Solved: Problem 2: (a) Determine the Fourier transform of the sequence r[n] =

Question

Problem 2: (a) Determine the Fourier transform of the sequence r[n] = [1, 0 ≤n ≤ M, 0, otherwise. (b) Consider the sequence w[n]= 2 0. 1 - cos 2nn M 0 ≤ n & M, otherwise. Sketch w[n] and express W(e)"), the Fourier transform of w[n], in terms of R (ej"), the Fourier transform of r[n]. (Hint: First express w[n] in terms of r[n] and the complex exponentials e/(2xn/M) and e-j(2xn/M)) (c) Sketch the magnitude of R (ej") and W (ej") for the case when M =4. Problem 3: A linear time-invariant system is described by the input-output relation y[n] = x[n] + 2x[n - 1] + x[n-2]. (a) Determine h[n], the impulse response of the system. (b) Is this a stable system? (c) Determine H(ej"), the frequency response of the system. Use trigonometric identities to obtain a simple expression for H(ej®). (d) Plot the magnitude and phase of the frequency response. (e) Now consider a new system whose frequency response is H1(e)") = H(ei(@+x)). De- termine h1[n], the impulse response of the new system.