Fast Fourier Transform - Algorithms and Applications

437 Pages·2010·11.21 MB·English

by  K.R. Rao

Signals and Communication Technology
For furthervolumes:
http://www.springer.com/series/4748
.
K.R. Rao D.N. Kim
l l
J.J. Hwang
Fast Fourier Transform:
Algorithms and Applications
Dr.K.R.Rao Dr.D.N.Kim
Univ.ofTexasatArlington Univ.ofTexasatArlington
Electr.Engineering Electr.Engineering
NeddermanHall NeddermanHall
YatesSt.416 YatesSt.416
76013ArlingtonTexas 76013ArlingtonTexas
USA USA
[email protected] [email protected]
Dr.J.J.Hwang
KunsanNationalUniv.
SchoolofElectron.&Inform.
Engineering
68Miryong-dong
573-701Kunsan
Korea,Republicof(SouthKorea)
[email protected]
ISSN1860-4862
ISBN978-1-4020-6628-3 e-ISBN978-1-4020-6629-0
DOI10.1007/978-1-4020-6629-0
SpringerDordrechtHeidelbergLondonNewYork
LibraryofCongressControlNumber:2010934857
#SpringerScienceþBusinessMediaB.V.2010
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Preface
This book presents an introduction to the principles of the fast Fourier transform
(FFT). It covers FFTs, frequency domain filtering, and applications to video and
audiosignalprocessing.
Asfieldslikecommunications,speechandimageprocessing,andrelated areas
are rapidly developing, the FFT as one of the essential parts in digital signal
processing has been widely used. Thus there is a pressing need from instructors
andstudentsforabookdealingwiththelatestFFTtopics.
This book provides a thorough and detailed explanation of important or up-to-
date FFTs. It also has adopted modern approaches like MATLAB examples and
projectsforbetterunderstandingofdiverseFFTs.
FastFouriertransform(FFT)isanefficientimplementationofthediscreteFourier
transform(DFT).Ofallthediscretetransforms,DFTismostwidelyusedindigital
signal processing. The DFT maps a sequence either in the time domain or in the
spatialdomainintothefrequencydomain.ThedevelopmentoftheDFToriginallyby
Cooley and Tukey [A1] followed by various enhancements/modifications by other
researchershasprovidedtheincentiveandtheimpetusforitsrapidandwidespread
utilization in a number of diverse disciplines. Independent of the Cooley-Tukey
approach, several algorithms such as prime factor, split radix, vector radix, split
vectorradix,WinogradFouriertransform,andintegerFFThavebeendeveloped.The
emphasis of this book is on various FFTs such as the decimation-in-time FFT,
decimation-in-frequencyFFTalgorithms,integerFFT,primefactorDFT,etc.
In some applications such as dual-tone multi-frequency detection and certain
patternrecognition,theirspectraareskewedtosomeregionsthatarenotuniformly
distributed. With this basic concept we briefly introduce the nonuniform DFT
(NDFT), dealing with arbitrarily spaced samples in the Z-plane, while the DFT
dealswithequallyspacedsamplesontheunitcirclewiththecenterattheoriginin
theZ-plane.
A number of companies provide software for implementing FFT and related
basic applications such as convolution/correlation, filtering, spectral analysis, etc.
on various platforms. Also general-purpose DSP chips can be programmed to
implementtheFFTandotherdiscretetransforms.
v
vi Preface
This book is designed for senior undergraduate and graduate students, faculty,
engineers, and scientists in the field, and self-learners to understand FFTs and
directly apply them to their fields, efficiently. It is designed to be both a text and
a reference. Thus examples, projects and problems all tied with MATLAB, are
providedforgraspingtheconceptsconcretely.Italsoincludesreferencestobooks
andreviewpapersandlistsofapplications,hardware/software,andusefulwebsites.
By including many figures, tables, bock diagrams and graphs, this book helps the
readerunderstandtheconceptsoffastalgorithmsreadilyandintuitively.Itprovides
newMATLABfunctionsandMATLABsourcecodes.
Thematerialinthisbookispresentedwithoutassuminganypriorknowledgeof
FFT.Thisbookisforanyprofessionalwhowantstohaveabasicunderstandingof
thelatestdevelopmentsinandapplicationsofFFT.Itprovidesagoodreferencefor
any engineer planning to work in this field, either in basic implementation or in
researchanddevelopment.
D.N.KimacknowledgesthesupportbytheNationalInformationTechnology(IT)
Industry Promotion Agency (NIPA) and the Ministry of Knowledge Economy,
RepublicofKorea,undertheITScholarshipProgram.
Organization of the Book
Chapter1introducesvariousapplicationsofthediscreteFouriertransform.Chapter2
is devoted to introductory material on the properties of the DFT for the equally
spaced samples. Chapter 3 presents fast algorithms to be mainly categorized as
decimation-in-time(DIT)ordecimation-in-frequency(DIF)approaches.Basedon
these,itintroducesfastalgorithmslikesplit-radix,Winogradalgorithmandothers.
Chapter 4 is devoted to integer FFT which approximates the discrete Fourier
transform. One-dimensional DFT is extended to the two-dimensional signal and
then to the multi-dimensional signal in Chapter 5. Applications to filtering are
presented in this chapter. Variance distribution in the DFT domain is covered. It
also introduces how we can diagonalize a circulant matrix using the DFT matrix.
Fastalgorithmsforthe2-DDFTarecoveredinChapter6.Chapter7isdevotedto
introductory material on the properties ofnonuniformDFT (NDFT) for the none-
quallyspacedsamples.NumerousapplicationsoftheFFTarepresentedinChapter8.
Appendix A covers performance comparison of discrete transforms. Appendix B
covers spectral distance measures of image quality. Appendix C covers Integer
DCTs. DCTs and DSTs are derived in Appendix D (DCT – discrete cosine
transform, DST – discrete sine transform). Kronecker products and separability
are briefly covered in Appendix E. Appendix F describes mathematical relations.
Appendices G and H include MATLAB basics and M files. The bibliography
contains lists of references to books and review papers, software/hardware, and
websites.Numerousproblemsandprojectsarelistedattheendofeachchapter.
Arlington,TX K.R.Rao
August2010
Contents
1 Introduction ............................................................... 1
1.1 ApplicationsofDiscreteFourierTransform .......................... 2
2 DiscreteFourierTransform ............................................. 5
2.1 Definitions ............................................................. 5
2.1.1 DFT ............................................................. 5
2.1.2 IDFT ............................................................ 5
2.1.3 UnitaryDFT(Normalized) ..................................... 6
2.2 TheZ-Transform ....................................................... 7
2.3 PropertiesoftheDFT ................................................. 13
2.4 ConvolutionTheorem ................................................ 18
2.4.1 MultiplicationTheorem ....................................... 24
2.5 CorrelationTheorem .................................................. 24
2.6 Overlap-AddandOverlap-SaveMethods ............................ 27
2.6.1 TheOverlap-AddMethod ..................................... 27
2.7 ZeroPaddingintheDataDomain .................................... 31
2.8 ComputationofDFTsofTwoRealSequencesUsing
OneComplexFFT .................................................... 32
2.9 ACirculantMatrixIsDiagonalizedbytheDFTMatrix ............ 34
2.9.1 ToeplitzMatrix ................................................ 34
2.9.2 CirculantMatrix ............................................... 34
2.9.3 ACirculantMatrixIsDiagonalizedbytheDFTMatrix ...... 35
2.10 Summary ............................................................. 37
2.11 Problems ............................................................. 37
2.12 Projects .............................................................. 40
3 FastAlgorithms .......................................................... 41
3.1 Radix-2DIT-FFTAlgorithm ......................................... 42
3.1.1 SparseMatrixFactorsfortheIFFTN¼8 .................... 46
3.2 FastAlgorithmsbySparseMatrixFactorization ..................... 47
vii
viii Contents
3.3 Radix-2DIF-FFT ..................................................... 56
3.3.1 DIF-FFTN¼8 ................................................ 57
3.3.2 In-PlaceComputations ......................................... 61
3.4 Radix-3DITFFT ..................................................... 61
3.5 Radix-3DIF-FFT ..................................................... 63
3.6 FFTforNaCompositeNumber ...................................... 66
3.7 Radix-4DIT-FFT ..................................................... 67
3.8 Radix-4DIF-FFT ..................................................... 73
3.9 Split-RadixFFTAlgorithm ........................................... 75
3.10 FastFourierandBIFORETransformsbyMatrixPartitioning ..... 78
3.10.1 MatrixPartitioning .......................................... 78
3.10.2 DFTAlgorithm .............................................. 80
3.10.3 BT(BIFORETransform) ................................... 82
3.10.4 CBT(ComplexBIFORETransform) ....................... 82
3.10.5 DFT(SparseMatrixFactorization) ......................... 82
3.11 TheWinogradFourierTransformAlgorithm ....................... 83
3.11.1 Five-PointDFT .............................................. 83
3.11.2 Seven-PointDFT ............................................ 84
3.11.3 Nine-PointDFT ............................................. 85
3.11.4 DFTAlgorithmsforReal-ValuedInputData ............... 85
3.11.5 WinogradShort-NDFTModules ........................... 87
3.11.6 PrimeFactorMapIndexing ................................. 88
3.11.7 WinogradFourierTransformAlgorithm(WFTA) .......... 90
3.12 SparseFactorizationoftheDFTMatrix ............................ 92
3.12.1 SparseFactorizationoftheDFTMatrixUsing
ComplexRotations .......................................... 92
3.12.2 SparseFactorizationoftheDFTMatrixUsing
UnitaryMatrices ............................................. 94
3.13 UnifiedDiscreteFourier–HartleyTransform ....................... 97
3.13.1 FastStructureforUDFHT ................................ 101
3.14 Bluestein’sFFTAlgorithm ........................................ 104
3.15 RaderPrimeAlgorithm ........................................... 106
3.16 Summary ........................................................... 107
3.17 Problems ........................................................... 108
3.18 Projects ............................................................ 110
4 IntegerFastFourierTransform ....................................... 111
4.1 Introduction ......................................................... 111
4.2 TheLiftingScheme ................................................. 112
4.3 Algorithms .......................................................... 112
4.3.1 Fixed-PointArithmeticImplementation ...................... 117
4.4 IntegerDiscreteFourierTransform ................................ 119
4.4.1 Near-CompleteIntegerDFT .................................. 119
4.4.2 CompleteIntegerDFT ........................................ 121
4.4.3 EnergyConservation .......................................... 123
4.4.4 CircularShift .................................................. 123
Contents ix
4.5 Summary ............................................................ 125
4.6 Problems ............................................................ 126
4.7 Projects .............................................................. 126
5 Two-DimensionalDiscreteFourierTransform ....................... 127
5.1 Definitions .......................................................... 127
5.2 Properties ........................................................... 131
5.2.1 Periodicity .................................................... 131
5.2.2 ConjugateSymmetry ......................................... 131
5.2.3 CircularShiftinTime/SpatialDomain(PeriodicShift) ..... 133
5.2.4 CircularShiftinFrequencyDomain(PeriodicShift) ........ 133
5.2.5 SkewProperty ................................................ 135
5.2.6 RotationProperty ............................................. 135
5.2.7 Parseval’sTheorem ........................................... 135
5.2.8 ConvolutionTheorem ........................................ 136
5.2.9 CorrelationTheorem .......................................... 137
5.2.10 SpatialDomainDifferentiation ............................. 139
5.2.11 FrequencyDomainDifferentiation .......................... 139
5.2.12 Laplacian .................................................... 139
5.2.13 Rectangle .................................................... 139
5.3 Two-DimensionalFiltering ......................................... 140
5.3.1 InverseGaussianFilter(IGF) ................................. 142
5.3.2 RootFilter .................................................... 142
5.3.3 HomomorphicFiltering ....................................... 143
5.3.4 RangeCompression ........................................... 146
5.3.5 GaussianLowpassFilter ...................................... 148
5.4 InverseandWienerFiltering ....................................... 150
5.4.1 TheWienerFilter ............................................. 151
5.4.2 GeometricMeanFilter(GMF) ............................... 154
5.5 Three-DimensionalDFT ............................................ 156
5.5.1 3-DDFT ...................................................... 156
5.5.2 3-DIDFT ..................................................... 157
5.5.3 3DCoordinates ............................................... 157
5.5.4 3-DDFT ...................................................... 157
5.5.5 3-DIDFT ..................................................... 157
5.6 VarianceDistributioninthe1-DDFTDomain .................... 158
5.7 SumofVariancesUnderUnitaryTransformationIsInvariant .... 160
5.8 VarianceDistributioninthe2-DDFTDomain .................... 160
5.9 QuantizationofTransformCoefficientscanbeBased
onTheirVariances .................................................. 162
5.10 MaximumVarianceZonalSampling(MVZS) ................... 166
5.11 GeometricalZonalSampling(GZS) .............................. 168
5.12 Summary ........................................................... 168
5.13 Problems ........................................................... 169
5.14 Projects ............................................................ 170