數(shù)字信號(hào)處理英文版課件：Chap5 The Discrete Fourier Transform (DFT)

Chap.5TheDiscreteFourierTransform(DFT)

contents§5.1Representationofperiodicsequences：DFS§5.2TheDiscreteFourierTransform(DFT)§5.3LinearconvolutionUsingtheDFT

§5.1Representationofperiodicsequences：

TheDiscreteFourierSeries(DFS)ReviewFT:aperiodic

continuousFS:continuousaperiodic

時(shí)域的連續(xù)函數(shù)-頻域是非周期的頻譜函數(shù)時(shí)域的周期時(shí)間函數(shù)-頻域的離散頻譜時(shí)域連續(xù)函數(shù)-頻域是非周期的譜時(shí)域的非周期-頻域是連續(xù)的譜aperiodic

continuousperiodicdiscreteDTFTperiodic

discrete時(shí)域的離散-頻域的周期延拓

時(shí)域的非周期-頻域的連續(xù)Question:howtogetasequencewhichisdiscreteintimedomainandfrequecnydomain?periodicTimedomainfrequencydomaindiscreteperiodic

periodic

discreteaperiodic

continuous:periodisN

RepresentaperiodicsignalbyaFourierseries

kthharmonicsequence、DefinitionofDFSHarmonicCoefficients

periodisN?

DetermineFourierseriescoefficientsSo,——periodisNSynthesis:Analysis:notation:

DFSrepresentationofaperiodicsequence

:periodisN:periodisNExample8.1,8.3p544二、PropertiesoftheDFS1.LinearityLet,bothwithperiodNThen2.Shiftofasequence2）(modulationproperty)IfThen1）Proof3.Duality(對(duì)偶性)IfThen

4.SymmetryProperties(p550:Table8.19-17)ProofDefinition5.PeriodicConvolution1)If

,then(2)If,thenproofProofand

isperiodicwithN2.ShiftofasequenceProof1:

Thatis,3.DualityProof2:

Thatis,3.DualityDefinitionaboutsymmetry1)conjugatesymmetry:或共軛偶對(duì)稱(chēng)conjugateanti-symmetry:或共軛奇對(duì)稱(chēng)2）toanysequence，conjugatesymmetrysequence:conjugateanti-symmetry4.SymmetryProperties5.PeriodicConvolution

1)Proof:Note：Thedifferencebetweenperiodicconvolutionandaperiodicconvolution

周期卷積的結(jié)果也是周期為N的周期序列周期卷積的求和只在一個(gè)周期[0,N-1]上進(jìn)行，將所得結(jié)果進(jìn)行周期延拓，就得到整個(gè)周期序列。periodicconvolutionlinearconvolutionExample:Supposewehaveand,calcculatetheperiodicconvolutionandthelinearconvolution1)2)三、TheFourierTransformofPeriodicsignalsDTFTExample8.5TheFTofaPeriodicImpulseTrainP552So,DFS四、RelationshiptoZ-transformorDTFTLetprincipalvaluesequence:compare

Conclusion

issamplingatNequallyspacedfrequenciesonunitcirclewithafrequencyspacingof1234567(N-1)k=0五、SamplingTheoreminfrequencydomainSamplingTheorem?recoverQuestion:ifsamplingatNequallyspacedfrequenciesonunitcircle,canbeuniquelyrecoveredfromthesesamples?IDFS

FrequencySamplingTheoremLet

N:thenumberoffrequencysamples

ComparedwithNyquistSamplingTheoremThen

x[n]canbeuniquelyrecoveredfromifandonlyif2.Reconstructionof(from)samples:NpointsInterpolationfunction：StructureofanFIRfilter---------Frequency-samplingForm頻率抽樣型結(jié)構(gòu)中，當(dāng)頻域采樣點(diǎn)有許多值為零時(shí)，結(jié)構(gòu)簡(jiǎn)單頻率抽樣型結(jié)構(gòu)特點(diǎn)它的系數(shù)H[k]直接就是濾波器在ωk處的頻率響應(yīng)。因此，控制濾波器的頻率響應(yīng)是很直接的。結(jié)構(gòu)有兩個(gè)主要缺點(diǎn)：(a)所有的相乘系數(shù)及H[k]都是復(fù)數(shù)，這樣乘起來(lái)較復(fù)雜，增加乘法次數(shù)、存儲(chǔ)量。

(b)所有諧振器的極點(diǎn)都是在單位園上,考慮到系數(shù)量化的影響，當(dāng)系數(shù)量化時(shí)，極點(diǎn)會(huì)移動(dòng)，有些極點(diǎn)就不能被梳狀濾波器的零點(diǎn)所抵消。（零點(diǎn)由延時(shí)單元決定，不受量化的影響）系統(tǒng)就不穩(wěn)定了。§5.2TheDiscreteFourierTransform(DFT)、DefinitionofDFT

：Npoints,n=0,1,…,N-1

infiniteandperiodicfiniteandaperiodicDFTExample：，question:？Solution1:fromDFTSolution2：fromZTPadzerotoLpointsThelengthofisL二、PropertiesP576table8.21.LinearityIfx1[n]haslengthN1andx2[n]haslengthN2,Thenthemaximumlengthofx3[n]isSo,bothDFTsmustbecomputedwiththesamelength2、Circularshift圓周移位或循環(huán)移位1）Definitionperiodiccontinuationprinciplesequenceshiftbym

ExampleN-pointcircularshiftinonedirectionbymisthesameasacircularshiftintheoppositedirectionby(N-m)，i.e.rightleft2)3.DualityIfThen1)Circularconjugatesymmetry:圓周共軛對(duì)稱(chēng)4.SymmetryPropertiesCircularconjugateanti-symmetry:Example

實(shí)部圓周偶對(duì)稱(chēng),虛部圓周奇對(duì)稱(chēng)CircularconjugatesymmetryCircularconjugateanti-symmetry實(shí)部圓周奇對(duì)稱(chēng),虛部圓周偶對(duì)稱(chēng)Proof:2)Symmetrya)b)Toanyfinite-lengthsequencex[n]——circularconjugatesymmetriccomponent——circularconjugateanti-symmetriccomponentThenExample:

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

LetSo,Then,5.ParsveltheoreminDFTformProof6.CircularConvolution(2)CircularconvolutiontheoremNNNote:Thecircularconvolutionofx1[n]andx2[n]istheprinciplesequenceoftheperiodicconvolutionofand.NN

圓周卷積的長(zhǎng)度:1）DefinitionExample5Example,n=0,1,…,N-1.showthatIf,thenIfNiseven,and,thenProof:a)b)Niseven,soExample：calculateSolution:Example：pad(r-1)N

zerostox(n),i.e.CalculatebySolution:

§5.3LinearconvolutionUsingtheDFT

一、Relationshipbetweenlinearconvolutionandcircularconvolution:

ThenLinearconvolution：Circularconvolution:LSoistheprincipalvaluesequenceof

L1)【Discussion】N2)N1N13）【Conclusion】When,L-pointcircularconvolutioncansubstituteforlinearconvolution.i.e.LWhen,from0to,circularconvolutioncannotsubstituteforlinearconvolution(i.e.therehave(N1+N2-1)–Lpointsisnotequaltolinearconvolution).

Example:(N1=5)(N2=3)(N1+N2-1=7)5(N1+N2-1=7)1)L=56(L=6)(N1+N2-1=7)2)L=67(L=N1+N2-1=7)(N1+N2-1=7)3)L=78(L=8)4)L=8(N1+N2-1=7)Example.Thelengthofx[n]

is6，thelengthofy[n]is15,Question:Whichvaluesinf[n]

areequaltovaluesofz[n]?Solution:,thenthelengthofisWhere

LSo,fromn=5to14,fromn=0to4,二、ImplementLTIsystemusingDFTZeropaddingBlockconvolutionLet

NThenNSo,zero-paddingwith(N-L)x[n]N-pointIDFTy[n]h[n]zero-paddingwith(N-P)N-pointDFTN-pointDFTX[k]H[k]Y[k]【Note】N-pointDFTandN-pointIDFTcanberealizedfastbyFFT1.ZeropaddingPpointsLpointsBlockconvolutionOverlap－addmethod重疊相加法Overlap-savemethod重疊保留法1)Theoverlap－addmethod重疊相加法Suppose：Ppoints:verylongThen

Let：overlap－addStep：a)h[n]padzerostoNpoints,b)xi[n]padzerostoNpoints,c)Calculated)Addtheoverlappoints2）Overlap-savemethod重疊保留法SupposeThenForm0toP-2

L：Ppoints:Lpointsh[n]padL-Pzeros,L-DFTzeroStep：a)

h[n]padL-Pzeros(length:L),calculate

b)First,augmentp-1zerostothefront-endofx[n].Then,dividex[n]intoLpointssequencesxi[n]andleteveryxi[n