113_1_Week08_print

113_1_Week08_print - 1 EE 113: Digital Signal Processing...

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1 EE 113: Digital Signal Processing Week 8 Fourier-Domain - continued 1. Discrete Fourier Transform (DFT) 2. Convolution with the DFT
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2 Discrete FT (DFT) ± A finite or periodic sequence has only N unique values, x [ n ] for 0 n < N ± Spectrum is completely defined by N distinct frequency samples ± Divide 0..2 π into N equal steps, { ω k } = 2 π k / N Discrete finite/pdc X [ k ] Discrete finite/pdc x [ n ] Discrete FT (DFT)
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3 ± Uniform sampling of DTFT spectrum: ± DFT: where i.e. 1/ N th of a revolution DFT and IDFT X [ k ] = X ( e j ω ) ω= 2 π k N = x [ n ] e j 2 k N n n = 0 N 1 X [ k ] = x [ n ] W N kn n = 0 N 1 W N = e j 2 N The higher the value of N, the better resolution we get in the frequency domain-> the more details DFT provides about DTFT
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4 IDFT ± Inverse DFT IDFT ± Check: Sum of complete set of rotated vectors = 0 if l n ; = N if l = n re im W N W N 2 x [ n ] = 1 N X [ k ] W N nk k = 0 N 1 x [ n ] = 1 N x [ l ] W N kl l ( ) W N nk k = 1 N x [ l ] W N k ( l n ) k = 0 N 1 l = 0 N 1 = x [ n ]
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5 DFT examples ± Finite impulse ± Periodic sinusoid: ( r I) x [ n ] = 1 n = 0 0 n = 1.. N 1 X [ k ] = x [ n ] W N kn n = 0 N 1 = W N 0 = 1 k x [ n ] = cos 2 π rn N = 1 2 W N rn + W N rn ( ) X [ k ] = 1 2 W N rn + W N rn ( ) n = 0 N 1 W N kn = N /2 k = r , k = N r 0 o . w .
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6 DFT: Matrix form ± as a matrix multiply: ± i.e. X [ k ] = x [ n ] W N kn n = 0 N 1 X [0] X [1] X [2] # X [ N 1] = 11 1 " 1 1 W N 1 W N 2 " W N ( N 1) 1 W N 2 W N 4 " W N 2( N 1) ## # % # 1 W N ( N 1) W N N 1) " W N ( N 1) 2 x [0] x x [2] # x [ N X =
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This note was uploaded on 11/07/2009 for the course EE 113 taught by Professor Walker during the Spring '08 term at UCLA.

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113_1_Week08_print - 1 EE 113: Digital Signal Processing...

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