Handbook of Real-Time Fast Fourier Transforms: Algorithms to Product Testing

484 Pages·1995·51.89 MB·English

by  Winthrop W. Smith

Handbook
of Real-Time
Fast Fourier Transforms
IEEE PRESS Editorial Board
John B. Anderson, Editor inChief
R.S.Blicq S.Kartalopoulos I.Peden
M.Eden P.LaPlante W.D.Reeve
R.Herrick J.M.F.Moura E.Sanchez-Sinencio
G.F.Hoffnagle R.S.Muller D.J.Wells
R.F.Hoyt
Dudley R. Kay,DirectorofBookPublishing
Carrie Briggs, AdministrativeAssistant
Lisa S.Mizrahi,ReviewandPublicityCoordinator
Susan K.Tatiner,Project Manager
Russ Hall, Senior AcquisitionsEditor
Ross A. McClain, Jr.,Joanne M. Smith,
and Winthrop W.Smith, Cover Designers
Technical Reviewers
VitoJ. Sisto
E-Systems, Inc.
James S.Walker
Mathematics Department
University ofWisconsin, Eau Claire
John C. Russ
Materials Science andEngineeringDepartment
North Carolina University
Handbook
of Real-Time
Fast Fourier Transforms
Algorithms to Product Testing
Winthrop W. Smith
Joanne M. Smith
+IEEE
The Institute ofElectricaland Electronics Engineers,Inc.,NewYork
mWILEY
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Wiley-IEEEPress ISBN0-7803-1091-8
LibraryofCongressCataloging-in-PublicationData
Smith,WinthropW.,(date)
Handbookofreal-timefastFouriertransforms I WinthropW.Smith
JoanneM.Smith
p. em.
Includesbibliographicalreferences andindex.
ISBN0-7803-1091-8
1. Signalprocessing-Digitaltechniques. 2. Fourier
transformations. 3. Integratedcircuits I. Smith,JoanneM.,
(date) . II. Title.
TK5102.9.S58 1995
621.382'2'0285416-dc20 94-12936
CIP
To our family and friends,
who encouraged us
Contents
Preface xxi
1 Overview 1
a
1. Introduction 1
1.1 Laying the Foundation 1
1.2 Design Decisions 2
1.2.1 Numberof Dimensions 2
1.2.2 TypeofProcessing 2
1.2.3 Arithmetic Format 2
1.2.4 Weighting Functions 3
1.2.5 TransformLength 3
1.2.6 Algorithm BuildingBlocks 3
1.2.7 Algorithm Construction 3
1.2.8 DSP Chips 3
1.2.9 Architectures 3
1.2.10 Mapping Algorithms onto Architectures 4
1.2.11 Board Decisions and Selection 4
1.2.12 TestSignals and Procedures 4
1.3 Types of Examples 4
1.3.1 Eight-Point DFT to FFT Example 5
1.3.2 Algorithm Steps and Memory Maps 5
1.3.3 Fifteen-Pointor 16-Point
FFT Algorithm Examples 5
1.3.4 Sixteen-PointRadix-4 FFT Algorithm Examples 5
1.3.5 Four-Point FFT and 16-Point Radix-4
FFT Algorithm Examples 5
viii CONTENTS
1.4 DesignExamples 6
1.4.1 DopplerRadar 6
1.4~2 PowerSpectrumEstimator 6
1.4.3 SpeechRecognition 6
1.4.4 ImageDeblurring 6
1.5 Conclusions 7
2 The DiscreteFourierTransform 9
2.0 Introduction 9
2.1 CommonUsesoftheDFf 10
2.2 EquationandBlockDiagram 10
2.3 Properties 10
2.3.1 FrequencyLimits 10
2.3.2 DFfFilterSpacing/Nulls 12
2.3.3 Linearity 12
2.3.4 Symmetry 12
2.3.5 InverseDFf 12
2.3.6 EaseofIDFfComputation 12
2.3.7 TimeandFrequencyScaling 13
2.3.8 TimeandFrequencyShifting 13
2.3.9 Parseval'sTheorem 14
2.3.10 ZeroPadding 14
2.3.11 Resolution 15
2.3.12 Periodicity 16
2.3.13 SummaryofProperties 16
2.4 RealInputSignals 16
2.4.1 Two-SignalAlgorithm 17
2.4.2 Double-LengthAlgorithm 18
2.5 Strengths 20
2.5.1 PeriodicSignals 20
2.5.2 RealorComplexInputData 21
2.5.3 SetsofData 21
2.5.4 CoherentIntegrationGain 22
2.6 Weaknesses 22
2.6.1 ComputationalLoad 22
2.6.2 QuantizationNoiseError 23
2.6.3 HighSidelobes 23
2.6.4 FrequencyStraddleLoss 23
2.6.5 TransientSignals 23
2.7 Conclusions 24
CONTENTS ix
3 The FastFourierTransform 27
3.0 Introduction 27
3.1 ImprovementstotheDFT 27
3.1.1 ComputationalLoad 28
3.1.2 QuantizationNoise 28
3.2 FFT-SpecificWeakness 28
3.3 Eight-PointDFTtoFFfExample 28
3.3.1 Eight-PointDFTEquationsinMatrixForm 29
3.3.2 180°RedundantComputations 30
3.3.3 90°RedundantComputations 30
3.3.4 45°RedundantComputations 31
3.4 Building-BlockConstructionofFFfAlgorithms 32
3.5 Conclusions 34
4 Weighting Functions 35
4.0 Introduction 35
4.1 SixPerformanceMeasures 35
4.1.1 HighestSidelobeLevel 36
4.1.2 SidelobeFall-offRatio 36
4.1.3 FrequencyStraddleLoss 36
4.1.4 CoherentIntegrationGain 36
4.1.5 EquivalentNoiseBandwidth 36
4.1.6 ThreedBMain-LobeBandwidth 37
4.2 WeightingFunctionEquationsandTheirFFfs 37
4.2.1 Rectangular 37
4.2.2 Triangular 38
4.2.3 SineLobe 39
4.2.4 Hanning 40
4.2.5 Sine Cubed 40
4.2.6 SinetotheFourth 41
4.2.7 Hamming 42
4.2.8 Blackman 43
4.2.9 Three-SampleBlackman-Harris 43
4.2.10 Four-SampleBlackman-Harris 45
4.2.11 Kaiser-Bessel 46
4.2.12 Gaussian 48
4.2.13 Dolph-Chebyshev 49
4.2.14 FiniteImpulseResponseFilterDesign
Techniques 52
4.3 WeightingFunctionComparisonMatrix 52
4.4 Conclusions 53
x
CONTENTS
5 Frequency Analysis 55
5.0 Introduction 55
5.1 FivePerformanceMeasures 55
5.1.1 InputSampleOverlap 55
5.1.2 SidelobeLevel 56
5.1.3 FrequencyStraddleLoss 56
5.1.4 FrequencyResolution 56
5.1.5 CoherentIntegrationGain 57
5.2 ComputationalTechniques 57
5.2.1 Nonoverlapped 57
5.2.2 Overlapped 58
5.2.3 WeightingFunctions 58
5.3 Conclusions 59
6 LinearFiltering andPattern Matching 61
6.0 Introduction 61
6.1 Equations 61
6.2 ThreePerformanceMeasures 62
6.2.1 NumberofComputationsperDataPoint 62
6.2.2 NumberofDataMemoryLocations 62
6.2.3 ComputationalLatency 63
6.3 DirectMethod 63
6.3.1 ComplexInputSignal 63
6.3.2 RealInputSignal 63
6.4 Single-StepFrequencyDomainMethod 64
6.4.1 ComplexInputSignal 64
6.4.2 RealInputSignal 64
6.5 Multiple-StepFrequencyDomainMethod 65
6.6 Overlap-and-AddFrequencyDomainAlgorithm 65
6.6.1 Introduction 65
6.6.2 ComplexInputSignals 65
6.6.3 RealInputSignals 67
6.7 Overlap-and-SaveFrequencyDomainAlgorithm 68
6.7.1 Introduction 68
6.7.2 ComplexInputSignals 69
6.7.3 RealInputSignals 70
6.8 LinearFilteringandPatternMatching
ComparisonMatrix 70
6.9 Conclusions 71