Discrete Fourier Transform-Digital Signal Processing-Assignment Solution, Exercises of Digital Signal Processing

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9.1. There are several possible approaches to this problem. Two are presented below. Solution #1: Use the program to compute the DFT of X[k], yielding the sequence g[r]. N-1 gin) = So X{ble#re™ io Then, compute 1 =n) = qell(N — 9x] N — 1. We demonstrate that this solution produces the inverse DFT below. Hale — nh] y Wat 1 7 & X{kJe PHN fora u afr tt ¥ US x1 X [klein k= Solution #2: Take the complex conjugate of X{k], and then compute its DFT using the program, yielding the sequence f{n]. N-1 fin] = xs X" [jew N £0 Then, compute } stn] = 5 S'tn) We demonstrate that this solution produces the inverse DFT below. z{n] zt {n) = 25 > X [k]e!xen/w kd 9.2. (a) The "gain" along the emphasized path is -W3. (b) In general, there is only one path between each input sample and each output sample. (c) x(0] to X{2]: The gain is 1 z(1] to X(2]: The gain is WR. x(2| to X{2]: The gain is -W = 2[3} to X[2}: The gain is -WR WR z[4] to X{2]: The gain is W% = 1. z[5} to X{2]: The gain is WW}, = z[6] to X{2]: The gain is -W2WE = 2(7] to X[2}: The gain is _wawaw, = -Wi, as in Part (a). Now 2 zfnjw?" an=d = f0] + 21]? + x[2ws + 2[3]W§ + 2fale + 2(5)WE° + 2{6]W27 + 2{7)3* = xfo] + 2[t)e + x[2](—2) + 213-2) + x[s](1) + 2[5]02 + 2(6)(~1) + 2{7(-We) Xf}