n d its r e p re s e n ta tio n in te rm s

o f (1.3.17) is c a lle d th e F o u ri er series e x p a n sio n fo r x a (t) T h e c o m p le x -v a lu e d

c o n s ta n ts a re th e F o u rie r se rie s co effic ie n ts a n d th e signal sk (r) is c a lle d th e fcth

h a rm o n ic o f x (l(t).

Discrete-time exponentials Since a d isc re te -tim e c o m p le x e x p o n e n tia l is

p e rio d ic if its re la tiv e fre q u e n c y is a ra tio n a l n u m b e r, w e c h o o s e f Q — 1/A' an d we

d efin e th e sets o f h a rm o n ic a lly r e la te d c o m p le x e x p o n e n tia ls by

sk (n) = e j2* kf" \ k = 0 ±1 ±2, (1.3.18)

In c o n tra s t to th e c o n tin u o u s-tim e c ase, w e n o te th a t

sk+Nln) = e J7* n'k+N,/N = e ^ s k (n) = sk (n)

T h is m e a n s th a t, c o n s iste n t w ith (1.3.10), th e re a re o n ly N d istin c t p e rio d ic co m p lex

e x p o n e n tia ls in th e se t d e s c rib e d by (1.3.18) F u rth e rm o r e , all m e m b e rs o f th e set

h a v e a c o m m o n p e rio d o f N sa m p le s C lea rly , w e can ch o o se a n y co n se c u tiv e A'

c o m p lex e x p o n e n tia ls , say fro m k = no to k — no 4- N — 1 to fo rm a h a rm o n ic a lly

re la te d set w ith f u n d a m e n ta l fre q u e n c y /(, = 1 / N M o st o fte n , fo r c o n v e n ie n c e ,

we c h o o se th e set th a t c o rre sp o n d s to no = 0, th a t is, th e se t

re su lts in a p e rio d ic signal w ith fu n d a m e n ta l p e rio d N A s w e shall se e la te r,

this is th e F o u rie r se rie s re p re s e n ta tio n for a p e rio d ic d isc re te -tim e se q u e n c e w ith

F o u rie r co effic ie n ts {q} T h e s e q u e n c e sk (n) is c a lle d th e /tth h a rm o n ic o f x ( n ).

Example 1.3.1

Stored in the m em ory of a digital signal processor is one cycle of the sinusoidal signal

( 2nn

x ( n ) = sin I + 6

(a) D eterm ine how this table of values can be used to obtain values of harm onically related sinusoids having the same phase.

(b) D eterm ine how this table can be used to obtain sinusoids of the same frequency but different phase

Solution

(a) Let denote the sinusoidal signal sequence

( 2 yrnk xk( n) = sin I - !

V N This is a sinusoid with frequency f k = k / N which is harmonically related to

x{n) But x k{n) may be expressed as

1.4 ANALOG-TO-DIGITAL AND DIGITAL-TO-ANALOG CONVERSION

M o st sig n a ls o f p ra c tic a l in te re st, such as sp e e c h , b io lo g ic a l sig n als, se ism ic sig n als,

r a d a r sig n a ls, s o n a r sig n als, an d v a rio u s c o m m u n ic a tio n s sig n a ls such a s a u d io a n d

v id e o sig n a ls, a re an a lo g T o p ro c e ss a n a lo g sig n a ls by d ig ital m e a n s , it is first

n e c e ssa ry to c o n v e rt th e m in to d ig ital fo rm , th a t is, to c o n v e rt th e m to a se q u e n c e

o f n u m b e rs h a v in g fin ite p re c isio n T h is p r o c e d u r e is c a lle d analog-to-digital ( A / D )

c o n v e r s i o n , a n d th e c o rre sp o n d in g d ev ices a re c a lle d A / D co nverters ( A D C s )

C o n c e p tu a lly , w e view A /D c o n v e rsio n as a th r e e -s te p p ro c e ss T h is p ro c e ss

is illu stra te d in Fig 1.14

1 S a m p l i n g T h is is th e c o n v e rsio n o f a c o n tin u o u s-tim e signal in to a d is c re te ­

tim e sig n al o b ta in e d by ta k in g “ s a m p le s’" o f th e c o n tin u o u s-tim e sig n al at

d isc re te -tim e in sta n ts T h u s, if x a (t) is th e in p u t to th e sa m p le r, th e o u tp u t

is x a ( n T ) = x ( n ) , w h e re T is called th e s a m p li n g interval.

2 Q u a n t i z a t i o n T h is is th e c o n v e rsio n o f a d isc re te -tim e c o n tin u o u s-v a lu e d

A/D converter

01011

"7

Figure 1.14 Basic parts of an analog-to-digital (A /D ) converter.

signal sa m p le is re p re s e n te d by a v a lu e se le c te d fro m a fin ite set o f p o ssi­

b le values T h e d iffe re n c e b e tw e e n th e u n q u a n tiz e d sa m p le x ( n ) a n d the

q u a n tiz e d o u tp u t x q (n) is c a lle d th e q u a n tiz a tio n e rro r.

3 Cod in g In th e co d in g p ro c e ss, each d isc re te v a lu e x q{n) is re p re s e n te d by a

6 -b it b in a ry se q u e n c e

A lth o u g h w e m o d e l th e A /D c o n v e r te r as a s a m p le r fo llo w e d by a q u a n tiz e r

an d c o d e r, in p ra c tic e th e A /D c o n v e rsio n is p e rfo rm e d by a sin g le d ev ice th a t

ta k e s x a (t) an d p ro d u c e s a b in a ry -c o d e d n u m b e r T h e o p e ra tio n s o f sa m p lin g a n d

q u a n tiz a tio n can be p e rfo rm e d in e ith e r o r d e r b u t in p ra c tic e , sa m p lin g is alw ays

p e rfo rm e d b e fo re q u a n tiz a tio n

In m an y cases o f p ra c tic a l in te r e s t (e.g., sp e e c h p ro c e ss in g ) it is d e sira b le

to c o n v e rt th e p ro c e ss e d d ig ital signals in to a n a lo g fo rm (O b v io u sly , w e c a n n o t listen to th e se q u e n c e o f sa m p le s re p re s e n tin g a sp e e c h signal o r se e th e n u m ­

b e rs c o rre sp o n d in g to a T V sig n a l.) T h e p ro c e ss o f c o n v e rtin g a d ig ital signal

in to an a n a lo g signal is k n o w n as digital-to-analog ( D / A ) c o n v e r sio n A ll D /A

c o n v e rte rs “ c o n n e c t th e d o ts ’" in a d ig ital signal by p e rfo rm in g so m e k in d o f in te r ­

p o la tio n , w h o se ac c u ra c y d e p e n d s on th e q u a lity o f th e D /A c o n v e rsio n pro cess

F ig u re 1.15 illu stra te s a sim p le fo rm o f D /A c o n v e rsio n , c a lle d a z e r o - o rd e r h o ld

o r a sta irc a se a p p ro x im a tio n O th e r a p p ro x im a tio n s a re p o ssib le , su c h as lin e a rly

c o n n e c tin g a p a ir o f su ccessiv e sa m p le s (lin e a r in te rp o la tio n ), fittin g a q u a d ra tic

th ro u g h th re e su c cessiv e sa m p le s (q u a d ra tic in te rp o la tio n ), a n d so on Is th e r e an

o p tim u m (id eal) in te r p o la to r ? F o r sig n a ls h a v in g a lim ited f r e q u e n c y c o n te n t (finite

b a n d w id th ), th e sa m p lin g th e o r e m in tro d u c e d in th e fo llo w in g se c tio n specifies th e

o p tim u m fo rm o f in te rp o la tio n

S am p lin g a n d q u a n tiz a tio n a re tr e a te d in th is se c tio n In p a rtic u la r, we

d e m o n s tra te th a t sa m p lin g d o e s n o t r e s u lt in a lo ss o f in fo rm a tio n , n o r d o e s it

in tro d u c e d isto rtio n in th e sig n al if th e sig n al b a n d w id th is fin ite In p rin c ip le , th e

a n a lo g signal can b e r e c o n s tru c te d fro m th e sa m p le s, p ro v id e d th a t th e sa m p lin g

ra te is sufficiently high to a v o id th e p ro b le m c o m m o n ly called aliasing O n th e

o th e r h a n d , q u a n tiz a tio n is a n o n in v e rtib le o r irre v e rs ib le p ro c e ss th a t re su lts in

Figure 1.15 Zero-ordcr hold digital-to-analog (D /A ) conversion.

th e a c c u ra c y , as m e a s u re d by th e n u m b e r o f bits, in th e A /D c o n v e rsio n p ro cess

T h e fa c to rs a ffe c tin g th e ch o ice o f th e d e sire d a c c u ra c y o f th e A /D c o n v e rte r are cost a n d sa m p lin g ra te In g e n e ra l, th e cost in c re a se s w ith an in c re a se in accu racy

a n d /o r sa m p lin g ra te

1.4.1 Sampling of Analog Signals

T h e re a re m a n y w ays to sa m p le an a n a lo g signal W e lim it o u r discu ssio n to

p e r i o d ic o r u n i f o r m s a m p l i n g , w hich is th e ty p e o f sa m p lin g u se d m o st o ften in

p ra c tic e T h is is d e s c rib e d by th e re la tio n

w h e re x ( n ) is th e d isc re te -tim e signal o b ta in e d by “ ta k in g sa m p le s ” o f th e a n a lo g sig n al x aU) e v e ry T se c o n d s T h is p r o c e d u r e is illu stra te d in Fig 1.16 T h e tim e

in te rv a l T b e tw e e n su c cessiv e sa m p le s is called th e sa m p l i n g p e r i o d o r s a m p le

in terva l a n d its re c ip ro c a l 1 / 7 — Fs is c alle d th e s a m p li n g rate (sa m p le s p e r se co n d )

o r th e s a m p li n g f r e q u e n c y (h e rtz ).

P e rio d ic sa m p lin g e sta b lish e s a re la tio n s h ip b e tw e e n th e tim e v a ria b le s t a n d

n o f c o n tin u o u s-tim e a n d d isc re te -tim e sig n als, re sp e c tiv e ly I n d e e d , th e s e v a ri­

a b le s a re lin e a rly r e la te d th ro u g h th e sa m p lin g p e rio d T o r, e q u iv a le n tly , th ro u g h

th e sa m p lin g ra te Fs — l / 7 \ as

A s a c o n s e q u e n c e o f (1.4.2), th e r e e x ists a re la tio n s h ip b e tw e e n th e fre q u e n c y

v a ria b le F (o r Q) fo r a n a lo g sig n a ls a n d th e fre q u e n c y v a ria b le / (o r co) fo r

d isc re te -tim e sig n als T o e sta b lish th is re la tio n sh ip , c o n s id e r an a n a lo g sin u so id al sig n al o f th e fo rm

x ( n ) = x a ( n T ) — o c < n < c c (1.4.1)

(1.4.2)