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Discrete Fourier Transform

: Signal ProcessingDiscrete Fourier Transform Discrete Fourier Transform (DFT) Relations to Discrete -Time Fourier Transform (DTFT) Relations to Discrete -Time Fourier Series (DTFS)October 19, 2021 Yet Another Fourier RepresentationWhy do we need another Fourier Representation? Fourier seriesrepresent signals as sums of sinusoids. They provide insightsthat are not obvious from time representations, but Fourier series onlydefined for periodic [k] = n= N x[n]e j2 kn/N(summed over a period) Fourier transformshave no periodicity constaint:X( ) = n= x[n]e j n(summed over all samplesn)but are functions of continuous domain ( ). not convenient for numerical computationsDiscrete Fourier Transform : Discrete frequencies for aperiodic Fourier TransformDefinition and comparison to other Fourier :X[k] =1NN 1 n=0x[n]e j2 kNnx[n] =N 1 k=0X[k]ej2 kNnDTFS:X[k] =1N n= N x[n]e j2 kNnx[n] = k= N X[k]ej2 kNnDTFT:X( ) = n= x[n]e j nx[n] =12 2 X( )ej nd DTFS:x[n]is presumed to be periodic inNDTFT:x[n]is arbitraryDFT:only a portion of an arbitraryx[n]is consideredRelation Betw

Discrete Fourier Transform De nition and comparison to other Fourier representations. analysis synthesis DFT: X[k] = 1 N NX−1 n=0 xn]e−j 2πk N n NX−1 k=0 j2πkn DTFS: X[k] = 1 N X n=hNi xn]e−j 2πk N n X k=hNi j2πkn DTFT: X(Ω) = X∞ n=−∞ x[n] e−jΩn] = 1 2π Z 2π (Ω) jΩndΩ DTFS: x[n] is presumed to be periodic in N DTFT: x ...

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Impulse Response and Convolution

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