数字信号处理英文版课件：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

时域的连续函数-频域是非周期的频谱函数时域的周期时间函数-频域的离散频谱时域连续函数-频域是非周期的谱时域的非周期-频域是连续的谱aperiodic

continuousperiodicdiscreteDTFTperiodic

discrete时域的离散-频域的周期延拓

时域的非周期-频域的连续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(对偶性)IfThen

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

,then(2)If,thenproofProofand

isperiodicwithN2.ShiftofasequenceProof1:

Thatis,3.DualityProof2:

Thatis,3.DualityDefinitionaboutsymmetry1)conjugatesymmetry:或共轭偶对称conjugateanti-symmetry:或共轭奇对称2）toanysequence，conjugatesymmetrysequence:conjugateanti-symmetry4.SymmetryProperties5.PeriodicConvolution

1)Proof:Note：Thedifferencebetweenperiodicconvolutionandaperiodicconvolution

周期卷积的结果也是周期为N的周期序列周期卷积的求和只在一个周期[0,N-1]上进行，将所得结果进行周期延拓，就得到整个周期序列。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频率抽样型结构中，当频域采样点有许多值为零时，结构简单频率抽样型结构特点它的系数H[k]直接就是滤波器在ωk处的频率响应。因此，控制滤波器的频率响应是很直接的。结构有两个主要缺点：(a)所有的相乘系数及H[k]都是复数，这样乘起来较复杂，增加乘法次数、存储量。

(b)所有谐振器的极点都是在单位园上,考虑到系数量化的影响，当系数量化时，极点会移动，有些极点就不能被梳状滤波器的零点所抵消。（零点由延时单元决定，不受量化的影响）系统就不稳定了。§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圆周移位或循环移位1）Definitionperiodiccontinuationprinciplesequenceshiftbym

ExampleN-pointcircularshiftinonedirectionbymisthesameasacircularshiftintheoppositedirectionby(N-m)，i.e.rightleft2)3.DualityIfThen1)Circularconjugatesymmetry:圆周共轭对称4.SymmetryPropertiesCircularconjugateanti-symmetry:Example

实部圆周偶对称,虚部圆周奇对称CircularconjugatesymmetryCircularconjugateanti-symmetry实部圆周奇对称,虚部圆周偶对称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

圆周卷积的长度: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重叠相加法Overlap-savemethod重叠保留法1)Theoverlap－addmethod重叠相加法Suppose：Ppoints:verylongThen

Let：overlap－addStep：a)h[n]padzerostoNpoints,b)xi[n]padzerostoNpoints,c)Calculated)Addtheoverlappoints2）Overlap-savemethod重叠保留法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