Digital signal processing computer based approach - sanjit k. mitra (2nd ed)

1. tor- Based Approach

Sanjit K. Mitra

2. DIGITAL SIGNAL PROCESSING
À Computer-Based Approach

Second Edition

Sanjit K. Mitra
Department of Electrical and Computer Engineering
‘University of Calfomia. Sana Barbara

‘McGraw-Hill

3. About the Author

Sanjit K. Mitra received his M.S. and Ph.D. in electrical engineering from the University of California,
Berkeley, and an Honorary Doctorate of Technology from Tampere University of Technology in Filan
After holding the position of assistant professor at Comell University uni 1965 and working at ATT
Dell Laboratories, Holmdel, New Jersey, until 1967, he joined the faculty ofthe University of Califor
Davis, Dr. Mitra then transferred tothe Santa Barbara campus in 1977, where he served as department
‘airman from 1979 to 1982 and is now a Professor of Electrical and Computer Engineering. Dr. Mita
as published more than 500 journal and conference papers, and 11 books, and holds 5 patents, He served
as President of the IEEE Circuits and Systems Society in 1986 and is curently a member of the editor.
Boards forfour journals: Mulidimensional Systems and Signal Processing; Signal Processing, Journal of
the Franklin Institute: and Automatiba, Dr Mitra has received many distinguished industry and sendeme
awards, including the 1973 EE. Terman Award, the 1985 AT&T Foundation Award ofthe American Soci
‘of Engineering Education, the 1989 Education Award of the IEEE Circuits and Systems Society. the 1989
‘Distinguished Senior U.S. Scientist Award from the Alexander von Humbold Foundation of Germany. he
1995 Technical Achievement Award ofthe IEEE Signal Processing Society, the 1999 Mac Van Valkenburs
Society Award and the CAS Golden Jubilee Medal ol the IEEE Circula & System Society, andthe TPE
Millennium Medal in 2000. He is un Academician of the Academy of Finlend, Dr. Mira is a Fellow of
{ie IEEE, AAAS. and SPIE and is a member of EURASIP and the ASEE.

4. Preface

‘The eld of digital signal processing (DSP) has seen explosive grown during the past tree decades,
35 phenomenal advances both in research and application have ben made. Fucling this grou have
een the advances in digital computer technology and software development. Almost every electrical and
computer engineering department in ths country and abroad now offers one or more courses in digital
Signal processing, wi the Hs course usually being offered at he senior level This book i rede for
3 cuo-semester course on digital signal processing for seniors or first-year praduato students. I is also
vain at a level suitable for self-study by the practicing engineer or sient

Even though the Art edition ofthis book was published bavely two years ago, based on the feedback
receive from professors who adopted ths book for their courses and many readers it Wa clear that a new
edition was needed 10 incorpore the suggested changes tothe contents, À number af new topics have
‘bee included inthe second edition. Likewise a mumber of topics thal are interesting but not practically
‘useful have been removed because of size limitations. It was alo Fel that more worked-0ut examples meto
needed 10 explain new and dificult concep

Te new topics included inthe second edition are: cakulation of total solution, ero input response.
zero site response. and impulse response of finite dicesional discrete time systems (Sections 2.6.1
26.3), correlation of signals and ls applications (Section 2.7), inverse syutems (Section 49), system
‘emification (Section 4.10, matched filter and is application Section . 1), sampling of bandpass signals
(Section 5-3, design ol highpass, bandpass, and bandstop analog filters (Section 3.5), effect of sumplerand
hold operation (Section 5.11). design or highpass, bandpass, and bandstop IR digital ter (Section 72),
design of FIR digital fiers with least mean-square cor Section 7.5), constrained least-square design of
FIR digital ler (Section 79), perfect reconstruction (wo-channel FIR ter banks (Section 10.9), conte
module L<hannel filter banks Section 10.1), spectral analysis of random signals Section 11.4), and
sparse antenna array design (Section 11-14), The topics that have been remaved from the fst elon
aro a follows: state-space representation of LT dicree-time systems from Chapter 2 signa flowegraph
‘epresenatin and state-space siuctures from Chapter 6, impulse nvanance method of IR filer desig
‘and FIR filter design based on the Frequency sanıpling approach from Chapter 7, reduction of produc
‘oundeoff errors from state-space sutures from Chapter 9, and voice privacy system from Chapter 11
The fractional sampling rate conversion using the Lagrange interpolation has been moved to Chapter 10.
Materials in each chapter ate nom organized more logical.

5. Pretace

‘A Key feature of this book i the extensive use of MaTLAw® used! examples that lustre he pro
ram's postertu capabihiy to solve signal processing probleuns. The book uses a three-stage pedagogical
Nructue designed t take full advantage of Matta and to avoid the pitfalls of a "cookbouk” approach 0
problem solving. Firs, each chapter begins by developing the essential theocy and algerthms. Second.
the material lusraed wath examples solved by hand cucultion. And third. solutions ae derived us
Mast an. Brom the beginning. Marian codes are provided with enough details 10 permit the students 10
repeat the examples on their computers. In addition to comemtiomal theoretical problems requring ana
Iyteal solutions cach chaper alo includes à large number problems requiring solution via MATLAB,
This book requires a minimal Knowledge of Martan. | believe students lam the intricacies of problem
Solving with MATIAR faster by using ted, complete programs, and then writing simple program to
Solve specii problems that ae included a the ends of Chapters 210 11

Because computer verification enhances the understanding ofa underlying eve and. in e fst
ten, a ace library of worked-ou! Mart. aw programs ar included inthe second edition. The original
Maras programs of th fist edito have been updated o run on the newer versions of MATIAS and the
Sienal Processing Toolbar. {n addition, new MATLAS programs aud code fragiient have been added in
Ai edition, The reader can run these programs to verily the results included inthe book. Allogether there
are 90 Matt. programs inthe text that have been tested under version 5.3 of MAtLAM and version $2
ofthe Signal Processing Toolbox. Some uf the programs listed in this book are not necessarily the Fastest
with regard to their execution speeds, nor are they the shoes. They have been writen for maximum
lai without detailed explanations,

A second attractive feature of this book isthe inclusion of 231 simple but practical examples that
expose the reader to real-life signal processing problems which bas been made posible by the use Of
compate in solving practical design problems. This book also covers many topics of eurent interes
0! normally (nd in an upper division tex. Additional topic ar also introduced he fender through
problems atthe end ofeach chapter. Finally the book conclues witha chapter hat focuses on several
important, practical applications of digital signal processing. These applications are easy to follow and do
or require knowledge of other advanced level courses

“The prerequisite fer this book isa JunorIevel course in linear continuous-time and disret-cime
em, whichis usually required in most universes. À minimal review of near systems and rnstorms
is provided inthe tex, und basic material rom linear system theory re included, with importan mate als
summanzedintables. This approach permis the inclusion. more advanced materiale without significan
increasing the length af the book.

‚The book sdividedinto chapters. Chapter 1 presents an introduction othe fc of signal processing
and provides an overview of signals and signal processing methods, Chapter 2 discusses the time-domain
representations of diseretestime signals and discrete-time systems ax sequences of numbers and describes
classes of such signals and systems commonly encountered. Several base discrete me signals that play
important roles in the time-domain characterization of urbrary discreto time signals und discrete-time
system are then introduced. Next, a number of basic operations to generate ether sequences from one OF
more ssquences ate described. A combination of these operations i also used indeveloping a diserte time
system, The problem of representing a continuous time signal by à discrete-time sequence is examined
fora simple ease, Final, the time-domain charaterzation of iserek-time random signals is discuss.

‘Chapter is devoted the transform-comain representations fa discrete me sequence. Specially
discussed are the discrete-time Fourier transform (DTFT), the discrete Fourier transform (DFD) and the

transform, Properties ofeach ofthese transforms are reviewed and a few simple applications oulined,
‘Th chapter ends with a discussion of the wansformm domain representation of a random Signal

This book concentrates almost exclusively om the linear time-invariant discreto time systems, and

MTS re ie

6. Proface w

‘Chapter discusses heirtrunsform.domain representations. Speeile properies of such ransformtomein
representation are investigated, and several simple applications are considered,

(Chapter 5 is concemed primarily with he discrete-time processing of comimuou time signal. The
conditions fr discrete-time representation ofa bandlimied cominuous im signal under a) sampling
Sod As exact recovery from the sampled version ate fs derived. Several interface circus are used for he
iserei-ume processing of continuoastime signal, Tarot these circuits are the an-aiasing filter and the
‘ccomsiruction ter, which are analog lowpass fiers. As a eau brief review of the basic theory behind
some commonly used analog ier design methods is included, and heir us ilustated with Marıan
Other merce circuits discussed inthis chapter are the sample and hold circuit the analog. digital
‘converter. and the digal-te-analog converter,

A sera representation using intrcomnected asie building blocks iste fis step in the hardware
er software implementation of an LTT digital er. The structural representation provides the relations
‘between some pertinent intemal variables with the input and the Output, which in um provides he Keys
10 te implementation, There are various forms ofthe structural representation of a digital filter and to
Such representation are reviewed in Chapter 6 followed by a discussion of some popular scheme forthe
‘calzaton of rea causal HR and FIR digital titer, In addon, i describes a method or ral ation ot
HR ta er structures that canbe used for he generation ofa par of orthogonal snusondal sequences

Chapter 7 considers the distal iter design problem. Fut, i discusses the issues associated with
the ie design problem, Then it describes the most popular approach lo IIR filter design, based on the
‘conversion a prototype analog transfer funcionto digital trame function. The spectra transformation
‘ one type of IIR taster Funcion ito another type I disomocd, Then a vey simple approach lo FIR
fer design s described. Finally the chaper reviews computer-aided design of bath UR and FIR digital
Hits. The we of MALAS In digital ler design i iustrated

Chapter is concerned withthe implementation aspect of DSP algorithms, Two major issues con
<ceming implementation are discussed fst, The satire implementations of dighal fering und DFT
algorithms on a computer using MATLAS ae reviewed to ustrate the main points. This is followed by a
“discusion of various schemes fr he representation of nunber and signal variables on digital machine
‘which is basic o the development of methods fo the amas of inte wordlengih ches considered i
Chapter 9. Algorithms used to implement addition and multiplication, the (wo key amie operators
in digital signal processing, we reviewed nex, along with operations developed to hundle overiow. Fie
mal, ie chapter outlines two general methods forthe design an implementation of tenable digital lies.
followed hy a discussion of alporthm forthe approximation of certain special funcione

‘Chapter 9 devoted o analysis ofthe effect ofthe various sources of quantization rom: describes
sets that are es sensitive to these ceci. Included here are discussions on the coc of coccion!
quantization

Chapter 10 discusses multirate discrete-time systems with unequal sampling rates at various pers
The chapter includes a review of he base concepts and properties of Sampling rate stration, design 04
\scomaton and Interpolation digital lies, and alate Hier bunk design

“The final chapter. Chapter 1, reviews a few simple practical applications of digital signal processing
10 provide a glimpse of ts potenti

‘The materials his hook have been used in a 1wo=quaner course sequence on digit signal processing
a the University of California, Santa Barbara, and have bec extsnively tested I the classe for over
10 years. Basically, Chapters 2 through 6 form the basis of an upper-division course. while Chapters 7
‘through 10 form the basis ofa graduae-fevel cour.

Many topics included in ih text can be omited From class discussion, depending on the coverage of
‘ther courses in the curriculum. Because a seniorjovel course on random signal amd systems b rege
fal lectrice and computer enpineering majors mont universities. materials in Sections 27. 8.10, and
4 com be excluded rom an wpperalivision course on dipl signal processing. However, these topics
are important ithe analysis of wordlength effect discussed in Chapter 9, and readers no Tailiae wi

7. xi Pretace

Anis subject ar encouigd 0 rviow thse sections before reading Chapter 9. Likewise. Section 8.4 on
‘umber represen nd Section 85 on met operations an may Be om fom cession
mo sen eng a Gl signal processing courte usually take course on digit hardware

“Piste contain 21 examples, 90 MATE programs, 64 problems and 186 Maruan exercises

earth been mado asar he aces ol materi ini book. lading ie Maras
programe. 1 werd, however, appeite rent Dunga to my Meco sy srs tat ay apa in
{he penca vero fr ran eyond my consol and thao he pusher These crs and ay thet
Women can be commit ome by mi adress: mitrageceased

Final: have ben parc frite to ave had the open to work wih the ousanding
students who were in my esearch group daring my teaching arc, which spans over 35 yeas, T hve
chi mens, and comin lodos, both roma snd pascal, om my easy and
asocian with thm, aná thom deat ts Pook.

Sanjt K. Mitra

8. Proface avi

Acknowledgements,

“The preliminary versions of te complete manuscript fr te frst edition were reviewed by Dr. Hole
Bab of the University of Zagreb, Croatia: Dr. James F Kaiser of Duke University: Dr. Wolfgang FO
Mecklenbräuker ofthe Technical University of Vienna, Austr and De. PP. Vaidyanathan of the Califo
Insite of Technology. A later version was reviewed by Di. Roberto H. Bambmerger of Microsoft Dr.
CChusies Boumann of Purdue University; Dr Kevin Buckley of the University of Minnesota; Dr. Jon A.
Flemming ofthe Texas AM University: Dr. Jerry D. Gibson of the Souther Methodist University Dr
John Gowy of Clemson University: Dis James Haris and Mahmood Nahv of the California Polytechnic
‘University, San Louis Obispo: De Yih-Chyun Jeng of Portland State University; Dr Troung Q-Neyucn of
Boston University: and Dr Andreas Spanias of Arizona State University. Various par ofthe manuscript
were reviewed by Dr. C. Sidney Burrs of Rice Univers Dr. Richard V. Cox ofthe AT&T Laboratories
(Dr lan Galton of the Univers of Califomia, San Diego; Dr. Nikil S. Jayant of dhe Georgia Institute
of Technology: Dr. Tor Ramstad of the Norwegian University of Science and Technology. Trondheim,
Norway. Dr. B. Ananth Shenol of Wright Site Universiy: Dr. Hans W. Schüssler of the University of
Erlangen Nuremberg, Germany; Dr. Richard Schreier of Analog Devices and Dr, Gabor C. Temes of
Oregon State Universi

Reviews forthe second edition were provided by Dr. Winser E, Alexander of North Carolina State
University: De. Sobail A. Dianat of he Rochester Institute of Technology: Dr. Sulush Dutta Roy of he
cian Insite of Technology, New Delhi, Dr. David C: Farden of North Dakota State Univesity: Dr
Abdulnasir Y. Hossein of Sultan Qaboos Univesity, Sultanate of Oman: Dr. James E. Kaiser of Duke
University; Dr. Ramakrishna Kakarala of the Agilent Laboratories; D. Wolfgang FG. Mecklenbräuker of
the Technical University of Vienna, Austria; De Antonio Ortega of the University of Southern California:
Dr Stanley J. Reeves of Auburn University: Dr George Symos of he University of Maryland, College Par
‘and Dr. Gregory A. Wornel ofthe Massachusets Institute of Technology. Various pats ofthe manuscript
{or te second edition were reviewed by Dr. Dimitris Anssassiou of Columbia University: Dr Rajendra
K Arora of the Florida State Universi: Dr, Ramdas Kumaresan ofthe Univesity of Rhode Islan, Dr
Upamanyu Madhow of the University of California, Santa Barbara; De. Urbashi Mita and Randy Moses
YO io State University Dr, Ivan Selesnick of Poyteehnie University, Brooklyn, New York: and Dr Gabor
©. Temes of Oregon Sale Univer

Thank al of them for their valuable comments, which have improved the book tremendously.

Many of my former and present rescoeh student reviewed various portions ofthe manaseipt of both
tions and tested a number of the MarLas programs, In particular, I would like 10 thank Drs. Charles
1D. Creuser, Rajeev Gandhi, Michael Lightstone, Ing-Seng Lin. Luca Lucchese, Debargha Mukherjee,
Norbert Strobel, and Stefan Thumhoter, and Messrs. Serkan Hatipoplu, Zhai He, E Leipnik, Michael
Moore. and Mylene Queiroz de Faria, Lam also indebted al former students in my ECE 158 and ECE.
258A classesat the University of California, Santa Barbara, for ther feedback over the years, which helped
eine the book.

"thank Goutam K. Mitra and Alicia Rodriguez for the cover design ofthe book, Finally 1 thank
Patricia Monchon for her assistance inthe preparation ofthe LaTeX files ofthe second edition.

9. a Protece

Supplements

All Matta programs included in this book are available vis anonymous file transfer protocol (FTP) from
the Inemet site iplser vce uesb.edu inthe directory [pubimitra/Book_2e.

A solutions manual prepared y Rajeev Gandhi Serkan Hstipoglu, Zhai He, Luca Lucchese, Michael
‘Moore, and Mylene Queiroz de Faris and containing the solutions toa problems and MATLAD exerelses
is avaiable o Instructor from the publisher

‘Acompanion book Digital Signal Processing Laboratory Using MATLAB by che sutborisalzoavailable
from McGraw-Hill,

10. Contents

Preface xiii

Signals and Signal Processing 1
1.1 Characterization and Classification of Signals 1
12. ‘Typical Signal Processing Operations 3

13 Etampes of Typicat Signals 1

1 Typica Signal Processing Applications 22
1S Why Digital Signal Processing? 37

Discrete-Time Signals and Systems in the Time-Domain
2.4 Discrete-Time Signals 42

22 Typical Sequences and Sequence Representation 53

23 The Sampling Process 60

24 Diseree Time Systems 63

125 Time-Domain Characterization of LIT Diserete-Time System 71
28 Finite-Dimensional LTT Discrete-Time Systems 80

27 Comelation of Signals 6R

28 Random Signals 94

29 Summary 105

210 Problems 106

211 Marian exercises 115
Discrete-Time Signals in the Transform-Domain 117
3.1 The Discrete-Time Fourier Transform 117

32 The Discrete Fourier Transform 131
33 Relation between the DTFT and the DFT, and Their verses 137.
34 Diserete Fourier Transform Properties 140

2S Computaion of the DFT of Real Sequences 146

36 Linear Convolution Using the DFT 149

37 TheseTranaform 155

AR Region of Convergence of Rational 2-Transform 159,

39 Inverse Transform (67

3.10 2’Transfom Propenies 173

311. Transfom Domain Represeatations of Random Signals 176

a

11. an
345
BM

LIT Discrete-Time Systems in the Transform-Domaín

Summary 179
Problems 180
Marian Exercises 199

a9

34 Finite-Dimensional Discrete-Time Systems 203
32 "The Frequency Response. 208

33 The Transfer Functon 218

3a Types. Transfer Functions 222

45 Simple Digital Fikers 234

48 Allpass Transfer Function 245

42 Minimum-Phase and Maximum-Phase Transfer Functions 246
43 Complementary Transfer Functions 248

19 were Systems 25%

110 System identication 256

411 Digial TwosPaies 29

412 Algebraic Stability Test 261

4.13 Discrete-Time Processing of Random Signals 267
Ans Matehed Pilar 272

315 Summary 275

S16 Probleme 277

217 Marat Eereises 205

Digital Processing of Continuous-Time Signals 299
51 Introduction 299

5.2 Sampling of Comtinuous-Time Signals 300

33. Sampling of Bandpass Signals 310

54 Analog Lowpase iter Design 313

55 Design of Analog Highpass, Bandpass, and Bandstop Fikers
36 AmicAlising Vier Design 335

57 Samplesand Hold Circuit” 337

58 Analog wo Digital Comerter 338

59 — Dipita-to-Analog Converter 14

510. Reconsrucion Filer Design 348

SA Btfect of Sample-and-Hold Operation 351

512 Summary 352

313 Problems 363

514 MananBxercses 356

Digital Filter Structures 359

61 Block Diagram Represemation 359

62 Equivalent Structures 363

63 Dasic FIR Digital Files Strcraes 364

64 Basic HR Digital Filler Structure 368

5 374
66 Alba Files 376

67 Tunable MR Digital Flers 387

65 UR Tapped Cascaded Lattice Structures 389

203

Contents

12. 69 FIR Casados Latice Suchen 395
$10 Palit Alpass Realizaon of IR Transfer Functions 401
SA Digit Sine Conve Generuor 205
$12 Computational Complensy of Digital Bite Snctures 408
ER Summary ate
614 Problem 409
CS Maras Beeener 421

7 Digital Filter Design 423

Preliminary Considerations 423
Bilinear Transformation Method of IR Fier Design 430
Design of pas HR Dil Pers 435
Design o High. Bandpass, and Bandsto IR Digital Files 437
tal Frarsfomtions ot IR Fer a
FR Fier Design Based on Windowed Four Series 446
Compu Aie Design uf Digital Filer 460
Design of FIR Digital Fier with Least Mean Square Eror 468
Constrined Least Square Design of FIR Digital Piles 469
710 Dia Fer Design Usa Masias. 72
Tt Summary 497
332 Problema 4%
71% ManarExerehes S10

8 DSP Algorithm Implementation SIS
AI Bmicteues 315
82 Sucre Simulation and Vericaon Using MATLAB 523
#3 Computation o te Discrete Fourie Tam 935
54 Number Representation 852
ES Anthmene Operations 856
$6 Handling of Overton 362
87 TunableDigial Flo 562
88 Funcom Approcimation 36%

x Samnay EN
#10 Pre 572
AD Marian berne $81

9 Analysis of Fi

Wordiength Effects 583
“The Quintization Process and Erors 584
Quantization of Fixed-Point Numbers 58S
Quantization of Floating-Point Numbers 587
Analysis of Coefficient Quantization Elects 588
AD Conversion Noise Analysis 600.

Analysis of Arithmetic RoundO Errors 611
Dynamic Range Seating 614

Signal-o-Noise Ratio in Low Order IR Fi
LLow-Senstivty Digital Filters 629
Reduction of Product Round-Of Errors Using Error Feedback 635
Limit Cycles in JR Digital Pier 639

os

13. x Contents

9.12 Round-Of Errors in FFT Algorithms 646
913. Summary 649

9.14 Problems 650

95 Matta Exercises 657

10 Multirate Digital Signal Processing 659
10.1 The Basie Sample Rate Alteration Devices 660
102 Piers in Sampling Rate Alteration Systems 671
0.3 MulUstage Design of Decimator and inerpolator 680
104 Tue Polyphase Decomposition 684
105 Arbirary-Rate Sampling Rate Converter 690
106 Digital Filer Bankr 696
10.7 Nyquist Fiters 700
108 Two. Channel Quadrature Mirror Filter Bank 705
109 _ Perfect Reconstructon Two-Channel FIR Filter Banks 714
10.10 L-Chanel QMF Banks 722
10.11. Cosine-Modulated L-Channel Fier Banks 730.
10.12. Mulilevel File Banks 734
1013 Summary 738
10.14 Problems 739
1015 Marian Exercises 750

41 Applications of Digital Signal Processing 753
Dual-Tone Multfeequency Signal Detection 753

‘Spectral Analysis of Sinusoidal Signals 758

Spectral Analysis of Nonstationary Signals 764

Spectral Aralysis of Random Signals 77)

Mosical Sound Processing 780,

Digital FM Stereo Generation _ 790

Discrete-Time Analytic Signal Generation 794

Sabband Coding of Speech and Audio Signals 800

‘Feansmabiplexers 803

Discrete Malin Transmission of Digital Data 807.

Digital Audio Sampling Rate Conversion 810.

Oxersampling A/D Converter 812

Oversampliag D/A Converter 822

Spare Antonna Array Design 826

Summary 829

Problems 830

MaTLAR Exercises 834

Bibliography 837
Index 855

14. Signals
and Signal Processing

Signals play an important role in our daly ie. Examples of signals that we encounter Frequently are
Speech, music, picture, and video signals. A signal is a function of independent variables such as time,
distance poston, temperature, md pressure. For example. speech and musi signal represent ir pressure
as a function of time ata point in space. A black-and-white picture is a representation of light intensity
5 a function of to spatial coordinates. The video signal in television conti of 4 sequence of images,
‘alle frames, ad is funcion of three variables: 1wo spatial coordinates and time.

‘Mos signals we encounter ae generated by natural means, However, a signal can also be generated
synthetically orhy computer simulation. A signal caries information. and the ojective signal processing
isto extract useful information curried by the signal. The method of information extraction depends om the
{ype of signal and the nature ofthe informatio being eariedby he signal, Thus, ough speaking, signal
processing is concerned with the mathematical representation ofthe signal and the algorithmic operation
Comme out on it to extract the information present. The representation of the signal can be in terms of
basis functions inthe domain of te original independent varable(s) or can Bei ters of bass functions
ina transformed domain. Likewise, he information extraction process may be carried out inthe original
domain ofthe signa or in a ransformed domain. This book te concerned with iserete.'me representation
‘of signals and their discrete-time processing

This chapter provides an overview of signal and signal processing methods. The mathematical char-
sctesivation of he signal it rt dicosced along with cassificaion of signals. Next, some ypc signals
ae disused in detail and the typeof information carried by them is described. Then a review of some
<onmaniy used signal processing operations is provided and illustrated through examples. Advantages
and disidvantages of digital processing of signals are then discussed. Final, brief review of some
typical signal processing applications i included.

1-1 Characterization and Classification of Signals

Depending on the nature of the independent variables and the value of the function defining the signal,
various types of signals can be defined. For example, independent vaiables can be continuous or dis.
‘rete. Likewise, the signal ean either be a continuous o a dncrete Function ofthe independent variables.
Moreover the signal canbe either a real-valued function ora eomples-valed function.

A signal con be generated by a single source or by multiple sources. Inthe former case, it ia sealor
signal and inthe later case isu vector signal, often called 4 multichannel signal

A one-dimensional (1-D) signal isa function of single incependen: variable, A two-dimensional (2-D)
signal sa function of two independent varables. A multidimensional (M-D) signal isa function of more
‘than one Variable. The speech signal i an example of a 1-D signal where the independent variable is time
‘An image signal, such asa photograph, isan example of 2 2.D signal where the we independent variables
fre the two spatial variables, Each frame of a lack and white video signal isa 2-D image Signal that i

15. 2 Chapter 1; Signals and Signal Processing

‘variables, with cach frame oscuring sequentially at discrete instants of
video signal can be considered an ample of a three dimensional (3-D)
‘Sigma where the thee independent variables are the two spatial variables apd time. A color video signal
{sa hree-channel signal composed of tree 3-D signals representing the tree primary colors re, green,
and blue (RGB), For vansonision purpose, the RGB television signal Wansformed ito another type of
ree.channe! signal composed of a luminance component and two chrominance components,

‘The value of the signal ata specific value(s) ofthe independent variables) scaled its ample, The
variation ofthe amplitude as a function of the independent vaiabI(s is called its waveform:

or à 1-D signal, the independent variable is usual labeled as time. I the independent variable is
‘continous. he signal called à continuous-time signal. Ifthe independent variable discrete. the Signal
ls called a diseer time signa À continuous-time signa is deine at every instant of ime. On the other
and, a discree-ime signal is defined at discrete instants of time, an hence, is a sequence of numbers

‘A continuous-time signal with a continuous amplitude is usualy called an analog signal. A Speech
signa isan example ofan analog signal. Analog sigals ate commonly encountered in cut daly ie and re
‘usually generated by natural means. A discrete-time signal with diserete-valved amplitudes represented
by afinite number of digits is refered to as a digital signal. An example of a digital signal ste digi
music signal sored ina CD-ROM disk. A discreto time signa! with coninuous-salued amplitudes ts called
2 sampled-data signal. This lst type of signal occurs in susiched-capacitor (SC) circuits. A digital signal
is thus a quantized sampled- da sigas, Fall, acontinuo.s-time sgnal with discrete valved amplitudes
hos been referred to a a quantized boxcar signal[Sic93), Figure 1.1 strates the four types of signals.

‘The funcional dependence of signal in ts mathematica representation soften explicitly shown. For
a continuous-time 1-D signal. the continuous independent variable i usually denoted by , whereas for
a diserete-time I-D signal. the discrete independent variabe is usually denoted by n. For example. #()
represents a continuou time 1-D signal and (v{n]) represent a discrete-time 1-D signal. Fach member
(a). oF discrete time signal is called sample. In many applications, discrete-time signals generated
from a parent continuous tre signal by sampling the later at uniform intervals of time. Hf the discrete
instants of te at which a discrete-time signal ie defined are uniformly spaced, the independent discreto
‘arable can be normalized o assume integer values

Inthe ease uf « continuous-time 2 D signal, he to Independent variables are the spatial coordinates,
which are usually denoted by x and y. For example, the intensity ofa black-and-white image can be
expressed ss u(x. y). On ihe ether hand, a digitized image is a 2.D discrete-time signal and os ive
independent variables are discretized spatial variables often denoted by m and. Hence, «digitized image
can be represented os vom, m]. Likewise. a black-and-white video sequence is a 3-D signal ard can De
represente ass. ».1) where x and y denote the two spatial variables and denotes the temporal variable
time. A color video signal ı a vector signal composed of three signals representing the tree primary

13]

"bere à another classification of signals that depends on the certainly by which the signal can be
uniquely deserved. A signal ha can be uniquely determined by a well-defined process such asa oath
‘ematical expression or le, or table look up, is called a determinate signal. À signal (hat is generated
in random fashion and cannot be predicted ahead of ume w called à random signal, I this text we ae
Primary concerned wih he processing of discrete-time deterministic signals. However, since practical
Aiseretestime systems employ finite wordlengths for he string of signals and he implementation of the
Signal processiag algorithms, itis necessary to develop 100k for the analysis of finite wordlengtn effects
‘on the performance of discrete-time systems. To this end it has been found convenient to represen erat
perinen signal as random signals and employ statistical techniques for hei analysis.
Some typical signal processing operation ae reviewed in the following section.

wa.

16. 112, Typica! Signal Processing Operations:

Aine

i
i

lo ©

À re E
Pr an

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igor 1.4: (9) continues ie signal (0 à diga sgn à sample ct signal, and (4) 2 quantize boxcar
sa

1.2 Typical Signal Processing Operations

Various types of signal processing operations are employed in practice. In the case of analog signals
‘most signal processing operations are usually carried cut in the time-domain. whereas. inthe case of
discrete time signals. both time-domain and frequency domain operations are employed In cher case.
the desired operations az implemented by à combination of some le mentary operations, These operations
are also usualy implemented in real-time or nea relie, even though in certain applications they may
be implemented fie.

1.2.1 Elementary Time-Domain Operations

The thie mos basic time-domain signal operations ar sealing, delay, an addition. Scaling is simply the
‘multiplication ofthe signal bya positive ora aegative constant In thecase of analog signal, this operation
is usally called ampliarlo K the magnitude ofthe mulüplying constant, called gain, 15 greater than
‘ome. I the magnitude of the muliplying constant i ess than one, the operation is called attenuation
‘Thus. x() 5 an analog signa, the Scaling operation gererates a signal DU) = exit) where eis the
‘multiplying constant, Two other clementary operations are integration and differemiation. The integration
fan analog signa E) generates signal vf) = fa (2) de, while its diereniaion resis ina signal
Wey decide

‘The delay operation generates a signal tha is à élayed replica of he original signal. For an analog
ital) 9) = AG Ho) isthe signa obtained by delaying x() by the amount 19 which ix asumed 10
bea positive number. IF mis negative, bon it is an advance operation

17. 4 Chapter 1: Signals and Signal Processing

Many applications require operations involving two or more signals 10 generate a new signal. For
‘example. yit) = Kun + Kalt) = va) i the signal generated by the addition ofthe thee analog signals
SHU). 210), and x3(0). Another clementary operation i he product of two signal. Thus, he product of
{Wo signal (7) and 224) generates a signal y(2) = (200)

“The elementary operations mentioned above are also caried out on discrete-time signals and are
‘discussed in later pare of this lex. Next we review some commonly used comple signal processing
“operations that ae implemented by combining two or more of the elementary operations.

1.22 Filtering

(One of the most widely used complex signal processing operations is filtering, Fileing is used 10 pass
rain frequency componente in a signal thraugh the system Without any distortion and to Bock other
frequency components. The system implementing his operation is called flier, The range of frequencies
thats allowed to passthrough he filters called the patchand, and the range of frequencies that is blocked
by the filter is called the stopband. Various types 01 Alters can be defined, depending on the nature of the
filtering operation. In most cases, the filtering operation fr analog signals is inca and à described by |
como integral

si = [U me one am

where (2) isthe input signal and +) isthe output of he titer characterized by an impulse response A),

A lowpass filter passes all low-frequency components below a certain specified frequency Jo. called
the catof frequency. and blocks all high-frequency components above fe. A highpass filler passes all,
high frequency components above a certain cof frequency f. and blocks al low-frequency components
below Je. A bandpass filter poses all frequency components between two Guo! frequencies /.1 and Ja
Where fos < Je, and blocks al frequency componeris below the frequency Ja and above the frequency
‘fa. A bandsop filter blocks al frequency components between two col requencios Je and fa. an
passes all frequency components below the frequency /.; and above the frequency fer. Figure 1.209)
hows a signal composed of thee sinusoidal components of frequencies 50 Hz, 110 Fiz, and 210 Hz
‘respectively. Figure 1.2(b) o (e) shows the results ofthe above Four types offering operations with
appropriately chosen cutoff frequencies,

Abandsto filter designed to block a single frequency component is calle a notch iter. A muliband
liz has more than one passband and more than one sopband. A comb filter is designed to block
‘frequencies tat are integral mutuple of a low frequency,

‘A signal may pet corrupied unintentonally by an interfering signal called interference or noise, In
many applications the desired signal occupies alo frequency band from de to some frequency. fu Mz.
and ti cormpied by a high-irequency noise with frequency components above fy He with Ja > fi
In such case, the desired signal can be recovered from the noise corrupted signal by passing the liter
throug a lowpass ker with acuta frequency J. where fi fe < fi. À common source of noise is
Powerlines riditing electric and magnetic fields. The noise generated by powerlines appears as a 80-He
sinusidal signal coerupting the desired signal and can be removed by passing the corrupted signa rough
noch iter with a nach frequency a 60 Hx.

1.2.3 Generation of Complex Signals

‘As indicated carie, signal canbe real-valued or complex-valued. For convenience the formeris usually
called a eal signal white the laters called complex signal Al natural generated signals are real-valued.
In some applications, iis desiable 4 develop a complex signal fom areal signal having more desirable

"Torin cease pve ie eae SOE ie

18. ]

|

19. 6 ‘Chapter 1: Signals and Signal Processing

properties. A complex signal can be generated from a real signal by employing a Hilbert transformer that
Is characterised by an impulse response hut) given by [Pre94],(Opp82)

anton À an
Tita med oia ao iu) wh comin ine Feuer stor
cori Ym tents 2

rua [Tavera as

292) cet sue la Ti magia san fe gal eii en ya

SM ts eS dt, TO EU
SoC Pence Leash ne

zum = num KR. as

where X 9(J22) isthe portion of X(¿22)occupying tre positive frequency range and X) isthe portion
of X(/G) occupying the negative Frequency range. If (2) is passed through a Hilbert transformers
‘ouput $) given by the inear convolution of 41) with hy):

202 [ane nites as

‘The spectrum À 0/0) of (0 is given by the product ofthe continuous-time Fourier trancforms of x) and
Mirto, Now the continvous-tite Fourier wansform Hi (/0 of AA rio of Ey (1 2) Eiven by

een

As the magnitude and the phase of X(JQ) are an even and xd function, respectively, it follows from
Ea. (17) that (Hi also eal signal, Consider the complex signal y(t} formed by the sum of (0) and.
EU

70) = 0 + RO as
‘The signals x(2) and 4) are call, respectively, the in-phase and quadrature components of y(t). By
‘making use of Eqs. (1.4) and (1.7 in the continuous-time Fourier transform of y(t). we obtain

YU) = XU) + UD) =2X,49) as
In ter wor. e complex sign). called ama igual, as ny posite eqn componen

A Bock agra ers othe scheme forthe sal sina generation fom Lo
sc ige LS

1.2.4 Modulation and Demodutation

For transmission of signals overlong distances, a transmission media such as cable, optical Aber, or
the atmosphere is employed. Each such medium has « bandwidth that Is more suitable forthe efficient
transmission of signals inthe high requeney range, As a rem, forthe transmission of «low-frequency

20. 12. Typical Signal Processing Operations. 7

Figure 13: Generation ofa aly sign using a ibn andren.

signal over a channel, itis necessary to transformo signal toa high-fequency signal by means of a
‘modulation operation. At the receiving end, the modulated high frequency signal is demodslated, and
the desired low-frequency sigral is then extracied by Further processing, There are four major types of
‘modulation of analog signals: amplitude modulation, frequency modulation phase modulation. and pulse

tude modulation. OF thee schemes, amplitude modulation ix conceptwally simple and ic discussed
ere res, [Opp

In the ampinude modularion scheme, the amplitude o high-frequency sinusoidal signal A costar)
called the carrier signal, is red by the low-frequency bandlimbed signal tt), called the modulating
signal, generating a high-frequency signal, called the modulated signal, () according 10

210 = Ax cone, a0

“Thus, amplitude modutation can be implemented by forming the product ofthe modulating signal with
the carer signal. To demonstrate the frequency-transtation property ofthe amplitude modulation process,
let 0) = cos) where {is much smaller than the carrier frequency Do, Le. D << fy From
Eq (1-10) we therefore obtain

a)
4 cos GR + mn + eos (Re 20, am

x0

“Tus, the modulated signal y( i composed of two sinusoids! signals o frequencies A, + and 2, = Di
“which are close to a5 y has been assumed o be much smaller han the earner frequency O

Tcisinsrtive to examine the spectrum of y(). From the propeitic of Ihe continuous tine Fourie:
‘wansform it follows tht the spectrum FR) Ov is given by

rm = Sr uaa Beta an a

‘where X JR) IS the spectrum ofthe modulating signal (1) Figure 1. shows the spectra ofthe modulating
‘signal and that of the modulated signal under the assumption tha he carrier frequency I greater than
Am. the highest frequency contained in (7). As sn trom this igure, 3) ie now a Dandlimiie high
Frequency signal with a bandwidth 22, centered at Du

“The portion ofthe ampliude-modulated signal betveen and Np + Sa i called the upper sideband
Whereas the potion between , and 25 ~ a is calle the lower sideband. Because of the generation
‘of two sidebands and the absence of a Carrier component inthe modulated signa. the procen 1 called
‘double sidehand suppressed carrier (DSB SC) modulation

‘Tue demodulation of 34), assuming Sa > Mn, is carried out in tuo stages. Fis, the product of (4)
ith sinusoidal signal ofthe same frequency asthe caries formed. Ths results in

FU) = YU cos Ret = ARE) cod Nyt das

21. Chapter 1: Signals and Signal Processing

xm)
— fla — y “
@

nm

BR) 0,0,
@

Figure 14 (3) Spectrum oft modulating pal «(and ( spectrum ofthe modulated signal y. Farconvenience
bth ject ae Sa rel functions

a

200,9.

29,0,

Figure 1.5: Spectrum of he product fe module signal and ie carter

Wich can be reiten as

ret) = yin coset = Sn + Arte 008) ass

2

‘This resul indicates thatthe product signal i composed of the orginal modulating signal sealed by a
factor 1/2 and an ampliude-modulated signal witha carrier frequency 252. The spectrum RUS) of #4)
is as indicated in Figure 1.5, The original modulating signal an now be recovered frm r(¢) by passing it
‘Urough a lowpass Aller with a ut frequency 9% satisfying the relation fq < Se < 20, — Aa. The
output ofthe te is then a scald replica ofthe modulating signal

Figure 1.6 shows the block diagram sepresenations ofthe amplitude modulation and demodulation
schemes. ‘The underlying assummcion in he demodlation proces outlined above is that sinusoidal signal
“dental to the cartier signal can be generated a he ceiving end. In general, ii fut ensure
(hat the demodulating sinusoidal signal has a frequency identical otha of the care ll he time. To get
around this problem, in the transmision of ampliude-modulated radio signal, the modulation procesa
‘is modified so hat the transmitted signal includes the carrier signal, This le achieved by redefining the
“uplitude modulation operation as follows:

YO) = ALI + meto costa), das

22. 41.2. Typical Signal Processing Operations 9

EB ur
Ce nl cs a

Ac: con:
@ a

igure 16: Schematic presentations ofthe amplitud modulation and demodulation sches () malt and
emos

A sinusoidal modsltng siglo frequency 20 Hz, nd (9) module carer Wh acai regucney
1 40 2 based om the DSD modo.

where m isa number chosen o ensure tat [1 + mo) is positive Fr ll. Figure 1.7 shows he waveforms
oF a modulating sinusoidal signal of frequency 20 He and the amplitude-modulaed carrier sbained ac.
ording to Eg. (-15) for a carer frequency of 400 Hz and m = 0.5. Note thatthe envelope of ie
‘modulated carriers essentially the waveform of he modalsting signal. As here the camier I NO present
ithe modulated signal. the proces is called simply double-deband (DSB) modulation. Atthe receiving
nd. Ih cater signal is separated Art and then used for demodulation,

1.2.5. Multiplexing and Demultiplexing

For an efficent utilization of a wideband transmission channel, many narrow bandwidth lov-frequency
signals are combined to form a composite wideband signal tht i wansmited as a single Signal, The
process of combining these signals is called mulplexing whichis implemented 1 ensure that a replica of
‘he orginal arrow ‘bandwidth low-frequency signals can be recovered at the receiving end. The recovery
proces is called demultiplexing

One widely used method of combining different voice signals in telephone communication system is
¡be requenci-division multiplexing (FDM) scheme Cou. (Opp83). Here, each voice signal typical
Bandtimited to alow frequency band of Width 207, i frequeney-transtated ino a higher frequency band
using the amplitude modulation method of Eg. (I-10). The carter frequency of adjacent amplitude
modulate signals i separated by So, with De > 202 o ensure that there ls mo overt in the spectra of
the individual modulated signals after they are added to form a baseband composite signal, This signal is
‘hen modulated onto the main cartier developing the FDM signal and transmitted. Figure 1.8 illustrates
the Frequency division multiplexing scheme.

23. 10 Chapter 1: Signals and Signal Processing

fh. bh.
mA
MAA.
a

Figure LA. lhsrauonof Be frequency division mulipleing operation (a) Secta of hee fo-frequeny signals
cn) speci he modus Compost ral

ww vo
Ze pan

tome pn
Ales

Figure 19: Single sideband modulation scheme employing a ibe stores

‘Atte receiving end, he composite baseband signal i Ars derived from the FM signal by demo
ulation. Then each individual frequency-translate signa is rst demultiplexed by passing the composite
Signal though 4 bandpass Aller witha center frequency of identical value as that ofthe corresponding.
amit frequency and a bandwidth slighily greater than 202. The output ofthe bandpass Aller i theo
‘demoduated using he method of Figure 1.(b) to recover a cal replica ofthe orginal vice signal

the case of the conventional amplitude modulation, as can be seen from Figure 1.4, the modulated
signal has a bandwidth of 20%, whereas the bandwith ofthe moduleing signal Is Sm; To increase the
capacity of the transmission medium, a modified form of the amplitude modelación Ts often employed
in which either the upper sidebaod or the lower sideband of the modulated signa is tansmited. The
Corresponding procedure is alle single sideband (SSB) modulation to distinguish it from the double.
Sideband modulation scheme of Figure GG.

‘One way to implement single-sideband amplitude modulation is indicated in Figure 1.9, where the
Wilbert transformer is defined by Eg. (1.6). The spectra of pevtinent signals in Figure 1.9 are shown in
Figure 1.10.

24. 1.2, Typical Signal Processing Operations “

CAMINA

o ©

igure 110. Spectra of prinen signals in Figure 19

1.2.6 Quadrature Amplitude Modulation

‘We observed arlier that DSB amplitude modulation i half as efficient as SSB amplinude modulation with
retard o utlization of the spectrum. The quadrature amplitude modulation (QAM) method uses DSB.
‘modulation to modulate two diferent signale so that they both occupy the same bendwith; thus QAM
‘only takes up as much bandwidth as the SSB modulation method. To understand the basic idea behind
the QAM approach, le 1) and 17(1) be to bandlinuted low-frequency signals wi à bandwidth Of Oy
as indicated in Figure LA). The two modulating signal are individually modulated by the wo camier
Signals A cos(Sqr) and À sini), respectively, and ate summed, resulting in a composite signal y(7)
nen by

MO = Any EE cosy) + Asa) sini) a0)
[Note that the two cater signals have the same carier frequency A but have a phase difference of 90°. In.
tetera. the carter À cou ar) called the in phase component and the camer A SGA) 5 called he
‘quadrature component. The spectrum Y1/2) ofthe composite signal 4) #8 now given by

YUM) = À GR A) + x GR + a)

HERAN 2 GE + D am
nd is seen w occupy the same bandwidth us the modulated signal obtained by a DSB modulation,

To recover he original movulating signals, the composi signa is multiplied by bath the in-phuse ond
‘the quadrature components of e carrier Separately. resulting in signals

TU) = yo.
CORNE

Sefstituing y e) from Ex, (1.16) in Eg, (1-18), we obtain after some algebra

am

I = BR + Er con 22.1) + Fra) RR. am
ato) = $1200 + Jarl sin) — $60) con).

Lowpass filtering of (1) andz5(1) by Bites with a utof at Da yields the two modulating signal, Figure
1.1 ws he Block dagram representations ofthe qua amplitude adulation and demoduaon

25. Processing

12 Chapter 1: Signals and Signe

wg

do) o

Figure 1.1: Schematic representations of the quads amplia modulation and demotion heres: (a
la and (br emo

Asin the caw ofthe DSB suppressed carrier modulation method, the QAM method aso requires tthe
reveiver an exact replica ofthe carter signal employed in the transiting end for accurate demodulation.
Its therefore not employed in the direct transmision of analog signals, but finds applications in the
leansmission of disretesime dats sequences and in the transmission of analog signals converted into
discrete-time sequences by sampling and analog-oigital conversion.

1.2.7 Signal Generation

‘An equally important pa of signal processing is symthetic signal generation, One of the simplest such
Signal generator is device generating a sinusoidal signal, called an osciltor. Such adevice x an integral
part of the amplitude-mdulation and demodulation system described inthe previous twa sections. also
has various other signal processing application

‘There are applications that require the generation of other types of periodic signals such as square
waves and triangular waves. Certain types of random signal with a spectrum of constant amplitude for
all Frequencies, called white noise, often And applications in practice, One such application, i in the
generation of discrete-time syn speech signals.

1.3. Examples of Typical Signals?

Te betr understand dhe breadth ofthe signal process
some typical signals and their subsequent processing

5 ink, we now examine a number of examples of
al applications.

Electrocardiography (ECG) Signal

‘The cecrical activity of the ears represented by the ECG signal (Sha. A typical ECG signal trace
is shown ia Figure Y 120}, The ECG trave is essentially a pariodic wavetonm. One such period of the
ECG waveform as depicted in Figure 2.120) represents one cycle ofthe blood transfer process fom the
in dhe sen, This par of the waveform is generated hy an electrical Impulse originating tthe
simoatnal mode in the sight atrium ef the heart The Impulse causes contraction ofthe ata, whic forces.
the blow in each arium to squeeze into i corresponding ventricle. The resulting sigral called the
Pare. The atioventicular node dlays the exciation impulse until the blood transfer from the strato
the vemnctes I completed, resulting in the P- interval ofthe ECG waveform, The excition impulse
‘Men eauses contraction OF the ventricles, which squeezes Ihe blood ino the arteries, This generate the

"Si stg ste ape ro Hand for Pe Sat Pc Sa lo a anes F Re o),
som iy Se antes yeoman oh Wey Se,

26. 1.3. Examples of Typical Signals 13

Matas

oor OF Os 04 Os 0
‘Seconds

©
Figure 1.2: (a) typical ECO usc, ad () one cycle of an ECG waveform,

QRS par of the ECG waveform. During this phase the atria are relaxed and filed with blood. The T-wave
ofthe waveform represents the relaxation ofthe vencles, The complete process I repeated penodicaly,
generating the ECG trace

Each portion ofthe ECG waveform cures arios type of information for the physician analyzing a
paticatshear condition [ShaB 1), Forcxample, the amplitude and ng ofthe Pand QRS portions indicate
the condition ofthe cardiae muscle mass, Loss of amplitude indicates muscle damage, whereas increased
amplimde indicates abnormal heart rates. Too long à delay in the auioventicular node i indicated by

‘very Jong P-R interval. Likewise, blockage of some oral ofthe contraction impuises is retested by

Intermittent synchronization between the P-and QRS-waves, Most ofthese abaormaliies can be tcated
With various drugs, andthe effectiveness ofthe drugs can again be monitored by observing the new ECC.
‘waveforms taken afte the drug treatment

In Practice, here are varios types of externally produced ardíics that appear in the ECG signal
(Tomi), Unless these interferences are removed, itis dial fora physician o Makes correct diagnosis
AA common source of noise is the 60-He power lines whose radiated electric and magnetic fields ae
soupled o the ECO instrument trough capacitive coupling and/or magnetic induction. Other sources of
intererence ac the electtomyographic signals that are the potemials developed by contacting muscos
‘These and other interferences can be removed with careful shielding and signal procesting lecinigues,

27. 14 Chapter 1: Signals and Signal Processing

A

A EN EN nent

A A one A Np A

A ER

PA
Figure 113: Mule EEG signal race

Electroencephatogram (EEG) Signal

“The summation of the electrical activity caused by the random firing of bilios of individual neurons in
the bain is represented by the EEG signal (Coh86, (Tomb. In multiple EEG recordings, electrodes are
iced at various positions on the sealp with two common electrodes placed on the carlohos, and potential
UiTerences been the various slectrodos are recorded. A typical bandwidth ofthis type of EEG ranges
foun 0.5 to shout 100 Ha. with she amples ranging from 2 10 100 mV. An example of muluple EEG
tonces is shown in Figure 1.13

Both frequency-domain and time domain analyses ofthe BEG signal have been used forthe diagnosis
ot epilepsy sleep disorder. psychiatric malfunctions, etc. To this end, the EEG spectrum i subdivided
Into ihe following tive bands” (1) the celta range, occupying the band trom 0,510 4 Hr; (2) the theta range
‘ecupying the band from 4 10 8 Hz; (3) the alpha range, occupying the band from 8 0 13 Hz: (4 the Beta
range, occupying the band rom 13 o 22 Hz; and (3) the gamma range, oceupying the band from 2 10 30,
ñ

“The deta wave is normal in the BEG signals of children and sleeping adits, Since ti not common
in aer adults, is presence indicates cenain brain diseases, The thea wave is usually found in children
even though it has been observed in alert adults. The alpha wave i common in all normal humans und is
more pronounced in relaxed and awake subject with closed eyes. Likewise, the beta activity ds common
in normal ads. The EEG exhibits raid, low-voltage waves, called rapid-eye-movemen (REM) waves.
ina subject dreaming during sleep. Otherwise. in a sleping subject, te EEG contains bursts ol alpha- Me
waves, called seep spindles. The EEG of an epileptic patient exhibits various types of abmormalitis,
depending on the type of eplepss thst ie caused by uncomtrcled neural discharges

Solsmic Signals

‘hese types of signals ar caused by the movement of rocks resulting fom an earthquake, a volcanic erup-
‘ion, or an underground explosion [BOIS3], The ground movement generates elastic waves hat propapate
‘through the body of the earth inal directions fom the source of movement, Three basic types of las
waves are generated by the earth movement, Two of thee waves propagate through the body ofthe cath,
‘one moving faster with respecto dc uber. The faster moving wave is called he primary or P-wave, wi

28. 13. Examples of Typical Signals 15

ine lower moving nc cal secondary or Sacar Th hind ype of wave is known a the surface
ware, which moves along the round surface, These scie waves ae comerte into elects signals
À à amograph and are evened on ap char recorder ora magno tapo
Beach ne die og tire of round CE, smog uly cans of
thee sepa recording instrument that provide information aba the movement nthe te horizontal
alzados aná ae arca icon snd Beaop tee code indicted a ue LA, Each sch
seconde aove-dncnsond signa. From ie record sig Ls possible wo determin the mata o
the eathguake or nuclear explosion ane cation al source ofthe original can movers
Same signal o pay an importan ol inthe geophysics! exploran for land ps TRebEO)
La ds type of eppicaion Docu say u sele roucon och as rene enploive are placed
{regular merral on We end surface. The explosion: cause sel wate 1 propagar Una
he subsorlace geolopcalnrctres and ret back tote surface fom interfaces bowen gealpial
‘Mata. The flected aes ae cavers int electric) signals by a composite ara, of geophones ad
tin cin pate and displayed as à tuo. dimensioal signal that uncon of te and space
fall a trace gather, a indented in Figure 1.15. Before these signals ae anlyred, some preliminar
me and amplitud comen: re: made the da o compensate for diferent physical Blenomene
From tie corta dat, he Une ferner between recto samc nas are weed 1 map stra
ceformatons whereas the amplitude changes usally indicate Ihe presence of hydrocarbons

Diesel Engine Signal

‘Signal processing is playing un important role in the precision adjusiment of diesel engines during produc
‘don JUS, Efheient operation of the engine requires the accurate determination ofthe topmost pont of
piston travel alle the tap decd center) inside the eylinde ofthe engine. Figure 1.16 shows the signals
{enerated by a dual probe inserted int the combustion chamber of 2 diesel engine in place of he glow
plug, The probe consists ofa microwave antena anda photodiode detector. The microwave probe captures
Signals reflecied fom he cylinder cavity caused bythe up and down motion ofthe piston While the engine
‘stunning. Interestingly, the waveforms of these signals exhibit a synmelry around the top dead center
independent! the engine speed, temperuure,cyinder pressure, oral fuel rao. The poin OF sync
ls determined automatically by a microcomputer, and the fol njetion pump position is then adjuste)
by the computer accurately 10 within 03 degre using the luminosity signal sensed by the photodiode
otto.

‘Speech Signals

“The near scout theory of speech production has led to mathematical models forthe representation
‘of speech sigma, A speech signal is formed by exciting the vocal tract and is composed of two type
of sounds: wiced und unvoiced {Rab78}. |Stußä}, The veiced sound, which includes the vowels and
onber of consonants such as B,D. L, M,N. and R, is exited by the pulsatile alow resulting rom the
‘bration of the vocal folds. On the other hand, the unvoiced sound is produced downstcar in the forward
ar ofthe oral cavty (mouth) with the voc Corde a rest and includes sounds lke FS. and SE.

Figure 1.12) depicts the speech waveform of a male ullerance "every salt breeze comes from the sea"
(Pl, The total duration ofthe specch waveform is 2.5 seconds. Magniied versions of the "A" and
'S"segmenis inthe word “ssl” uve sketched in Figure 1.174b) and (€), respectively. The slowly varying
low:trequency voiced waveform ot “A” and the Mi Frequency unvoived freatve waveform of "8" are
evident rom the magnihed waveform. The voiced wavetorm im Figure 1.17(b)is seen tbe quasi perioaie
and can be medeled by a sum ofa finite number of sinusoids. The lowest frequency of one ain ln this
represemation sealed the fundamental frequency r puc feequenes, The unvoiced waveform in Figure
ESC has mo regular fine sitar and Is more Rose lie.

29. 10 Chapter 1: Signals and Signal Processing

cro th wa ee

3 od) a nee
3

RL

i Te ae

EEE EEE

Figure, rap recondof he None ersboc, January 29,194, Recorded Stone Canyon Rete
Los Angels, CA (Courtesy of Instat for Crt Reach Univesity of California Sant Barra, CA)

30. 1.3. Examples of Typical Signals

Figure 1.15 typical sem iaa trace ger. (Courtesy of Insti for Cl Research, Univesity of Califo,
Sana Bartra CA)

(One ofthe major applications of digital signal processing techniques isin the general ares of speech
Processing. Problems in this area are usually divided into thre groups: (1) speech analysis (2) speech
Synthesis, and (3) speech analysis and synthesis [Opp78]. Digital speech analysis methode arc used ima.
tomatic speech recognition. speaker verification, and speaker identification. Applications of digital speech
symihsis techniques include reading machines Tor the automatic conversion of writen text into speech,
and rewieval of data from computer in speech form by remote access through terminal or telephones
(One example belonging othe third group I voice scrambling for secur transmission. Speech data com
pression for an efficient use ofthe transmission medium is another example ofthe une of speech analysis
followed by symhesis A typical speech signal after conversion into a digital form contains about 64.000
its per second (bps). Depending on the desired quality of the synthesized speech, the original data can be
‘compressed considerably... down Lo about 1000 ps.

Musical Sound Signal

‘The electronic synthesizer is an example ofthe use of modem signal processing techniques [Moo77),
Lex), The natural sound generated by most musical insument is generally produced by mechanical
bruions caused by activating some form of ocillator that then causes eer Parts ofthe instrument lo
vibrate. Al these vibration together in a single instrument generate the musical sound, I a violin the
Primary oscillator is a stretched piece of sing (ca gu. is movement is caused by drawing a bow across

this sets the wooden body ofthe violin vibrating, whic inten ses up vibrations of he air inside os
ells outside the insrument. Ina plan the primary oeillator 15a stretched ste! wire chat is st no
vibratory motion by the biting a hamıner, which in tun cause vibrations inthe wooden body (sounding
board) ofthe piano, In wind or bras instruments the vibration occurs In a column of at and a Mechanical
change inthe length of the ar coluran by means of valves or keys regulates the ate of vibration”

31. 18 Chapter 1: Signals and Signal Processing

Figure 116: Diese engine signal. (Reproduced with person from RK. ungen Deri Bes celtics 10
ni Jap, ER Spectra. val. 1 Jl 1981 pp. 29-32 0198) IEEE)

‘The sound of orchestral instruments can be clasifica ino two groups: quasi-periodic and aperiodic.
Quaspeñtodie sounds can be described by a sum of a finite number of sinusoids with independently
‘varying amplitudes and frequencies. The sound waveforms o! wo different instruments the cello and the
bass drum, ae indicate in Figure 1-18(a) and (), respectively. In each figure, the top waveform isthe
plot of an entire isolated note, wheress the boom plo shows an expanded version of a portion ofthe note
10 ms for the cello and 80 ms for the Bass drum. The waveform ofthe noe from a cello i scen to be
‘quasi-periodic, On the ther hand, dhe bass drum waveform is clearly aperiodic. The tone of an orchestral
instrument is commonly divided ino three segment called the attack par, te steady-state par, and the
‘decay pan. Figure 1.18 ihustrats this division for the tw tenes. Note hat che bass drum tone of Figure
1.180) shows no steady-state par. A reatonable approximation of many tones 1 obtained by splicing
together these pars. However, high-fidelity reproduction require a more complex model

Time Series

The signals described dus fa are cominvous functions wi time as he independent variable. In many
«asec signals of ees re natural serte funcion of ne independent variables. Often such signs
a one duration. Examples of such signals are the yearly average numb of snspos, daly sock
Prices. the vale of wal monthly expors of county. the sexy population of animal pci ina onan
Ecopraphical aes, the annual eds pr sc of cop in à Gout, dnd the monthly {ia One ot
‘its sen recen prs Te ét xr nl. bly ele ine veri cs
in buche, economic, piel sciences, oe stones nginefing medicine and any oler Bld
Plots of some typical time series are shown in Figures 1. en ds

32. 13. Examples of Typical Signals 19

Figure 1.17. Speech waveform example: (a) sentence Je segment, 9) magi version the woe segment
¡heee A, and 6) magniied versio ofthe unvoiced segmen hele S). (Reproduced wih permisos fom à
1. Manoganet al, Speech coding, IEEE Trons an Cammanicvons, vol. COM Apa 1979,97, 10-177 ©1975

‘There ate many reasons for analyzing a particular time series (Box). In some application, there
may dea need to develop a model ta determine the nature ofthe dependence of the data on the independent
‘rable and use i to forecast the future behavior of the series. As an example, in business planning,
feasonably accurate sales forecasts are necessary. Some types of series Passcss scasonal of periodic
‘components, and his important fo extract these components. The study of sunspot numbers i important
for predicting climate variations. Invariably. the time series data ae noisy and tei representativos require
modes based on their statistical properties

Images

As indicated eases, an image is 2 two-dimensional signal whose intensity at any point i a function of two

peta variables. Common examples ar photographs, stil video images, radar and sonar images, and chest

and dental x-rays. An image sequence, suchas that cen in a television, is essentially a tree dimensional

Signal for which the image intensity a any point isa function ofhree variables: two spatial variables and
he. Figute 1.22(a) shows the photograph of a digital image

33. 20 Chapter 1: Signals and Sgnal Processing

Figure 18: Wineforms oa) the ello and (b) the bas um. (Reprodiced with permission from. À Mose. Signal
procesa aspects of Computer mune: À survey, Precedings of the IEEE, vol, 6. August 1977. pp. 1108-1137
rea

M

Figure 1.19: Scasonally junta quately Gros Nation Post of Und Stasi 1982 dlls from 1976 10
1996. (Adapted vom [LU

‘The basic problems in image processing are image signal representation and modeling, enhancement,
restoration construction from projections analysis, ond coding 12189.

‘Bach picture clement ina specific image represents a certn physical quant: a charactriation of
‘the element is called the image representation. For example a photograph represents the luminances of
various objects as seen bythe camera. An infrared image taken by a satellite or an airplane represents the
temperature profile of geographical area, Depending on the type of mage and it appicaion, various

34. 1:3. Examples of Typical Signals a

SE me

Figure 1.20: Mombly mean SL Louis, Mis temperature in degres Celsius forthe years 197540 1978. (Adapted

from (Ma

Et D

en AM fei el

EE rd
¿AUN ALLAN)

nen RE DA

Figure 1.21: Monthly sine demand in Ontario, Canada (in mlions o ono) fom January 197 to Dscernber
1998 doped rom [ARE

"pes of image models are usually defined. Such models are also based on perception, and on local or
‘hobs! charactersties. The nature and performance of the image processing algorithms depend on the
Image model being used,

Tmage enbancement algorithms are used to emphasize specific age features to improve the quality of
the image for visual perception o to aid inthe analysis ofthe image for feature extraction. These include
methods for contrast enbancement. edge detection, sharpening, linear and nonlinear Aleing, 200ming.
and noise removal. Figure 1-22) shows the contrast-enhanced version ofthe image of Figure 1.220)
developed using a nonlinear iter | ThuDO)

‘The algorithms used for elimination or reduction of degradations in an image, sich as blurring and
comes distortion caused y te imaging system andor is surround are known asumage retoraron.
Image recensiruction from projections involves the development of a two-dimensional image slice of à
tires dimensional object fom a number of plarar projection obtained from various angles. By creating a
number of coatiguous slices, a three dimensional image giving an inside view ofthe objec is developed.

Image analysis methods are employed to develop a quamitative description and classification of one
‘or more desired objects in a image

For digital processing, an image needs tobe sampled and quantized using ananalog-o-digital converter,
A rexcomable sie igitalimage init orginal form akes considerable amount of memory space for tora.
For example, an image of size 512 x 512 samples with 8-bit resolution per sample contains over 2 milion
bis, Image coding methods are used to reduce the tota numberof bits in an image without any degradation
in visual pereeption quality ss in speech coding. eg. downto about 1 bit per sample on the average

35. Chapter 1: Signals and Signal Processing

2

‘gure 122: (a) A gil image, and (it comrt-enhance version. (Repaduced wi permission fom Nonlinear
Image Procestng, S.K Mrs apd G,Sicorasa, Be. Atem Pret, New York NY 02000)

1.4 Typical Signal Processing Applications’

‘There are numerous applications of signal processing tha ween encounter in our dil life without being.
are of thom. Due t space limitations isnot posible 9 discuss ll ofthese applications, However
an overview of selected applications is presented

1.4.1. Sound Recording Applications

“The recording of most musical programs nowadays is usually made in an acoustically inert studio. The
Sound from each instrument is picked up by is own microphone else placed tothe instrument amd ie
recorded on single track in a multitrack tape recorder containing as many as 48 tacks. The signal rom
individual tracks nthe master recording ar then cited and combined bythe sound engineerin amix-down
system to develop a (wo-uack stereo recording. There area numberof reasons for folowing this approach,
First, the closeness of each individual microphone to its assigned instrument provides a high degree of
separation between the instruments and minimizes the background noise inthe recording. Second, the
sound par of on instrument can be rerecorded later if necessary. Third, during the mix-down process the
Sound engineer can manipulst individual signals by using a variety of signal processing devices 1 alter
‘the musical balances between the sounds generated by the instruments, can change the timbre, and can
‘add natural oom acoustics effet and other special effects [Ble78}, [Ear]

Various types of signal processing techniques are ublize inthe mix-down phase, Some are used to
modif the spectral characteristics of the sound signal and toed special effects, wheresscthervare used o
improve the quality ofthe transmission medium. The signal processing circuits most commonly wsed ae:
D compressors and limiters, 2) expanders and noise gates, (3) equahiers and filter, (4) noise reduction
systems, (5) delay and reverberation systems, and (6) iris for special effects [Ble78,[Ear76l, (Hubs,
[Wor89]. These operations are usually performedonthe original analog audio signals andare implemented
using analog crust components. However, there isa growing rend toward all digital implementation and
its use in the processing of the digitized versions ofthe analog audio signal [Mi

Fr cn his tee ag oy Hon Dit Sal Presi, Sn K Mia ae Raise Ed, 195%,
‘ti O pesca oe a SH

36. 1.4, Typical Signal Processing Applications 2

ES

spain

Figure 1.23: Transfer catsteisti opa compres,

igure 1.24: Parneters characterizing typical compresor

Compressors and limiters, These devices are used for he compression ofthe dynamic mange of an
aio signal. The compresor can be considered stan amplifier with two gaia levels the gin sty for
‘input signal Ives below a certain threshold and les than ity for signals with evel above the treshald
“The tieshold level fs adjustable over a wide range of the input signal, Figure 1.23 shows the transfer
characteristic ofa typical compressor,

“The parameters characterving a compeessor ares compression rai, threshold evel, attack ime, nd
release time, which ae illurated in Figure 1.24.

‘When the input signal level suddenly sex above a prescribed threshold, the time taken by the com-
presor to adjust its normal unity gain othe lower value i called the attack ime. Because ofthis effect,
the output signal exhibits a slight degree of overshoot before the desired output level is reached. À zero
tack time is desirable to protect the system from sudden high-level transients, However. in this Case the
impact of sharp musical attacks is eliminated resulting in a ul “ietos” sound (Wor89). A langer attack,
{ime causes he output 10 sound more percussive than normal

Simular, the time taken by he compressor to reach ts normal unity gain valve when the input level
suddenly drops below the thresholds called the release ine or recovery time. I the input signal actuate:
rap around the threshold in à small region, the compressor gain also Buctates up and dows, In such
situation the rise and fll of background noise reali in an audible effec called breathing or pumping.
hich can be minimized with «longer release time for (he compressor ain.

‘There are various applications ofthe compressor unt in musical recording [Ear76]. For example, it can
e used o elimina variations inthe peaks of an electric bas output signal by clamping them to constant
evel ths providing an even and solid bas: lie. To maintae the original character ofthe instrument

37. 2 Chapter 1: Signals and Signal Processing

Oapainan

Lace

pad

Figure 125: Tenafrcharaceisi of «typical expander

necessary to use a compressor with along recovery ime compared (othe natural decay rate of the electric
bass. The device is also useful to compensate forthe wide variations inthe signal level produced by a
Singer who moves frequently, changing the distance from the microphone.

"A compressor wih a compression ato o 10-0- or greater i alle a limiter since is output levels
are essentially clamped 1 the threshol level, The hier 5 used to prevent overloading of amphfers and
other devices caused by signal peaks exceeding certain levels.

Expenders and noise gates. The expander’ function is opposite that ofthe compressor. ti also
‘an amplifier with two gain level: the gain is unity for input signal levels above a certain dvesbold and
Tess than unity for signals with levels below the theshold. The Ihreshol level is again adjustable over à
‘wid range ofthe input signal. Figure 1.25 shows the transfer characteris ofa typical expander, The
‘expander is used to expand the dynamic range of an audio signal by boosting the high-level signals and
Stfcnuating the low-level signals, The device can alo be used o reduce piso below a threshold level

The expander is characterized by its expansion eto, tyeshold level, attack time, and release Ue.
Here, the time taken by the device 19 reach the normal unity gain for a sudden change inthe input signal to
evel above the threshol is defined asthe ack ie. Likewise, the time required by the device to lower
the gain from is normal value of ene for a sudden decrease inthe input signal level is called the release

“The noise gate is u special type of expander that heavily attenutes signals with levels below the
threshold. Is ud, for example, to totally cut off «microphone during 2 musical pause 30 as not o pass
the noise being picked up by the microphone

Equalizers and fers. Various types of fltersare use to modify the frequency response ofa recording
‘or te monitoring channel. One such iter, called the shelving iter, provides boo rise) or cu (drop) in.
the frequency response at either he low or at the high end of the audio Frequency range while no affecting
the frequency response in the remaining range o the audio spectrum, as shown in Figure 1.26. Peaking
Jitters ae used for midband equalization ard ae designed to have ether a bandpass response to provide a
‘boost or a bandstop response to provide a cut, as indicated in Figure 1.27.

“The parameters characterizing low-frequency shelving fier are the two frequencies Jar, and far.
"Where the magaitude response begins tapering up ar down from a constant leve and the Jow frequency
gain levels in dB. Likewise, the parameters charactrizing s high frequency shelving filer aro the tuo
frequencies fi and fan, where the magnitude response begins tapering up or down from a constant level
and the high-frequency’ ain levels in dB. Inthe case of a peaking Alter, the parameters of interest oe the
enter frequency fo, he 3-48 bandwidth Af of the bellshaped curve, ad the gain level at the center

38. 1.4, Typical Signal Processing Applications 25

pres em ein er

mente Pepe Me

igure 1.7: Peak ter frequency response

frequency. Most offen, the quality factor O = {6/2 is used to characterize the shape of he frequency
response instead of the bandwith AF

A typical equalizer consist of a cascade of a low-frequency shelving fier, a high-frequency shelving
fier, and thee or more peaking filters with adjustable parameters to provide adjustment of Ihe overall
eaualirer frequency response over a bread range of frequencies in the audio spectrum. In paramere
tler, each individual parameter of ns Constituent er blocks can be varied independently witht
‘efecting the parameters ofthe other Alters in the equalize

“The graphic equalizer consstsofacaseade of peaking ites with fixed centr frequencies but ajustable
ain levels that are controlled by vertical sides in Ihe front panel. The physical position ofthe lides
rcosonably approximates the overall equalizer magnitude response, as shown schemaically in Figure L 28

‘ther types of filters that also find applications in the musical recording and transfer processes ae
9e owpass. highpass, and notch filters. Their comesponding frequency responses are indicated in Figure
1.29, The nich filters designed to attenuate a parucular frequency component and has Bartow notch
id so x no 10 affect the Ft ofthe musical program.

‘Two major applications of equalizers and filters in recording are 10 correct certain types of problems
that may have occurred during he recording or the transfer process and to alter the Harmonie ra
contents ofa recorded sound purely for musical or creative purposes (Ear76), For example, dre transfor
‘of a musical recording from old 78 rpm disks toa widebend playback system will be highly noisy des
19 the limited bandwideh of the old disks. To reduce this noise, a bandpass fier witha paschand match
Ing the bandwidth of the old records is tilized. Often, older recordings are made more plening by adding

39. 26 Chapter t: Signals and Signal Processing

ain

Evo o 1m

Fe
© co

Figure 138: Graphic equlizer: (a) contol panel stings, ax () corresponding frequency response. (Adapted fom
Con

Pure 1.29: Frequeney reponses of ther yes of ln (a) ons ir () gas ler, snd) note ter

‘broad high-frequency peak in the So 10-kz rango and by shelving out some of the lower frequencies.
‘The noch filter is pariculry useful in removing 60.2 power supply bum

Increatng aprogram by mixing down amullichanneh recording. the recording engineer usually employs
‘qualization of individual racks for creative reasons [Ear761. Fo example. “ullness eflect canbe added
10 weak instruments such asthe acoustical guitar, by boosting frequency components inthe range 10010
300 Hz. Similarly, by boostingthe2- to HE range, he eransiens caused By the Fingers against the sing
‘of an acoustical guitar can be made more pronounced. A High frequency shelving boos! above the 1. 1
2-kHz range increases the “crispness” in percussion instruments such as the bonge or sare drums,

40. 1.4. Typical Signal Processing Applications 2

Tilers
eta — Compress

Fiera

ete | compresor >> u
Tier

Ea, 1 compresor

{shea

Fer

pr)

igure 1.30: The Dolby À iype nis edition scheme for the cording mude, (a) Block gram, and frequency
resp ofthe our Fite ih cu fequreics a Bown

Noise reduction system. The overall dynamic range ofhuman heating is over 12048. However, mest
‘cording and transmission mediums have a much smaller dynamic range. The musi tobe recorded must
be above the sound background ur noise. Ifthe background noise is around 30 dB, the dynamic range
avale for the music is only 90 dB, requiring dynamic range compression for noise eduction,

A noise reduction system consists of two parts, The int part provides the Compression during the
‘scoring mode while the second part provides the complementary expansion during he playback mode
‘To is end. the most popular methods in musical recording are the Dolby noise reduction schemes of
‘which there ar several types (Kar76], [Hab#9},[Wor89].

Inthe Dolby A-type method used in professional recording, forthe recording mode, the audio sign! is
splitint four frequency Bands by a bank of four Alters; separate compression is provided in each band and
{he outputs af dhe compressors are combined, as indicated in Figure 1.109), Moreover, he compression
in each band is restricted to a 20-48 input range from —40 to — 20 dB. Below the lower threshold (0)
88), very low level signals are boosted by 10 dB, and above the upper threshold (-20 dB), the system
‘has unity gain, passing the high-level signals unaffected. The transfer characters fr the record mode
ds thos as shown in Figure 131

In the playback mode, the scheme is essentially the same as that in the recording mode, except here
the compressors are replaced by expanders with complementary transfer characteristics, as indicted in
Figure 1-31. Here, the expansion i limited to 10-48 input range from —30t0 -20 dB. Above the upper

41. 2 Chapter 1: Signals and Signal Processing

pati 8
Figure 131: Compresor and expander anor charcierih for the Dolby Artyp mise ection scheme:

Aeshold (20 4B). very high level signal are ut by 10 dB, while below the lower threshold (—30 dB,
the system has unity gain passing the low-level signale unaffected,

Note that foreach band, a 20-1 compression is followed by a 1-10-2 complementary expansion
such that the dynamic range of the signal atthe input ofthe compressor is exactly equal o tha at the
‘expander output. This pe of overall signal processing operation is often called companding. Moreover,
the companding operation in one band has no efect on Signal in another hand and may often be maskod
by other bands with no companding.

Delay and reverberation systems. Music generued in an inert studio does not sound natural com-
pared tothe music performed inside a room, such as a concer all. Inthe latter case, the sound waves.
propagate in ll directions and reach he tener fom various directions and at various times, depending
fon the distance traveled by the sound waves fom the source tothe listener. The sound wave coming
‘iret tothe listen, called the direct sound, reaches firs and determines the listeners perception ofthe
location, size, nd nature ofthe sound source. Ths followed by afew closely spaced echoes, called early
reflecions. generated by reflections of sound waves from al sido ofthe rom and reaching the listener
at rrulas times. These echoes provide the listeners subconscious cues as fo te sizeof the room. After
these early reflections, more and more densely packed echoes reach the listener due 10 multiple reflec:
tions. The later group of echoes is referred to asthe reverberaion. The aroplitde ofthe echoes decay
exponentially with time as seul of ation at each retection. Figure 132 illustrates this concept
‘The period of time in which the reverberation falls by 60 dB is called the reverberatien time. Since the
absorption characteristics of diferent materials are not the same ar different frequencies. the reverberation
time varies fom frequency to frequency.

Delay systems with adjustable delay factors are employed to atificaly create the early reflections.
Electronically generated reverberation combined with artificial echo reflections are usually ade 10 the
récordings made in a studio. The block diagram representation of typical delay-reverberation system
a monophonic system is depicted in Figure 1.33.

here ae varios other applications of electronic delay systems, some of which are described next
(art

42. 1.4. Typical Signa! Processing Applications

igure 132: Various types of cchocs poney à single sound sure in à room.

Diesen

Ben LR

Figure 1.3: Block diagram oa on system ins monoamine

a a
car Re

Let ight

Figure 1.34 Localization of und source wing dele systems and tenaton network:

Special effects. By fecding inthe same sound signal through an adjustable delay and gin control, as
indicated in Figure 1.34, itis possible w vary the localization of the sound source from the left speaker
o the right fora listener located on the plane of symmeurs. For example in Figure 134, a 0-48

the left channel and a few miliseconds delay in the eight channel give the impression of «localization of
the sound source atthe ef. However, lowering ofthe left.channel signal level by a few-dB los resul
in a phantom image of the sound source moving toward the center. This scheme can be further extended
ta provide a degree of sound broadening by phase shiting one channel with respect tothe other though
aps nero a6 shown in Figure 1.35.

pa tsa ae pe pen wich eg Wane fr algues

43. 30 Chapter 1: Signals and Signal Processing

Pens | munie
(Als) [spas

eet nor

17 ms

Figure 1.35: Sound boadening un llpas toc

Figure 1.36: À possible application of dela systems and reverteaion in a srsophosis stem.

Another application ofthe delay-everberaion system i in the processing of a single crack into a
pseudo-stereo format wile simulating a natural acoustical environment, illustrated in Figure 1.36
‘The delay system can also be used to generate a chorus efec from he sound of roots. The basic
scheme used is Mustated in Figure 1.37. Each ofthe delay units has a variable delay controlled by a
Tow-frequency pseudo-random aise source to provide a random pitch vacation [Ble 781
shui be pointed cut here that additional signal processing is employed ta make the stereo submaster
‘developed by the sound engineer more suitable or he record.cuing ane or the cassette tape duplicator.

1.4.2 Telephone Dialing Applications

Signa) processing plays a key role in the detection and generation of signaling tones for push-button
telephone dialing (Dar76. In telephones equipped with TOUCH-TONE® dialing, the pressing of each
button generates a unique set of two-tone signals, called dua-tone muinfrequenc» (DIME) signals, that ae
processed at tc telephone central office 1 kiemity the number presse by determining the two associated
tone frequencies. Seven frequencies are used to code the 10 decimal digits andthe two special buttons
marked * > * and * 4”. The lowsband frequencies ace 697 Ho, 770 Ma, 852 Mz, and 941 He. The
remaining tec frequencis belonging to the highband are 1209 Hi, 1336 Hz, and 1477 Hz. The Fourth
igh-band frequency of 1633 Hz ts not presently in use and has been assigned for future applications to
Berit the use of four additional push bons for special services. The frequency asignments used i the
‘TOUCH-TONE® dialing scheme are shown in Figure 1.38 [TUBS]

44. 1.4. Typical Signal Processing Applications a

Era fd

Figure 137: scheme for implementing chon eee

Fu Lo ll
orm 3 HA
T Le i
mw 4 sk
I I
sa 7 [9 He
E
so «Hp

Figure 1.38: The one frequency asignments for TOLCH-TONER dialing.

‘The scheme used to identify the two freguensies associated with the button that has been pressed is
shown in Figure 1.39. Her, the two tones are ist separated by a lowpass and a highpass filter The
passband cutoff frequency of the lowpass filer is slightly above 1000 Hz, whereas that of the highpass
fier is slightly below 1200 Hz. The output ofeach iter is next converted ino a square wave by à imiter
‘and then processed by a bank of bandpass Alters with narıowr passbands, The four bandpass fiers inthe
low-frequency channel have center frequencies a 697 Hz, 770 Hz, 852 Hz, and 941 Hz. The four bandpass
{hers in the high-frequency channel have center frequencies at 1209 Hz, 1336 Hz, 1477 Ha, and 1633 He
“The detector following each bandpass ker develops the necessary de switching sign ifs input voltage
is above a certain ro.

‘All te signal processing functions described above are usually implemented in practice in the analog
domain. However, inceasingly, these functions are being implemented using digital techniques”

45. 32 Chapter 1: Signals and Signal Processing

JA Lo group

J seine J

om]
fie

Ea ET)

oe an
Imre

lm terra |

Figure 1.39: The tone detection scheme (or TOUCH-TONE® ding.

1.4.3. FM Stereo Applications

For wireless transmission of a signal occupying a low-frequency range, such as an audio signal, t is
necessary to transform he signal © ahigh-frequency range by modulating it onto à high frequency crie.
[At the receive, the modulated signal $s demodulated to recover the low-frequency signal. The signal
processing operations used for wireless transmision are modulation, demodulation, and fiterig. Two
‘commonly used modulation schemes for radio are amplitude modulation (AM) and frequency modulation
m.

‘We next review the basic ida behind the FM stereo broadcasting and reception scheme as used ithe
United States (Couß3). An important feature of this scheme Is that atthe receiving cd, the signal can
be heard over a standard monaural FM radio with a single speaker or over a stereo FM radio with two
speakers. The system is based on the frequency division multiplexing (FDM) method described earlier in
Section 125.

The block diagram representations of the FM stereo transmitter and the receiver are shown in Figure
1.40(a) and (0), respectively. At the transmitting end, the sum and the difference ofthe left and ight
‘channel audio signals, 5207 and sg), are fst formed. Note thst the sammed signal sy (0) + 280)
used in monaural FM radio. The difference signal 5,1) ~ 52() is modulated using the double-sideband
“suppressed carrier (DSB-SC) scheme using a subcamer frequency f. of 38 KHz. The summed signal,
(he modulated difference signal, anda 19-kHz pilot one signal are ten added, developing the composite
baseband signal sa). The spectrum of the composite signal is shown in Figure 1.400). The basebond
signal is next modulated omo the main caier frequency f, using the frequency modulation method. At
te receiving end, the FM signals demodufsted o drive the baseband signal ae) whichis then separated
into the low-frequency surumed signal and the modulated difference signal using à lowpass fier and à
bandpass filter The cutof frequency ofthe lowpass filter is around 15 Ki. whereas the comer frequency
‘of the bandpas filter is at JE. The 19-X217 pilot tone is use inthe roaciver to develop the 38.kllz

46. 1.4. Typical Signal Processing Applications.

Le cames 2)

©
ber Senna

EL re Ba

14 mer

©
Figure 140: The FM seco system: a) anse, (D) tceve ná Cc) spectrum of the composite baseband signal

sa)

reference sigral fora coherent subcarrier demodulstion for recovering the audio difference signal. The
sam and éifference of te wo audio signals crest he deste left audio and right audio signals,

1.44 Electronic Music Synthesis

‘The generation ofthe sound of a musical instrument using electron circuits is another example ofthe
pplication of signal processing methods (AUS, {Moo77]. The basis of such music synthesis is the

47. s (Chapter 1: Signals and Signal Processing

Figure 1.1: Perspective plo othe amp function Aa 0) for an ta note from clarinet. (Reprodoced with
permission tom A. Me. Signal processing aspect uf computer mus. A suey, Proceedings ofthe IEEE, wo
Raps 1972.09. 108-137 OUT EEE)

Following representation ofthe sound signal s(t:

40

Daunen am

‘where Auf) ard io) we thetime-varyingampitadcand frequency ofthetheomponentof the signal. The
Frequency función Ja vaies sion with ne. Foran instrument playing a oltedtone, Ja) = fos
. 200 = ALO sin Oke) az

where fa is called the fundamental frequency: In a musical sound with many tones, al other frequencies
re usualy integer multiples ofthe fundamental andare called partial frequencies, aso called harmonies,
Figure 141 shows, for example, the perspective pot of the amplitude functions as a function of time of
17 parta frequency components of an actual note from a clarinet. The aim of he synthesis sto produce
electronically the Ag) and f4(0) Functions. To this end, the two mest popular approaches followed are
described next

Subtractive synthesis. “This approach, which nearly dupicats the sound generation mechanism of a
musical instrument, ishased onthe generation of a period signal containing all required baxionice andthe
se of iter to selectively attenuate (.,subract) unwanted pario] frequency components, The frequency.
‘dependent gain ofthe filters can also be used to boost certain frequencies. The desired variations in the

nade functions are generated by an analog mlplicr or a voltage controlled amplifier, Additional
variations in the amplitude functions can be provided by dynamically adjusting the frequency response
haracteristics ofthe fer

48. 1.4. Typical Signal Processing Applications as

ee ro lew line opos othe ample nca Figure 14, (Reproduced with permis
E ann mec of copay nana Ka A of the ER vos,
gas 1977 pp. 1108-1157 ©1079 aE

145 Echo Cancellation in Telephone Networks

[Doy tems work the cenual offices perform the necessary Switching 10 connect two subscribers
anes lS [MOL For economic reasons a tine coeur PL connect a subscriber to
tal

49. 36 (Chapter 1: Signals and Signal Processing

sen DT,
ees a anna Re
Loy 4

gare 143: Basic 24 vie interconction scheme

n

Ds EE

Al

=

©

Figure 144: Vaio signal paths in telephone net. (a) Transmission path fom ker A io tene D (9) obo
‘ph for tlker A, nd (=) echo path fr listener.

‘The effect ofthe echo can be annoying tothe talker, depending on the amplitude and delay ofthe echo,
ie. on the length ofthe trunk erclt. The effect ofthe echo is wort for telephone networks involving
geostuionary satelite circuits, where te echo delay is about 540 ms,

‘Several methods are followed o reduce the effect of he echo, In rk circuits upto 3000 km in length.
adequate reduction ofthe echo is achieved by introducing addtional signa lors in bth directions of the
four-wire circuit. Ja this scheme, an improvement in the signl-toesho ratio is realized since the echo
undergoes loss in both directions while the signale are attenuated only once.

For distances greater an 3000 km, echoes are controled by metas afin ch suppressor inserted inthe
swonk circuit, as indicated in Figure 1.5, The device is essenualy a vat activated switch implementing
‘wo functions. it first detects the direction of the conversation and then block the opposite path in the
four-wire circuit. Even though it introduces distortion when both subscribers ar talking by clipping parts
of the speech signal, the echo suppressor has provided a reasonably accepable solution for terestrial
transmission,

‚For telephone conversation involving satellite circuits, an elegant solution is based on the use of an
cho cancele. The circuit generates a replica ofthe echo using the signal in the receive path and subes
‘fom the signal inthe transmit path as indicated in Figure 1,46. Basically itis an adoptive filer cie
‘whose parameters ae adjusted using certain adaptation algorithms ont the residual signa is sausfactory

50. 15. Why Digital Signal Processing? 37

mn —
Lia] [3

igure LAS: Echo suppression scheme,

Figure 1:46: Echo cancelacion scheme,

a les
Te ange, a an
= a | Ges rn fe Pe |

igure 147: Scheme or he dpa pocesing of an analog ie

mininized.? Typically, an echo reduction of about 40 dB is considered satisfactory in practice. To eliminate
the problem generated when both subscribers are talking, the adaptation algerih is disabled when the
signal in the transit path contains both the echo and the signal generated bythe speaker coser tothe
hybrid colt

1.5 Why Digital Signal Processing?"

Insome sense, the origin of digita! signal processing techniques canbe traced bak othe seventeenth cen
tury when nie difference methods, numeral integration methods and numerical interpolation methods
er developed to solve physical problems involving cainuous variable and functions, The more rece
interes in digital signa processing are inthe 1950 with the avait of age digit computers
Inial applications were primary conceme withthe simulation of analog sign! processing methode.
Arta the beginning ofthe 1960s, rares Began o consider ial sul POLE a a separie
fay sit. Since then, here have been significant and myriad developments a reakthrongh inet
on and aplicado of aia signal procesa

Diga) processing of an analog ige consis basica, of re tpt: conversion of the analog signal
sna diga form, proceso dig version, an tally cameron of te proue aig sal
back nt an analog form, Figu 1.47 shows the overall che iva DICK lagra form

“for a eses of apie ering menda so Cio),

ck a bce om Hanon or Dp Sigel Press Sn K Miva and Js F Ke. ls. 0199,
Sune iy Sens. Aaa ype a ey Se

51. 38 (Chapter 1: Signals and Signal Processing

Figure 1.48: ypical waveforms of signals searing at various sages in Figur 147, (a) Analog input signal 0)
‘utp ofthe SA el, (9 A/D comener out (utp othe dial processor. () DA converter tpt und
{Dig pa gn (and Gh Stat MON LON ol shown poi má eso ut
for lat

Since the amplitude of the analog input signal varies with time, a sample-and-hold (S/H) circuits used
firs to sample the analog input at periodic interval and hold the sampled value constantat te input of the
‘anatog-o-digtal (AID) converter to permit accurate digital conversion. The Input to the A/D converter
is stircas-typo analog signal ifthe S/H citcuit Holds the sampled value until the next sampling instant,
‘The output of the A/D converter ia Binary data Stream Ut le next processed by the digital processor
implementing the desired signal processing algorithm. The output ofthe digital processor, another binary
“data stream, is then converted into a staircase-type analog signal by the digital-10-analag (D/A) converten
“The lowpass filter atthe ouput o he D/A converter then removes all undesired high-frequency components
and delivers atts ouput the desired processed analog signal. Figure 1.8 illustrates the wavetoeme of
‘the pertinent signals at various stages in the above process. where for clarity the two levels of the binary
signal are shown as a postive and a negative pulse respectively.

In conrast 1 the above, adiectaralog processing ofan analog signal is conceptually much simpler
since i involves only a single processor, as Musiated in Figure 1-9. 1 is cheefore natura to ak what
the advantages ae of digital processing ofan analog signal,

‘There are ofcourse many advantages in choosing digital signal processing. The most important ones
are discussed next (Bel84),(Pr092)

52. 15. Why Digital Signal Processing? 39

Analog | Analog
input 7] Processor ot

igure 149: Analog processing oF analog signals.

Unlike analog circuits. the operation of digitar ciscuis does not depend on precise values of the
dira signals. Ax aan, a diia) crete les sensitive 0 tolerances of component values and i fairy
independent of temperature, aging. and most other extemal parameters. À digital circuit can be reproduced
‘easly in volume quantities and does not require any adjusmmenas either during construction or ater while in
(se. Moreover, it samenable to full integration, and wih ihe recent advances very large scale integrated
<VLSD circuits. has been posible o integrate highly sophisticated and complex digital signal processing
Sisters on a single chip.

Ina digital processor. the signals andthe coefficients desribing ihe processing operation are represented
as tinary words, Thus, any desirable accuracy can be achieved by simply increasing the wordlength subject
Lo cost limitation. Moreover, the dynamic ranges for signal and coefficients can be increased sti further
by using Mostng-point arithmetic I necessacy.

Digita processing allows the sharing of à given processor among a numberof signals by timesharing
tus reducing the cos of processing per signal, Figure 150 lustres the concept of timesharing where
{wo digital signals are combined nto oe by time-diviston multiplexing. The multiplexed signal can then
be fed into a single processor. Ry switching the processor coeflcients prior tothe ara of each signal
‘a the input ofthe processor, dhe processor can be made to Look like two different systems. Finaly by
Sematiplening ine vuıpu of ths processor, the processed signals can be separated.

Digital implementation permits easy adjustment of processor characteristics during processing, such as
that needed in implementing adaptive Alters. Such adjustments can be simply carved out by penodicaly
‘hanging the cociients of the algorithm representing the processor characterises, Another application
ofthe changing of cocficient is in the realization of systems with programmable characters, such 22
ftequeney selective filters with adjustable eut fequencis, Filter banks with guaranteed complementary
Frequency response characteristic are easily implemented in digital form.

Digital implementation allows the realization of certain characteristics not possible with analog imple-
‘mentation. such as exact linear phase and multirate processing, Digital circuits can be cascaded without
any Joading problems unlike analog circuits Digital signal cam be stored almost indehmiely without any
loss of information on various storage media such as magnetic tapes and disks, and optical disks. Such
Stored signals can later be processed of-line, such as in the compact disk player, the digita! vdeo disk
Player, the digital audio ape player, or simply by using a general purpose computer as in seismic data
processing. On the her hand, stored analog signals deteriorate api as time progresses and cannot be
ecovered in their original forms.

‘Another advantage ste appicabity of digital processing to very low frequency signals, such as those
‘ccurting in schunie applications, where inductors and capactiors needed for analog pmeessing would be
Physicaliy very large in size

Digital signal processing fs also associated with some disadvantages, One obvious disadvantage is the
"increased system Complexity inthe digital processing of anglog signals because of he need for sditiona
pre and postprocessing devices such as the A/D and D/A converters and their associated fers and complex
it circuitry.

‘A second disadvantage associated with digital signal processing is the limited range of frequencies
Available for processing. This propery limits its application paniculaıy in the dighal processing ef analog
signals, Ax shown late, in general, an analog continue signal must be sampled a frequency that

53. 40 Chapter 1: Signals and Signal Processing

Figure 150, trono the ene shrig concept. The signal showin (has ben blaine by me-mpleing
gris shown in (and

atleast (ice the highest frequency component present in the signal. If this condition is ot satisfied, then
Signal components with requencies above ha the sampling frequency appear ai signal components below
this particular ftequeney totally distorting te input analog signal waveform. The avalable frequency range
‘of operation ofa digital signal processor i primarily determined by the SH circuit and ne A/D converte,
And as results limited by the tae of the art ofthe technology. The highest sampling frequency reported
in the lterature presently is around 1 GHz {PouB7). Such high sampling frequencies are not usually used
in practice since the achievable resolution of the A/D converter, given by the wordlength of the digital
‘equivalent ofthe analog sample, decreases with an increase inthe speed ofthe converter, For example. the
reported resolution of an A/D converter operating at | OHz is 6 bits [Pouß7]. On the other hand, in most
applications, ih required resolution of an AD converter i from 12 bit o around 16 bit. Consequently,
a sampling frequency of at most 10 MHz is presently a practical upper limit. This upper Limit, however, 18
getting lager and larger with advances in technology

“The third disadvantage stems from the fact chat digital systems are construct using active devices
that consume cictrical power. For example, the WE DSP32C Digial Signal Processor chip contains
‘over 403,000 wansistors and dissipates around Y wait. On the oer hand, à varity of analog pressing
Algorithms can be implemented using passive circus employing inductor, capacitors, and resistors that
o not need power. Moreover, active devices are les reliable than passive components;

However, the advantages far outweigh he disadvantages in various applications, and withthe continuing
decrease in the cost of digital processor hardware, applications of digital signal processing ave increasing
rapidly

54. Discrete-Time Signals
and Systems in the
Time-Domain

‘Thesignalsavising in digital signal processing ar basically discret-sime signals, and discrete tin system
reused to process hese signals. As indicate in Figure. (c)adserete-time signal ints most basi form
is defined at equally spaced discrete values of time. the independent varinbe, with the signal amplitude
al these discrete times being continuous. Consequently, a discrete-time signa) can be represented ar a
sequence of numbers, with the independent lime variable represented us an integer in ıhe range from
“20 +00. Diserte time signal processing then involves Ihe processing of a discrete-time signal by a
diserretime system to develop another discrete-time sigial with more desirable properties or lo extract
‘certain Information about the original discrete-time signal

In many applications. is increasingly becoming more attractive to proces a continuous-time signal
by discreto time signal processing metheds. To in end, te con first converted nto
an equivalen” discreto time signal by periodie Sampling; the discrete-time signal s then proceed by à
Aiscrete time system to generate another discrete-time signal, and th laters converted imo an equivalent.
connus time signal. if necessary. As we shall show Tater in this Book, under corta (ideal) conditions,
the conversion of continuous time signal canbe carried out such thatthe disrete-time equivalent ha al
{he information contained inthe orginal continuous-time signal, and sf necessary, can bo converted back
‚to the original continuous-time sign) without any distortion,

“Thus 10 understand the theory of dial sigaal processing and the design of diserete-time systems,
swe need io know the characterization of discrete-üme signals and systems in the ime domain, a subject,
swe discus in his chapter. I tums ou ha i soften convenjent to characterize the discrete-time signals
and systems in a transformed domain. Tais altemative representation is considered inthe following two
‘haps.

Ta tri chapter, we ist discuss the time-domain repre
numbers and fs various classifications. We then describe several base discrete-time signals or sequences
(hat play important roles inthe re domain characterization of arbitrary discrete-time signals an discrete
time systems, A number of bask operations that generate other sequences From une or more sequences
ate described next. As we show later, a discrete-ime system is composed of a combination ofthese base
‘operations. The problem of representing continuous signal by a discrete-time sequence is examined!
for simple cue. A more thorough mathematical wealment forthe general cas is defered until Chapter 5
since its based on a transform domain representation ofthe discrete-ime signal discussed in Chapter 3.

Inthe late half ofthis chapter. we introduce the yenerl concept ofthe processing ofa discrete-time
signal by a disrete-time system and the classification of such systems. Ofthese systems, the class of linear.
‘ime-invariant pe is of excluiveiterestin tis book, and we describe sme domain chatacieiaión la
several different forms. We als introduce (he concept of eros-coreltlonbetwsen a par of discrete
‘Sequences which provides a measure o the degree of similanty between the pal

ion oa discrete-time signalasasequenceot

a

55. se Chapter 2: Discrato-Time Signals and Systems in the Time-Domain

ae
Figure 21: Graphical representation of» discreto sequence tan:

Most ofthe book deals with the processing of signals that are deterministic in nature, However, in some
instances, the signals encountered could be random, and a discussion of the Une-domain representation
Of isectetme random signals i also included in tie chapter.

"Throughovtihis chapter and successivechapters, we make extensive use of MarL.an illustrate through
computer simulations the various concepts introduced.

2.1 Discrete-Time Signals

2.1.1 Time-Domain Representation

As indicated carr, in digital signal processing. signals ar represented 28 sequences of numbers called
“samples. A sample value of atypical discrete-time signal o sequence demoted as sn] with the argument
being an Integer in the range ~20 and oo. It should be noted hat ein] is defined onl for integer vales
Of n and is undefined for noninteger values of n. The discrate-time signal iy represented by (sl), Ira
iserete-üme signal is written as a sequence of numbers inside braces. he location of he sample value
‘associated withthe time index m = 0 is indicated by an arrow } under it The sample values ois right
are for positive values of, and the sample values tit left are for negative values of An exemple of a
‘iscrtetime signal with real-valued samples is given by

KM = (2,085. -02,2.17, 11,02, 367,29, -08,4., 00 an
4

For the above signal, xÍ—11 = ~0.2,xf0) = 2.17, xf1] = 1.1. and soon, The graphical representation of
sequence {ujn]] with ea-valued samples i illntited in Figure 2.

Insomeapplicationsadiscrewetimescquence xn]) is generate y periodically sampling aconinvous-
time signal 2, at uniform time interval

nd = calar AT), eo BNO, es

as ilustrated in Figure 22, The spacing 7 between two consecutive samples in Eg. (22) is called the
sampling interval vs sampling period. The reciprocal ofthe sampling interval 7, dented as Fr. is called
the Sampling frequency.

Frat es
Te uri o sampling Feqveney is yes er scan, o het Hs) ihe sampling edi in seco

56. 4

‘igure 22: Sequence generate y sampling a conimous me signal).

should be noted that, whether or not a sequence {x[n]} has been obtained by sampling, the quantiy
xs calle the nok sample of the sequence, Por a sequence (x(n[l the nth sample value xn] can, in
general, take any real or complex value, If a] is real forall values of, then (XI) 1 a real sequence
‘Gn the other hand, ifthe nth sample valve is complex for one or more values of, then ii a complex
sequence. By separating the real nd imaginary pars of xn), we can write & complex sequence (x {a} us

(extol) = Exell) + Cru es

‘where tre] and sin] are the tal part and the Imaginary part of x nl, respectively, and a thus real
sequences. The complex conjugate sequence of {ar} is usually denoted by [x"(n]) and written as
Wen] = Ela] = j lint} Often the braces are ignored to denote a sequence if there is no
ambiguiy.

As defined inthe previous chapter, here are basically two types of discrete-time signals: sampled-data
sigrals in which the samples are continvous-valued and digital signals in which the samples are discret
valved. The pertinent signals in a practical digital signal processing system are digital signals obtained by
Quanizing the sample values either by rounding or by truncation. For exarple, the digital signal (SI
bained by rounding the sample values of the discrete-time sequence x of Bq. (2.1) 0 the nearest
integer values ls given by

FON = Es 1 0, 2, 1, A al

Figure 2.3 shows u digital signal with ampltades taking discrete integer vales in the range from — 2 103

For digital processing ofa continuous-time signal, its frs converted into an equivalent digital signal by
means ofa sample-and: hold circuit followed by ananalog-to-dgital converter, The processed digital signal
is then converted back into an equivalent continuous-time signal by a digtal-to-analog convene followed
bby an analog reconstruction fille. Chapter $ is concerned withthe digital processing of continuous-time
signals. It develops the mathematical foundation of the sumpling process and describes the operations of
various interface iris between he continuous time domain and the digital domain, Chapter conside's
Ue effec of discretization ofthe amplitudes

The discrete-time signal may be a frite-lengih or an infrite-lengdh sequence, À Anite-lengih (also
called fnte duration o&finteextent sequence is defined ony fr a Anite time terval

Menem

es

here 00 < Ni and Ms < oo with Nz 2 Ny. The length or duration N of the above finite length
Sequence is

. Kemmer as
Fie comp cond pen decid es

57. “ Chapter 2: Discrete-Time Signals and Systems in the Time-Domain

»
Figure 24: (a À iii sequence, and (b) lied sequence.

A length. discrete-time sequence consists of Y samples and is often referred to as an N-point sequence,
‘A fic Jength sequence can also be considered ax an infiit length sequent by astiging 220 vales to
Samples whose argument, are outside the above range. The process of Ienihening a sequence by ding
Zero valued samples is called appending with zeros E ero-paddin.

“There ate Ue types of intinie-lengih sequences. À vight-ided sequence xn] has zero-valucdsamples
forn «Nie.

TS an

AN Isa nie integer that can be positive or negative. I M > O, a right-sided sequence is usually
called a causal sequence. À Likewise. a ef-sidod sequence x|n] has ro valued samples forn > Nie

x 20 form > es

‘where Nz Is a finite Integer which can be postive or negative. If Nz 0, a lftsided sequence is usually
called an anticausat sequence, A general wo-sided sequence is defined for all values of nin the range
“00 < n < 00, Figure 2.4 lusts Ihe above two types of one-sided sequences,

For simplicity for fnt-length sequences defined or postive values ofthe time index n beginning.
1 = Othe fist sample inthe sequence will always be assumed tobe he one atsociated withthe me indes
‘2 = 0 without the arrow being explicitly shown under

2.1.2 Operations on Sequences

A single-input single-output diseretestime system operates on a sequence, called the input sequence,
according to some prescribed rules and develops arodher sequence, called the ouput sequence. usually
‘with more desirable properties. For example, the input may be a signal cormptod by an additive aise. and
the ciscrete-time system is designed to remove the noise component from the input. In some applications
the diseceto-ime system can have more than one input and more than one output, An Mrinput, N-
‘output discrete-time system operas on M input pra generating N output signals, ‘The FM stereo

Firm cor agence a ri ay coin of dci stem a dace in Section 234

58. 2.1. Discrate-Time Signals 45

sin ES
ai seein
w o
Nora m 7
tele Atal sale alent
© w
nn en
sean ay
© o

Figure 2: Schematic represento o bss operations on sequence: () modular, (2) ae, (6) meli, (9)
{i dla, eon nas and) PON o,

transmision system isa two-inpur, single-output system since here the left and right channel audio signals
are combined into a high-frequsncy composite baseband signal. In most cass, the operation defining a
particular discrete-time system is composed of some basic operations that we describe next.

Basic Operations

Let x bn) and (n] be two known sequences. By forming the product ofthe sample values ofthese two
sequences at zach instant, we form a new sequence w [nl

vite = in] yt es

In some applications the product operation is also known as modulation. The device implementing the
modulation operation i calle a modulator and its schematic representation is shown in Figur 2 Sta)

‘An application ofthe product operation is forming a finite length sequence from an inte length
sequence by multiplying the Later witha fniteclength sequence called a window sequence. This process
0 forming the fnie-length sequence is osually called windowing, which plays an Important role the
“design of certain types of digital tes (Section 74),

‘The second basic operation i the adiuion by Which a new sequence main] is obtained by adding the
sample values of to sequences x1n) and ln}

2.0)

‘The device implementing the addition operation i called an adder and its schema
shown in Figure 2.50)

representation is

on is the salar multiplication, whereby a new sequence is generated by mull
quence xln] by ascalar À

sta = Asin em

59. 46 ‘Chapter 2: Discrete-Time Signals and Systoms in tho Time-Domain

The device implementing the multiplication operation is alle multiplier anditeschemati representation
is shown in Figure 2 Se).
The rime-shjfing operation ilusrated below in By. (2.12) shows the rel
time-shifted version wen]
si. en

here N is am integer. 1 > 0. is delaying operation and if N < 0, vis an advancing operation,
‘The device implementing the delay operation by one sample is called a wnt dela» and ts schematic
représentation à shown in Figure 2.5), The reason for using the symbol 1 will be clear after we have
reviewed the <-ransform of sequences in Chapi 3. The schematic representation 0 the un! advance
‘operation is shown in Figure 2.30),

‘The time several operation, aso called the folding operation, is ancther useful scheme to develop a
new sequence. An example is

jon berween fn] and ls

veal) = xin

7

mt atom
‘ie sed en fe sa
A PPP
UNLZ oni on a ll tia
nos ah 96-13, 0
wisn, Tran
Brenn pe Sr men a
minis dal Da 203 0 78 O
na ln ma can
sine Tan ja 1 aos m

‘As indicated by te above example, operations on two ot more sequences to generate a new sequence
san bs carried out if all sequences are ofthe same length and defined forthe same range of he time nex
1 However. in some stations, this problem can be circumvented by appending serosalued samples to
the sequences) of smaller lengths 10 make all sequences have the same range of the time index 7 This
process tlusrated im he following example

EXAMPLES Co sue a AG une
MA
vont tart Pa Le peo lod = 7
9 = 0 5 A y appending e ab ud oem lud samples: ua
CREER"
ae e enc ni i felt A le revit cp wate i

eats fn nl L249 2. 615. 108, 0. 0)
shell den ad FAT ABS. M 208 0

lo by

60. 2.1. Discrete-Time Signals. 47

Figure 27, Dire time sytem of Example 24

Combination of Basic Operations

m most applications, combinations ofthe above bask: operations as used. stration of such combina
tions are given in the following two examples,

bond — Male =.
= om

ar

ln Socks erect the senses sin — [Funds — 21 cas e o io day Beca dry e

a the sequence hotel bebe = ban 2 Paye 1) ados o a)
nas hd gain 2 > I) don la = 2)

Hal = tel + ity loba od ah = au
‘Sampling Rate Alteration

61. 48 CChaptor 2: Discrete-Time Signals and Systems in the Time-Domain

ae su in) tis bein
@ o

Figure 28: Representation of bass suming ae allrtion devise: () upper ond () dewe-sampler

A

wee i |
i ll

il

En aus

A & 2 1, the proces is called inrerpodato and tesu in a sequence witha higher sampling rate. On the
Other hand, iC = I he sampling fate i decreased hy a proces called decimation,

"The basic operations employedin the sampling ate altersiom proces ar called up sampling and down
sampling. These operations play important roles an multirate discretetime systems and are considered in
Chapter 10,

In op-sumpling by an integer factor £ > 1. L — 1 equidistat zero-valued samples are inserted by
(ie up-sampler between each two consecutive Samples of the input sequence x(a] to develop an output
sequence yin] according to the relation

AIRE n= 0.4L, 220.
D cn

sant ean

Note that the sampling rate of yn] Z mes age hu tht ofthe original sequence x(n].
“The block-diggram representation of the wp-samper, alo called à sampling rue expander, 6 shown
in Figure 2.8), Figure 2.9 illusates the up sampling operation for an up sampling factor of 2.
Conversely, the dowa-sampling uperation by an integer factor M > | on a Sequence «(a consists of
Res men Misael nan Mit samples, ee an sue
Hla} according tothe relation

sin) = xin 2.18)
This results in a sequence yPr] whose sampling rate is (1/10 that of ln). Basically, al input samples
‘ith indices equ o an iteger multiple of M are retained a the output and all thers ae dise ar
The schematic representation of the down-sampler 00 Sampling rate compresser à shown In Figo
we 2.8). Figure 2 10 sites the down sampling operaien for a down sapling facet OF M = 3

62. 2.1. Discrete-Time Signals

Figare 2.11; (Am even sequence nd (0) amo sequence,

2.1.3 Classification of Sequences

A discrete-time signal can be classified in various ways. One classification discussed cater is in termas
‘ofthe number of samples defining the sequence. Another classification is with respect 10 Ihe Symmetry
‘xhibited by he samples with respect tothe time index m = 0. A discrete time Signal can aso be classic
in terms ofits her properties such as peridiciy summabili, energy, and power

Classification Based on Symmetry

A sequence fn] ix called a conjugate-symnctric sequence if xn] = x°[—n], A real conjugate-symmeteic

led an even sequence. A sequence ala] is called a cunjugate-antinymmetrc sequence tt

1. A real conjugate-amisymmetr sequence is called an dd sequence. For a conjugate:

meet sequence x(al, the sample value at = O mus be purely imaginary. Consequently, fora
‘oda sequence 0} = 0. Examples of even and od sequences are shown in Figure 2.1,

Any complex sequence xn) can be expressed a à Sum of is eonjogate-Spmmerie par con and its

‘conjugate-anisyrnmetne pat,

Anl = sel) + ri en

63. 50 ‘Chapter 2: Discrete-Time Signals and Systems in the Time-Domain

where

b= (stale tim) 220)

CS) 2206),

‘As indice by Egs (2.200) and (2209), the computation of the conjugate-symmeuic and conjuga
acomete pars ofa sequence involves conjugation, time-reversal addition. and moliplication pera-
lions, Because of ths ime-reversal operation, he decomposition of a finite-lengh sequence into sum of
‘onjugate-symmetrie sequence anda conupate-anieymmetrie sequence is possble, i the parent sequence
{sot add lenvth defined fora syrameie interval, M3 0 = M.

EXAMPLES. Comes the fi Sngth sequence of feng? dei for 3
PR

PEN

demi caps ymenté pat gael ad jo any par poll, we Fo
eje a

CAMES STE
?

rhea need wenn the oe
Ulm 2 mag De

‘eng (3-208) ae at
testa

US 084m. 384 Hh 4 3 4, 05-1

kein 230) we pet
tn 1-1 09>), SA ya 1 fd, 93-7) 13)

le can been erie but ol =m) we

tri ay elses st een a sm of ve arn ad io pat
xa) Ku) + aude am)
where

1
esla] =F tal + alm am

nl = Gin) ln > 02m

For a lengths sequence defined for © < m < N ~ 1, the above definitions of symmetry are not
applicable. The defniuons of symmetry in the case of fnite-length sequences ae given instead using a
‘modulo operation with al such symmetric and antisymmetric pants of alength-W sequence being aso of
Sength N and defined forthe same range of values ofthe time index n. Thus, alengtb-N sequence x(a]
can be expressed as

an) = perth + Kein 050 NA, en,

64. 2.1. Diserete-Time Signals st

here pln and ol] denote, respecte the peri onpgate-smmeile part and he pride
Conjugateantisymmetrc part defined by 3

A À (aa +10 =) 6
Zell RM) y (ein + IN m). an
1 à 1 a

À ein Lt) 3 el EN — 1) 22%)

Fora real sequence xn. the periodic conjugate symmetric part is a eal sequence, calle the periodic
en par and denoted y pain], Likewise, for areal sequence (e), the periodic conjugate-antisymmeiie
artis also eal sequence, called the periodic odd part, and denoted BY xy].

‘A lenglb-N sequence xja], defined for O = n = N ~ Lis said 10 be peniodic conjugate-symmetic
if alu] = Em] = HIN — m), and is said 10 be periodic conjugate-amisymmneri 21m
LAC] = AIN = nl. A Ancient real periodic conjugate-symmetc sequence is called a
symmetric sequence and à Faite eng real period Conjugate-antnymimete sequence s called an and
Smmetric sequence

ERAMPLEZA: Condor fun eng sequen flog dead ter 0 €
(blot) = 4 je 3479. 4— 0
“To determina print conor: rn u pri sone ut pale
bi (nt 144, 2 44 340
‘Te compa falo te prendes Lieja! char tn
le le CL al dl
DEN tt 46 3446, EE
‘lng 5224) wei ae at
pain tts 334748 4 4
Leni. ing E, (2248) we
natal = tit 15415. =, 13119
can ety vei that pind =

4) = WO) 1 74
Memes,

lt part eg ah

‘The symmetric propeties of sequences often simplify tic respective frequeney-domuin representa
ions and can be exploited in signal analysis. Implications of the symmetry conditions ar considered in
Chapter 3

Periodic and Aperiodic Sig
A sequence Al] satisfying

Fin) ln + LN) foraln 225)

is called a periodic sequence with a period N where N is a positive imeger and kis any integer. An
example ofa periodie sequence that has period N = 7 samples is shown in Figure 2.12. A sequence is
called an aperiodic sequence if is not periodic. To distinguish aperiodic sequence from an aperiodic

65. 82 ‘Chapter 2: Discrete-Time Signals and Systems in the Time-Domain

Figure 2.12: An example of erode sequence.

sequence, we shall denote the former witha" ontop. The fundamental period Ny ofa periodic signal
Sine smallest value of 3 for which Bq (2 29) bold

Energy and Power Signals
Theta energy ofa sequence s(n is eine by
em D wit 225

‘An infrite-ength sequence with nit sample values may or may not have finite energy, as illustrated in
the following example

[EXAMPLE 21 Die img sequence 1) sie y

PORTE em
nen eps 2

-2()
“which converses to 2/6 indicuung that sits] has finite enersy However the indie hong sequence sul]
aay

pea karen =>

ne ot ar a

mis
CR EC ARE
“The average power of nape seque an) seine by

im 7 À Il. 29)
Pom din aps E ht am
“The average power of a sequence can be related to its energy by defining its energy over a finite interval
UE reeks

fur =D bem? eso

66. 22. Typical Sequences and Sequence Representation sa

Theo

Pam gin pt es

The average power ofa periodic sequence X] with a period N is given by

as)

“The average power of an intinte-Ieng sequence may be finite or infinite.

EXAMPLE 28 | Condé hr cae sequence defined by

MP, 420.
if Sea

6 os from By 26) ox] nie emery. Oo he oes Man Moat Ex (229) I average pre
shea dy

+ E AR
natn (hy) ito
aed

An infinite energy signal with nie average power i called a power signal. Likewise, a finite energy
signal with zero average power is called an energy signal. An example of a power signal is a periodic
sequence which has a fine average power but infinie energy. An example of an energy signal is a
finte-Length sequence which has finit energy but zero average power

Other Types of Classification

A sequence af] is sad 10 be bounded i each ofits samples is of magnitude less than or equal toa finite
postive number By. i.
Intnl] < Be < 0. ess
‚The periodic sequence of Figure 2.12 is abounded sequence with abound 2, = 2.
“A sequence win] is said to be absoluey summable if

Dieta < se. a)

A sequence is sid tobe square-summable if

$ tant? ce. ass

A square-sumunable sequence therefore has nie energy and isan energy signal it aso has zero power.

2.2 Typical Sequences and Sequence Representation

‘We now consider several special sequences that play important ole inthe analysis and design of discrete
‘ume systems. For example, an arbitrary sequence can be expressed in terms of some of these basic

67. sa Chapter 2: Discrete-Time Signals and Systems in the Time-Domain

w o

11711]

© o

PLE:

Figure 2.14: a) The un sep sequence ri, nd de sd unit step sequen Lalo +2)

sequences. Another fundamental apliction tha isthe key behind discrete-time signal processing isthe
representation o a clas of discrete-time systems in terms of the response ofthe system to certain Base
Séquences. This representation permits the computation of he response of the discrete time system to
bay discrete-time signals Ihe later can be expressed in terns of these basic sequences,

2.2.1 Some Basic Sequences

‘Me most common basic sequences are the unit sample Sequence, the ni Sep Sequence, the sinusoidal
sequence, andthe exponential sequence. These sequences are defined next

Unit Sample Sequence

‘Thesimplestand one of the mostuseful sequences isthe unc sample sequence, on cated the discrete-time
“pal or the uni impulse. as shown in Figure 2 Sta). ru denoted by Sa] and defied by

AS 236)

“The unit sample sequence shifted by k samples s thus given by

ml int

Figure 2.146) shows Sin — 21. We shall show Inter in this section that any alta sequence can be
represented as a sum of weighted time sed uni sample sequences. In Section 2.61 we demonsrate
‘hata certain class of uiseret-time systems is completely characterized in the time domain by 5 output
response 10. unit impulse input. Furthermore. Knowing tis particular response of the system, we can
compute its response to any arbitra input sequence

Unit Step Sequence

A som basi sequence I the unit step sequence shown in Figure 2.14(3) I is denoted by jin) and is
Genet by

68. 2.2. Typical Sequences and Sequence Representation ss

a0.

[hno 0
ami (Y, ex
“The uni ep sequen shied by A samples i tus given by

IA {2h

inure 2.14) shows uf 2),
‘The unit Sample andthe unit sep sequences are related as follows (Problem 2 19)

mine À dite ste) wnt in th ex

‘Sinusoidal and Exponential Sequences
A commonly encountered sequence iste real sinusoidal sequence with constant amplitude of the form

Ant = Acos(ayn +9), 90 <n = 00, es

where A. and ae real numbers The parameters Ay, and $ are called, respectively, Ihe amplinude,
(he ongular frequency, and the phase of th sinusoidal sequence x(n)
Figure 2.18 shows diferen types of sinusoidal sequences, The eal sinusoidal sequence of Eq. (2.39)
‘ca be writen altematvely as
Hla) = nmi + sgl 24)

wets nn and xy nar, respectively the phase and the quadrature components cf x(n} and are given

sun] = Acospeostenm), xml = —Asing into), ean

Another set of basic sequences is formed by taking the mh sample value tobe he nth power of a ral
or complex constant. Such sequences are termed exponential sequences and their most general form is
piven by

AIM Aa, oo <n <0, am

where A anda’ an reel or complex numbers. By expressing

andere, 4e taie
eu rente Eg (2.423 as

mn AE men tans ess

= AI concn +0) + JAI" san + 6). ea

Le amve atun ahematve general form ofa complex expunential sequence where do, and a, are nom
real numbers If we write xl] = aren] + emf then han: Eq. (2430)

xl) = [Ale conten +0)

ibn] = [Ale sina +),

‘Thos the real an imaginary parts ofa complex exponential sequence are real sinusoidal sequences with
constant (a = 0). growing (o, > 0), or decaying ta = 0) amplitudes form = 0. Figure 216 depicts a

69. 56 Chapter 2: Diserote-Time Signals and Systems in the Time-Domain

Cu u gore

AP Te

© m

Peu 23, amy of notó ns e y ae) = Sc (a = (ne = m
Po ear nent en ART RT SOR

70. 22. Typical Sequences and Sequence Representation or

: |
|
EN
# teca
®

igure 2.16: À complex exponential sequence xn) = e-1/1247H/8%, (a) Rel pr and.) imaginary par

Tos 1 =

ial EMO

E «Em

tt me)
= m

Figure 2.17: Examples of real exponential sequences: (a) xin} = 0.201.237.) sla} = 2000.99,

‘complex exponential sequence with decaying amplitude. Note thatinthe display of acomplex exponential
‘sequence is real and imaginary parts are shown separately.

‘With both A and a ral, te sequence of Eq (242) reduces toa real exponential sequence. For n > 0
sucha sequence with Ja] = 1 decays exponcatally a m increases and with al > 1 grows exponential
sn increases. Examples of real exponential sequences obtained for wo values of a are shown in Figure
zu.

‘We shall show ler in Section 3.1 that a large class of sequences can be expressed in terms of complex
exponential sequences ofthe form 2/4.

[Note tha the sinusoidal sequence of Eq. (2.39) and the complex exponential sequence of Eq. (2.438)
sites = Oare periodiesequences of period N aslongas we isan integer maltipleof23,€.ainN = Zur.
Where N and ave positive integers. The smallest possible N satisfying this condition is the fundamental
period ofthe sequence. To verity this, consider the two sinusoidal sequences x:(n] = costas + 6) and
"all = costasin + N) + 6). Now

xl = eo en + A7 à 6)

cosa + coca = sina + 9) ina

hich willbe equalto costusn +9) = 211] only sine = 0 andcos 9 = 1. These tuo conditions
are sais if and only if e an integer multiple of ie.

WN an ta)

71. se Chapter 2: Discrete-Time Signals and Systems inthe Time-Domain

A 2/0 is» ponintger rational number, then th period will be a multiple of 2x aay. 121/05 is not
‘rational number then the sequence is aperiodic eventhough it ha sinusoidal cevclope. For example
In] = cos(v/3n-i-@) is an aperiodic sequence

EXAMPLE 29 La ur aie the pert pl maca secs if 213 th Figs 2 15a
‘ey = PU (2444) std with = nod any vac vt Dem De Ama valo al te
{tenn Fare 3:15) ay = 0 Ir end ee, pin E, (2.44) we dl at N= 20 Pola ima
en we rive at N= 0 oe Pte 3 50 N = St que 2 182 0 = 30 fa Pigeon es
Topo ZAS 8 2010 Ple 215g) nod N = Sr Figure ZN, Tate meh ae sno ese re
tps

‘The number in he two sequences of gs, (2.39) and (2.33) is called the angular frequency. Since
{he time instant is dimensionles, the uit of angula frequency a and phase le simply radians, W the
‘nit of is designated as samples, then the unit of a, and e radans per sample. Ofen in practice the
angular frequency is expressed as

oa 4s)

ete fis frequency in cycle ger sample,

‘Tuo interesting properties ofthese sequences ae discussed next. Consider two complex exponential
sequences al) =e" and rain] =P with 0 < a < 2x and 2k = ay = RE D where kn
ny positive or negative integer. TE

ey = wy +20, (246)
ten
al = efit ¿li dea os,
‘Thus these two sequences ae indisinguishabe. Likewise (wo sinusoidal sequences 21{ = costun +)

and sain) = costunn + #) with O < oy = 2m and 2k = uy < 2a(k + D where kis any postive oe
negative integer, re indistnguishable from one another nt Dek,

‘The second interesting feature of disret-tine sinusoidal signal can be seen from Figure 215, The
Frequency of osilation of the discrete-time sinusoidal sequence {a} == Acostann) increases os og
increases from 0 to x, and then the frequency of oscillation decreses asin increases (rom = to 2
‘AS rest ofthe fist propery. a frequency 4, ia the neighborhood of = 2h is indstinguishable
from a frequency a — 2 in the neighborhood af « = 0. and a frequency wu, in the nelghberhoed of
© = (2k 41) is indisinguishabe from afeequency ay — (2k + 1) In the good of oe for
any imeger value of k. Therefor, frequencies in the neighborhood of a = 2x are usually called low.
frequencies, and frequencies i the neighdorhood of a = (2K + 1) are called MEA frequencies. Ror
‘example, in] = cos(0.17n) = cos.) show in Figure 21506) is a low-frequency signal, whereas,
vola) = cos(O.8x7) = cos(2xm) shown in Figure 2.154) and Cb) is a high equency signal

Another application of the modulation operation discussed earlier in Section 2.1.2 is to ransform à
sequence with low-frequency sinusoidal component to a sequence with hiph-requeacy components by
modulating the former witha sinusoidal sequence of very hig requency, a usted in the flowing
exam

EXAMPLE 210 thn} = sonia) ant sie] = emia)
que yin} = sits thes phon by

Mol = 2émio 0 «contaros

i a pa

72. 2.2. Typical Sequences and Sequence Representation 5

Leg à piment enr ine
MA ma em) + costa =.

‘The mew sequence ya] I ur compose ol o said sequence of Fannin wn + and >

PUS Be nated a ca he pepe ven by Ba (2 tay sae E fee

toquen coli + un) apa arg nn gar ck = an ea) a
Tran aoe wa ann

2.2.2 Sequence Generation Using MATLAB

Matan includes a number of functions that car be used for signal generation. Some ofthese functions
‘oF interest are

exp, sin, cos, square, sawtooth

Fox example, Program 2,1 given below can be employed to generate a complex exponential sequence of
‘he form shown in Figure 2.16,

complex exponential sequence

+
a - input type in real exponent = 1);
D = input(‘type in imaginary exponent =");
caes

E

© Input "Type in the gain constant = 1
- input ("Type in Length of sequence
an

x = Kexp(ctnl

Stentn, real 6 À à

xlabel ("Time index n°} ;ylabel (/Amplitude’):

tátle("Real part");

Gisp(" PRESS RETURN for (maginary part}:

pause

Stentn, imag 60);

xlabel ("Time index n°) ;ylabel ("Amplitude");

title('Inaginary pare”)

Likewise, Program 2. listed below can be employed to generate areal exponential sequence ofthe fora
shown in Figure 2.17.

© Program 2_2
Generation of real exponential sequence

= inpue (type in the gain constant =);

A
:

8 = inputittype in argument = +);

Ns input ("Type in Length of soquence ©");

REIT e en + en tne sea ~~ ced ag

73. 60 Chapter 2: Discrete-Time Signals and Systems in the Time-Domain

igure 218; An ablar sequence ln

a = où
Sram:

label "Time index n‘);ylabel (’Anplitude”);
titletl alpha = “smingstr(a))}7

Mari is given later in Example 2.14

‘Anoiber type of sequence generation si

2.23 Representation of an Arbitrary Sequence
Anarbitrary sequence canbe represented in he time-domain asa weighted sum of some basic sequence and
Ws delayed versions. A commonly used basic sequence in the representation iste unit Sample sequence
For example, the sequence x[n} of Figure 2.18 can be expresse as
a(n} =0.56(n +2} + 1.361n = 1) fn —21-+ Sn — 41 + 0.7851 — 61 am

An implication ofthis type of representation is considered later in Section 25.1, where we develop the
{general expression for ealeuating the output sequence of cenain types of discrete-time systems for an
atbitary input sequence.

Since the unit step sequence and ie unit sample sequence are simply related through Eg (2.38), iis
also possible to represen an arbitrary sequence asa weighted combination of delayed unit step sequences
(Problem 224)

2.3 The Sampling Process

We indicated earlier tha often the discrete-ime sequence is developed by uniformly sampling a continuous
time signal x4 (1), as state in Figure 2 2. The relation between the two signal is given by Eg. (2.2),
‘where the time variable ofthe continuous me signal i related tothe time variable n ofthe discrete-time
signal oly at diseretestime instants te given by

n Onn
Fr” ar
with Fr = 1/T denoting the sampling frequency and Ar = 2x Fr denoting the sampling angular
frequency. For example, if the continuous-time signal is

AO fot +9) = Acos(Rot +0), (2.49)
‘the corresponding disereic-time signal is given by

aa

lt

sla] = A cos +4)

Ac (ARR +6) = acotar, em
Ren 40) = acta +8), es

74. 23. The Sampling Process st

gure 2.19. Ambiguity a ie discrete-time representacion of continuous time signal. 4) shown with he sid

126) is shown with the ashe line si) ds mim with the dashed don In. ad he Sequence chained by

where

es”

is the normalized digital angular frequency of the discreto simo sign al]. The uni of the normalized
gia angular frequency cx, is radians per sample, while the Unit ofthe normalized analog angular fe
‘quency ©, is radians per second and he unit of the analog frequency fy is bert if the uni ofthe sampling
period T isin seconds,

EXAMPLE AUT Com tee sms send y ay sampling oe cone noes of

fogueo 1) Ha, gira, ent) = sna (ert ao sp) emt

a a einen rn nn 7
A cm.

‘ro of tn en nhs wih ke) nd pro e fc te ie la Fig 2.18 ae He

‘ene plats a ach quen has ely he me mpl alu aa pen Ti con eid 1
a

alo) = cot An) on = ae) = en em)
in| = onen ea +) = een
‘carte sq sve recae ica mc ru anc
neque nia ais end GE
Fi sas pie a sil

‘nthe general case, the family of continuous-time sinusoids

Faut) = ACOSO + 6) LAR, AOL, ess

leads to idenical sampled signal

Atl OT) = AOS UR, +A TNT + 9) =

ana

+0) = mers 0 at es

75. ‘Time-Domain

& Chapter 2: Discrete-Time Signals and Systems in

“The above phenomenon of a continuous-time sinusoidal signal of higher frequency acquiring the
identity of a sinusoidal sequence of lower frequency after sampling is called aliasing. Since there are an
infinie numberof comiavous-tiue functions that can lead toa given sequence when sampled periodically,
‘tddtonal conditions need to be imposed so tha re sequence {x} = [xa(17)) can uniquely represent
the parent continuous time function vale). In which ease, xa(£) can Be fully recovered from a knowledge
of ite.

EXAMPLE212._Datertie he ocre die gd fo) caine y wifey pl, pls ee
Ma à ominous sgl) composed of ig num of anal anal cl eqs NI
A 250 a a poem thw
sal) = Seale) + Dane) +2 conc)
+ ace) + units,
‘Thesampling period a 1/700 = 0.00 sec. Hence the gence dci sign 11
Hin = Sera.) + LA à 2 Tr 4m an)
+ nn Xen)
= nan) a 3 = inn) 20 (Dd
4 confia 0.50 ie) + nr ~ dam)
= trou) = Yoni Sem) + 2e Bon) + Lct)
— 007
= banni en) + Sont Sen + 0.0459) — 1 de an)

© See CE SEE CR
SA He ds Wl AAN
SSeS en

Ne sot) om à Seon te « 0.605) — Bia He).
Alber eagle of een gal pera same dort tm sequen
mt = Zn 4 AOL + Hat + ca 4 ner,

omaha meted a irri ped gm 320110

follows fom Eq. (2.51) tat Dr > 27, then the corresponding nonalied digita! frequency oy
the discrete-time signal x(n] obsined by sampling the paren contnuoar-me nal 2,1) will be in
the mange — < à < 7. implying no lang. On de oer haa, I Sy 28, the normale digita
Frequency wi fl imo lower dg frequency n= (aro hax a ie ange ot cw r Becher
fling, Hence, o prevent aliasing he sampling frequency fy should be greater an 2 mes the
frequency O, ofthe sinusoidal signa being sampled. Generling the above res. we observe at if
e have an bar) continous te signal) tha canbe represented sa weighed sum of à amber
Of sinusoidal signa, en x) can also be represente uniquely bys sampled version (xn I Ihe
‘angling ete fy sch be rer an nes en ety cnn) The
onto to be sa Sampling frequency to preven! ain al ore
‘cn somal rer Sos D PIERRE ae de ms de

76. 2.4. Discrete-Time Systems 63

a se Je m

Input sequence ‘Opa sequence

Figure 2.20; Schema representation of a discreta.

“The siscree-ime signal obtained by semplieg a continuous-time signal 1 () may be represented as
sequence salaT)). However, we shall use he more common notation (rfa]} for simplicity (with T
Sioned to be normalized o 1 se), Te should be noted that when dealing with the sampled version of à
Tontinuous-tine function, itis essential vo know the numerical value of the sampling period 7.

2.4 Discrete-Time Systems

“The function ofa discrete-time system ito process given input sequence to generate an output sequence
In most applications the discrete-time system used isa single-input, single-output system. as shown
Schemmaticelly in Figure 2.20, The cutpot sequence is generated sequentially, beginning with a certain
Vale of the time index m, and thereafter progressively increasing the value of, If the beginning tine
Indes is ma, the output Uno] ie computed, then Une + 1] I computed, and so on. We restrict our
‘tention inthis texto this clas of disoet-time systems with certain specific properties as described ater

in a practical discrete-time system, all signals ase digital signals, and operations cn such signal also
lead to digital signals. Such a discrte time system is weually called a digial fier However. if there i
no ambiguity, we shal refer to a ascete-time system also as a digital filter whether or not i has been.
implemented using finite precision arithmetic.

‘Simple Discrete-Time Systems

“Thedevices implementing the basic operations shown in Figures 5 and2.8 canbe considered aselementary
ire time systems, The modulator andthe adder are examples of two-input single-output discrete-time
Systems. The tomaining devices are examples of single-input single-output discrete-time systems. More
Complex discrete time systems are obtained by combiniag two or more ofthese elementary discrete-time
‘ystems as ilostrted in Figures 26 and 2.7, respectively. Some addtional examples of discrete-time
sms are given below.

EXAMPLELIS A very imple ccoo af dere me tem i fhe arm din hy the pa
ut réa

mis D an
ps
= Y are sai a

oe em te uth mee an ao me a he pe a

a tie sta nic the sae a geri pu tope sal om 0 fo dm man
A ne. scowl pu spe aes = 0"

77. ss Chapter 2: Discrete-Time Signals and Systems in tne Time-Domain

De

mens
1 e
Figure 221: (a) Te original acompte sequen sr], and (the noise sequence día,

‘We nox illustrate the operation ofa discrete-time system by two examples

A een crates

va

ee

78. Figure 222 mp 1 athe sig om lions
{nail ated ne apa eve mon eg ee vee

79. ss Chapter 2: Diserete-Time Signals and Systems in the Time-Domain

Q
ae

Note tat in Figure 2.200), the output [1] of the 3-point moving-average filter is nearly equal to the
“desired uncorrupted input sr, except tha ts delayed by one sample. We shall show later in Section
42.6 that a delay of (M — 1)/2 samples is inherent in an M-point moring-averge Alter.

mento

ol am
able E: AR GES es
A Lam

80. 24. Discrate-Time Systems or

2.4.1 Classification of ime Systems.

“There are various types of clasificaton of discrete-time systems that are described ment, These clasifica
‘Gon are based on the input-output relation ofthe system.

rete.

Linear System

‘The most widely used discrete-time system, andthe one that we shall be exclusively concerned within
this tent isa incar system for which the superposition principle always hols. More precisely for a
linear dietete-time system. if yıln] and yo(n] ate the responses tthe input sequences an] and nal.
‘respectively, then for an input

sin) = axila) + ral

the response ls given by

D = ayılm + Byam
‘Te superpsition property mus hold for any arbiary constants, a and 3. and fr al posible inputs, fn
and zoll. The above property makes it very easy to compute the response to a complicated sequence
that can be decomposed asa weighted combination of same simple sequences, suchas the unit sample
sequences or the complex exponential sequences. I this case, dhe desired output i given Dy silly
‘weighted combination of the ouput to e simple sequences,

XAMPLICZI6 | Come cette sah of Exp 2.13, From ingot elton
‘9B St un chars map 0 ol al o lps] loo e BO
nú > site.

e
a Ena.
SS
Te put y) ue o apt d+ o m ren by

PI mega
E ¡da

M ee ua
A ne

ns Et
o = pal) + Y
Men, cs y) a input ola) + Bsn ft oa

Host

+ Leena scien = neue Sains Sa as

81. 68 Chapter 2: Discrete-Time Signals and Systems in he Time-Domain

can be easily vere that the discrete time systems of Bas. (2-14), 215). (2.17), 0.18) (26). and
(258) ae linear systems (Problem 2.28). However, Ue linearity ofthe discrete-time system of Eq. 2.19)
‘depends on the type of input being applied.

‘The shfcimvarionce property is the second condition imposed on most digital files in practice. For a
shift-invariant discrete-time system, if] isthe response to an input [a] hen te response to an input.

al) = xl = no]

iesimply
yin} = pile mol,

82. 24. Discroto-Timo Systems. 69

hera am psoe or nave eg. Th cha sente pu nl st ms hol fo y
ta Input sequence nd A coresponding opt. In the case ol sequences and ystems with indios
(sea to dere tan of time, the above reifen is more commenly called tb tme-isarance
Property. "ho time invariance property ensures ut fra specified Input, the oupu of the Systems

Endependent of the tine the inp e being pica
‘A linear time-invariant (LT) discrete time system sais Both the linarity and the dne-invariance

properties. Such systems ase mathematically easy to analy7e, and curacterize, and asa consequence, easy
o design. In addition highly weful signal processing algorthns have been developed wilzing this class
of systems aver the last several decades In this text, we consider almost entirely this typeof discrete-time

EXAMPLE 218 Te upper ol, (2.7 o lem. To show th we ler fo,
FE (17) tat dg fo] e Igo 2 x] = plo ie y

ink. = 021 220,
ruse

nv [à
a gir cn naar
Di pairs

tm 2.17)

Men LE # = maine A Lim kb,
ns ern

N
ania

Likewise, it can be shown tat the down sampler of Eq, (2.18) is a Ime-varying system.

Causal System

In addition tothe above to properties, we impose fr practical. sdtiomal restrictions of causality and
Stability on the cass of diseretetime systems we deal within this text In a causa? discreteSano system.
the math output sample sn] depends only on input samples af] form = mo and does not depend on input
samples for > na. Thus. if yi[n] and ya) ae the responses of a aura diseetetime system tothe
inputs ua] and wala, respectively, then

ant = ln) forn <

implies also that
yale = yale) for < N.

Simply speaking. for a causal system. changes in output samples do not precede changes in the input
samples. I should be pointed out here tht the definition of causlity given above can be applied only 10
Aisciete-time systems with the same sampling rate fr he inpu and he output

an be easly how that he discrete-time systems of Eqs. (2.18) (2.15), (254) (235) and (236)
are causal systems, However, the discrete-time systems deined by Eqs. (258) and (2 9) are noncausal
systems, I should be noted that these two noncausal systems can be implemented as causal systems by
simply delaying the output by one and two samples respective.

Fi mp en ah sth dfn oot a tbe med

83. 70 (Chapter 2: Discrete-Time Signals and Systems in the Time-Domain

‘Stable System
‘There are varius definition of stability. We define a discrewe-time system 1 be stable and only if. for
every bounded input, he output ir also bounded. This implies ha, he response to xla] is the sequence
yl and it

Bla] < Ba
foc all values of, thea

Dal < By
for all values of n where By and By are finite constants. This typeof stability is usually refered 10 as
‘bounded input, bounded. output (BIBO) stability

RE AT RÉ CS En Rire 20
PR

rt Eno E a

Passive and Lossless Systems
À discrete-time system is said 1 be passive if, fr every Sle energy inp sequence slo, te ouput
Sequence In] has, at mos the Sane ener, Le

Y via? s Y beim? < ce. aa

Ifthe above inequality is satisfied with an equal siga for every input sequence, e discree-Ume system is
sd 10 be lossless.

AENA 238 Cont be sci im tod EA
o te
E rr À tse.

Hence, pas ap jo} | aa ou pl

‘As we shall se later, in Section 9.9, the passivity and the Joslesmess properties are crucial to the
design of discrete-time systems with very low seasiviy to changes inthe iter coefficients.

2.4.2 Impulse and Step Responses

‘The response ofa digital te toa uit sample sequence [Sis called the unit sample response, orsimply.
the impulse response, and is denoted as (n]], Comespondingiy the response ofa discrete-time system
10 aunt step sequence {u[m]] denoted a (e) is unit step response ar simply, the sep response. AS
¡we show next, a near time-invariant digital ble i completely characterized inthe le domain by fe
Impulse response or lt step response,

84. 25. Time-Domain Characterization of LT! Discrete-Time Systems ”
EXAMPLE 231. Temps moe (Ml of be sereine ten 2.1) oa sg
nh = Ai rai
Me) =o} Haste 1} + bl 2 4 bl
“Te imp regen hatte length seque bach 4 given by
Orth = fay. 07. où ad
y

EXAMPLE 222 Die impute reponse [la] ofthe came cumin bg. (2:50) à ti by
‘sting a} = Ala erating i=

üi= He
‘ich ro 38 in rc bri ep sequence al

EXAMPLE223 The ings respons ct iti (29 ene y ot
Sse) = diel den l
M = Bal Fate + a + N
“Te impute pee in een t e à nie ag sequence o nth ac be et ly
tetes 1.091

2.5 Time-Domain Characterization of LTI Discrete-Time
Systems

In most cases an LT discrete-time system is designed as an iterconnecton of simple subsystems. Each
subsystem in tum is implemented with the aid of the basic building Block discussed carr in Section
2.12. In order o be able to analyze nich systems in the lune domain, we ns to develop the pertinent
tions between the input andthe output ofan LT! diseretetime system, andthe characterization of
{he interconnected sytem,

2.5.1 Input-Output Relationship

‘A consequence of the linear, tme-invariance property is hat an LTI discrete-time system is completely
specified by its impuls response: Le, knowing the impulse response, we can compute the ouput of the
system to any arbitrary input. We develop this relationship now.

Let Mn] denote the impulse response ofthe LTT discrete-time system of interes, i. the response
Lo an input An]. We fist compote the response of this Aller to the input x{n] of Eq. (2.47), Since the
isrete-tme system is time invriam, its response to Sin — I] will be Ain — 1. Likewise, the responses
to 81m +2), Sin — 4], and ¿ln — 6] will be, respectively, Mn + 2), lr — Al, and Al — 61. Because of
Fineariy the response of the LT discrete-time system tothe input

ln) = 058in +2)-+ Lila — 1) Sn —2}+ 31m — 4) +0758ln ~ 61

85. 2 Chapter 2: Discrete-Timo Signals and Systems inthe Time-Domain

sll be simply

AT OSA #214 LS 1 = Al = 214 la — 85 4 075% n — 61.

Ie follows from the above result hat an arbitrary input sequence xl] can be expressed a a weighted
insar combination of delayed and advanced unit sample sequences inthe form

a À stein in. am

‘where the weight [A] on the right-hand side denotes specifically the Ath sumple value ofthe sequence
Ain}. The response of the LI discrete-time system to the soqucnce aloja = 41 will be lé im 6)
AS resul the response sI] of the discrete time system to 1 willbe given by

y= Ye ie Kl. esto
Which canbe altemately weiten as
ani À ln a as

by a simple change of variables. The above sum in Bg. (2.642) and (2.640) is called the consolation sum
‘of the sequences} and NA], nd represented compactly as

sla} = An ess
where the natation @ denotes de convolution sun

“The convolution sum operation sales several useful properties. Fest he operation i commutative,

lO rial = Ole 1200)
Second, the comolution operation, for stable and single-sided sequences. is amaciative. ne.

MMS ala] MO LEO la), es
and lst, the operacion is diiribuave, i.
MO Gala] + lad =D ele] + ti @ ft as

Proof of these properties is tetas an exercise (Problems 2.37 to 2.39)
‘The convolution sum operation of Eq, (2.64a) can he Interprete as follows. We fist time reverse the
sequence hlk ariving a hk). We then sift A to the right by sampling periods fn = 0, or othe
lett by n sampling periods if» « 0. to form the sequence ln ~ A), Next we form the product sequence:
(kl = stk}hin — E. Summing al sample of ik] then yicks the mth sample of s(n] ef te comelution
sum. The process of generating e] 5 illutrated in Figure 224. This process is implemented for each
value of in the range _00 < m < 00. The representation of the allerate form ofthe coavoletion sum.
Operation given by Ea, (2.645) is obtained by imercaanging the sequences «lé and hid in Figure 2 24
sai the complex congo operation, tv ext we have opted de bol © un denne he convoy sae o

86. 25. Time-Domain Characterization of LTI Discrete-Time Systems 73

none ag thE om

a
igure 2.24: Schematic represento th conoltion sum operation.

A is clear from the above discusion that the impulse response (hnl} completely characterizes an
LET discrete-time system in the timo domain because, Knowing the impulse response, we can compu,
in principle, the output sequence y{n] for any given input sequence x(n] using the convoltion sum oF
Ea @ Gta) or (2.648), The computation of an cutpu sample is simply a sum of products involving fly
simple arithmetic operations such as additions, mulplcaions, and delays. However, in practice, the
‘convolution sum can be employed to compote te output sample at any inant only if either the impulse
response sequence and/or the input sequence is of nit length reauting i a nite sum of products. Note
(hat both the input and the impulse response saquences are of fine Length, the ouput sequence i lso of
finite length. In the case of a discrete-time system with anifinitelength impulse response, is obviously
o: possible to compote the output using the convolution sum i the input is also of infinie length. We
Stall therefore consider alternative time domain descriptions of such system hat vole only finie sums
of products,

EXAMPLEZZA _ Wespsemneai in gend by coat of te nef
teng morose eo] nd a) aca bee tome pe a ne en
A fis 3 Toc vas spe den A e 4

of she | me mt ace fay À
Je re = 4 ae moder gi a or ay

Sa teen 20,

aan iw ton 2240) D mn
Pr entree E sort alme nomacro wumple for & = O. s1OJAJO] Thun

Eee

no A ns
Pie 2351 A shown gare paseo ample or =

AD A
Fra it MO FINO! 44 = 4
Per = eo ated ste is Figaro 2.259), The erat prac sence (c(h M12

TON = LOIRE Y = 0404
“Te poes con, eng e
[erre llos Es
u
ue tes
pren

Lannion) nado ro Leal nf en 0 np Am mt

os

87. 7a Chapter 2: Discrete-Time Signals and Systems in the Time-Domain

»
a
te kal
>
m
©

3
_—
EEE CIO
a
oe a
immo
7 M

2
AAA loa

tata

tp ne noe
han: lion À
m

o
‘igure 2.25; Nuttin othe comoluion process

HALO kan.
‘Tre sequoce {rll attained sie e kid in Pgs 226
Ik should be noted thatthe sum of the indices ofeach sample product inside the summation of either

Eq, (2.642) or Eg. (2.648) is equal 10 the index of the sample being generated by th convolution sum
‘operation. For example, he sum in the computation of yin the above example involves tbe products

88. 2.8. Time-Domain Characterization of LTI Discrete-Time Systems 75

Figure 226: Sequence generated by the courts.

LMI AI CAM nd SOLO. The sm of We indices ac ofthese four products egal
3 which is the index ofthe sample V3].
CO an be seen or Krane 224. the comlution of two aie-lengh sequences ress in ne
length sequence, In this example, the convolution of a sequence {x{n]] of length 5 with a sequence (hin})
‘of length 4 resulted in a sequence (y{n]} of length 8. In genera, if the lengths af the two sequences
eng coocived at 4 an an srl sean aos comedo OÙ og MN — 8

Tn MATLAB, the statement © = conv (a,b) implements the convolution of two fnitetength se-
‘quences a and E, gonorating the fnite-length sequence c. The process is illustrated in the following

89. 78 Chapter 2: Diserte-Time Signals and Systems In the Time-Domain

Figure 227: Sequence geerte by conoluion using MATIAR

Pres

Figure 228; The ask conecten

2.5.2. Simple interconnection Schemes
Teo widely used schemes for developing complex LT diserete-tmie systems from simple LT discrete-time
system sections ate deseibed next.

Cascade Connection

In Figure 2.2, the output of one filter s connected to the input of à second filter and the two fillers ace
sald to be connected in cascade. The overall impulse response hn] of the cascade of tw filters of impulse
responses hin} and haba) given by

at

MID hal), as

"Note at, general, Ih ordering ofthe filers in Ue cascade as no effet on te overall impulse response
because ofthe commutative property of convolution.
can be shown tha the cascade connection of two stable systems is stable, Likewise, the cascade
connection of two pasive (lossless) syatema fe pasive oases.
“The cascade connection scheme is employed inthe development of an inverse system, Uf the two LT
systems in the cascade connection of Figure 228 ae auch that

Him hala) = Sin. 2.70

(hen the LTI system fpf ssid o be the inverse ofthe LT system hal, and vice versa. As resul of
fe above relation, fie input othe cascaded system isl} is output also xfn). An application of this
concept in the recovery of a signal fom its distorted version appearing atthe ouipul ofa tansssion
hannet. This is accomplished by desigaing an inverse system ifthe Impulse response of te channel Tt
Knaus,

Tre following example illustrate the development of an inverse system.

90. ‘The connection scheme of Figure 2.29 is called the parale connection, and here the outpus of the two

‘ers are added o Form the new output while the same input i fed to both fiers. The Impulse response
ofthe overall filter is given hereby

An) = Asta) + ha). am
lisa simple exercise to show that the parallel connection of two stable systems is stable. However,
‘he parallel connection of two passive (lossless) systems may or may not be passive (lossless).

91. 7 Chair 2; Discos Time Signals and Syste In o Tine Doran

Figure 230: Te dcr time system of Example 227.

Nee
at = ta = te — mate = ai — Bts
in ADM = (datei tie = 11) © (—2(3)" nimi)
(Y) tet ("on nu (3) ter (4)" or
(Y am
im

Mn + Ba 4 Hain = Hy

2.5.3. Stability Condition in Terms of the Impulse Response

Recall from Section 24.1 that a discrete-time system is defined tobe stable, or precisely, bounded-input,
‘bounded-output(BIBO) stale, ifthe output sequence yin} ofthe system remains bounded for al outed
Anpat seauenees xin}. We now develop the stability condition for an TI disree ime system. We show
hat an ITI digital filler is BIBO stable if and only ft impulse response sequence (Ant) aboli
summable, ie 7

S= Y) inmi < oo. em
We prove the above statement for ral impulso response Ala. The extension ofthe proof for a complex
Impulse response sequence is left as an exercise (Problem 2.59). Now, ifthe input sequence la] it
bounded, ie. {ll = By < 00, hen the output amplitude, from Fa. 2.640), de

|= mas À moe a

te

=m D mures <2 am

‘Buus, $ = oe implica Ip{nil < By < 00, indicating that y] Is also bounded. To prove the converse,
assume y(n] is hounded, Le. [yn] < By. Now; consider the input given by

ea eared

at wan) So,

er

92. 25. Time-Domain Characterization of LTI Discrete-Time Systems. 7
‘where sento) = +1 ife > Oand sento) = = ie < 0. and X] < 1. Note hat since bn] 1. (lad)
ús obviously hounded. For this input. „Im at = Os

or = E eno

52, <0. es

‘Therefore In] < B, implies S < 0%,
EXAMPLE 25 Cinder saul LT ductos otr sith to spore ney
Mi = tanto

E ple rate
Tito 0 a | fo wc bone A O me er 1
Tron oe ina

EXAMPLE229 Now an LT intime sy va nuse cepo py
tiny = [ete M se 2 A
| 2 am

Pas only à aie number uf eo Iguloe espe slo or le ales of Nia. Heme, he imp
resp segui ely sale intra the Value fm og a tne Mace. he
‘em of, 277) BEBO ee

2.5.4 Causality Condition in Terms of the Impulse Response

We now develop the condition for an TI discrete-time system 1 be causal. Let zul] and az] be two
input sequences with

ab = tal form = ny es
From EQ. (2640) the corresponding ouput samples at n = na of an LTI discrete-time system with an
{impulse response (Mn]) are then given by

A)
able À talas — 4 E i. Gt)

Wf the LTT discrete-time system is also causa, then Jılma] must be equal to yaIno]. Now, because of
Eg (278), the firs sum on the right-hand side of Eg. (2 79a) is equal fo the fet sum on the right-hand
side of Ba. (2790). This implies tha he second sums on the right hand side ofthe above two equations
must be equal. As x[n) may not be equa to xala) for # > no, the only way these two sms vil e equal
{Sif ey are each equal to peo. which at st

En]

fork <0, es

93. so Chapter 2: Discrete-Time Signals and Systems in the Time-Domain

[As a result, an LT! discrete-time system i cava if and only is impulse response sequence (Ala) ia
‘canal sequence satiatyin the condition of Eq (2.50)

Te follows from Example 2.21 thatthe dscretetime sytem of EQ. (2.14) is a causal system since
is impulso response satin the causality condition of Ka. (2.80). Likewise, from Example 222 we
observo ha the discrcte me sccomulator of Eg. (254) i also à causal system. On the oe hand, From
Example 22% u can he seen tat the factor of 2 linet inte-polatordelned by Eq. (258) isa norcausol
system becuune is impulse response does no satisfy the causality condition of Eg. (280), However, à
oncausi diserete-time system wath a fnt-length impulse respon can ofen be realized ay a causal
Sytem by inserung delos of an appropriate amount, For example, a causal version of the discrete-time
factor? linea interpolator obtained hy delaying the output by ne sample pero seh an pur Sup
relacion given hy

pind ale 174 Svat = 29 + stn)

2.6 Finite-Dimensional LTI Discrete-Time Systems

AD important subclass of LT discrete-time systems is characterized by alinear constant cocfficien differ
ense equation of the om

Day 1= Y prin = 41 ean
where sl] und ae, respectively, be inputand the outpute the system. and dy) and px) re constants.
"The onder ofthe discrete-time system is given by max(N, M), whichis the order oF he difference equation
hiracteizing the system, It possible 10 implement an LTT system characterized by Eq. (231) since
the computation here involves two finite sum of products eventhough such a system, in zenerl, as sn
impulse response ol inline eng.

‘The output sal con then be computed recursively from Eg. (2.81). IF we assun
‘causal thon we can rene Bq. (2 81) to express yn explicas a function of adel

the system 19 be

Y SE
o Zain, es
provided dy #0, The output va] can be computed for al =, koowing «In and te initial condicions

re = ene = 2h ere = NI

2.

“The procedure tor computing the solution of the constant coefficient difference equation of Ex. 12.81 is
very Similar to hot employed in Solving th constant eoefcient differential equation in he case of an LTT
continuous time system. Inthe case ofthe discrete time system of Eg, (2.81) the output response |) also
consists of te components which are computed independerily and then added a yield the tora sation

1 Total Solution Calculation

= dm + sn es
< equation te components the station of Ey (281 wth he np a) = ets
he bomogeneous dicen equation

Davina ess

94. 2.6. Finite-Dimenslonal LT! Discrete-Time Systems. a

“and the component yin) is à solution of Eg. (2.81) with xfn} # 0. yele is called the complementary
“olin. while yas called che particular solution resulting from the specified input x {a}, often called
ihe orcing function. The sum of the Complementary and the particular soltions as given by Eg, 2.83) is
called the foal soln.

We int deserve he meihod of computing the complementary solution yt}. To this end we assume
that tis ofthe form

PE am
Substituting the above in Eq (284) we ave at
Das a
ANA 42 dd) = 0 ex

“The polynomial ZiLodiAN is called the characteristic polynomial of the discret
Eg, (281), Let Ar À Ay demote N oct. I these roots ae al stint, hen the genera form af
‘the complementary solution is given by

yan) = ead Er too bani. em

where a 2, … av are constants determined from the specited initial conditions of the discrete-time
system. The complementary solution takes a different form inthe case of multiple rots. For example.
if is of multiplicity Land the remaining N — L 004s. Ad Age... A are disc, then Eq. (2.87)
takes the form

RD nat na 2)

Next. we consider the determination of the particular solution yp(a] of the dierence equation of
Ba (281). Here the procedure l o assume thal the particular solution also of the same form asthe
specified input JE (n] has the formagO x hod = 1,2. N) forall. ‘Thus, if xl isa constar
then yp] 1 also assumed to be constan. Likewise, if xin} i sinusolda sequence, then y ln] is alo.
assumed tobe a sinusoidal sequence, and so on.

‘We ire below the determination ofthe total solution by means of an example.

XAMPLE 230 Lac Germinal san fora > Of doce item charmed by the
ing Bree as
nat in 1) Orta = 21 ah, am

or nie ing sl = ul) und with at comes 1 fam 31 =
"We fn uri he frm af be complement ston Seg al = m yn} in E A) we

Te man QE ya

Uno-e

tee rc fü cha pinot. À ar Ay = 2233 = 2 eno cop
‘tion eg sn

Ala) = CE Haar am

95. ss Chapter 2: Diserete-Time Signals and Systems in the Time-Domain
he pale soi we sc
mele.
‘Sabin de swe 2.97 e pu
CPE ELA

me en =O il
“Phe etl wan eh of he frm

Ma sa -2 ee a
“pe nu. conch ately speci nl nn. Prom ig 2.09 nnd 2.91) a

HN mtr
Ar EST
‘Solving thee two uations we cea
sent
Tran de las given by

LIN RAR wo

fie input excitation sof the same form as one ofthe terms in the complementary solution then iis
necessary 18 modi the form ofthe particular solution as ¡ustatd in the following example.

EXAMPLE We eine te oa satin ten = Of rete ation En, CS) fe np
EN = a te ae att canny nthe fori ample,

"it ie ru wi de meca a cms ete ae
ue sa he ap tp, ner ve need o ec oi or he parca san whit iis and dees
te ina ern la ca us ope many acto. We aa

pint fai

ata bine me
paca + eae! EN = Pa

rn Oe can fomi steve ua = 04, The dtl oto hmm Oe form

sh att RA ET

‘emi faa ne pg te

A

AA pee! nach

ob hen dpi — O. rz 0 06 Thre A lan 8

he 50-37 AO + A, me x

96. 2.6. Firite-Dimensional LTI Discrete-Time Systems ss

2.6.2 Zero-Input Response and Zero-State Response

An alternate approach to determining the total solution ya) of re difference equaion o Eg. (21) dy
computing its zero-input response ln] and zero-state response ¥.,(n}. The component y 1n] is obtained
by solving Eq, (281) by setting the input x(n] = 0. and the component Jul) is obtained by solving
‘Eq. (2.81) by applying he specified input witha inl conditions st to zero. The total solution fs ten
ven by yl] + ml

"This approach sllusuaed by the following example

EXAMPLE 2.2 We deine the Wl chin of te cre pana of Exp 20 by compo
‘he erm epee und ero topo

‘Die seo pl espe le) of Eq (2.09 ges by te complementa sae el C20) were te
Oman ang ae cone sty te pst nl condi, Nom Do E, (29) me ge

MOI = = 1091-31» —
MM +671=11= 7 +62 0.

Me rm Eg 290 es
+
AU = in + Bay
Solingen wo eo quan we tie nt) ==, ay = = Tht
ganz st a,

ics a ops ene ri (2.91) y crm con a sth ees
its condiciona. From Eq (2-30) we pre 2

HOt su =
Hie att) =>.

het fom E. LP) an he stave set litio wea = = 6.4 Th the eon rept
foc = 0 wah iil coins a Yank 11 OS neh ey

este) = A 0400)" 2
ence te ou ati 1 on by te sm ali im
“4 ne

CERTES
lc semi al died in Exar 230 a expe

2.6.3 Impulse Response Calculation

‘The impulse response hn) of causal LT discrete time system is the output observed with input {ni
Sin. Thos, is simply the zero state response with ala] = Sr, Now for such an Input rl] D
forn > 0, and thus, the particular solution ts zero, Le, sn] = 0. Hence the impulse response can be
computed from the complementary solution of Eq, (2.87) nthe case of simple roots ofthe een
<auation by determining he constants a, 10 Saf, the zer initial conditions, À similar procedure ca,
be followed inthe case of multiple ross of the characteristic equation, À system With fll reo iii
conditions soften called a land system

97. al (Chapter 2: Discrete-Time Signals and Systems in the Time-Domain

EXAMPLE 218 tous example we eine input spss Mn ca acetone sy of
aang 230. Frm Eu (290) wg

Mal

rete above. ware at

‘Nest tho By 2.89 wth aa} = ae

op
ANS NOI = 0

Sali te svete gi ya = A, 0 = 04
“Th, he pte ssp a ven dy

MIDA, m

follows from the form ofthe complementary solution given by Eg. (2.88) that the impulse response
‘of a fint-dimensional LT system characterized by a difference equation of the form of Ea. (2.81) ie of
infinite length. However as illustrated by the following example, here exist infinite impulse response LTE
‘iserete-time systems that eannot be characterized by the difference equation form of Eg. (2.81),

EXAMPLES The qu dened dpa epa
(EEE

Ss at han represent ne ne of neat con ech iene natin I sow be note
(ate save yarn ncaa an BO ve

Since the impulse response Ala] ofa causal discrete-time system i a causal sequence, Eg. (2.92) can
also be used to calulat recursively the impulse response form > 0 by sctig initial conditions o zero
values, Le. by seting y[=1] = yl—2] ==>: = [1] = 0, and using a un sample sequence Sn as
‘the input x(n}. The step response of a causal ET system can similarly be computed recursively by sting
Zero inital conditions and applying a unit step sequence asthe input. should be noted that the causal
diseretetime system of Eg. (2.82) ls linear only for zer intial conditions (Problem 2.49)

2.6.4 Output Computation Using MATLAB

‘The causal LTI system ofthe form of Eg. (2.82) can be simulted in MATLAB using the funcion £1 er
already made us of in Program 24. In one of ts form, the function
y= filter pa)

processes he input data vector x using the system characterized by the coefficient veetors y and à to
‘Renerate the output vector y assuming zero inital conditions. The length of y isthe same asthe length of
x Since the function implements Eg. (282) the coefficient do must be nono,

‘The following example illustrates the computtio ofthe impulse and sep responses of an LT system
described by Ea (2.82),

98. 2.6. Finite-Dimensional LTI Discrete-Time Systems 8

Y - $5

ql

ry

o ne m 5 Tor
© >

Figure 231: () Impulse response and (1) ste response ofthe system of Eq (299).

=

tacks The ogra bn | "To haere da ¿e desen

TT ete
Ha y 11 O45sln 21 Dto 3

= Onn) Satin — 114 DMA le + den SE: ess

erige 26
Stliecrarion of Impulse Maspofa Computatión:

= AnpueltDeeired Impulse coepcción length = 1)
2 Lput type in the vector ="!

{rout (ropa in the sectaria ds
wenn

Eo
L'art
ET
aba! ime Laden nt) 7 vinos ('ampti ide 1

Tr input eme re

wevaunese

2 où
+ (hwy use

a Lester one enge pe re td

function Laps (Prater MZ 11) =
Ge socia app we lic in be iri dit = 14 Sera
Te ce A ms of he a pen m nd

2.6.5 Location of Roots of Characteristic Equation for BIBO Stability

Is should be note that the impale response samples of a stable LTE system decay 10 zero values as the
time index n becomes very large. Likewise, the step response samples of a stable LTT system approach

99. ss Chapter 2. Discrete-Time Signals and Systems in he Time-Domain

an value 1 n becomes vers largo. om the plat of Figure 2318) and (B) we can concludo that
‘mont likely the LTT system of Hu, (293) is BIBO sable. However es impossible to check te sabi
Sr system just by enamining only finie segment or is Impulse cr ssp responses in these figures.

“The BIBO stability of causal LTT system characterized by a constant coefficient difference apuation
‘of the form of Fa (2.81) can be inferred trom the vales the roots A of is choractenste polyoma!
Te establish the stay conditions, recall that the form of he pulse response the same as tat of he
complementary solution. From Eg, (2.87), assuming ll the sats tobe init. we have

A)

cr in the above expres

Tn = Ejes] Saar, ess
1 flog rom the above equation US 0 L Jr a als Of then FA 00 anda à
Tet gill eo, Le, de mps respon 1 able come EU ISO eae ot
ie ui UM dncresime stem, However the pulse space ee D oe ty eee
{tone or more ofthe tetas maga ear hare coe oe head oe ee a
“ope ine stem of Example 2.30 nero Ex (230) eal an sa u
ofthe care cation hve up reac as one

Inte case of rape rota the hater aller e impale response vil contin rms of
the form nf Ara eal presio tr Fog Heal nid cone ee

are determine to wtisty zero inital condi. From

Erro
sich comers) € 1 ete 2
Sabie
Summarig. x ous LT tem characte ana constant cocci free equation of
the om of Ex (2.8) BIBO sb, ine magn o acho the tor chaste aan
¡SIS than one This condon o cosa and sunt

and as a result. here slo che impulse response I absolutely

2.5.6 Classification of LTI Discrete-Time Systems

{Linear time-invariant (LTD diserei-tme systems are sully classified either according o the length of
their impulse response sequences or according to the meth of calculation employed to determine the
‘oat samples

Classification Based on imp
EA is of init ene

Response Length

Ain) =0 formen anda > Ys with Ny Na eo

then it known ax imite impulse response UR) discrete
reduces to

system, for Which the convolurion sum

stat Y Adela al esn

100. 26. Finite-Dimensional LI! Discrete-Time Systems. 87

Note that the above coavolution sum. being a finite sum. can be used 10 calculate y(n] directly. The
sic operations involved are simply mullipiction and addition, Note tha the calculation ofthe present
value ofthe output sequence involves che value ofthe input sample at = Mand Nz = Ny previous
vales ofthe input sequence along wih Ihe N = A + | impulse response samples describing the FIR
discrete-ime system,

"Example of FIR discrete-time systems are the moving-overage system of Eq. (2.56) and the linear
imterpolates of Eas. (2.58) and (2.59).

IAG] sof ante length, then ii known as an finite impulse response (UR))distet-time system.
For a causal HR discrete-time system wih a causal input x(n}. Ihe convolution sur can be expressed in
te form

vial = Yan #1
‘which an be wsedtocompute the output samples. However, fr increasing nthe computational complexity
Increases caused by he growing numberof terms in he sum.

‘The class of IR filters we are concemed within this ces the causal system characterized by the Linear
constant cotficientdiference equation of Eg, 2.82). Note that here also the basic operations needed in
the output cleslations are malipication and addition, and involve a nite sum of terms For all valcs of
(m. An example of such an IIR system isthe accumulator of is. (2.54) and (2.55). Another example is
described ne

EXAMPLE 236 Toe lam mere imegetion forradas tar ed 1 merical otre aeg of the
vom [ande
ante ow to be chars ia consti eect fc equations nd, hs, ae examples u

IR yams I we ive enter negation np eal par eth 7, Es the Seve nega cam ot

minero-tn [conde om

tee we hap et 7 a a ie ci

ur”.

hing the tnpevoial metal we integra the egal om the band ide i 2:9) a ee dh

Hr.

en ee

Desing 7007) wy nd nn) sl a ori air wa

ee. am

‘hen regie a dieron equation serrations Faser I et an

Classification Based on the Output Calculation Process

lt he output sample canbe calculated sequentially. knowing only the present and past input samples, the
‘lier i salto be @nonrecursive diserete time system. I, on the other hand, ce computation othe op
Involves past urpat samples in addition 1 he present and past input samples i fs known as recursive

101. es Chapter 2: Discrete-Time Signals and Systems in the Time-Domain

discrtetime system. An example ofa nonrecursive system i the FIR discrete me system implemented
‘sing Eq. (297), The IR diseretetime system implemented using the difference equation of Eq. (282) is
an example of a recursive system. This equation permits the recursive computation of e output response
beginning at some instant n = na and for progressively higher values on provided the nits] conditions
Jlrs = Y) through ying — N] are known, However, tis posible to implement an FIR system using à
‘ecursive compatational scheme and an IIR system using a nonrecursive computational scheme [GoIS8]
“The former case ls used inthe example below,

JEKAMPLE.ST. Conder the FIR cr ne ym pren bythe apa esp

1 Dane mt boy

Ml soin

as

‘The shove doctrine tur carey cn mth ng am Ar Cata ale
Er

‘ve cher a abe rng nu Bl mel 2 pl ng ema er wa
Slat are ig 127) we et pat ne ehe
vera Sain au

“Wo develop mme ern fn ee rome at

wem Ems Ba

oh = ala) a r+ eM. ao
‘which mp mu of the form of En, GA). Toe renato nf Ha (2102) a Do aid a
crie ran su (RS ip tier LA

Classification Based on the Coefficients
A third classification scheme is based on the real or complex nature of the Impulse response sequence.
“Thus, a discrete-time system with a eal-valued impols respons is defined asa real discrete-time system.
Likewise, for a complex dicree-ime system, he impulso response is x complex-valued sequence.

2.7 Correlation of Signals

“There are applications where it is necessary to compare one reference signal wih one or more signals to
determine the similarity between the pair and to determine additional information based cathe similarity.
For example, in digital communications a st of data symbols are represented by a se of unique discrete
time sequences. Ione ofthese sequences i transite, the receiver has to determine which particular
sequence has been received by comparing the received signal with every member of possible sequences
‘rom the set. Similar, in radar and sonar applications, the received signal reflected from the targets the

102. 2.7. Correlation of Signals E

¿etaycd version of the runsmined signa and by measuring the delay. one can determine the location ofthe
target, The detection problem pets more complicated in practico, often he ceived signal coupe
boy aio random mise

2.7.1 Definitions

A meisure of similarity between a pair of energy signals, ala and in]. is given by the coss-corretation
une rss} dened by

roti E mind costa. co
he param cole ag nites th time sh beten ie par, The tie sequence tn} sail toe
sted y 7 samples wth espeso e trece seque the ah or postive wes ff ad
“ley E samples the Tok or note ales of
“The ang of he subscripts in En (2103) species ia xa] ithe reference sequence which
reims incl tne ren the sequence SÍ) tong shied wäh ropen 10 slap Ihe Wao
‘rake ithe reference sequen and lin sequence with respect a, hen the Como
{rowel sequence en hy

taltl= D nisin à
E sum + ét = rain 2.105

Thun. 1) is obtained by time reversing he sequence rue]
“The turocorretanon sequence of is given by

rate El aimara 2105

‘sine y setting yf] = in] in Eg. (2.103). Note from Ex (2.105) that rl} = ES: ll = ES
seen the sgl cn. Fc (2.10) oo ta gf = ramp at an
‘vom function for real sn,

An examination of Eg, 2.105) revea that the expression for the cmss-corelation ook quit simi
‘w that he convohition given by Eq, 2649). This similarity s much clearer i we rewnte Ea (2103) as

À sete aroma aus)

‘Thesshowe rest implies ha the crose-comreation ofthe sequence vn withthe reference sequence xIn) can
‘he compote by processing +] with an LTT diserste-time system of umpuhe response 11, Likewise,
run Ie canbe deine y pain og an Taree km fue
response sat

103. so Chapter 2. Discrete-Time Signals ang Sysiams in the Time-Domain

2.7.2 Properties of Autocorrelation and Cross-correlation Sequenc:

We next drive some basic properties of the auocorcitn and cros-cocution sequence (Pr092}
‘Gomi two lit ener sequences slr] and vin. Now. he energy ofthe combined sequence ui
SE ao Anl sn ponnegaive: Mats

À eaters sine aa? Y rs ar À rena

TN]

Wwietera1Ol = E, > Oand rol = És > O are encres ofthe sequences al] and ste respect
We can one Eq 0.107) a
ve nett) cta]

Fate) 2300
for any init value of. Or in wher words he matrix

Cut rt]

Fel) rel 0

oc equivaienty

08)
‘The shove inequality provides an upper bound for the eros-correltion sequence samples. IF we et

= VER.

Intel = DT

ln] = fre above reduces o
edel S rec} = E, 2.109)
‘This i a significant resalta sas that at zero lag {€ = ON. the semple value ofthe autocon ton
sequence has maximum vals,
To derive an additional property of the cross comelatin sequence consider the ese

bby Ny

here N san integer and > Dis an arbirary member. In ihis case Ey = BE, and therefore

VEZ, = te

Using the above result in Eg, (2.108) we get
hr l0)< rate he do)

2.7.3 Correlation Computation Using Maras.

‘The cs-corelation ad Ur nocortelacin sequences can be easily computed using Mats Ana strate
in the following two examples, “ PERS a

104. 27. Correlation of Signals st

@
Figure 232: a) Crom-comlition sequence and () ocean sequence.

CANAL 28 nia eto ann qe
ana! à à D 2 MERE
wea 1 à 13 m +
Mata we determane and plot the cress cometation «queme 73310)
o eo cnc ete eg omen

e reegrás 2.7
Convurarion of Cepas -everelación soquenen

À - tnguecaype Le the reference: semionom » 414)

PIN
Bibel ("tag index 11: lanas Amps (rue!
A ma

EXAMPLE 239 In thie comple. me era and phot the autoporretation x
a
‘Program 2.3 cam atmo be anes to compete the ad anten hee
Me es
et
CR erg en 4
errata
‚femanstranng the fast Aa the: cremy-cortstation can he sraptoyed Ww compu 4

105. 92 Chapter 2: Discrete-Time Signals and Systems in the Time-Domain

poe

e rl et - Pa Letltel ate

@ ©
Figure 2.33: (1) Delay estimation from crss-corrltion sequence and (9) anoceelaon sequence of nois-
compet perdi sequence

ick enn in po cm y eg
‘hac Fo, ule mecca e mb coved si le Pd gr A
‘ope teen ai ons emcee pon oe

1 sould te noted tht the uote anders creation sequence ca als be compac
using tbe Martas function sor However, e conan pence fered ng tis fun
ar de merece version of ose generat ng Propane 2 and 28 Te roca rel
Of evo sequencer slo nd nl can Be computed tinge de © = cost ery) mile
Raconter semanal ug de med © eee

2.7.4 Normalized Forms of Correlation

For convenience in comparing and displaying, normalized forms of autocorrelation and crosscorrelation
given by
2110)

CIN)
independent of

soften used, It follows from Es. (2.108) and (2109) 1€] < 1 and {psy
‘the range of values of (2) and fn

2.7.5 Correlation Computation for Power and Periodic Signals.
In the case of power and periodic signals he autocorrelation and cross-<orrlsion sequences are defined
slighty differently.

For a pair of power signals, xn] and sn], the cross-comlation sequence is defined as

CAC]

Ve
Jr Ebo. eu

and ie autocorrelation sequence of xn] i given by

CAC]

1 «
ri E soe ann

106. 2.7. Correlation of Signals. 99

Likewine 3 and bn un two period ina with period. then tht eross- correlation sequence
by

Limit ann

anche aunocorrehnion sequence os given by

y Ein «e aus

1 Kinos tom ah une de

dal ane also periodie sequences with a period

Te peialiciy properties of sion sequence can be explited 10 determine the period
à parie sil that may have Been conupted hy an additne random disturbance, Let Zn] be a
ive periodic signal corrupted by the an noise dir] resulting the signal

win = à + ain

Which is observed for << M = | nero MN: The autacomeltion of vi i ven by

Y vino

sana, E ara
main À ali ni

RE ral rele rado, eno

Now inthe above cquation, 7.1] a perindie sequence with a period M und hence ü will have peaks at

IN AN... wih the Came amplis un € approaches M4. As in] and din} are not corelated,
Hsoteronsoreltion sequcnces Fad rs Jaretkely the very mal relative wie ampltades
OF 10), The autncomelation ofthe diswrbance signal din] shows à peak at € = O with other Samples
having rapidly docrersi ample with increasing values of [e]. Hence the pesks of ray 10] for € > 0
are essentially due to he pes of | van be used o determine whether f/m] ia parodie sequence
il pen the peaks oscar u period tervals

2.7.6 Correlation Computation of a Periodic Sequence Using MATLAB
We seat next he determinaron ofthe period ola none -corrupta peri

Sequence using Mar an:

107. ss Chapter 2: Discrete-Time Signals and Systems in he Time-Domain

Figure 224 (a) Autoconelation sequence ofthe noie <omuptd sinsoid and () correlation sequece ofthe

EXAMPLE 240° UE Seon the pid afte simil sequence af} = COSA) 0 m = 98
cape y a ative wary totes tandems pobre of spite te ng (20 02]

oti end we ua Progmen 28 1 Compute coe ación sequen Ute fhe or mail
Es

À Frogean 24

Bozen

2 FoBipirO.2Sen): Densnate the .inutolds). sequence
RIO Le nalen

Ya Xe a Veneta Sida) anguen
e are ©

Kanne

TOA
babel Lap Ande > yaa ("ampittuner à

‘Ti lo nera by mig hs prog it shen Fl (a. As an sem plo hte ie
sar pa. ot ero tg. Homero heme ne dot penis a ag shat ae sal otage eras

‚land wersnce 1 be ls expec” Fog 2 0) shams e pt € ie amour seta
ad te pl component. As cm be sen From Has po. a] M à ny m eek ely we a
Les esa als fh lg ame va oe De Le a

2.8 Random Signals

‘The underlying assumption on the discrete-time signals we have considered so far is that they can be
uniquely determined by well-defined processes such as a mathematical expression or à rule or a lnakup
table. Such a signal is wally called à deterministic signal since all sample values of the sequence me

Fine econ pe fi prs ep nen € tata dete ie eng pe een which
co CG nan of nora pe sain Be oss

108. 2.8. Random Signals 95

well defined forall values of the time indes. For example, the sinusoidal sequence of Eq. (239) and the
‘exponential sequence of Ex (2.42) are deterministic sequences.

Siemals for which each sample value is generated in a random fashion and cannct be predicted uhend
‘of ume comprise anesher class of signals. Such a signal, called a random signal ova stochastic signa
Cannot he reproduced at wil, ot even using Ihe process pensräing the signal, und therefore needs to pe
‘modeled using statistical ffounation about the signal, Some common examples of random signals ac
Spesch, music and aciomc signal. The mor signal pnersted by furming the difference between the ideal
Sample version of à comtinvous time signal and is quantized version generated by a practical analog-to
¿ligue converter is usually modeied ss random signal for analysis purposes à The roise sequence di)
of Figure 221(t generated using the 23:50 function of Marat is alo an example os random Signal,

The discrete time random sud or process consists of a typiealy infinie, collection or emsembe of
isco tne sequences [XI]. One particular sequence inthis collection {x(n} is called realization
fof he random process. Ata ¿nen time index nthe observed sample value xn 5 the value taken by the
random verichle Kin). ‘Thus, à random process isa family of random variables [Xlo]}. In general, the
Farge of sample values cominaum. We review in thi section the important statistical properties of che
‘Fandom variable and the random process

2.8.1. Statistical Properties of a Random Variable

‘The statistical properties of à random variable depend on is probability distribution function or, equive
‘coy. mis probaiiy density Funcion, which ate defines next. The probably that e random variable
taken value in a pocihed range From ~ 30 0 u is given hy ite probability distribution function

Pate = ProbabiiyiX = a am

“the probability density Juno

on of Xs defined by

pxla) eno

LE X can assume à continuous unge of values. From Kg. (2.118) the probability distribution fonction i
therefore given by

nator = fd au

“The probability density fonction sats

ie Following, to properties
Pla) 26, 2.1200)
free 2.1208

Likewise. theprohabiltyditabution function satisfies the following properties, which fallow from gs. (2119)
an and 2

02 Pela) < 2.1219)
Petes) < Peles). forattas > ay cum)
E] 2.1210)

Probuhiiglas < X San) = Prien) = Ext), ens)

109. 96 ‘Chapter 2: Discrete-Time Signals and Systems in the Time-Domain

Figure 2.35; robb density funtion of E (2125).

A random variable is characteized by a number of statistical properties. Forexample the rth moments
are defined by

sac) [rad em

where r is any nonnegative integer and EC) denotes the expectation operator. _ A random variable is
‘completely characterized by all ts moments. In mosteases al such moments are not known a prion or are
iil evalute. Three mote commonly used statistical properties characterizing a random variable
are the mean ot expected value mx, the mean-square value F(X"), and the variance a} as defined below:

me £00= [7 apta, aızm
cutie [eine an
AR (mer) = [emt rxlaide 1%)

‘These e properties provide equate forma about a random variable a mat pia caves.
Pre
= ER mt, eno

The square oo of the variance, called the sonar des oc random vrs X. (os
from E. 212) tate vance andthe mes square alee equ fora aoe are wee

ean be shown tat th mean vale mg i the bes constant representing a random vale X in
a minimam mean gun enor an. ter EN a) fo min fers ok de no.
inca ae crore given by i ance of (Potion 278). Ts lps ati ie aan soa
Ben he ac assumed by ily Bock omy and ihe variance se te val source
Mi ely tobe fr on

"We asas concep intoduced so aby means of a camp

FAMILIAS a do eX e cnt ay y ans
POS b
2 np 238. pr tn ci a nly
ea co (ame
no pis ue Lie

110. 2.8. Random Signals 7

19 an 2:20) we cat compote probably hat Ni nn speed ang, For emp, he
cx res Pieper
Eh

Prot E
eh aan I are 235. Pam De, (2.1256 he mean wae Gaine a

erh m

‘Two probability density factions, commonly encountered in digital signal processing applications
are the uniform density function defined by

mass

nan [po ge: eum

st. ann ery tn a ma ft ld by
Era an

POLE

‘where the parameters my and oy are, respectively, he mean value andthe standard deviation of X and ie
in he range —00 < my < oo and oy > 0. These density functions are ploted in Figure 2.36. Various
‘other density functions are defined in the Literature (Problem 2.79).

AMEL Di u men ne rc la it en i fete

ya
Pree gy 2-12) an (2.1290 we av at

aim
mi
(0) [en Ett
ang sve wae 2 2 ah
20
ate uo

Inthe case of two random variables X and Y thei joint stitica! properties as well as her individual
satstical properties are o practical interet The probability that X takes a value in a specified range from.

111. Time-Doman

9 Chapter 2: Discrete-Time Signals and Systems

»

‘igure 2.6: (a) Uniform an (1) Gausiangrbabıay density tunis

010.5 and that Y takes a value in a specified range from —00 16 Bix given by their odas probably
distribution faction
Pry ft) = Probability IX a. ¥ = Pl con

1 equivalent, by their Joins probably density Junction

pave: pm cua

“The joint probability distributios funcio is thus given by

note fff prince 159
Tasty day sion ni nn re

tf. eu

[ofr vor mana à pe

The joint probability distribution function satistes the following properties, which ar direct consequence
of Eqs (2.133), (2-13), and (2 134b),

De Preta, P< I. 2.1350)
Perla Bi) = Pres. fe) fora: 2 ay and Pa > Bi. 2.1330)
Pert=30. 00) = 0, Pyy (+00, +20) = 2.1386)

‘The joint statistical properties of two random variables X and Y are described by their ross-coreation
“and cross covariance, as defined by

on rare f° [annee deso eno

112. 2.8. Random Signals ry

Figure 2.7: Range of he random varies with he jor probability density faction of Eg 2139).

ve EUR met nn
Soano noma nés

where mx and my are respectively, dhe mean th random variables X and Y. The two random variables
X and Y are sid 10 be linearly independent or uncorrelated if

Fann)

ERDE. (2.1388)

and staustically independent i
Parla f) = Prta)Priß) a.)

Jean be shown dat random variables X and ar statistically independent he ey are al near
independent (Problem 280). However, i X and Y are inary independent, ey may no be Sinistca
Independent.

“he saistcal independence property makes iceair to compute e statistical properties o random
‘variable thts function of several independent random variables, For example ad ae tal
independent random variables with means mx apd my, respectively. den ican be how hat the mean of
the random viable Y = aX + BY, where aad arc constants given by my = am x + bm, Likewise
ifthe variances of X and Yao and of. especvely. he variance of is given by of = aie} + bof
(Problem 283.

EXAMPLE 243 Conder D two rade veias X and desc by «fey dei fine
route deity func piven by

nran=é dsestes#st ‘am

Drame a a econ Ad ech po aX iY ea es =
1.0.5 # = Dome by the shaded region le Figure 237. qe

the = He ti 2 ot
DT get
roms oros fanal

113. 100 Chapter 2: Discrete-Timo Signals and Systems in the Time-Domain

Figure 2.38: Sample sclizaiuns he andom sins) signal of (2 140) or = 0.067.

2.8.2 Statistical Properties of a Random Signal

As indicated eater, the random discrete-time signal is à sequence of random variables and consists of

‘typically ininite collection or ensemble of discrete-time sequences, Figue 2.38 shows four possible
Of à random sinusoidal signal

(tai) (A contour à 0) e

0.061, where the amplitude A und the phase @ are statistically independent random variables
uniform probability distribution inthe range = a = 4lorthe amplitude andinthe range < @ = 27
ore phase

‘The statistical
properties of he random variable X}. Thus. the mean or expected value of (XU) at i
sien by

roperties of the random signal [XV] at time index n are given by the statistical
index nis

renin EXD = f px 1) da. eur

The manger vue of (X) time index ns given by
# (xin?)

‘The variance a, oF ¡Xln]) atime index is defined by

[otra eus

= E (tn — mare

E (tn) = (ma. eu

In general the mean, mean-square value, and variance of a random discrete-time signal ae functions of
‘the time index and can be considered as sequences.

So far we have assumed the random variables and the random signals to be real-valued. eis straight
{forward to generalize the treatment o complex-valued random variables and random signals. Forexample.
the mth sample of à complex-valued random signal (Xa ofthe form.

tn

Keel) + Hint eus

114. 28. Random Signals. vor

‘where {Xl} and {Xum{r) are real-valued sequences called the real and imaginary pars of (URI),

Tinsley "here lad o's EN wise enn the sven
mat = ECD = BODA JE Kin = ms + Iman QM

Lewis, vn o o ated hen by
in = (Lx ml) = (xt) = ri) ano

‘Often, tc statistical relation of the samples of a random discrete-time sigo at two different dime
indices m and n is of intere, One such relation is Ihe aulocorrelation, which for a complex random.
Aiserete-time signal (Xa) is defined by

xxl.) = E (XX nl). eun

‘where * denotes complex conjugation. Substiing Eg. (2.144) ia Eq, (2147) we obtain the expression
{or the autocorelation of X(n}

Perl ml = Ox, .% en] + xxl.)

= FO Ral + 6x mon), aus
where
Ox. men = Em en aaa
Pratl) = E nlm Xml 2.1490)
xxl ml = E XebmXinlnD + 2.1496)
Prax] = E (Ktm Xe D + CN

Another relation is the auocovariance of (Xia, defined by

vesting] E (Xt) = mc GT — mx"
exxlen.)— myx)” 2.150)

‘As canbe seen from the above, both the autocomelaion andthe amtocovariance are function of two time
indices m and n and can be considered a two-dimensional sequences,

SRE, on tomes ae erg an it 1
mn AE Bu
m
mis [fe ae. am
pre Si rue ara etc indepen! tr joe perle Seite Action

ssosesm ausm,

il e tanzt

115. 102 Chapter 2: Discrete-Time Signals and Systems in the Time-Domain

Time rl ite ton process {Xa that

ame

EC en)

aise,

‘The iu a pen à
BP) tan rn

ars

wich ho te variance tm e rad pecera roman.
‘he auerrslanon face u gen by

dad, ED
A nen + bonne sonen
= ett 0). ae)

‘The correlation between two different random disree-ime signals (X[n)) and (Y n]) is described by
the erver.correlarin function

Perlis} = E (Xiti)

[Let rmumemanaen am
sn cm cone cin
van m= te mn)
enim marine ais)
wis ayia) e e bi ey nt] Bom
EEE Sd ee ha eg:

‘two random discrete-time signals [Xl]) and (F{n]} are uncerelated if yx yim, n) = 0 forall values of
the time indices m and 7

2.8.3 Wide-Sense Stationary Random Signal

ln general, the statistical properties ofa random disrete-tim signal (X nl). such the mean and variance
‘ofthe random variable Xin}, and the autocorrelation and the autocovariance functions, are mc vario
functions, The class of random signals often encountered in digital signal processing applications athe
‘so-called wide-sente stationary (WSS) random processes fos which some of te key staustcal properties
‘te either independent of time or ofthe time origin. Mor specifically, for a wide Sense stationary random

116. 2.8, Random Signals 103

process LKtal) the mean EX) has he same constant value mx for al values af the time inde and
Te autocorrelation and he autocovarince functions depend only un the difference ofthe time indices m
admise

mx = EXD oral, as»
calé = oexin + Eon) = ECXI + EX "Un, for alm and 6, 2.160)
vaxlel = ala + Coat EQ + € mx Xt) m0)

= dx) = lx forall n and e een
[Nowe that in he case of WSS random process, he autocorrelation andthe autocovariance functions are
‘one dimensional sequences

“The mean-square vale of à WSS random process (tr I} is given by

E (km?) = onto es
O

of = reel) = PIO] — imac. 2.163)
io fon ap. (215) an (2156) ae und pco (214) whee sone

The ross correlation and ems covariance functions between two WSS random processes [Xe] aná
(Yin are piven by

srl = E (Xin + ED) easy
ere) E (On 4 0 mel — mn)
MIDEENERS ass)

The symmetry properties saßsfid by the autocorreation,autocovariace,cross-corctaion, and cru:
oxrianee Functions ae

exa 121660
veia (2.1660)
oxvi-el 21660)
zuteil (2.1660)

From the abone symmetry properties it canbe seen that sequences Oy xf). yraléle dx 16 and 16
are always two sided sequences.
Some additional useful properties concerning these functions are

Axio > Ier (67a)
PxxlO}yrvi0} > Iya fel? am)
CHU Liso
FO] > rte na

A consequence of the above properties is thatthe autocorelation and autocovariance functions ofa WSS
random process assume their masimum values at € = 0, in addition, can be shown that fora WSS
signal wit nonzero mean, i. xy 7 0 and With no periodic components,

in Ort = main? 2168)

117. 108 Chapter 2: Discrete-Time Signals and Systems in he Time-Domain

If XI] has periodic component, then 6x x[£] wil contain he same periode component as illustated in
Example 240.

ES E O ann
ha
perpen a

‘pst En. 0.16 où cn nt” 7, ds. cl ta A, Met a
ig.) un (LO me un De eur ae 104 and. anc al =
pa

2.8.4 Concept of Power in a Random Signal

‘The average power ofa deterministic sequence x(n] was defined eater ad is given by Eg. (229). To
‘compute the power associated witha random signal (X(n]) we use instead the Following definition:

1 = à
pam e (an mp À mu) am

In most practical cases, ihe expectation and summation operator in Eq. 0.170) can be interchanged,
resulting in a more simple expression given by

Pr = lin a E, Elton). am

In addition, i the random signal ha a constant mean square value fr al values of , asín the caso of a
WSS signal. hen Bq, (2.171) reduces o

Pe = E (ame) em
From Bis (162 and 2.16 fos at oa WSS sig he sera power shen by
Pee bo) 2 08 +m em

2.8.5 Ergodic Signal

In many pcia stations the random signal o nterestcanotbe desrbein em of asimple
expresion, asin Eq (2.140), 1 peri computo oft taste proper which ane iors
the evaluation of deine integras ur summations, Often a Anl ar of a gle enter de
random signal avala, from which some estan of esta properties ensemble mus
ts made, Such an approach cn lead meanigfu resus the xpd onen dr sahen Mens
precise, a statonar random signals defined lo be an erg sign al lo tapetes Co
Be estimated from a single realization of sffcinty large Ant long
Foran egodi signal ime averages equal ensemble averages dered via he expectaion operator in
the lint as the Len of the realization goes o into, Par example, fora mal cd apa ae ron
compute the mean value variance, and sulscoverance se
E
an En es

118. 105

29. Summary
a

oF = hm xin] = mai? 174b)

4 rs nme ar)

pale Jim gr DL O = mo al ma ee

“The limiting operation required Lo compute the ensemble averages by means of time averages is si not
pracdical in most situations and therefore replaced with a finite sum to provide an estimate of the desired
Statistical properties. For example. approximations to Eg. (2. 174a)-(2.174c) that are often used ae:

eos

Pod 37 Dnt = mo) ln (1 mx) ans)

29 Summary

In is Chapter we intuduced some important and fundamental concept regarding the characterization of
diseret-time signals and system in the time-domain. Certain basic diseret-timo signal that play impor-
tant roles in disrete-ime signal processing have been defined, along with sie mathematical operations
used for generating more complex Signal and systems. Te relation between a coninuons-time signal snd
the discrete time signal generated by sampling the former at uniform time interval is been examined

‘This text deals almost exclusively with linea, time invariant (LTT) diseretetime systems that find
‘numerous applications in practice. These systems are defined and their convolution sum representation in
the time-domain is derwed, The concepts of causality ard stabliy of LTI systems are introduced. Ao
discussed ¡can imponan: class of LTT Systems described by an inpat-output relation compose of a linear
«onstantcociicien difference equation and the procedure for computing ie output for 2 given Input and
inal conditions, The LTI discrete-time system usually classified in terms of the length of ls impulse
response. The concepts ofthe atocomelation ofa sequence and the ross-corelation between a pair of
sequences are introduced, Finally, the chapter concludes with a review ofthe me: domain characterization
fa discreteime random signal in terms of some a its statisteal properties

or further details on discrete-time signals and systems, we refer the reader to the texts by Cadzow
{Cal73), Gabel and Roberts (Gub87}. Haykin and Van Veen [Hay99]. Jackson (8091), Lat (L198),
Oppenheim and Willy (Opp), Sum and Kirk [Stu88).and Ziemer ei al. (24683). Additional materials
fon probability theory and statistical properties of random discrete-time signals can be found in Cadzow
1Cx187),Papoulis [Pupé5), Peebles [Pee87}, Stark and Woods (Sa. and Themen FThe92]

‘Further insightscan often be obtained by considering the frequency domain representations of discrete
time signals and LTI discrete-time systems. These ae discussed in he following two chapters

119. 106 Chapter 2: Discroto-Timo Signals and Systems in the Time-Domain

2.10 Problems

24 Coosider the following length sequences toed for 3 == Y

201452
071-3492

“ets 436 25 00m

e following sequences: (a) ur] = xl] + ye (9) u) = All wir), (est = yla = ul. ae)

sin)

vio

Figure P21

23 Dermine the even and odd part of he sequences aa, pl], and u) of Problem 2.

24 Lat ll and in be even are 0 eal suce, respecto, Fo each te flaming sequence, determine
(a) xin} = aint) win) = eA) int = Ala)

28 Let tn Ain and sin be the perdi sequences with fundamental pig Mi, and Ns spectively.
a ieur combination ofthese ce perl agua a prod sequence? it whats namo pod

24 Decoine he pete conga mie pecs ae aime pars fhe ewig so

(ela = (407), =N <n < N, where A anda are complex number

O) in (245 443 5436 344 7432,
1

120. 2.10. Problems 107

27 ich one fhe felling sciences ate bounded sequen

13) Le) = [Aa where A and ar complex umber and lo] = 1
(0) [sil = Aufl. ne À andar ne complex sum and la = 1
16 [Ala m CHa where € aed ar comes numbers. ad >,
lé = nt
Le Lela = Son (em

2 dan Show that a canal rea sequence nn can be fly recovered om is eve part rfl fer all > 0.
here can he ren rom heed at nll for alm 0

(0 Is posible 1 ly recover a causal complex sequence ln) fom i conjugate animent pst ean?
Can Mee ty recovers fom us compar Sit pan sc]? Dy your mor

29 Show thatthe even and od pat ral sequence ae, pet ven and ed unes.

2.10 Show hate per conjugate syne par tp) an the pee coma asymmetric par sp
of lengli sequence tla). = = A = 1.2 debi Eg. 2 2a) ar 24D) can be altomaney expres ae

soln] = sole) bxain= ML Osa SN ans
yell = tab Male A EME NL es

ehe net and (e. especia tbe conjugate sy and comugat anymmetric par of af

14 Show thatthe period: comte yet par cl ad the peine conjegate antisymmetric par pal}
EM sequence in VS A > Las el Ugh 2 Za) and 248) can ala Pepe as

solr Le + IN ab. Len eat ama
SO Gas am
spate) Jens nd. Lem am) amo
seal) mal. ema

2:12 Show het an absolutly somo sequence has it ents, ha

ft nergy sequence muy note bucle

243 Show thatthe square suma sequence lm] = ol Bq (227) isnot absolutely summable

214 Show tat the sequen ln] = AE = w 4 le square summable bu ot sbilly soma

246 tell nd yin dt rps th een ns par ae sae sees) Pe
the tli
E emo Y dino À tr

2216 Compute he energy of the ns sequence

Ante
y

nee (ER). ocren

217 Determine the are poner und teeny ofthe flog sequences

121. 108 Chapter 2: Discrete-Time Signals and Systems in the Time-Oomain

a) au = ant
0) soln) = nan
(0) ase = Auclair,
(a) au = Asie +8),
248 Expres séquence cfa = 1, =20 <a < 90 nes of hewn ep quence nl
249 ity ie celation between be un sample Sequence Sr] an te ui tcp sequence an given in E, (2.38)
2.20 The following sequence epesent ne period el aisla! sequence of te Fo fn] = Acosta +9)
an - VE -2 - vio 22 Va,
OVE VB YE Y
on
0 150 15,
Determine the values oF the parameters A und ar each ease

2.21 Determine the udn pei ofthe fling perc ne
{a Ange oe,
(0 ala] = xt én + 06m)
(©) Kill 2eou te 057) + 250007701,
(8 Gala «Saint 3) — 4 cou Sa + 045m,
(©) isin] = Sant mn + 065) à OE — ou,
© in = modulo 6
2222 Determine the fondamental perf th pesca sequence rn = À oso) forthe following als the
aa raquo
(WORK. (6) 0.248, (10.34, (9/0 8, (690.75,
2.28 A comimoutaime smo signal g(t) co a à Sample at = nT. 00 = m < 20, generis ine

discrete agence ln] = data?) = can?) ox wha aes oF 7 laa pedis cuenca? Wa the
anal peed of s(n} De = 18 diam and F = 3/6 comes?

2.24 (a) Express the sequences in. >), amd wht Probl 2 a a liner combination of delaye un sample
science

(0) Express the sequenes Al. in), and win) of Problem 21 as a Hncar combination of delayed eit step
sequences

228 Show that he discrete cme systems desenbed by the following cation are ica systems:
E CIA (Be IS Wlan. E IO, CASO, EY OSS), and ip)
Es

226 For ach ofthe flowing discrete me sytem, where ln) ad xi ae. especie, the vtput and impot
sequences detemine wheter or ote system ar (2) cas (3) table, aC hf mann

(as stoi = aaah,
stele ating.

122. 2.10. Problems 109

I si = #4 Ehosin =. Bi a mance cons,
Ad) sin} = 8+ Ea vin Pi a none conta
ir sio] = anal 64a monde com
D sta) = cn sl
2227 Te sen derivative a] a sequences at me san uni apprit by
Mal = an + = Zeb ste
A nl a su denote the utp and inp of discreto tie sy, he system near Ei in variant? I

228 Trent ersten usedorhesamcthingf signal coman by impulse Rep. Kisimplecenneé
by sting window OF oe Length over the Input sequence xn a sample at à time At he ot insta he input
angles nnd be wind are ek der from de anges lo he smallest valu, a he Sample the de
‘He median valve, The output sin ofthe mean er then given

Shak media = Rh cecal Hani ele A A

For example, med 5, 10, 5, —

2. fr the mean ls a linear or noalinea dere sytem? fi

2259 Consider the discretos system characterized bythe input lation (Ca)

here and sn] re respect the pet and output gun. Shu tat the op yn o the ave system
Foran tat rl = aut] wh 1 © Tcaoerges o Vass n> 20 when ie a pa rame Ts he above
Se nar ar monica? oi ne nil Joy pour onset

2230 An algo or he callan ofthe square rot of a number gine by [MIKO2]

. en

sitz sin} = he A sl

here ce = ee wih < ae < LL lo] and pl] are cumsdered a te input and output of a dre me
“yer is the sso near or ponia? 1 u messin? AS = 00, show al Ja] > Va Nowe ia vl
as inital approximation to VA.

21 Develop a general npreton fo the capt vin of an LT it
‘he ip reap sn sen

tye in terms os put at a

1232 A petiole sequence a] with period Xi appli sun input oan LTL discrete-time syst cratic by
impulse response A generating an up stn). si) a pero sequence? Ii what wt pened

248 Conder the following sequences: D le] = ln — 210.5 = 3, Gi) gla] = dal = 1} à dl 421,
‘sy hla) = Aal ln 1 Sin am Ag = cn 21 08d Ar, Derm ie
Following sequences obtained by a tar coton oft po he bone sequence: (o 9181 = a PANE At (O)
al = ial O sd AR

234 Let ato] te a fte eng sequence tine for My <n = Mp ih Np > M, Likewis. I ln] be a ie
Jeng sequncedetind fr My om = My. math Ma > M. Deine nd = Ala bla, (a) What tthe eng of
‘Mn Cov Whats the ange ofthe de for which sl da

123. vo Chapter 2: Discrete-Time Signals and Systems in the Time-Domain

295 Let pla} = recto ada = aya — lao — N. Expreso) in erm of

296 Let lal (ODA e] and hal arta MO Nel Das — Nab Expres Alain ems of
si

237 Prove tha the comvolion sum operations commative ad dire

2238 Considershefoitoming ce sequences

(comin, tale a tn)

ln =

Show tha nil Ou bn À OO ate
2.99 Prove tha the comolation option x acia or subi ad single sides sequen,

240 Show tht the comolution af à lenge MT sequence with length’ sequence leds 10 a sequence of et
nn,

BAR Ler xl be a emg séquence given by

PTE

DS ETS

Determine yl = nr ad show titi ais sequence with maximus sample value Determine
helos oft amples wih te following values N/A. N72 aad M

242 Let an) and hn Be two eng sequences given by

aaa LE
DA RUE ES
Bei ot ee jo a = ei

toring the

2.43 Conder imo real sequences ln and ei] expressed as a sum uf thee respective even and ol pats. he

Mal = hes] + agin. and al = gal + al), For euch ofthe following rence demie iit even ot

A hotel Ko hala Doel
44 Lt nj te the sequence obtained bya ner comojation of two casa inten sequences al and fo
For cn prof yi] and At] Inked below, demie lo) Te rs sample i each egueme ola a =D.
(a) Isi = (1, A, 3, 20,28, 48) tha à EL 2. 3 th
0 LW = 1.3, 6, 10, 18.14, 12,9, SJ (la = 11.2 2 4 Sh,
(©) Vinh = JS 317, 2438. 978 4 1128. 875.67. tm

CRE

245 Consider acusa rete system characterize bya fonder ica, consint-oetice ference equ
thn piven by

Hla} =ayin~ Hb) m2 0.

‘ste sn] and al are, espace. he op and np sequences. Compte epson Fete out sample
‘Meioterme ofthe nial condi 1] and he put samples,

124. 2:10. Problems a

das ee tem tie-in if sl] = E Ith system lar 1 = 17

(bs Rep Wii = 0

des Gonaaize the rests f pnts (a anh) to he ase of am Nih-oder cal discrete me system given by
Er

246 4 causal LT ini system i sald o have an overhao! int ep respon de response exhib an
silty behavior wa decaying amplitudes rund al comi value. Show Oat be siem has no overt
Int step response e npulse response A ofthe System is negative fo alle = 0.

1247 The sequnse of Fitonasl numbers Fin] ix a al sequence dei by
fim) Jin U4 Sm 22

‘sith 10 = ane JIM
(2) Develop aa enact ora oculta ur any a
10) Show that trie impulse responce of a anal LT ste deste bythe ference equi

onen

AIR = Sin yt Are N

248 Considers test order complex digital er characterized by a difference equation
yin) aye = 11 a

aber sin enim sequence la} = Jl + JY iste comple ouput sequence wth sel and ml)
ening ie real and imaginuy pa, and a+ jb compen Conan, Develop an eyeval uo oo
Singh spat va ference car representation ofthe above compen pale, Show ta ie ange put
Sinetron y(t o ala] is Geri by» Second andr donc equation,

249 Determine the expression forthe impale response of he Factor of inca nepolatar of a, (29)
250 Describe expession forthe impale response of he fet oF nea interpolator.

SU Let MO, Ai, and 2) denote te fire Impuls response samples ofthe fro causal LTE system
Problem 256, Show thatthe cocon ofthe difeence agua Share this sytem can be quel,
eternos from these pale expose samples

2.52 Let causa) IR etal ter be desert he ference cation

Erin D rate eu

tee le] ad in} dene the ouput and the input sequences, respective. A) denotes its impale response,

nee Ei Oe

Fre above res how tat pa = Al Sd

2.83 Comer «casado of two cal sable LT syste characterized by impulse responses yn and Buln
Were0 <a Vand 0 8 = 1, Determine te expression forthe impalse espa Nl of U cite

2284 Determine the Impulse response gl ofthe inverse ya of the LT discrete-time sytem of Example 2.28

125. 1 Chapter 2: Discrete-Time Signals and Systems in the Time-Domain

255 Deermine the impulse response glo] harten the inverse assem af the LT rte lime sytem of
Prot 245.

2.56 Consider ihe casa LT system described bythe erence equation
Mad poste + past = divin N
eher andy denote. espetively. saut and por. Deter the difference equation representa of

257 Determine he expession for he mpuls response ofeach of the LTT sytem shown in gue P22

ate

Dacia x 1 E a

ri]

Figure P22

2238 Determiae tie oneal nue response o th system of Figure P23, whee
DN
ate) = Sim = SI Tan 3 4 2

art ini “is

Figure 23

ere 4 D

12.59 Prove tie BIRO salt condition of (273) ao holds for an ET digital fer with a complet impulse
sponte

2240 he cascade connection of uo stable LT systems slo sabe? Jas your anser
241 ts he pari conneion of two sabe LT systems al sab? Just your answer.
2462 Prove tht tc cave comet of to passive (loses) LTT systems algo passive (oes)

7263 nthe pale connection of two passive Costos) LT systems uso pasivo (loses)? Justify our answer.

2.64 Considera causa FUR le oe +1 with an imple respon given by ela. = 0.1. L. Develop
Ube difference eg epresentatin of the for oF Bu, (LI) whore MY N = o cial te diia!
‘Spal fier with an Impulse exponen such that ie] = pl form OL cn

126. 2:10. Problems 118

268 Compute the output of the accumulate of Eg, 0.55) for an input slo] = mul andthe following iit
conditions (a) (=I) = 0, and) 1

246 In in cetanglar med of numerical integration, the integral he gard sie of By. (2.98) exprese.

Ll, ¿ao mr xl om. au
Deep dence qaion repro fie tanga meh mame gran.
247 Deep ec pensation of manga rei me yuem cani y
“JADE ata o.
aac „20

248 Determine he total solution form 2 Oo he diesen equation
TN
wie ini condition 1 = 2
1249 Determine he ol solution form = Oo he derence equation
APCE TEE PET

‘wih he intial contin 2 1 and {21 = 0.

2270 Determine the tal slain fen 0.0 the difference egaton
tn) +0 sta 10.06 — 21 = lol Zen = 1
wih he inital conti yf = 1, nd {21 = 0, when the ering fun

sate

Fu)

2:71 Determine the impulse respon A] ofthe LIT sytem described by the Grece equim
yin] + 0.5916 = 1 = al

22 Determine he impulse response M fe LT system described by he difference equation
Yin} + 0.1910 = 10 06m — 21 = ala) Zen =)
273 Show thac the sum Loin Gs)" convenes if il = 1

214 (a) Brahate the automation sequence ofeach ofthe sequence of Problem 2.

4) Braune he cou cocon Sequence ry) between the sequence x] and yla), andthe cost-sorlation
sequence ra] Between the sequences a] and win] of Problem 21

2228 Determine ihe autocorelaion sequence ofeach of the following sequences and show tht is an evi sequence
mach case Whats the location of the maximum value ofthe autocorrelation toque in each case?
a) af = eu)

L Osnsw
6. eve

CE

276 Determine ie atocerlaton sequence and its period ofeach ofthe alain periodic sequences.

127. 16 Chapter 2: Discrete-Time Signals and Systems in the Time-Domain

cost Mo, where 30 pose ites,
DETENTE

1277 Let X and? be sworando variables, Show that E(X Y) EX + EN) and B10) = CEUX) where ie

28 Determine the value ofthe constant « tha má
‘minimum vive ofthe mean aqua ero.

es te mean-cque enor EUX =, und then nd the
2279 Compa th mean vale andthe variance of the random variables with the probability density funcions listed

ca (Pa)
(a) Cauchy dsribaion: px) re

9) Laplcin distri: px
(©) Boma dit. px) ag (DAE — pF =.
(0) Poisson distribucion: py x) = II, FPS — 0.
(© Rare drain: pute) = 2e SP a

ne above quis, 5 ie Di ea function a) à uni tp Anton

2.80 Show that fhe in random variables X and Y ae statistically independent then they are sso ine indepen
de

201 Prove Bg 224)

zu ı

ln] and nl be eo statistically independent stationary random signals with means my and my and
repasse. Consider he random signa un} obtained by inet consta laa] and

nee nl = ale) +o were a and a constant, Stow Dean ny ad he variance 2 of va are
Ben me mamas by an ae = ale? + oP repent

2283 Let ni and yin bee independent eo-mean WSS cado sigas wih atocorelaios atl ad fl.
respectively” Consider the random sina en] aimes by à Inca sombinauon ck sla] and ya, Le. Wel
ln) + bol, where a and are constants. Express he auocorcinton ad ros cocisico, desma
esta), stem of te and yn}, Wok would e de esl either sls} o in] was aero men

2284 Prove the symmetry properties of Eqs (2.1663) hough 2.1655,

1285 Veiy the inequalies of gp. (21678) tough 2.1678).

246 Prove Eq (2168)

1247 Determine the mean and varisce fa WSS rel signal with an autocomelaton faction given by

A a

128. 2.11. Manan Exercises 15

211 Maras Exercises

NCA Wee a MAL As program to generate the flloing sequences and plot them wing Ue fonction aor (a)
sample sequence nk) un tp sequence ule, nd () rang sequene man). The op parameter species
byte user reihe desire ego Lf he sequence ande sampling frequency Pin Hz. Using Mis program genera
Th fst 100 Samples of each ofthe above sequences wi sampl rte ol 20 KH.

AM 22 The square wave and the samtoth wave are two periodic soquences as sketched in Figure P24. Using he
nets saweooth and square write a MATLAR program to penes the above two vequences and plo them
(ng the función £ Com. The input ta speci bythe ser a desired length L ofthe sabence pak vale A
rhe period 4. For the square wae sequence an anal pe parameter is he ¿y ci, which 3 de
een fe paid for which the signal ste Using this program generat he is 100 samples ofeach of he
ove sequences with camping ate of 20 Ae. à peak Yale 617.2 pene ol 13. a a day elo 60 lo he
Pres

MAS (a) Using Program 2.1 genre the sequences shown in Figure 2.16 and 217.
(©) Generate and plot he complex exponent sequence 2. "PAR for) =n = 100 wing Program 21

MZA a) Wet 2 MATLAB program 1o generate à sinusoidal sequence la) = A cola + 9) and plot the
sequence usage se uncon. Te inp ata speed Ihe user ar desire eng amplitud
he angular requeney a, a ne phase 9 here = ay < andO= @ = Zn. Using hs program ger
Ie sinusoidal sequences shown in Figure 215

19) Generate susi! séquences withthe angular frequence give in Problem 222, Demis the peo of
sch sequence fromthe la and very the esol eo

MAS Generate he sequences of Problem 2210) to 2216) sine MATLAB.

M2 Writes Maran program plot a ominocus time modul signal and sample version ad verify Figure
2:19: You needa une the 101 fanton fo Rp both ps " =

129. 116 Chapter 2: Discrete-Time Signals and Systems in Ihe Time-Domain

M27 Usage program developed in the previous problem, very experimentally hat the family o coninuous sine.
sinus given by By (253 10 0 deta sampled sia

MZA Using Program 24 svete the lec of signal cil by a moving average lle of ent 3 7 and
3 Dres he tana smociting mprov wi an increase in the tngth? What the eet ofthe length othe delay
ren the ad ouput ad o nosy tap

29 Wee a MALAS program impementin the discreto tne system of E. (2.178) a able 229 nd how
uae vipat ye) oF is system for an input ale] = ale] with yf] =U converges to asm > 20.

AM 210 Wake a MATAN program to compute he square oot sing the alg of Eg, (2.178) Problem 2.0 and
show tha ost > yate or an input] ela wih 1] = comento JE sen > so. Pi
the rm as à fun of for several deren vals o. How would you compute the ua eo of à mar a
‘tha vale preter than one?

AM 213 Using he function impz write a MATLAB program to compute and plot the impulse response of causal
fntecimenstona discee ie system characterized by 3 diference equation ofthe form of Ea, (281) Tre pat
wo the program ae te desired length ofthe impulse response. andthe constants [pu] ar of he diferen
‘equation. Generale an pl the fir A] samples ofthe pate response ofthe system of Eq (293)

(M212 Using Program 2 ermine the autocorelatio and de eros orlaion sequences of Problem 294. Are
ur rl same as hose determined in Problem 2.245

M213 Modify rogram 2.7 to determine the nstocoelaion sequence f a sequence compel with unio
Sri random Signal generated sig the Meluneton ando. Using de moses propio demote tat
"uoconctation sequence 0 a aise-comupeed signal exhibits speak at 76 ag

M218 (a) WriteaMart as program to generate the random sinusoidal signal og (2 140) and pot four possible
realizations of te fandom signal, Comment on your ult
is Compuso the mean and vance of à ingle razo ofthe above random sigma using gs. (27) and
(21170, How clove ie your answer 1 hoe give in Example 2.447

M215 Using de Mctunton rand gener a nil distrbtedHength-1000 random sequence in the range
(1D. Using Bas. 6.174) and 1740), compat the mean and variance othe Fandom sena

130. Discrete-Time Signals
in the Transform Domain

ln Section 22.3 we pointed out that any arbitrary sequence can be represented in the time-domain as a
‘weighted linear combination of delayed uni sample sequences [Sia — K)]. An imporant consequence of
this representation, deved in Section 251, i he input-output characterization of an LT digit) Brin
the time-domain by means ofthe convolution sum describing the output sequence i terms o a weighted
linear combination of its delayed impulse responses. We considec in this chapter an alternate description
‘ofa sequence in terms of complex exponential sequences o the form fe-/%"] and 15") where € is à
omen variable. Ths leds 1 hr particulary useful representations of dsrete-tine sequences and
[Erldiseretesime systems in a ransform domain.) These tansform-domein representations ae reviewed
ere along with dhe conditions for their existence and their properties. Mar A8 has been used extensively
toillustrate various concepts and implement a number of useful algorithme. Applications of tase concepts
are discussed in the following chapters.

“The ist ansform domain representation of a diserer-iime sequence we discuss isthe discrete-time
Four transform by which a ime-domain sequence is mapped into a continuous function ofa frequency
variable. Because ofthe periodicity of the discrete time Fourier wansform, he parent diseretetimo se-
‘quence can be simply obtained by computing is Fourier series representation. We then show that for
A lengtb-N sequence, equally spaced samples of is discrete-time Fourier transform are suficient 10
describe the frequency-domain representation of the sequence and from these N frequency samples, the
original N samples of he inerte time sequence can be obtained by a simple inverse operation. These A
Traquency samples constitute the discrete Fourier transform f à leggth-N sequence, sorond ransform-
domain repreentation. We next considera generalization of the discrete-time Fourier transform, called
the z-transform, the Id type of transfom-domain representation ef a sequence. Finally, the transfor
domain representation ofa random signal is discussed. Each ofthese representations isan important tool
in signal processing and is used olten in practice. A thorough understanding of these three tas forms is
therefore very important 10 make best use ofthe signal processing algorithms discussed in this Book,

3.1 The Discrete-Time Fourier Transform

“The discrete-time Fourier wransform (DET) or, simply, the Fourier transform ofa discrete-time sequence
xs a representation of the sequence in terms ofthe complex exponential sequence [e-/="} where is
the real frequency variable. The DTET representation of a sequence if it exis, is unique and the orginal
sequence can be computed from its DTFT by an ievere transform operation. We fist define the forward
teanaform and derive us inverse transform. We then describe he condition for is existence and summarize
it important properties.

"oa agences can te episod De icy domi by mei of ate Fares ere Poble 3.8)

ur

131. 18 Chapter 2: Discrete-Time Signals in the Transform Domain

3.11 Definition
‘The inerte time Fourier wansform Xe) of a sequence x1n] is defined by

Emm an

x)

In general X(e!#) is a complex function ofthe real variable and can be writen in rectangular form as
KA = Niele) + Kimi), 62

where Xuete/9) and Xi(e/®) are, respectively the real and imaginary parts of X(e!*), and are real
functions af o, X(£/”) can alterately be expressed inthe polar form as

(el) a IX (eye, as
where
Bu) = art ei} as

‘Ths quantity Gel) i calle the magne function and the quantity 0a) i called the phase function
‘with bot fünctons again being real funcions of «In many applications, Ihe Fourier transform scaled
the Fourier spectrum and, likewise, IX (| and (e) a referred 1 s the magnitude spectrum and phase
spectrum, respectively. The complex conjugate of X(2/9) is denoted as X(e/9), The relations between
the rectangular and polar forms of X (e/®) follow from Eqs. (3.2) and (3.3), and are given by

rele!) = [X(e/*)| cos Bla),
Kane) = 1X (ein 90),
IX = XR) + Kandel),
Ame)

tan ga = Xe)
es)

Ik can be easily shown that for a real sequence al] [X\e/)| and Xate/0) are even functions of,
‘whereas a) and Xumlel®) are odd functions of w (Problem 3.1),

Note from Eq. (3.3) that if we replace (0) with 8) + 2k, where k is any integer, X(e!) remains
“unchanged implying tha the phase function Cannot be uniquey specified forany Fourier transform. Uatess
‘otherwise stated, we will assume that the phase Function) estricto to the following range of values,

sa) <

called the principal valve. As ilustrated in Example 3.8, be discrete-time Fourier transforms of some

sequences exhibit discontinuities of 27 in thei phase responses. In such cases, is often useful to consider

Analtemate type of phase function thats continuous function of derived fromthe original phase function

by removing the discontinuities of, The process of removing the discomtinuitis is called "unerpping

the ahead th nen page faction wil be dented as (a) wih subscript" nica ha 5
We illuswate the DTFT computation in he following two examples

insane cae Gioia f= yal Be res pn warping TA o name.

132. 3.1. The Discrete-Time Fourier Transtorm ty

@ ©

sem.

SAMPLE Core ng I rn a LA pen

au Sante i =,
sabre we have wend he ampli property the ingle fon.

EXAMPLEAZ | Conte x cool sequence
An) = elek p<
or wand 44/0) ng 4 0.0 ren by

ee

as

pu ae An ol |The sonido an pe of be stove ir imac ane Boon pt Fl 3.
Mamas

‚should be noted here that for most practica discrete-time sequences, their DTETS can be expressed
In terms of a sum of a convergent geometric seres which can be summed in à simple closed form as
¿llusrated by the above example. We take up the issue of the convergence of a general UTFT later in this
‘Ascan be seen from the definition andals from Figure 3.1, the Fourier transform X (e/¥) of sequence
isa continuous function of. Is also perce Function in w witha period 2. To verify nl latter
Propeny observe that

itty $ apt E tn a

=D amen = xt,

133. 120 Chapter 3: Discroto-Timo Signals in the Transform Domain

I os dE, pe Fe pro le pos in
SSN En co loe TS
Ba

ann Ef stew" da on
cst he Imre dr in Ri oom Kan.) ad sea ie âne
ieee al te

To verify tht Eq. (3.7) s indeed the inverse of ig. (3.1) we substitute the expression For Xe") rom
9.0.1) in Bq, ON amving at

O ca

sie ofthe above equation can be interchanged
der this condition we

“The oder ofinegraion and the summation onthe sight
if the summation inside the brackets converges uniformly, Le. if X (0/9) exist.
et From the shove

a 1 weg) = Fo ape LD
Zul wu) = E Y BANS
Now,
Lane
rin lo mee,
da.
Hence, .
Ÿ «a

sing he sampling property of

3.1.2 Convergence Cond

Now, an infinite series of the form of Ba. (3.1) may or may not converge. The Fourier transform X (el)
‘of x(n] sado exist if he series in E (3.1) converger in some sense. It we denote

xr $

as

then for uniform convergence of X (el), the absolute value of the enor (X¢e!™)
roach zero foreach value of ax K approaches 0. e+

ese!) must ap

im, rue rte

134. 31. The Discrete-Time Fourier Transtorm 121
Now fx san abeoiuelssummabe sequence, i.

DITES on

pres] =

Ener e Fone

{or all values of. Thus, Eq (3:9) is a suficent condition for he existence of the DTET X(e!) ofthe
sequence xn}. Note the sequence xn] = aul of Example 3.2 absolutely summable as

and its Fourier transform Xe!) therefore converges to 1/(1 = ae”) uniformly
Since

aye

E wt = ($ m) ‘
an soy summed sequen hs away a Ani egy. Hones, fit cep sequence not
mn) oleo semis, Tee quence nfs of ample vu aaa rt
nee y tame outer Came ls era comio me square Com
SEX wie cs the al ee al er (RC) eel) man apron sr ach
Be pentose

mf

In such a case, the emor |X (ei) — X (e/¥)| may not goto ero as X goes to oo, and the DTFT is no
Jonger bounded ic. ie absolute suramablity condition of Eq (3.9) does not hold. The following example
‘considers such a sequence.

free — xxi du 0.0)

om

sow im Pig 3.2 Ths DTT ndo apto ia Ai ade al cor gi a Section 4
Theloverse DIET of, pe ei by

mit. née co eu

‘ea how ar Es 7h cen te ;
Tete taps mee tm a Aca en

135. 122 Chapter 3: Discrete-Time Signals inthe Transform Domain

Mute)

ure 32: Frequency spose plot of Eg...

ae

Elim =

mc story comer pe) 21) fr al val fa bo congo My pele
ann ae

“The mean-square convergence propery of the sequence he fn} discussed in the previous example can
e further illustrated by examining the plotof the function

Mupxie®) = cas

for various values of K as shown in Figure 3.3. can be seen om this figure tha, Independent ofthe
numberof terms inthe above sum. there are pest the plot of Hr p(e/4) around both sides ofthe point
to = x, The number of ipples increase as K Increase with the height of the largest ripple remainin the
ame forall valves of K. As K goes o Infinity, he condition of Eq. 3-10) holds indicating the convergence
Of Her (2%) o Hy pielo), The oscillatory behavior in the plot of A p.x (e!®) approximating a DIFT
Mae) in the mean-square sense at a point of discotimay, as indicated in Figure 33, is commonly
now asthe Gibbs phenomenon. We shall return to hs phenomenon in the design of FIR filters based
on the windowed Fourier series discussed in Section 7.6.3

“The DTET can be defined for a certain class of sequences which are nether absolutely summable nor
square-summable, Examples of such sequences are the Unit step sequence of Eg. (2.37). te sinusoidal
Séquence of Ea. (2.39) andthe complex exponential sequence of Eq. (242) which are nether absolutely
amiable nor square-summable. For thistype of sequence a discrete-time Fourier transform represento
ls possible by uring Dirac delta functions. A Dirac delt function ¿(a) is a funcion of «with infinite
olga zero wid, and unit area. I is the limiting form of unit area pulse function p(w) shown in
Figure 3.4 as À goes to O satisfying

‘The discrete-time Fourier wansforms resulting from the use of Dirac delta functions are not continuous
unetions of

KXAMPLIE4 mide the complet exponen ence

136. 3:1.. The Discrote-Time Fourier Transtorm 123

pri

Figure 33: Frequency sponse plu of (LU Fr various values or Y = 24.

Figure 24: Uni rex pte fonction.

ADT ge ty

eme D Sata a 2, os
rd) pase fun ad 2 ou = Te lio th sigh ew te sows
“Spin de period funcion of wh prod Zr an aida perdi im Pr.
Ts Ge aa mal. we corte e ta ITT of (318%

E me
RR A ie

be etre wed pl proper mp son ark

137. 124 Chapter 3: Discrete-Time Signals in the Transform Domain

“ale 3.1: Commonly used serie te Four vane pis

‘Sequence Dieree-Time Fourier Transform

aim) '

0 E ans + 20

un

pe E anti ms
E

lado a cha

“Table 3. its the disrete-time Fourier transforms of some commonly encountered sequences,

A full-band diserete-time signal has a spectrum ocevpying the whole frequency range O < Jal < x. Ifthe

¡mite o a portion ofthe frequency range O = lu! <7. itis called a Bandlimited signal. A
Tompass discrete-time signal has a spectrum occupying the frequency range O < jal = ep < m, where
(05 1 called the bandwidth of the signal. A bandpars discrete-time signal as spectrum Occupying the
frequency range O < az £ Jol © au <x, where oy ~ ois ts bandvidth.

3.1.4 Discrete-Time Fourier Transform Properties

There ae a number of important properties of the discrete-time Fourier transform which ae useful in
ltl signal processing applications. These are listed here without proof. However, their proofs are
quite sualghtforward and have been left as exercises. We ist the general properties in Table 3.2, nd the
Symmetry properties in Tables 33 and 3.4.

‘The following examples ilusrate some applications of a few ofthe properties of the DTFT.

EXAMPLES Denain TFT atthe serene
Fatal whet
Lala) = tite ho: ecm here ie

Had = matt atm)
ror Te 3 te DTT ofa ie by

rie

138. 31. The Discrete-Time Fourier Transtorm 125

ing freien property o TT pve in Ta 2, we none tat TT O a) ia by

rire) om

ang neu prop 0 he IT gen in Te 3.2 aie at he To na

FXAMPLESA Dit DET Y lo ) athe sueno ll re
erin) dial) nl + mile Ah. ai
0 Ep 2 a Ta DIT oH py Net Table 3.2 ing ng
a ls DEFT we net be DTV oft” TT e- ed DEPP la ne iO
ate eit repay of Tate 2 we tens rm, dota en
EV due MV ebm po mare

Solving te ae uti We are

Mattie rer
‘The expresion fr the DIET given above is a rtional function in ei, Le. a ato of polynomials in

e/e. The two polynomials are each of frst order. Inthe general cate the DTFTs we shall encounter in

(his book are ratios of polynomials of higher order and are ofthe form,

+ pen

+ dye

x(n) = FE

3.1.5 Energy Density Spectrum
‘One important application of Parsevals relation given in Table 3.2 sin the computation ofthe energy of
‘nite-nergy sequence. Recall from Eq, (2.26) that the total energy of a fnite-enery sequence ein] is
tiven by

con Em
HC = nan Pama ran we ee
e À int [cdo om

“Thus the energy ofthe sequence g[n} can be computed by evalutinghe integral on the ight. The quantity
Sue) = 1G)? em

{is called he energy density spectrum ofthe sequence gn). The area under this curve in the range — =
(© 5 7 divided by 2 is the energy ofthe sequence

139. 128 Chapter: Discrete-Time Signals in the Transform Domain

Tole 32: General properties othe net me Fourier rs of sequences.

Linsey ont ran)

Times an nab

ueno]
Diseno

in on are
Convolution tei Gel miei
Modulation Soin AE, Gott ee a

Pare relation state =

Geel Hie

Table: yan rai fhe ice me rer reson ofa comple sequence

Kent sie
cio xem)

Relat) E A) bracers arten
Heise) Kater) = Levee) = arte hen
co Kieser

slot Inte)

Note: Never) and Kosei ae the conjgate-symmetic and conjuga omite
Dats Kiel) respectively. Likewise. alot and erin re the conjugate sacs aná
Some anse pats lalo, especies

140. ;

3... The Discrete-Time Fourier Transform 127

Table 34: Symcuy rl of the discret ie Fourier amor of rel sequence

Sequence Discreto: Time Fourier Transform
an) tel) = Neale + rate)
soir Kiel)

Er Eine)

Kiely = tei)

Kiel) Ale Je)

Symmetry rom Mint) =» Klee)
DR Ge
arm = = apte

Pen 20 xl dete the even und dpa of nl respective

EXAMPLES. We compute the energy f the sequence pl OL Fri gp we ee

E mins

Lf Mania
= [a

Recall from Eg. 2.105) th the autocorrelation sequence rep] of gl] can be expressed as

Mene A li à a eme sequen

ral = Y einige) = MAO EI -0 fe

Now from Table 33, the DTFT of gl-P1 is Gle~). Therefore, using the convolution property ofthe
DTFT given in Table 32, we observe that the DUET 0: ¿[LG 1€) is given by Gte/=)G(e =")
IGte/®3R, where we have used the fact that fr a real sequence gin], Ge 19) = G*(eJ9). As a resul,
(he energy density spectrum Seg (e) of areal sequence gf] can be computed by taking the DTFT of is
avtceoeelation sequence sale,

Sut D rate av

141. 128 ‘Chapter 3: Discrete-Time Signals in the Transform Domain

Analogous), the DIET Sysie/=) ofthe cros-corelation sequence rzal€] of two sequences fn] and
dn is called the erossemergy density spectrum

Él tutto 02

3.1.6 DTFT Computation Using MATLAB.

‘The Signal Processing Toolbox in Mar an includes anumber of.
disrete-ime signals. Specifically, he functions that ean be weed are £a, abe, angle, and uniera.
In addition, te built-in Matias functions roa 2 and ¿ag are alo useful in some applications.

‘The function £ reg can be used 10 compute the values ofthe DTET of a sequence. described as a
rational function ine in the form of Eg, (3.17) ata prescribed set of discreto frequency points = wr.
For a reasonably accurate plot, a fairly large number of frequency points should he selected. There are
various forms of his function:

den, £. 0),
2 (nun, den, PT),

4 = Frogz tnum.den.w),
Din] = £reaz (num sden, k)
ale) 2 fear num.den.k

IDE) 2 frage trum. den ic; whole’ ,FT), Eregz(num, den)

The function freqz retums be frequency response values as a vector of a DIET defined in terms
of the vectors mun and den containing the cocficients (pi) and |d,) respectively, a a presenbed se of
Frequency points In > Erogz (num, don, ). te prescribed set of frequcnsis between O and 2 are
given by the vectorw. In = Cregz (num, den, £, PT) the vector £ s used o provide the prescribed
frequency points whose values must be in the range 010 #7/2 wi #7 being the sampling frequency. The
‘otal number of frequency points canbe specified by k in he argument of £ ez. In this case the DTFT.
values H are compoted at equally spaced points between Y and and returned us the ouput ata vector
wor computed at equally spaced points Between O and 2, and returned asthe outpul data vector Y
For faster computation, its recommended thatthe number k he chosen as a power of 2, such as 236 06
512. By including *whiote” inthe argument of Freq2, e range of frequencies becomes Oto 2x or 009
Fr, ashe case may be. After the DTFT values have been determined, hey canbe plowed either showing
‘heir real and imaginary pars using the functions 2021 and imag or in term of their magnitudo and
‘hase components using the functions abs and agi e. The function ange computes the phase angle in
fans. If sited the phase can be unwrapped using the function unsrap. freq? (nun, den) with no
put arguments computes and plots the magritude and phase response values ssa function of frequency
inthe coment gure window.
‘We illustrate the DET computation using MATı AB inthe following example,

EXAMPLAN Praga cade teenie var fe DIET lol sequen deeb
a e

cura 32

DiscetocTime Poursar Fratiefate Computas lon

Y Head in the desired jenath of rr
Y Head in the mumtoror and denominarar coateichenta

Au > pue Mmezater coed ties

142. 3.1. The Discrete-Time Fourier Tanstorm 129

PES
Citer Reel par
ES
Hein. laa thts said
Simi" imag loary part")

(Bloc (MB alle (00440
Uitlet whew Spectra)
Alabeli*\onmgn pi 1s label Phawe, rasianarı

‘Teeter gun ehe me cy plik wich be DTT sb
‘seven pd de ca! He coto de ner ande ocn DTF eager
ening pemen fe" They ath ened e rc, Te pe
CET vc pied ur Jn dp lev a lit pad le gd ou
he cams. hl be nd Und Do sl e sync et fe DIET e ol o
AT Tih 3, eT atm cl oa uc a ren a y

‘Ava ene ve ca vs ing e de fc FT

inne. AIR DIE + 008er Fir nine" de + one
TS Bie Tes a Te
‘The np a oe forthe DTT given aoe ar flows:
x ane
‘on =

os progre eit computes he TFT a tap 234 cae legacy pois may puce Donna
y ben compute the al and tines yor nd Be nie ul pú of I TFT a he regen
are posed Aj u Figure 1S. As Con e een resi ine. pls ato pay. cc
(m noun wm 072. TW ige an be moved wing he fect tira the uta)
han pd sonra 9 Figure Le

2 7

Ds “pen /6qr
ta oi

3.1.7 Linear Convolution Using DTFT

‘An important property of the DTET is given by the convolution theorem in Table 3.2, which states hat he
DIET Ye!) of a sequence Yin] generate by the linear consoluion of two sequences, ala] and fk à
Simply given by the product of ther respective DTFTs, Ge”) and Pe"). This impice tat the Vicar

143. 130 Chapter 3: Discreto

ime Signals in the Transtorm Domain

seve ew

© @
Figure 3. Poof thereat and imaginary prs andthe magie ond phate spectrums ofthe DIET of Example 38,

‘igure 46: Unwrapped phase apctnum of the DIET of Example 38.

‘convolution sta] of two sequences, gín] and Al. can be implemented by computing frst their DIFTS.
Go!) and H(e) forming the product ¥(e!") = Ge!) He!) and hen computing the inverse OTT
‘ofthe product, In some applications, particularly in Ue eae of inf length sequences, this DTP: based
Approach may be more convenient to cary out tha tae dire! convolution.

144. 32. The Discrate Fourier Transtorm 181

3.2 The Discrete Fourier Transtorm

ln the case ef a finitelength sequence rin}, O = m < N — 1, there i a simpler relation between the
sequence and its discrete Fourier wanstorm Xe). In fac, fr a lengh:N sequence, only N values
SEX (GIE). called the frequency samples, at N distinct frequency points 2 = 0,0 = k= NÓ
Sufficient to determine xn. nd hence, Xe"), uniquely. Ts leads o the concept of the disrct Fuer,
ans, à second wnsform domain representation tt applicable only toa fit length sequence. In
is section we define the discrete Fourier transform, usualy known as the DFT. and develop the inverse
transformation, often abbrevisted as IDFT. We then summarize is major properties und study especia
wo fins unique properties, Several important applications ofthe DPT, such 2 the numerical computation
‘ofthe DTT and implementation of linear cosvolution, ae also discussed her.

321 Definition
“Toe simplest relation between a int eng sequence sin}. deine for 0 < m < N — 1, and its DTFT
KE 1 onaincd by uniformly sampling Xtc") onthe an Dti Ow = Zr al = INR
ASS NI Proa O

X= amas Dane, OS NI am

Note hat X(A] is also a finite-length sequence inthe frequency domain and is of length N. The sequence
XI] is called the discrete Fourier transform (DRT) ol the sequence x(n) Using Ihe commonly used.

or
EN was

To verify the above relation we multiply Both sides of Eg. 13.26) by WA! und sum the result from n = 6

A resulting in

umwelt Em) me

Y umge cm

25 rene of De DT cre rr dcr Fi mom (DT) tied y ong de DT
cd ced py ps (ae, Tae NDP gd Pi Ws En

145. 192 Chapter 3: Discrete-Time Signals inthe Transtorm Domain

An interchange ofthe orde of summation onthe ightchand side of Eg, (327) ielés
pare
bon
um

ufr]

‘Theright hand side fe above cquationreducesto X|£] by vire fe following identity (Problem 3.30}

Sapte do rer nimes 62

‘tus vertying Bq 3,26 is indeed the IDFT of XI
“The DIT computation is ¡lustrated in he following (wo examples.

EXAMPLE Condé the gi een dele fo D € = N=
Planet. a

Le point DET tsb by applying (0.28) ati in
= 0

oi cis ng pc defies fo =. =A

lexi,

0 | gare om
AT npn by
r= wn van
EXAMPLE 130 Compe be jit DT bee nc
Anim unanrayW OS OSI on
= (à sin) (pora 0) a
hing m ii (3251 me ae 1 DT,
eh Y
A 3 wg Est] as;
king ne feet og 0.20 mE 5) wen be Tot length seg sno 3.29
NA rt =
PDT es

0) Gene

146. 32. The Discreto Fourier Translorm

193

As can be seen rom Eqs (3.25) and (3.26) the compataion of he DFT and the IDFT require, respec

nei appronimatls N complex mul

1 complex additions. However, elegant
ets have been developed o reduce the compatavonal complexity o about N log; M) operator

‘hose techniques ate usually called fst Fourier transformo (HPT, stgo and ae discussed in Section,

AAA, Aca rol of the avalablty of those fast shoes the DFT and the IDF and thie varia

ase on used in digital signal processing application: or various pupae.

3.2.2 Matrix Relations

‘The DET samples defined in Hi. (3.25) an be expressed in mates Form as
X= Dax
where X isthe vector composed of he N DFT samples.

KIKO XI MIN,

xi the vector ol A input Samples,

Shel) 00 TENTE

and Dy isthe N x N DET mtrs ison by
aa 1

0 mi 0 a
Liria ee

ee Sn
un. Luro

here Dj isthe NN IDET marre given by

a wg gio ug laws |

2 floes fern Eqs (340) an (3:42) that

„a ST]

am

eax

039

eo)

ean

am

Gay,

147. 194 Chapter 3: Discrete-Time Signals in the Transform Domain

3.2.3 DFT Computation Using Maras
‘There ate four bin functions in Marı a for he computation ofthe DFT aná the IDF:

FELIPE EL atroce

(AN of these functions make use of PET algorithms which are computationally highly efficent compared!
10 the direct computation of DFT andthe verse DFT.

‘The function ££¢ (x) computes the R point DET of vector, with R being (he length of. For
competing the DFT ofa specific length N. the fonction £ 2 (9) isused. Here PR > Nitin cae,
to the rat N samples, whereas, if RN, the vector is zero padded at the end to make ¡tinto a length
Sequence. Likewise, the function £1 (22) computes the R-point IDFT of a vector %, where R is the
Jeng of x, while £££ (X. 0) computes the IDFT of x, with the size N of the IDET being specified by
the user AS before, if R > Nw ic automaticaly treated tothe frst N samples. wherens R= Ms the
DFT vector is zero-padded a the end by the program to make i into length DIT sequence.

In addition, the function amt (2) in the Signal Processing Toolbox of Matt aa can be used to
compute the Ax DFT matrix Dy defined in Eq (3.40). To compute the imerse of the N 2 N DET.
matrix. one ean use the function con} «afm (NI) /N

‘We lista the application of the above M-fies inthe following three examples

EXAMPLE) Wing Marian we deen poe DP te lg Np e

aint (3; PENE oa
Yosh nt wean gt eh et, Dug execute reg rep hing at and
mn ce DT aM nen e reer a co» Ae hy me an emp te Moe

PT et he col Wp seque magna: cod paw of FT Me eek a
Fe 37 for Nm Sand At = 16 =

Program 32
Altlaeratien oF 197 Compltat ion

N Had in the Length a oF sequence ait Li desired
à eat of tae Der
Ww input (ye An the length of the sections = +1
MO imgut (ape in the Songeh ot the Der sy
Y Ommerate the Lectio cima-domsn eequenco
AA
a Sta poi or

oad: serbes}
Pa aap ete!)

149. 136 Chapter 3: Discrete-Time Signals in the Transform Domain

4 Compute see N-point oF
u = Sef CR

N bist te DPT and ito 10Pr

‘seen x 2,0)

übel (Frequency index X°); ylabel (‘Amplitude )
Sielei’originei Dr? samples’)

pa
Suplor (2, 1.2)

ne ON

Stenin,real(u))

Eitie(’Real part of the cime-donain camples")
Mabel ("Time index n')s Ylabel Amplitude”)
Subplot (2,12)

Stemn,imag(u) )

Eitlel* Imaginary part of che tine-domain samples”)
Hbabel Time index n°11 yiabel ("api ieude")

‘Aste program run, cal for he opt at coming ofthe eng ofthe DET and the lng fe DET. tt
‘ee competes the IDET ofthe rap DPT sequen o£Eg (145) and lots orignal DFT sequence and ts DFT
dias in Figure LA. Note at even though he DPT seguro i sel, IDET i comple me domain,
e a expected.

EXAMPLE AS Lair 3 and 16 fo De ite engin sequen Ala of Eg CAD Frum Eg. (136)
{pot DFTs terete given by

8, fork =

au [Y tre 213,

ces bernie

‘We ermine DTFEX(e™) 0 een comp +512 pint DI wing te Mart progra
Dre

Y Program 34
N Minerical Computation of TFT Using Der

:
"N Generate) the 1angth-16 simisción1 sequence
Il
ESA
N Gooute Aa s12-poine ort
ren
teas:
Slot the frequency response
Leese
ON
POE 6. ae (7). 01
‘abel /Nömalizcc atgular frequency")
brother)

3 be plot of DTFT Xl) along with the DET samples XU} As indicated in this he
fe RR pe eileen er ort

150. 3.3. Relation between tho DTFT and the OFT, and Their Inversos 197

ria DET wpe

"oe
©

igure 3.9. Te magnitudes ofthe DIET Xie!) and the DET XI] o ih sequence ln] of Eq (2.33) with» = à

sn = 16. The DIET i plted a said ine snd the DET amples are shown by sles

3.3 Relation between the DTFT and the DFT, and Their
Inverses

‘We now examine the explicit ation between the DET and dhe Aoi DET of a length-W sequence,
and the relation between the DTFT of a length A sequence and the N-point DET obtained by sampling
he DIF,

151. 138 Chapter 3: Discrete-Time Signals in the Transtorm Domain

3.3.1 DTFT from DFT by Interpolation

As indicated by Eq, (323) the A-point DFT XIA] of a length-N sequence ln] is simply the frequency
amples ofits DTET Xe") evaluated at N uniform spaced equency points, = wx = Zuk/N viano,

<N = 1. Gwen the N-point DFT XI) of a length sequence, I i also possible Lo determine
its DTFT Xe") uniquely To this end, we fist determine x] using the IDF relation of EQ. (3.26) and
then compute ls DTFT using Eg. (3.1), resulting in

won En SPE oe

here we have used Ed. (3.24), Now the right hand summation in the above expression can he eviten

inet)

Banane oan

as

3.3.2 Sampling the DTFT

Consider a sequence (x(t) with a diseete time Fourier transform (DIET) X (0/9), We sample Ne!) at
N equally spaced points au = 27k/N. 0 =k = N = 1, developing ne N frequency samples (Xe!)
‘These N frequency samples can be considered as un N-point DFT YIK] whose N-point inverse DET i 4
Length-N sequence {y{nl).0 =n = N= L
Now, Xe!) isa periodic Function of with u Fourier series representation given by Eq. (3.1. ls
Fer seine xl) are given by Ea, 52. 11 imac o develop the elation tween la) and
From Eq. (3D.

FIR x = x 0100010) Segue am

1870", An inverse DFT of (KI vies

es

ym es

152. 33. Relation between the DTFT and the OFT, and Their Inverses 139

Substituting Ea. (349) in Eq, 0.50), we get

mE ame

Recall rom Bg, (3.28) that

LE wen a [he ter an tm a:

Making use ofthe above identity in Eg. (3.51), we finaly arrive atthe desired relation

vila D damn, Osnsw—1 os»

‘The above relation indicates that lr] is obtained from xin] by adding an infinite number of shied replicas
‘of ln) lo xn), with each replica shifted by an integer multiple of N sampling instants, and observing e
sum only forthe interval O = n = N — 1. To apply Eq, (3.53) o Brite lengih sequences we assume the
samples ouside the specified range ae zeros. Thus, if ln) isa fnite-ength sequence o leogth Mest
‘han or equal to 2. en yin} = sn for O = n < N — 1, otherwise there i a time-domain aliasing of
samples of In] in generating yin] and x(n] cannot be recovered from yn) as stated inte following
example

EXAMPLEAIA Let fo} be eng ene pren by
bir ie 1294 si
t

E gio ou crete Foe water X (eS) La] 4 ely oped pot pve by a = ri
LEA ae pling «pint nee DPF ese ap, ne ae a ee
Ba OSH pm

lalalala 0 23,

o Latin 6.2.3.4 6
1

3.3.3. Numerical Computation of the DTFT Using the DFT

‘The DFT provides a practica! approach 10 the numerical computation of the DTFT of a Aaite-length
sequence, particulary if fast algorithms are avilable forthe computation of the DET. Let X(e/2) de
the DIFT of a lengih-N sequence xin). We wish to evaluate C2

= 2A/MLO LE = M 2 Le where M >= M

153. 140 Chapter 3: Diserete-Time Signals inthe Transtorm Domain

Emma Y certe ass
Defines new semence si chine fo abel y aug wih 0 — aol ample:

Ki

sim), O2 ne NI

sante [air Er EN ss
Making use of ell in By. 1354) we arrive at

xtc) = Y vue

which is seen to be an Asia DET X [E] of the longt-42 sequence aula]. The DFT X 141 can be
ompuicl ser eficient using Ihe FFT Sigrid ı san inieper power of 2

The Matin function "2042, desenbed in Secon 3.16. emplays Ihe above approach o evaluate
Ihe frequency response ul rational DTT expressed as tinal function in €" a prescribed set oF
ire tieqivacies. computes the DET ofthe numerator snl thedenominatorseparaely by considering
cal a ite length seqeencen and ren expresses the ratio ofthe DFT Samples at cach frequency point
local the DTFT.

3.4 Discrete Fourier Transform Properties

{ike the DTET, the DFT alo sais a number of properties th ne usetul in signal processing ampli
cations. Some ofthese properties are csemially Ment to those of the DET while some others ae
Somzsha diferent. À summary of the DFT properties are included in Tables 35. 36, und 3.7. Their
proc ar again quite taightforward and nave been lef us excises, Most of exe properties cam ast
he void using MATAS. We discuss nex inose pruperies that ae dire tom tir Countess Tor
the Dirt

3.4.1. Circular Shift of a Sequence

This propery is analogous tothe ümeshifing propery of the DTIT as given in Table 32. ut with a
subte diflerence. Let us consider len sequences defines for 0 < n © N = I, Such sequences hve
sample valves equal 10 zero form =O and > Al. Esla] is such a sequence, hen. for any ariary
Fate he sifted sequence tal] = al — ms 0 longer defined forthe tanga 0 =n = N
‘We therefore need o define another type ofa shi that will always keep he aid Sequence in Ihe range
03m 8-1. Tmsi uchieved hy defining anew (ype of shit. called he cinsdar shift. ing a modal.
‘operation sccording to

sl an

ann.
For > 1 (ight circular if, he above equation implies

Mindo foray en 2 N= 1

A= EW ine bh, fos ne ne

as

The concept of cicular shift o à Anite-lengıh sequence à illustrated in Figure 310. Figure 3.1060)
¡shows a length sequence an]. Figure 3-10) shows it iculaly shifted version shite to the ight by

154. 3.4. Discrete Fourier Transiorm Properties 10

"Table 3: General properties ofthe DEE,

nei axe anim cies pt
cresta nln whom
Great wat cra
copan ing sl on
Ouah ai DE
Emi mt cumin)
rin! E enana mx

Eo

Lemgtt- Sequence pont DEF

an xia

cin iby)
lens) a
E)
O pull JON ER
pe] exten,

Systm] Am

Noe: see] and ple] ar peroo conjuate-symanete and
peroo Conjate-aninmmetiepancolste tespecivey, Llene.
Kie] Kalk ace podi compac aymmetie and predic
gate at ma pur of Xie pate

155. 142 ‘Chapter: Discrete-Time Signals in the Transform Domain

Table 37: Synmery propensa the DFT oF mal sequence

Rex
Dr

DORE]
Re XD = REX)

Symmevy relations Um] = tow X14} ]
PU = ren

AU = =e X(N)

Note: pel npn] ate the period even and periods a pas

Figure 310: Moston fs cris ti oa inke-enthsequece. (a) all. (D = Hig) = al + Soh 000
(tla gh De

| sample period or, equivalently, shifted the let by $ sample periods. Likewise, Figure 3.10) depicts
arly hited version shifted othe right by à sample periods ar equivalently, shied tothe eh by

‘As can be sen from Figure 3.100) and (0), aright Oncular shift by ma is equivalent toa left circular
shitty X = sample periods. should be noted that Circular shift by an integer number preter
than is equivalent 0 à circular shift by (no)

li we view the length-W sequence displayed on the ciscumference of a cylinder at Y equally spaced
Points, then the circula shit operation can be considered asa clockwise or anticlockwise was ot the
Sequence by no sample spacings on the line.

156. 34, Diserete Fourier Transtorm Properties 143

te APTA

igure 311: Two length sequences

3.4.2 Circular Convolution

“This propery is anslogous to the linear convolution of Eg. (2.64), ut with a subtle difference. Consider
two lengthier sequences, en] and hf]. respectively. Tes linear consolation results ina length. ~ 1)
sequence yuln] given by.

sala = D gimhin mt On 520 —2 am

where we have assumed har both N-lengih sequences have been zero-padced w extend thet Jens 10
2N — 19 The longer length of y) results rom the time several of the sequence A] and its linear
Shiting to she right. The fist nonzero value of fn] is (0) = g(01A(0], andthe last nonzero value of
MAO ZN = 21= GUN — MN — 1

To develop a convolution lke operation resulting in à Tengib-N sequence yeln], we need to tine
‘circular time-reversal and then apply a circular ime-chift. The resulting operation, called a circulo
‘onlution 5 dened below

sein Y etme m] 00
Since he above operation involves tuo length sequences, salen fered 0 as an N-point culo
Somiten, dented ss

yell = M0 ht an
Like the linear convolution, the circular convoluton i commutative (Problem 3.65), Le,
AS he) = An] sin 66

‘We ilusa the concept of circular convolution through several examples

EXAMPLES termine de pou cular oval ofthe me Je sue nd ven

met 201), O
' di cm

an ch. Thea ng een sc) ge
reel = sink) À mis 0 am)

From 0160; re ren

te Sein 25 te sn ro mie uni ae
eect em on ech ge rad mie suma dl tl he ind apt

157. 144 Chapter 3: Discrete-Time Signals in the Transform Domain

set, Mrs, alle, e
ott TH LEE
o o © ©

Figure 412: Te circulate reversed sequence ans ical shied versos: 1) Mma
(MO mshand A mad

DA mel

sets

oy o
Figare 113: Rest of convolution ofthe to sequences of Figure 3.1, a) Cirurcomvelation, and (9 incar
Coral.

a es

ve = Y si
=

“The rent te evened sequence A bon Figure 3.128) By era the pc a ih
‘tat M ma cc val of mi the ange O <= 3 and aa pres, ate

ac OM + AC + ACM +A
DADO UI e aa)

ex, tve E, (3.64 we compete ls

ctth= E simio — mah am
Te men ML} ml baby cry me ing Ae) eighty on peri ae
Figo 3120) Siting de pets gM ~ ja) for ch van m, wen

ye ODA + MEN + Ba} EA
PAD Ont RT am

tng this Roce, we determine the einig 0 caps el Yea) at
Jet ORI + af AU NO + SIA

HUXM4 RRIF OX D+ (eee won
ve} = MEN + SIA RL + MO
UPA AD RDA am

i
|

Jin po ilar sen fe 0 ng gen sí)
Mad fies en ee rod

158. 3.4. Discreto Fourier Transform Properties oe

“The circular comoltion ofthe two length-N sequences an lso be computed by forming the product
ofthe Nopoin DFTs and ten aplyng an N-point IDPT, as indicated Table 35. The process 3
‘stated nthe following example

EXAMPLEI6 We pi he cali common of eit equ FRA 8
Er 3
Tt pt Ge eg quin ny (3495 eb
am eons ie eae MA aap
Jn He. Amann om

Gio = 1426
Umi

HR am

+ tra pe
+ AUST A os
A RE

er D ihn mat hi on OFT oc nC) y

EEE =

Ayo po IDI ih prot Yh = GUIA

se. +

Ta
H ] om

Bike.
2

IE
3

E

=>
a | tn
A

red} A
ch Ka oe en rn im Eee AG, SAN (20)
‘ih ee sr sacd agen 2 ay fon
ONE or
we Ce ET

For LE
DIA
tes Net
fi tee

159. 146 Chapter 3: Disorete-Time Signals in the Transform Domain

pinks 0s ES
We eu dti the 7 ob ala cai of 21) and a

me Ÿ gate mi om
Frec
ON MU GA + AA
‘cli SAA + OI com
Soin ve 17) an 1.78 180

HO} O) = 8
Next ne sump 1 om Ea 17)

HL = eth Y= ee ALO $
A + leid

ca

(OI + 11 ae
ing ES 0.77) ad (AT. Coming the ae per. ware eri sangle of yin
OA + DM AO LL AE 4 90
OI coo
=the +
DAT DEN ELBA] LIAM) = 2 «De OTE Dee
AI + AH BE =
NOF= «DIN = U «De
ti kt fon the we ha Un ph he ed yp nich ge 346 hil by ine
ono of elon Mo
‘The N-point circular convolution operation of Eq. (3.61) can be writen in matrix form as

velo) HO MNT MN ce BT tos
seit ao Ma A| at)
wee) || a Mm a api} | 52) cra)
cl =, INN AA won Lei)

Tre tement i ach diagonal ofthe A> N aun of ag, (3.82) ar equal. Such a mati i led à

3.5 Computation of the DFT of Real Sequences

{n most practical applications sequences of intrest are eal. In such cases, the symmety properties of the
DFT given in Table 3.7 ean be exploited to make the DFT computations more efficient

160. 2.5. Computation of the OFT of Real Sequences 147

3.51. N-Point DFTs of Two Real Sequences Using a Single N-Point DFT

Let ln) and Aln] be two real sequences of tenth N each with GIk] and MIK] denoting tel respective
point DFTs, These two point DFTs can be computed efficiently using a single N-poim DFT NIE]
of a comple length sequence A|] defined by

do} = ein) + tn a

From the above, gf) = Refa nl} and Ala} = kts]
row Table 3.6 we arrive at

CONTRER ox
l= [01 — raneı) om

Note that (A

“UN = Rx

EXAMPLESIN Le bi ess ne tin de compo of te pi UT of the Ye mal arme
Example 3,4 wd ingle 4 pola OPE
Fri, 3 Ge compa sue alo] = la] + lon lee by

24m À tel
yn Orie
x ou 128% Tas
zu || ts ma) use)
x a N = J
x 1 #1 ñ
om she.
COTES a
Therein
Wale ye AR 2 3 om
Sang Ba, (396) and AT Ex D AG AIS) qe
Ce) ith
muvee i=) 0 ben os

fing the rel bind ia Beagle 3.16.

3.5.2 2N-Point DFT of a Real Sequence Using a Single N-Point DFT

Let ln] be areal sequence of length 2N with VI de
¿lr and h(n] of length cach as

ain)

V2n hin) = van +

Osn< 6.89)

‘wih GE) and H denoting her N-point DFTs. Now define acompleslength-N sequence xa] acconding
10 Eq. (383) The DFTs Glk] and HUE] can be computed from the point DET XL) of the sequence
tr] by means of Equ. (3.84) and (3.85)

161. 148 Chapter 3; Discrete-Time Signals nthe Transtorm Domain

Now,
vue Y winds Y opt Y vn + yan

AN
= =
= E amv + why E no

"Note thatthe frst sum on the last expression is simply an X-point DFT GIA] ofthe length sequence
‚Kl, whereas the second sum isan ¥-point DFT MIk] ofthe length-W sequence ln. Therefore me can
Express the 21¥-point DFT VIA] a8

VUI= GLEN + Wu HIER OSKS2N-t, 43:90)

where we he used te identity WZA, = Wand used the modulo sig forthe argument ofthe two N-point
DFTs on the right-hand side since here isn the range 0 <A = 20 = 1

HXAMPLEESI9 Lan eine the int DET VÍA te lng ea sec Ha gv bone
Malet 2,230 11 1),
+

We fie sg ea
TAN 2 ONL Me 2 1
rm Bg £3.90,

Pees

MI Os where O58 57.

‘hs uo D a Ml nn ec in) an fl ee emp a prions
il ml sn PL era fe

Por Gi à MS 64 6 10,
VIS edo pr) ren ae
VO nm Rn

EN ETES
= + ao m4 À m2
DO SCUEN UE EN ET EEE Ju
Min I + mo = en
Y= Orbe mI = (he A ma

162. 38. Linear Convolution Using the DFT 149

ny POTRO ea [ett

a PA a
CES
0 af EE ht l Barmen
A e man

Figure 3:14: DET bass implementation lec comolution of to nie eng sequen

3.6 Linear Convolution Using the DFT

Lineas convolution is a Ley operation in mos signal processing applications. Since an N-point DET can
be implemented very effiienly using approximately Cog, A) arithmetic operations, interest to
Daga methods forthe Implementation ofthe linear conveni EDF. Earn Example 3.17,
Shave already llusrated fora very specifi ca how to implement a ines convoltion using a circula
‘SSovelotion. We frst generalize this example forthe linear convolution of two finie length sequences of
“cava lengths, Later we considere implementation ofthe linear convolution oi lengú sequence
‘with an infinite length sequence.

3.5.1 Linear Convolution of Two Finite-Length Sequences

Let gn) and hin be finit-tength sequences of engins N and A, respectively. Denote L = M4 N
Deine two length sequence,

sam [a am

An). 0 <n< M1

LORS a Mee set, Em

obtained by appending gl] and fn with zeo-valcd samples. Then
ini). css

To implement Fa. (193) using the DFT, we Ar zero-pad gín] with (Mf — 1) zeros to obtain gel,
and zero-pad hn] we (N — D zero t0obtsin Auf}. Then we compute the (+ M ~ D-point DFTs of
{elo} and hen), respectively resulting in GE and H,{é], An (N + M — Depot IDFT ofthe product
EE ress in yz]. The process involved is sketched in Figure 3.14.

"The folowing example uses MATLAR wo Mustate the above approach,

all = alin = yetn

EXAMPLE 228 Progen 35 canbe wo eine te lin main ot seg pieces
Va ds DIT mai ape nd omer the rea ing «rc mar eben. The lng a» te
‘Sherr er Gat nie ute ais ate the to sequence cooled. The pogo pl era
US twee coum oben wing the DET bad approach un fen Utes he al ed du
source obama) aby dc ina coreo ing e Mle coe Using as pagar me vente e
‘Example X17 as demas u Figur 118

À Linge Cmejucion wai ee apr

163. 150 Chapter 3: Discrete-Time Signals in the Transtorm Domain

o ©
gure 3.15: Pos of th out and te enor sequences

othe two, sequencór
Sips type La the ficar sequence « “11
A > Sepet "type im the second sequence > "1;
Y betetmine the larges of tho rarait of conrolisson

4
Nipetacnine the tür St the product
Yan nee eth

‘riot the muituonce generated by IBPT-bises
Convolution ap the ersur from dbrect inner.
à Cometarios
EN
Subp tae 2. 1
eee iyi)
Mabel Elmo Inden af sy label engl vinta’)
Eitknt’nanda of SeT-beyes Linder sou it dont)
ae corte
Lace
HORS
Abal (+ rime Inder ir) 1 Lame tute |
EHRE khonttade oe werse seauence”|

3.6.2 Linear Convolution of a Finite-Length Sequence with an Infinite-
Length Sequence

We consider now the DFF-hased implementation of

tn Y lala A = inl@ tr. 699

164. 36. Linear Convolution Using the DET 151

where ni a finite length sequence «eng M and ln] is oFinnite length (or a inite-ength sequence
Flog much greater than A. There are two diffrent apprasches to solving is problem, as described
below [Sto66]

Overlap-Ada Method

In chic method, we fst segment xa. assumed o be acausal sequence here without any los of generality.
im a set of contiguous te Jenzih subsequences zu) of Len N cach

sin) = El sut mil css
whee
vain = [lem On LE E
Sunstwing Ba, (395) in Eg. (394) we get
sto = Y sain mv, am

ln] = MIO xn 1). Since Aim is of engt M and mn] sof length NY, te near convolution
‘nk nl eof length (N + 44 — 1) AS rest, the desired linear convolution of Eq, (3.98) has been
broken up no à sum of aninfite number of short-length Kear coovolution of length (N + M ) ech,
Each of Dese soc convolution can be implemented using the method outlined in Figure 3.14, here now
the DFTs (and the IDE) are computed on the bass of (X + M — 1) points, There is one more subtlety
to take care of before we can implement lg. (3.97) wsing the DET-basad method,

Now the fist short convolution in Bq. (0.97) given by A{n@ zur], whichis of length (N + Mf = 1).
defined for0 Sn = N+ M2. The second short convolution in Eg (397), given by Mn} ail ls
‘ot length (A + A4 — 1) but is defined for N << 2 + Mf ~ 2, This implies that there isan overlap of
24 — 1 samples between these tuo shor near corvoluion inthe range Nin = N + M 2, Likewise,
the tid con olution in Eq (357), given by MAJO (a), defined for 2N = m < SN + M — 2 cavsing
an overlap beireen the samples of A {and hm) saint for2N < n < ZN + M 2, In genera,
(here sil bean overlap of M 1 samples between tho samples of he sort convolution Min} ala
adi rN = n= PA À M à

‘Tas process is late in Figure 316. Figure 3.164) shows the fist thee length-7 (N = 7)
segtiens sql] of the sequence x(n] of Figure 3.160). Each of these sepments Is convolved with à
fengih-s (Mf = 5) sequence Hin. resulting in length 1 (N 4 M — À = 11) shor linear comvolutions
Sal shown in Figure 3.1600). As can be seen fom Figure 3.166), the lat M = Y = 4 samples of ln}
‘overlap with the fis 4 samples of ya (a) Likewise. the last M — 1 = 4 samples of yıln] overlap with be
firs samples of ya, and so on. Therefore, the desired sence sn] obtained bya linear convo},
‘oF xn and) is ven by

sh sole,
Mal = soln) y,
sia] = sun,

allen 18), hem 217,
Sin valor = ME EEE

165. 152 Chapter 3: Discrete-Time Signals in the Transform Domain

lll
I m IT

Il: _ _

fi tal
‘ong
©

Figure 3.46: (2) Orginal xt 0) gens fn) of ef and () ica convoi of a) thin)

166. 2.6. Linear Convolution Using the OFT 153

Figure 3.17: Uncmpted input signal Le] (sown with sold line) and he filtered noisy signal yin show ith
ashe ie)

‘The above procedure is called the overap-add method since the results ofthe short inca convolutions
‘overlap and the overlapped portions are added to ot the correct final resul

The M le ££¢£-1¢ can be used te implement the above method. There are two differen forms of
is fonction:

A)

here ss the impulse response vector of the FIR ler, x i the input vetar segmented into successive
Scions. amd y is the fered output. In he Art form the input data xis segmented into successive sections
of length 312 cach, whereas in the second form the input i segmented into sections of length à specified
bby ie user, We illustrate Is use inthe following example.

EXAMPLE 121 Ve cc he ating She mone compte sio € ane 2.14 ing Jeri
Iris ne he Ton, ne Program 2 ven below The mp rected ee need

Y Progr 34
À ilustracion or Derio add jo
N Genacate the noise sequia

N Genoese

bar ae
Blah = Aetmttetio,01 oma) de
Mal cata oui)

Loa
N head in the Length of the movteg avecagh! Cll car
M = Inut(/tampeh of moving average filter = ‘hy
À omerate the moving avwcage filter contficlents
bean ah ay

Pestura the overiap-add fiiteriag operation
Pro 81)

be seules
Poca,

2 label tant Leo)
nt

167. 154 Chapter 3: Diserete-Time Signals in the Transtorm Domain

Overlap-Save Method
In imptementin the previous method using the DFT, we need to compute two (N + M — 1)-point DFTs
and one (N 4 M ~ D-point IDFT since the overall linea convolution of ES, (3,94) war expressed as à
sum of shor-engt linear coavotutions of length (N + M 1) each, Lis posible to implement the linear
convolution of Ed. (3.94) by performing instead circular corsolutions of length shorter than (Y + Mf
“Tots end itis necessary to segment xx) ito overlapping blocks |, keep the tems ofthe circula
convolution oF An] with xml that corresponde (othe terms ubtained by linear convolution of ln] and
ala), and throw away the other parts ofthe eieular convolution.
nd the correspondence between the linear and cxcular convoiutions, consider a lengih-t
sequence xl] and a lengh-3 sequence Ala]. Let y(n] denote the result ofa lincar convolution of ala]
‘wilh Ain). Te six samples of yr In] are given by
MORD).
MOT + A1 J4O!
MIO I2I + A1) + A120},
HORAS ZIERT
AU} + AWE,
PORTE 308)
If we append Al] with a single ero sample and convert i into à Jengtra sequence h(n) the 4-poine
sireular convolution yela] of iin] and ln] is given hy
LOL = MODO + MLD + M2
SU MODE + ALTO + M2.
vel2}= HO} 2) + ARE) + AL2Is10).
el = MOD + AU I2 + ARI. 09)
‘Comparing Far. (398) and (3.99), we observe thatthe frst two terms of the circular convolution do nor
correspond to he first (wo tenns of the linear convolution, whereas the lat two ttre of he eraular
Sonvoiution are precisely the sume a the third and fourth terms ofthe linear convolutón ie
PO # ye1O), elt TU
AR MISC
Inthe general cae ofan Y-point circular convolution of alengih-Mf sequencehla] with a length-W sequence
lr] with X > M4 the first M ~ I samples ofthe eireular convolution are incareet andar reject while
the emaining N = AF 4 1 samples correspond tothe correct samples ofthe linear convolution of Ala] ond
EN
Now consider an infinitely ong or a very long sequence x(n]. We break it up as a collection of smaller
length (length-4) sequences xml] as indicated below

sable sin tm. 05053, OSme 00 6.1005
Next. we form
walt] = HO.

‘or enivalenly,
‘Ohl + atten) + HI.

Ol (11+ AL 1) + AIT

O2 4 EL + HO en

AO ES + Al }enl2] © PT

168. 87. The 2-Transtorm 185

Computing te above oem = D, 142, and substituting the values oF set] from Eg. (3.100), we

OL 2 MOLSTO] + ALI AZ), + Reject
Lt) = AGO ALO] + 2! Skee
MATES = Save
wald) = OWE) AUT PORT = Save
alo} = AOL LAN + AUX ES) 2 42x41 = Rejecr
M = MOL LB + ART + HORS] Reject
mall MOLL] € AE + Mm MAL Save
BA = OST + AN pets € A2 = Save
v0] = AO TA) + AIS + 2161. = Reject
wall = MODS + Mela] + 23917 = Reject
Walz) = HOMO + MIS) + AZRA) = 916, Save
aD] = MOBIL AO] + ARIS = SIT. Save

"should be noted that deerrine M0) and 3111, we need 0 form eh

100) 10}, 2

0. « at (D

and compute wf] = Mole fer D
CORTE

Generali above. lt An] be a sequence of length Af, and x an}, the mth section of an infinitely
og sequence In} dened by

< rec 110) and w (1 and save 1112) = 9

toll mV MEN Osa Nm 102)

be of length N with M = NH uf] denotes the N-point circula convolution of Ala] and xfa).
des malo = MANO rh then we reject the Ast M — E samples of um} and “abut the remains
NA 1 saved samples 6€ il] (0 form yz] the linear convolution Of Ha] and xn]. He denote
the caved portion of wal as snl be

soled = (8 igh eee SHA, (3.10)
ten,
sun min MEDI sul. MIS EN 1 on

“The above process is used in Figure 3.18, The approach i called the averlap-save method since
(be input is mented into overlapping sections and part of the resul of he cicularcomoluton are saved
and abuned 10 determine the Imcar convolution resul

3.7 The z-Transform

The dette ine Fourier wunsfrm provides regueney domain representation of discrete ie siglo
and LT sem. Becascof theeomergece conn many casein cra tne Fos canons
of a sequence may ot ens na are tm at pose to make Use of sch ren) don
cheracteriationnhesccasec. A gencrlzaton of he dsc tm Fourertenaform defied 0
io castor, which may ext for many enc for lr lime En ee,

169. 156 Chapter 3: Discrete-Time Signals in the Transform Domain

SS

Bi ee Hs

Orpea sement of he sequence ala Figur 3.1640)
ation, an () sequence Shaina hy rejecting the He fou

170. a7. The 2-Transtorm 187

(does nok exist, Moreover, the use of z-transform techniques permits simple algebraic manipulations
‘Consequently, the 2 transform has become an important too in the analysis and design of digital ter.

"We tint define the z-ransform of» sequence and stud ls properties by Weating tas a generalization
of the discrete-time Fourer uansform, “This lea tothe concept ofthe region of convergence of à 2
transform hati vestigated in detail We then desenhe the inverse transform operation and point out
‘sro sraphtforward approaches forthe computation ofthe inverse o a eal ational transform. Next, tbe
properies ofthe x aransform are reviewed

37.1 Definition
For a given sequence ln), its z-transform Giz) is defined as

Go = Zune À sin”. 10)

here 2 = Rele) + fe) isa comples variable. we let
expression seduces to
Y stew

‘which can be interpreted asthe discret time Fourier transform of the modified sequence {gln}r="}. For
r= Ue. le = 1. the 2 transfer of gr] seduces toi diseret-ime Fourier transforme provided the
Tater exist. The contour [al 1 ea cicle nthe 2-planc of unit radius and is called the unt bel.
Like te diserteime Fourier transform, there are conditions on the convergence of the infinite series
of £4 (3.103), Fora piven sequence, the se R of values of: for whichis zransform converges is called
the resion of convergence (ROC). e follows from our earlier discussion on the uniform convergence of the
sbseretetims Fourier transform that the series of Eq. (3.106) converges if nr" is absolutely summable,

‘rol, then the ight-hand sie of the above

Gee)

2.106)

letnr="| < oo. 3.107

In general. the region of convergence R-of a z-transform of a sequence gl] is an annular region ofthe

AS no

he 0 < Ay- < Rye = 00. Nshouldbe noted ha the transforma defied by Ea. (3.105 a Fom
à Laurent Seis ats an analyte faction at ever point the ROC. Tis a a pls tat ie
Strato un als derivates re continue functions o the compl sara ln ROC
EXAMPLE 122 Lt us erie ihe reto: oe ail RSE a
reerpence. Applying the defintion of Eg, (1.105), we obtain er

Kiam À it Dares im
Tr ar uw sen come m

Brio

171. 158 Chapter 3: Discrete-Time Signals in he Transtorm Domain

“able: Some commonly use raton paire

Sequence Roc
int r Allyaesof¢
aon est

eu | 1e > tal

OF est)

nant pe eee,

‘Toe 2-ransform (ofthe unit step sequence ni canbe obtained from Eg, 3.1 10)by setting = 1

for

ae eu

1
weds

‘The ROC of (e) ss the anlar region 1 < el < oc. Nut that the unit step sequence i not absolutely
Summable, and. asa result its Fourier transform does not converge uniform.

‘EXAMPLE 23) Cmdr the mc sic af
penton ho à tans

ifn Ua CS ne ee a te

un

‘hee nent LOC testa rio e <

A should be noted that in both ofthe above examples, the z-transform are identical even though hei
parent sequences are diferent. The only way a unique sequence can be associated with a z-transform 1s
bby speciing its ROC. We shall discuss further the importance af the ROC in de following section

It fotlows from the above tas the Fourier uansform Ge”) of sequence fla] converses uniformiy if
and only if ie ROC ofthe transform G (2) ofthe sequence includes the unirte On the oer hand, the
‘existence ofthe Fourier transform does not always imply the existence ofthe = tansfor. For exampl
finite-energy sequence A pln] of Kg, (3.12) has a Fourier transform he ple) given by Eg, (3.11) whieh
converges inthe mean-square sense. However, this sequence docs not have a transform shear”
is not absolutely summable for any value of 7

Some commonly used z-ransform pair re lated in Table 38.

172. 3.8, Region of Convergence of a Rational 2-Transtorm 159

37.2 Rational z-Transforms

Inthe case OFLTI alscrete-time systems that we are concerned with in this ext, ll pertinent transforms
ar rational functions of 2". Le, ae ratios of tv polynomials in 2

Pi) _ pot pe
De)” ay edie

‘where the degree of the numerator polynomial P(2) is M and tha of the denominater polynomial DL) is
IN. An alternate representation of a ational :-tfansform is asa rai of two polymomalsin =

Ge) au

PT ET piel +

6 ens

“The above equation can be atenately ven in factored form as

ee) = O82) yay TL to STE

Alu” TP ea)
‘Ata wot = & ofthe numerator polynomial, G(&) = 0, and asa result, these values f are known asthe
zero of Gte). Likewise, a ao < = hy of the denominator polynomial, GCAL) — ec and these paris
tn the plane are calle the pores of GC). Observe from the expression in Ea. (3115) that here are Mf
Site zero and finit poles of Gz). It also follows from the above expression tha! there are atonal

(8 — M) zeros at 2 = À (te origin in the > plane) iE N > M or additional (M = N) poles at = Dif
N < M. For example, the -tansform (2) of E (3.111) can be rewritten as
foetal >. ens

which has a zero at = O and a polea
A physical interpretation of the concepts of pols and zeros can be given by plating the log-magritude

2010810166). Now 2010p |G (2) sa two-dimensional function of Reiz) and Im). Hence splot

will describe a surface in the complex -plane as ilostatd in Figure 3.19 forthe rational ztrnsforen

lt can be seen from this figure that the magnitude plot exhibits very large peaks around the points £ =
Did 40.6928 which are the poles of Ge), and very narrow and deep wells round the location ofthe
vores at = 2 LD

3.8 Region of Convergence of a Rational z-Transform

‘The ROC of zransform is an important concept for a variety of reasons. As we shall show Isr
without the knowledge ofthe ROC, theres no unique relationship between a sequence and is ans om
"once, the z-transform must always be specified with its ROC. Moreoven if the ROC ofa eiransform of
sequence includes the unit cre, Ihe Fourier transform of the sequence ls cbiained Simply by evaluating
{he z-transform on the uit cil. In the following chapter, we shall point ou be relationship between the
ROC of the z-transform of the impulse response of «causal LT system and ity BIBO A Iris Ue
‘of merest 1 investigate the ROC more thoroughly.

173. 160 Chapter 3: Discrete-Time Signals In the Translorm Domain

y

Figure 349: The2-D plo 2010210010) a à funcio of Reiz) and Im).

igure 320: The pole zero plot andthe eon of comergece of Zn.

Now; the ROC ofa rational transform is bounded by the locations ofits poes, To understand this
relationship between the pales and the ROC, iis lnstrctne examine Ue plot o the ples and the zeros
‘of a zransform. Figure 3.20 shows the pole-zero plot ofthe z-transform wu) of Ea, 1.110). where the
Location of the pot 3 indicated by across" and the location ofthe eros indicated bya circle =". In
this figure the ROC, shown as the shaded area, is the region ofthe 2-plae just ouside the circle centered,
‘al the origin and going through the pole atx = L and extending al he way ole = 00.

EXAMPLICADA, Determine tee ROC of te ome 12) niece Mk = LU me
En OM pete i
CRE. om

prende Jel > Qi. Thin ais ai te AOC la Ja a on circle goin mg the = 6 and,
Seas ee

174. 35. Region of Convergence of a Rational z-Transform 161

Pigure 324: Rae er plot of 210 6) an

In general. the ROC depends on thetype ofthe sequence of intrest as defined etter in Sesion 2.1.1
‘We examine inthe next four examples the ROCs of the z-transform of several diferent pes of sequences.

EXAMPLE LAS Come «Hei sequen gl] seine Yor a < 10 ha At and ae
ete ges ae = 20 To aan given by

ñ AS
= $e peg EE ae MAD, am

=
‘re MO ptm Eos = nn m er:
DE te nmr ts gdh one age ern rn ie po eh

rare

XAMPLE226 A juste sequen slo] wäh user sole ses oly Trt = a sometimes cli
cana sequen I uma pen y

a = Emi" on

Hebe yeti eee Mean pe Epoque run
DS Tee Se EE Eine. ro]
DS SA os my ur ee ce A Ce

AE ei A i si i a <a i
cal sequence a à trans given bY ys

wie À nie a

E
Somerset ta Teen
SEHEN EURER

175. 162 ‘Chapter 3: Diserete-Time Signal inthe Transtorm Domain

ie carefree sce Win] can be pres

ede ecw A
pepe ey omer TER.
Sn es pyre
AL La ela
=

EXAMPLES Tatonka define y
llanos.

he a be y el comple mu. ot bave a non, ese be sche var a, Th
lis by stag Qu the uncer eeprom cae res 2

Lehm Dreta D ee ou

“The ru er oh al de ol Tp 3.223 comer So L > Wo. on x econ tr ones for
IN Di. and Anne. de ns reap a pa ROCK

Fora sequence with a rational z-transform, the ROC ofthe z-transform cannot contain any poles and
is bounded by the poles. This property of such ztransforms can be seen Gom the z-transform of a unit
‘step sequence given by Fa (3.116) and ¡lstrate by the pole zero plot of Figure 320. Another example
isthe sequence defined in Example 3.24 and its z-transform given by Eq. (3.117), with te corresponding,
Pole zero plot given in Figure 321

To show tat is Boanded bythe poles, assume thatthe -ransform X) has simple poes ata and 4,
with al = IB ithe sequence à also assumed to be a right aided sequence then i i of the form

(029 +0089") ala = Na

Ga)

L love] < 00

forsome z. N cam be seen hat he above holds for | > | but no for] ly. The right-side sequence
‘of Eq, (3.123) has us an ROC defined by JAI < [2 < 90. A similar argument shows that sf X.) isthe
transform of left-sided sequence ofthe form of Eq (3.123) with uln Ne) replaced y yel~n ~ Ne).
then is ROC is defined by 0 la < lal. Finally fora two sided sequence, some of the poles contribute
to terms forn = 0 and the others lo terms for > 0. The ROC is thus bounded on the outside by the
pole with the smallest magnitude that contributes for m < O and onthe inside by the pole with the largest
magnitude that conbutes form = 0.

igure 3.22 shows the three posible ROCs of a rational z-transform with poles 5 =e and = Band
‘with each ROC associated with unique sequence. In general, ft rational 2 tansform has Y poles with
A distinct magnitudes, thea ithas 1 ROCS and, as a resul, R + 1 distinct sequences having the sume

176. 3.8. Region of Convergence of a Rational 2-Transform 188

Figure 322: The poleo lt af a rational ans with ve poule ROCs corresponding to ce diferen
Sequences. (a Right vequenc. (1) tone sequence, ná (2) ede sega

‘ational -ransform. Consequently, a rational -tnnsform witha specified ROC has a unique sequence as
its inverse z-transform. A rationa z-transform without a specified ROC is thas not meuningful

Matas can be uted to determine the ROCs ofa rational z-transform. To this end, several functions
‘need tobe cd, The statement (z.p,k) = ©22P (num, den) determines the eros, poles. and the
sin constant ofa rational <-tansform expressed as arab of polynomials in descending powers of = as
in Eg, 3.113. The input arguments are the row vectors an and den containing the eseieints ote
‘numerator and the denominator polynomials in descending powers of 2? The ouput les are e column
vectors z and p containing the zeros and poles ofthe rational z-transform, andthe gan constant, The.
Statement (nur, den] « 2p2t€(=,p, k) is used to implement the reverse proces.

‚From the zero pole description he factored form of te tanner function can be obtained using the
function sos = 202805 (2, D, k). The statement computes the coefficients ofeach second-order actor
given as an 1. x 6 matrix soe, where

dor bir bay au au an
soon | 22 ir an an om

bu Dir be a a ane

were the kth row contains Ihe coeficiets of he numerator and the denominator ofthe kth second-order

fn cdi one of

177. 164 ‘Chapter 3: Discrete-Time Signals in the Transform Domain

factor of ie z-transform GC)

fo Por + bus + bu à
ne a

OT an Fa TF ene

The pole-zero plo of a rational z-vansform can also be plotted by using the Melt plane. The
_=-ransform caa be described either i terms oft zeros and poles given as vectors ¿ezos and poles or
in terms ofthe numerator and the denominator polynomials entered as vectors nus and dr, containing
‘oetficients in descending powers of 2

<plane(zeros,poles), _zplane (num, dem

should be noted thatthe argument Soros and paleo must be entered as column vectors, wheres the
arguments num and den need to be entered as row vectors, The function zpLanc pts the eros and
poles inthe cuen igure window. with the zero indicated by the symbol “o” and the pole indicted by
the symbol "x." The unit circle i also included inthe plo for reference. The automatic sealing included
inthe funtion can be overwritten i necessary ®

‘The following two examples illustrate the application ofthe above functions.

FXAMPLEI29 quese folowing ono it frre ir plo plea er. sl om rine

Bett 100) y ate? y Se #38
ME EF ho
o y eran et ining MLA pa
y pare a

À Deinsalnation of the Factores Form
Sot « Rational. e-tranatoes

Due: impar (type in the minmeratar cont tieseate + #14
dea © cope ("Type in He Sensminater corttistasts » *)y
sepia = eezep tau dan à

aloe

Siapl'2aro are atts sataptes:

babe Polen are ae") dimpiplı

isp {Guin eonazant baimothbs,

Lap í“hadius of goles) AG mnIR)

foe = cplaneig.psi)

api "Second. ceder. sect lana’) dlmpérent (noe)

plane iman. den)

ate ne arm ks cs O pe domine loi Te
cal tera cero ern re Lg TE ET
‘the secado section an indices bom? a7 gi

al

one wanna ae Tine Prag Toles or
Noten MATLAB, di wl nota o =! nad

178. 33. Region of Convergence of a Rational z-Transform

179. 166 Chapter 3: Discrete-Time Signals in the Transform Domain

4
E ox
pr wo =
al

RE

Figure 323: Pole-zer pot ode 2 ras org 3.126.

180. 59. Inverse 2-Transtorm 167
3.9 inverse z-Transform
‘We aow derive the expression forthe inverse ¢-tansform and ouline (wo methods fr is computation.

3.9.1 General Expression

Recal that, for = re! the transform (2) given by Eg, (3.105) is merely Ihe Fourier transformo he
modifica sequence gta". Accordingly, by the inverse Fourier tansform relation of Eg. (3.7), we have

payers = sf Gare el de. 126)
a faena u

unchanged when C is replaces with any contour C encircling the point = 0 in the ROC of Gc). The
Sontour integral in Bg, (3127) ca be evaluated using the Cauchy's residuo more CHU} resulting in

stn = Y [resides oF or! we pes inte C] om

Note hat Eg.(3.128) ceeds tobe evaluated a ll aies of and isn pursued here, Two simple methods
forthe inverse wansform computation are reviewed next

3.9.2. Inverse Transform by Partial-Fraction Expansion

‘The expression of Fg, (3.127) can be compute in a number of ways. À rational z-nansform Gi) with
se transform gl] has an ROC that i exterior to a cite. In ths case its more convenient

te express Gc) in a paralfraction expansion form and then determine gIn] by suraming the inverse

‘eansforms ofthe individual simpler tent in the expansion, A rational Ge) can be expressed ax

cos? 129)
= De) e

where PG) and 42) are polynomials in 2 as indicated in Eq. (3.113). Ihe degree M ofthe numerator
polynomial FC) is greater than or equal to the degree N of the denominator polynomial DIS). we can
vide P42) by D(z) and re-express Gi) an

See Ets

6.30)

where the depre ofthe polynomial (+) is ess than that of D(z). The rational function Py(s)/D(2)
called proper fraction.

181. 168 Chapter 3: Discrete-Time Signals in the Transtorm Domain

Rassen

1s cm be men ha der os namen poy nomial 3. berms he gr a demon poy mi
1 Sine De meses epee pester ha Re denna degen, he seve foal obey mn a
rer fasion We can pre 4 num aa pele n« ud a rept Pace in nm E. DO)
E rang u mn y desc hat te amero ft rer a he pre

‘Tota cod we sy the ma mim wich th polyol cert ere ner ct eats

‘Simple Poles

ln most practical cases, G(s) Iss proper fraction with simple poles. Let the poles of Ge) beat z
OSH = N, where ar distinc. À paria) fraction expansion of G(2) then is of the form

An

a ken aus
eee conta te sor error al he dea ie y

pe = UAG

E Gas
Each erm of he sum on the rghchand side of Eg. (3.131 has an ROC given by 2 > Pe] and, hs, an
inverse transform of the form 94)" adn], Therefore. the inverse transform 41) of G(s) is given by

Emoorum. any

Nowe te ab ppm visigodos canaiobewed ae ter stom

rn open wi ate eae
A ea emai tinge

poles cms nde fl name

(CAMEL ea pe hg

2007
TFT u

al

2,120
De #0)
A pasa ram pain Hc) fe for

He

ans,
hing A Fe obli

182. 3.8. Inveree 2-Tranaform 160

pam 14047
Sabeis howe 13.135) we ar

278 pes

Mo apt
‘The ete te of stor le gro y
inh = 279027 ao) = 175-047 a CP

ra

1 G) has muhiple poles, dhe prtia-fraction expansion is sighily of diferent form. For example, if the
poleatz = vis of multiplicity £ andthe remaining N — L pole are simple and st = Au Eo Mh
then the general partial fraction expansion of Gi) takes the form

+S tarda oun

where the constants}, (0 longer called the residues for + 1) are computed using the formule

oa la | sin. 659

and the residues pr ar calculated using Eg, (3.132). Techniques for determining the inverse z-transform
‘of terms like y /(1 = v2") are described later in Example 3-41

3.9.3. Partial-Fraction Expansion Using MATLAB

The M file res iduez can be used o develop the parti fraction expansion of a rational =-transform
and to convert a z-transform expressed in a partial fraction For L ls rational form. For the former
case he statement is [r,D,k} = residuos nur, den). where the input data ae the vector mn
amd des contining the coefficients of the aumeraor and the denaminator polynomial, respectively,
expressed in descending powers ofc, andthe output fies are the vector = ofthe residues and he mumeratr
constants the vector po corresponding poles, andthe vector k containing the constants y. The statement
nur, den) - vesiduez (r.p.k) isemployed to cary out the reverse operation. The applications
‘of these fonctions are considered in the following two examples.

EXAMPLE S9 Lg MATLAM we dei he pri tión espais cli ra GU)
C2

ou
we:
‘otis de ne ie flloing MATLAR program:

Y Fartial-Fract on Enjanslön of Rational, »-Transtarm)

Tun» Ant Type in mueracor SAFE lien = the

183. er &: Discrero-Time Signals in te Transtorm Domain

170

den = input ("type in denoninacor coefficients = +)
(r.p.k] = raoidues (men, den);

Lap’ Residues’) dep

Step (Poles) ¡ala (pr)

Sisp(Conatants') .disp(k)

During execution the progra cal for he ip dat ht are ie vector m and ef he cocino he
‘numerator nd he denominator pin respectively, in detending powers of 2 These data re entre ne
gare racket a lows

nun = (18)

Gene 3 4 11
‘The coto ta re he resides nd be const, he poles ofthe expansion oe) ne fom of a. (2.1372.
[orca above cape, hse ae ven lo

609 0.2400 9.4000

Poles
0.5000 0.511 -0.3333

constants
"ote sno hat tutor fF (3.139) hs double pest = —1/3 = 0.333, The fa ty ba Ue
‘eset un poles given above corenpoga o slo po nce (1 = 0827), De eo

ty conde
es à
ee Ea

a | be
T7031 * Tos * TO

Go

EXAMPLE 334 env conser de deteminaio of the ron! fom of a ana foc para
incon expunion representan 2 pre by Ei (3.10), The MATLAD rogram that can be sed tnd the
‘onl form tives bon

A Program 3.10
Y Partial fraction Expansion to Rational +-franatorm,

y

E = input (“Type in the residues + +);

Dom input ("type in the poles = “Ja

KS input(‘type in the constante = +) ute
(man, den) = rericuer ir.pıkl

dipl Nemacator polynomial coefficients"); disp (num)

‘disp Denominator polymanial cosfficionte"11 disp (dea)

During execution, the program requests the input dain vector r of residues, the vector p of pote locations, and the
sector kof comas, with ach enced using square bracts Slows

E = (0.4 0.26 0.361
© 0.3333 “3.3393 0.5)

184. 89, Inverse z-Transtorm m

en pren oc
ea a
‘sch cote I

manent and dere polymenah, res. eae pee:
‘fe seen tha he cick ue ey hese in E ELO) we nay

3.9.4 Inverse z-Transtorm via Long Division

For causal sequences, the c-ransform GE) can be expanded ino a power series in 2". In the series
‘expansion, the coefficient multiplying the term +=" is then the nth sample gin}. For a rational Gt),
A convenient way to determine the power seres is to express the numerator and the denominator 38
polynomials in 2 , and then obtain the power seis expansion by Jong division.

EXAMPLICIIS Camaño ct chf Example 3.22 gem. ie mme
ni osos ve

Mit og,

sa
Tovar sal

Lt
Be
spose} ma
Tee a=
Tags oa
rue
“OS "Om? nde
Bar bebe
tos +0 10 + — oi
3 Fo

A le am her

ee Lo tal sa ue = names

le “6 04 02M) fenzo,

185. 172 Chapter 3: Discrste-Time Signals in the Transtorm Dora

3.9.5 Inverse z-Transform Using MATLAB

‘The inverse of a rational c-ransform can also be readily calculated using MATLAB, The function impz
can be uilzed fortis purpose. Three versions ofthis funtion area follows:

Shee) = Impztnum,den), [MEL + ape (numaen, 1),
init) = Impe(num don, La PT)

‘where the input data consist ofthe vectors num and den containing the coefiientsofthe numerator andthe
denominator polynomial ofthe z-transform given in descending power: of 2, he output impulse response
Vector h, and the time index vector €. In ho fst form, he length 1-0f fis determined automatically by
the computer with © = 0:21, wheres in the remaining two Fons iis supplied by the user through
‘the input datum 2, Inthe Jas form, the mo intervals scaled so tha he sampling icra is equal o the
reciprocal of #7. The default value of #7 i I

‘Another way of arriving at thisresultusing Maran is by making use ofthe M fe £12 tex asindicatet
below

y + Filter (nun, den, x)

where y is the output vector containing the coeficiente of the power series representation of (2) in
increasing powers of 2-1. The numeraor and denominator cocíficiens of #/() expresed In ascending
Powers of 27} are woof the input data vectors num and don. The length of x i the same as that of y.
And its elements are all zeros except forthe fist ote which sa)

‘We present next two examples 1 sat the use of Dot funcions,

SAMPLE tre te the in Say ems et soscicn
en mao of (3134 aan end. we ete folio ng MATA proa

ÓN
Y Power dering Kapannion of a national x-vanifore

DAN + input (type inthe mismestor cdot ticience + +}
den : input (“type in the denomisator sonffirtenen = Mr
Eonpul the denied sunber of Inverse’ tensions

+ coutticteste

feet) = es tie ent;
Gisp\‘coetfielents Of the fewer sécian
Cet

pans té}

A
A A

fom = 13 a
den = Da ua

186. 3.10. z-Transtorm Properties 173

“Tee opos das wee dees nee? Uno secc, 2 ces teow

ootticiente of the power mari
usa D'éneugn >
3.9908. 1,0000 2:59, 2.4000

at 6.00

‘rh seem ae ke rap 3.3 vom Long rem apg,

‘EXAMPLE AS? We eta ie nr E CA) MASA pan cn a
16 terrine me anti ls sl! > adi len los

kon of a Nacional + remet

of fsvèrue s-trmateen soatttatonce:

= stgueltype in the length of output vectar « 434
end {5 the rueerator and deremipatos Costiicionta Of
che a eranatnn

x
À 22 be compet

in the mimerator eoattioients = À
im the denominator copeticianca + fly
Fad Roger of inverse tramntcen

N couttiateate
ad. u

#2 Filterinun. den 2
Sisptitoattictants of the power serien qxpansién’);
inp!

‘Ax program in un so cr te esi! engi of he capt veto y ele bythe vec Sting
‘the muster wl dencia coefficient enter id tae bashes and Im msn fren oo, AÑ
opus th ouput oc. pr herent a ae oia a pcs expe ot ea
sudo The progr sienes that > aaa à ger facto he pee te dei
‘being eae th e ap m are mer

‘As api of he above pero the 2 ae of fy, 0.130) ye the lowing cpu

Codttictente of the pote envies expansion
40900 1.8080 ~pvaang 0.0000 0.2388

‘ch so Be ital wo hat rive in Example 35 wang the Longo apre

3.10 2-Transform Properties

Ne summacizs in Tale 3 some specie properties ofthe z.ransarm. An understanding ofthese prop
ies makes de aplicain of he = ns techiques ts he anal and con ol gl las en
casio Te readers therefore encouraged to vr ese proper, We conside e optan of some
‚of these properties in the next several examples. es

187. 174 Chapter 3: Discrete-Time Signals in the Transform Domain

“Table 39: Some wel properties ofthe stone

Pope Seqoence rato Roc

q a

a m3 Ei
Carson ci or Ro
mer 42) ours um
Lacio ain] + Ahi) GING) de RARA
Tinesbiting ena tow e cepo

Bed

Maps by
Pr) Ste) ta
a ne
Dire sae cep psy
40 vin a Peer
Cool tak owe Andes Rey
Madson Hint by fe GO do Mes Rey
Parera! elation E tante = Le OOM ay

OP AZ
= Py Ses,
TETE pe

188. 3:10. 2-Transform Propertes. 175

189. m Crapor3: Der: Tme Sal hh Tansorn Doman
Te ducemine the rene armor of 1/64 + OLD A we ctra trom Table 3:9 an the imei of
MIP al] em ay

i 2
no Cs) ea

pm he amare ea u 1 + (DEE i given by 34 = MLS ie I Thee
ne rar of GC) OÙ Ha ie

taj fois (-4)" + 09687] uta + 03600 —14(-4)" ate 1)

‘The time several property and the convolution theorem of Table 3.9 can be employed to develop the
expression forthe z-tunsform of he cross-correlation sequence ré] of two sequences g(a) and Ala]
An terms of their transforms. Let the z-transform of gín) be denoted by OL) with an ROC Re, and
Jet the z-transform of gin] be denoted by PIC) with an ROC Ra. Now, recall fram Eg. 2.106) ha! the
‘ross-corvlaton sequence ryu[£] can be expressed in terms of convolution as

ral) = (OM.

Using the timereversal property. we observe thatthe z-transform of Al] is simply (271). Therefore,
using the convolution theorem we obtain from the above equation that

2 (ral) = Game

With the ROC given by atleast Ry Ra

‘Asin the case ofthe discrete-time Fourier transform. the Parsevals relation forthe z-transform given
in Table 39 can also be used to compute the energy ofa sequence. To establish the required formula, we
Jet lal = Ale, where gin) real soquence in the expression given in Table 39. This leads to

deme

here € is a closed comour in the ROC of G(e)65(e-1). Note that ifthe ROC of Ge) includes the uit
ir, then that of G(e" will also include the unit circle. Ya fact, for an absolutely summable sequence
ln}, tbe ROC of its z-transform Ge) mus include the unit ciclo In this cae, we ean let 2 = edo in
Ea. (3.144), which then reduces to Bq (3.18)

a1

face ets ous

Er

3,11 Transform-Dom:

Representations of Random Signals

‘The notion of random discrete-time signals was introduced in Section 2.8 along with thie statistical
characterization in the time-domain. These infnite-length signale have infinite energy and do not have
‘wansform-domain characerizations like te determinisuc signal. However, the autocorelation and the
autovovariance sequences of stationary random signals, defined by Eqs. (2.147) and (2.150), are of Bite
energy and, in most practical cases, their ransform domain representations do exi. We discuss these
représentations inthis section

190. 3:11. Transtorm-Domain Representations of Random Signals w7

3.14.1 Discrete-Time Fourier Transform Representation

The diseret-time Fourier transform ofthe autocorrelation sequence pxxlt] of a WSS sequence Xin],
‘defined in Eq. (2.147) i given by

Oxxtel»

Ÿ exter, japon Su)

nd is usually referred to asthe power density spectrum or simply, the power spectrum 1 of Kin]. Is
denoted by Pxx (0), The above relation between Ihe amtocorclation sequence and the power spectrums
‘more commonly known as the Wiener-Khintchine iheorem. slfcint condition oc the existence of te
power spectrum P(e) is hat the autocorrelation sequence dx x be absolutely summable. Likewise,
the isret-time Fourier ransform ofthe autocovariance sequenoo yrx(lofs WSS sequence str], defined
in Eg, @.150),is given by

rare e À pant ji en aus

I fen nn fr tenso xt) dt sauce sequence ye te ab.
coll surmb AIN de us im Pot ann E OO a de
notation Py x(a) = xx (e!*), we arrive at
eg
Ef penne en
ont = L [7 Prater aus
fotos om Es. Aaa (2.162) hat
DE PP.
2 (xis?) mort À [paré ess)
Ths Or o era por ae ado siga Xi) Sindy. een want ok
PS ane
route 3 fax it a au

From Bas. (3.149) and (2.163) we get

oF = rel

1

Ef ranas

[roto mr 0150)

E

Applying the discrete-time Fourier transform to both sides of EQ, (2.1668), we can show thatthe power
spectrum Pr () of aWSS random discrete-time signal (Xin) is aesi-valed function of, nadas,
if (Xn) is à real random signal, Pyx(a) is an even funcion of a Le, Pula) = Prt a). We sal
demonstrate later in Section 4.132 that or à teal-valued WSS random signal, Pyx(a) = 0

191. 178 Chapter 3: Discrete-Time Signals in the Transform Domain

Likewise the disrete-time Fourier wansform of the eross-comelaton sequence 4x [6] of two jointly
Stationary randorn sigeals (X(r)) and Ta}, given y

Sate) À onrtele olen ausm

is usually refened to us the cross-power special densi or erus-power spectrum. It is denoted by
Pro) and, in general, sa complex function of >. À sufiient condition for Ihe existence of Py (o)
is thatthe crose-coreltion sequence @xy[e] be absolutely summable. Appiying the discrete-time Fourier
transformo both sides of Eg. (2.1666), we can show that Pro) = Pry(a). Similarly, we can define
the discrete-time Four transform ofthe cross-covarance sequence yxy E] as

Pire À yerttle wien casa

tence of Pl is rat the cross-covarianos sequence yxy(£] be

absolutely sunmable.
The reaion between the discrete-time Fourier transurms of the autocemelation sequence and the
tutocovariance sequence can be derived from Eq (2.161) and is given by

Partei) = Paxlo) — 2x img). lai < a, 015

where we have used te notation Pyxlu) = Oxx(e/%). Likewise, he relation between the discrete-
time Fourier wanstorms of the crss-correltion sequence and the cross-covariance sequence follows from
Eg, (2.165) and is given by

Parce

Part) —2nmxm Sw, loi <x, 0158)

‘where we have used the notation Pay (o) = Dre.

3.11.2 z-Transform Representation

Ascan be seen from Eqs (3.183) und 3.154), the Fourier transforms ofthe sequences yal£) and yale)
‘contain impulse functions. As a result, ei z-tansforms do not exist in general, However, for zero:
‘mean stationary random signals. the z-transform of the aLtocorelaon sequence, rx (2) and tat of
the crost-coelaion sequence, x(), may exit under certain conditions. Since the aulocorrelaion
and cross-corelation sequences ae two-sided Sequences, their region of convergence must bean anlar
region ofthe form,

1
Rickey 015)

Wi can generalize some of the results of the previous section ifthe z-transform exit. For example,
fomthe symmetry properties of the power spectrum Px x (0) and eros power spectrom Per (wit follows
‘that xx (2) = Dix (1/27) and @ xy (2) = 05 (1/2. also follows from Es. 2.150) and G 153) tat

fort

Wire C is closed counterclockwise contour inthe ROC of Ole).

de, 0.156)