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HWIX - ECE 3250 HOMEWORK ASSIGNMENT IX Fall 2009 1 Let x1...

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ECE 3250 HOMEWORK ASSIGNMENT IX Fall 2009 1. Let x 1 be a 1024-point signal and x 2 be a 1025-point signal. Note that x 1 * x 2 is a 2048-point signal. (a) Show that it takes a worst-case 512 . 5 × 2 11 multiplications to compute x 1 * x 2 directly. (Suggestion: it takes one multiplication to compute x 1 * x 2 (0), two multiplications to compute x 1 * x 2 (1), etc. You may use the fact that P n k =1 k = n ( n + 1) / 2.) (b) Suppose you use N -point DFTs to compute x 1 * x 2 . First of all, what is N ? Second, assuming you use the standard decimation-in-time FFT to compute the DFTs, find how many multiplications it takes to compute x 1 * x 1 . You may assume that it takes the same number of multiplications to compute an inverse DFT as it takes to compute a DFT when you use the FFT. 2. Find the z -transform (formula for X ( z ) along with (ROC) X ), if the transform exists, for each signal x specified below. If the transform fails to exist, explain why. (a) x ( n ) = 2 - n u ( n + 1) for all n Z .
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