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III FastAlgorithms andStructures P.Duhamel ´ EcoleNationaleSup ´ erieuredesT ´ el ´ ecommunications(ENST) 7FastFourierTransforms:ATutorialReviewandaStateoftheArt P.DuhamelandM. Vetterli Introduction • AHistoricalPerspective • Motivation(or:whydividingisalsoconquering) • FFTs withTwiddleFactors • FFTsBasedonCostlessMono-toMultidimensionalMapping • Stateof theArt • StructuralConsiderations • ParticularCasesandRelatedTransforms • Multidimensional Transforms • ImplementationIssues • Conclusion 8FastConvolutionandFiltering IvanW.SelesnickandC.SidneyBurrus Introduction • Overlap-AddandOverlap-SaveMethodsforFastConvolution • BlockConvolution • ShortandMediumLengthConvolution • MultirateMethodsforRunningConvolution • Convo- lutioninSubbands • DistributedArithmetic • FastConvolutionbyNumberTheoreticTransforms • Polynomial-BasedMethods • SpecialLow-MultiplyFilterStructures 9ComplexityTheoryofTransformsinSignalProcessing EphraimFeig Introduction • One-DimensionalDFTs • MultidimensionalDFTs • One-DimensionalDCTs • Mul- tidimensionalDCTs • NonstandardModelsandProblems 10FastMatrixComputations AndrewE.Yagle Introduction • Divide-and-ConquerFastMatrixMultiplication • Wavelet-BasedMatrixSparsifi- cation T HEFIELDOFDIGITALSIGNALPROCESSINGgrewrapidlyandachieveditscurrentpromi- nenceprimarilythroughthediscoveryofefficientalgorithmsforcomputingvarioustrans- forms(mainlytheFouriertransforms)inthe1970s.InadditiontofastFouriertransforms (FFTs),discretecosinetransforms(DCTs)havealsogainedimportanceowingtotheirperformance beingveryclosetothestatisticallyoptimumKarhunenLoevetransform. Transforms,convolutions,andmatrix-vectoroperationsformthebasictoolsutilizedbythesignal processingcommunity,andthissectionreviewsandpresentsthestateofartintheseareasofincreasing importance. ThechapterbyDuhamelandVetterli,“FastFourierTransforms:ATutorialReviewandaStateof theArt”,presentsathoroughdiscussionofthisimportanttransform.SelesnickandBurruspresent c  1999byCRCPressLLC an excellent survey of filtering and convolution techniques in the chapter “Fast Convolution and Filtering”. One approach to understanding the time and space complexities of signal processing algorithms is through the use of quantitative complexity theory, and Feig’s “Complexity Theory of Transforms in Signal Processing” applies quantitative measures to the computation of transforms. Finally, Yagle presents a comprehensive discussion of matrix computations in signal processing in “Fast Matrix Computations”. c  1999 by CRC Press LLC . Transforms,convolutions,andmatrix-vectoroperationsformthebasictoolsutilizedbythesignal processingcommunity,andthissectionreviewsandpresentsthestateofartintheseareasofincreasing importance. ThechapterbyDuhamelandVetterli,“FastFourierTransforms:ATutorialReviewandaStateof