Let hd[n] denote the impulse response of an ideal desired system with corresponding freq...

Let h_d[n] denote the impulse response of an ideal desired system with corresponding frequency response H_d (e^jω), and let h[n] and H(e^jω) denote the impulse response and frequency response, respectively, of an FIR approximation to the ideal system. Assume that h[n] = 0 for n < 0 and n > M. We wish to choose the (M + 1) samples of the impulse response so as to minimize the mean-square error of the frequency response defined as

(a) Use Parseval’s relation to express the error function in terms of the sequences h_d[n] and h[n].

(b) Using the result of part (a), determine the values of h[n] for 0 ≤ n ≤ M that minimize ε².

(c) The FIR filter determined in part (b) could have been obtained by a windowing operation. That is, h[n] could have been obtained by multiplying the desired infinite-length sequence h_d[n] by a certain finite-length sequence w[n]. Determine the necessary window w[n] such that the optimal impulse response is h[n] = w[n]h_d[n].