An Introduction to Parametric Digital Filters and Oscillators

263 Pages·2003·3.63 MB·English

by  Mikhail Cherniakov

An Introduction to
Parametric Digital Filters
and Oscillators

An Introduction to
Parametric Digital Filters
and Oscillators
Mikhail Cherniakov
University of Birmingham, UK
Copyright2003 JohnWiley&SonsLtd,TheAtrium,SouthernGate,Chichester,
WestSussexPO198SQ,England
Telephone[phone]
Email(forordersandcustomerserviceenquiries):[email protected]
VisitourHomePageonwww.wileyeurope.comorwww.wiley.com
AllRightsReserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystemor
transmittedinanyformorbyanymeans,electronic,mechanical,photocopying,recording,scanningor
otherwise,exceptunderthetermsoftheCopyright,DesignsandPatentsAct1988orunderthetermsof
alicenceissuedbytheCopyrightLicensingAgencyLtd,90TottenhamCourtRoad,LondonW1T4LP,
UK,withoutthepermissioninwritingofthePublisher.RequeststothePublishershouldbeaddressed
tothePermissionsDepartment,JohnWiley&SonsLtd,TheAtrium,SouthernGate,Chichester,West
SussexPO198SQ,England,[email protected],orfaxedto([phone].
Thispublicationisdesignedtoprovideaccurateandauthoritativeinformationinregardtothesubject
mattercovered.ItissoldontheunderstandingthatthePublisherisnotengagedinrendering
professionalservices.Ifprofessionaladviceorotherexpertassistanceisrequired,theservicesofa
competentprofessionalshouldbesought.
OtherWileyEditorialOffices
JohnWiley&SonsInc.,111RiverStreet,Hoboken,NJ07030,USA
Jossey-Bass,989MarketStreet,SanFrancisco,CA94103-1741,USA
Wiley-VCHVerlagGmbH,Boschstr.12,D-69469Weinheim,Germany
JohnWiley&SonsAustraliaLtd,33ParkRoad,Milton,Queensland4064,Australia
JohnWiley&Sons(Asia)PteLtd,2ClementiLoop#02-01,JinXingDistripark,Singapore129809
JohnWiley&SonsCanadaLtd,22WorcesterRoad,Etobicoke,Ontario,CanadaM9W1L1
Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappears
inprintmaynotbeavailableinelectronicbooks.
BritishLibraryCataloguinginPublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
ISBN0-470-85104-X
Typesetin10.5/13ptTimesbyLaserwordsPrivateLimited,Chennai,India
PrintedandboundinGreatBritainbyAntonyRoweLtd,Chippenham,Wiltshire
Thisbookisprintedonacid-freepaperresponsiblymanufacturedfromsustainableforestry
inwhichatleasttwotreesareplantedforeachoneusedforpaperproduction.
To my wife Irina
and our sons
Pavel, Alexei and Andrei

Contents
Preface xi
1 Introduction: Basis of Discrete Signals and Digital Filters 1
1.1 Discrete Signals and Systems 1
1.2 Discrete Signals 3
1.2.1 Time-Domain Representation for Discrete Signals 3
1.2.2 Presentation of Discrete Signals by Fourier Transform 4
1.2.3 Discrete Fourier Transform 9
1.2.4 Laplace and z-Transforms 11
1.3 Time–Invariant Discrete Linear Systems 16
1.3.1 Difference Equation and Impulse Response 17
1.3.2 DLS Representation via Transfer Function 20
1.4 Stability and Causality of Discrete Systems 22
1.5 Frequency Response of a Discrete Linear System 23
1.5.1 Properties of the Frequency Response of a Discrete Linear
System 25
1.5.2 Transfer Function versus Frequency Response 25
1.6 Case Study: Low-Order Filters 27
1.6.1 Purely Recursive Filters 27
1.6.2 Effects of Word Length Limitation 37
1.6.3 Transversal and Combined Filters 37
1.7 Summary 41
1.8 Abbreviations 42
1.9 Variables 42
1.10 References 43
Part One Linear Discrete Time-Variant Systems 45
2 Main Characteristics of Time-Variant Systems 47
2.1 Description of a Linear Time-Variant Discrete System Through
Difference Equations 48
2.2 Impulse Response 49
2.3 Generalized Transfer Function 52
2.4 Signals Analysis in Frequency Domain 55
viii CONTENTS
2.5 Sampling Frequency Choice for Linear Time-Variant Discrete Systems 59
2.6 Random Signals Processing in Linear Time-Variant Discrete Systems 61
2.7 Combinations of Time-Variant Systems 63
2.7.1 Parallel Connections 63
2.7.2 Cascade Connections 64
2.7.3 Systems with Feedback 66
2.7.4 Continuous and Discrete LTV Systems 68
2.8 Time-Varying Sampling 70
2.8.1 Systems with Non-Uniform Sampling 70
2.8.2 Systems with Stochastic Sampling Interval 75
2.9 Summary 77
2.10 Abbreviations 78
2.11 Variables 78
2.12 References 79
3 Periodically Time-Variant Discrete Systems 83
3.1 Difference Equation 83
3.2 Impulse Response 84
3.3 Generalized Transfer Function and Frequency Response 85
3.4 Signals in Periodically Linear Time-Variant Systems 86
3.4.1 Bifrequency Function 86
3.4.2 Deterministic Signal Processing 86
3.4.3 Random Signals Processing 89
3.5 Generalization of the Sampling Theorem 91
3.6 System Stability 95
3.6.1 General Stability Problem 95
3.6.2 Selection of Stability Criteria 96
3.6.3 Stability Evaluation 97
3.6.4 Stability of Parametric Recursive Systems 99
3.7 Stability of Second-Order Systems 100
3.8 Stability of Stochastic Systems 107
3.9 Summary 114
3.10 Abbreviations 114
3.11 Variables 115
3.12 References 116
Part Two Parametric Systems 119
4 Parametric Filters Analysis 121
4.1 Non-Recursive Parametric Filters 121
4.2 The First-Order Recursive Parametric Filter 123
4.2.1 Impulse Response 123
4.2.2 Generalized Transfer Function 126