dsp2純課件數(shù)字信號(hào)處理chap8第2講

18.6PropertiesoftheDFT1.LinearityIfx1[n]haslengthN1,andx2[n]haslengthN2Thenthemaximumlengthofx3[n]isSo,bothDFTsmustbecomputedwiththesamelength2.CircularShiftofasequence(i)definitionx(n)e.g.N-pointcircularshiftinonedirectionbymisthesameasacircularshiftintheoppositedirectionby(N-m)(ii)Thetheoremofcircularshiftintime-domain(iii)Thetheoremofcircularshiftinfrequency-domain3.DualityIfThen(i)Proof4.SymmetryProperties(ii)Toanyfinite-lengthsequencex[n]——periodicconjugate-symmetriccomponent——periodic

conjugate-antisymmetriccomponentThenExample:x1[n]andx2[n]aretwoN-pointrealsequence,computetheirDFTsbyusingaN-pointDFTonlyonce.

Solution:

LetSo,Then,5.CircularConvolution(ii)CircularconvolutionTheorem(i)DefinitionNNNote:Thecircularconvolutionofx1[n]andx2[n]istheprinciplesequenceoftheperiodicconvolutionofand.NN

圓周卷積的長度:(iii)Theprocedureofcircularconvolution57.SummaryofPropertiesofDFT(p576:Table8.2)6.PasveltheoreminDFTformProve:118.7LinearConvolutionUsingtheDFT兩個(gè)序列時(shí)域圓周卷積在頻域上相當(dāng)于它們的DFT相乘，因而可以采用DFT的快速算法——FFT，與線性卷積相比，計(jì)算速度可大大加快。一般實(shí)際問題都是線性卷積運(yùn)算。如果信號(hào)以及系統(tǒng)的單位沖激響應(yīng)都是有限長序列，那么能否用圓周卷積運(yùn)算來代替線性卷積運(yùn)算呢？12(i)Linearconvolution

Let

(ii)Circularconvolution

length:(L+P-1)

Nlength:Relationshipbetweenlinearconvolutionandcircularconvolution13RelationshipbetweenlinearconvolutionandcircularconvolutionN點(diǎn)的圓周卷積是線性卷積以N為周期進(jìn)行周期延拓所得周期序列的主值序列14N(i)DiscussionN(ii)L(iii)

N<L+P-1LLetL>P,N=L(iv)Conclusion

When,N-pointcircularconvolutioncansubstituteforlinearconvolution.

N18Eg(L=5)(P=3)(L+P-1=7)5(L+P-1=7)6(N=6)(L+P-1=7)7(N=L+P-1=7)(L+P-1=7)8(L+P-1=7)(N=8)23ImplementingLTISystemsUsingtheDFTLet

NThenNSo,zero-paddingwith(N-L)x[n]N-pointIDFTy[n]h[n]zero-paddingwith(N-P)N-pointDFTN-pointDFTX[k]H[k]Y[k]N-pointDFTandN-pointIDFTcanberealizedfastbyFFTZeropadding(i)Inmanyapplications,theinputsequenceisofindefiniteduration.Theoretically,wemightbeabletostoretheentiresequenceandthenimplementtheprocedureusingaDFT.But,suchaDFTwithalargenumberofpointsisgenerallyimpracticaltocompute.(ii)Nofilteredsamplescanbecomputeduntilallinputsampleshavebeencollected,soalargedelayiscausedinprocessing.

Probleminpracticalapplications:

Solution——blockconvolution(i)TheinputsequenceissegmentedintosectionsofshorterlengthL.Eachsectionareconvolvedwiththefinite-lengthimpulseresponseandthefilteredsectionsarefittedtogetherinanappropriateway.(iii)ThelinearconvolutionofeachblockcanbeimplementedusingDFT.

Blockconvolution

Theoverlap-addmethod(重疊相加法)——implementedby(N=L+P-1)-pointDFTsFilteredsectionsareoverlappedby(P-1)points,andareaddedtoconstructtheoutput.Theoverlap-savemethod(重疊保留法)(L>P)(P-2)P-1——implementedbyL-pointDFTsInputsectionsoverlap,e