Challenge15_Convolution%20Theorem

Challenge15_Convolution%20Theorem - This means that each...

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Digital Signal Processing Dr. Fred J. Taylor, Professor Lesson Title: Convolution Theorem Challenge Challenge: [15] Challenge Two 4-sample signals, x[k] = δ [k-1] and h[k] = δ [k-2] are to be convolved using the convolution theorem and a radix-2 FFT (i.e., FFT length is N=2 n ). Q. What is the minimum number of zeros that would need to be added to x[k] and h[k] prior to computing their FFT (do not consider using the overlap and add method). If both signals are padded out to 8 samples, to form x’[k] and h’[k], what is Y[n]=X’[n]H’[n] and y[k]? Response The linear convolution length is N 1 +N 2 -1=4+4-1=7. The nearest radix-2 FFT length in N=8.
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Unformatted text preview: This means that each signal is zero padded by 4-zeros. Observe that if X’[n]=FFT(x’[k]=x[k]&[0000]) = [x[1], W 8 1 x[1], W 8 2 x[1], W 8 3 x[1], W 8 4 x[1], W 8 5 x[1], W 8 6 x[1], W 8 7 x[1]] = [1, W 8 1 , W 8 2 , W 8 3 , W 8 4 , W 8 5 , W 8 6 , W 8 7 ] …… H’[n]=FFT(h’[k]=h[k]&[0000]) = [h[1], W 8 2 h[2], W 8 2 h[2], W 8 6 h[2], W 8 8 h[2], W 8 10 h[2], W 8 12 h[2], W 8 14 [2]] = [1, W 8 2 , W 8 4 W 8 6 W 8 8 , W 8 10 , W 8 12 W 8 14 ] Y[n]=X’[n]H’[n] = [1, W 8 3 , W 8 6 W 8 9 W 8 12 , W 8 15 , W 8 18 W 8 21 ] = [1, W 8 3 , W 8 6 W 8 1 W 8 4 , W 8 7 , W 8 2 W 8 5 ] IFFT(Y[n]) = (00100000) => δ [k-3] DFT - 1...
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This note was uploaded on 08/21/2010 for the course EEL 5525 taught by Professor Yang during the Summer '09 term at University of Florida.

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