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# lec28 - DFT and IDFT X(ej = k= x[n]ej n 1 The spectrum X(ej...

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DFT and IDFT 1 X ( e j ˆ ω ) = summationdisplay k = -∞ x [ n ] e - j ˆ ωn The spectrum X ( e j ˆ ω ) is sampled in frequency domain over its period with stepsize Δ ω = 2 π N so X [ k ] = X ( e j 2 π N k ) = N - 1 summationdisplay n =0 x [ n ] e - j 2 π N kn This is the Discrete Fourier Transform (DFT) . DFT is a 1 1 relation. The Inverse DFT (IDFT) is x [ n ] = 1 N N - 1 summationdisplay n =0 X [ k ] e j 2 π N kn Lectures 29-33

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Basis vectors 2 X [ k ] = X ( e j 2 π N k ) = N - 1 summationdisplay n =0 x [ n ] e - j 2 π N kn Set ω N = e - j 2 π N . Construct vectors w T k = bracketleftbig ( ω k N ) 0 ( ω k N ) 1 · · · ( ω k N ) N - 1 bracketrightbig = bracketleftBig ω k · 0 N ω k · 1 N · · · ω k · ( N - 1 N bracketrightBig x T = [ x [0] x [1] · · · x [ N 1]] Then X [ k ] = N - 1 summationdisplay n =0 x [ n ] ω k · n N = w T k · x Lectures 29-33
DFT matrx 3 F N = w 0 · 0 N · · · w 0 · l N · · · w 0 · ( N - 1) N w 1 · 0 N · · · w 1 · l N · · · w 1 · ( N - 1) N . . . . . . . . . w k · 0 N · · · w k · l N · · · w k · ( N - 1) N . . . . . . . . . w ( N - 1) · 0 N · · · w ( N - 1) · l N · · · w ( N - 1) · ( N - 1) N Set X = ( X [0] ,X [1] ,...X [ N 1]) T Then X = F N x Lectures 29-33

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Properties of DFT matrix 4 F N =
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lec28 - DFT and IDFT X(ej = k= x[n]ej n 1 The spectrum X(ej...

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