Question

Problem 1 (11 pts): The commonly used windows presented in MODULE-06 can all be expressed in terms of rectangular windows. This fact can be used to obtain expressions for the Fourier transforms of the Bartlett window and the raised-cosine family of windows, which includes the Hanning, Hamming, and Blackman windows. (a) Show that the(M + 1)-point Bartlett window, defined in Page 621 (MODULE-06), can be expressed as the convolution of two smaller rectangular windows. Use this fact to show that the Fourier transform of the ((M 1)-point Bartlett window is WsCe/a) -e-lai(or M evn. Wa(ej") = e-jal/2(2/M)(sts n(www)(strin( /24 ) in[(M+1)/4sin(M-1)/4] for M odd (b) It can be seen that the (M + 1)-point raised-cosine windows (Hanning, Hamming, and Blackman windows) can all be expressed in the form where wR[n] is an (M +1)-point rectangular window. Use this relation to find the Fourier transform of the general raised-cosine window. c) Using appropriate choices for A, B, and C and the result determined in part (b), sketch the magnitude of the Fourier transform of the Hanning window.

Answer 1

To obtain (M + 1) point Bartlett window for M even, Convolve the following rectangle windows r_1[n] = {squareroot 2/M, n = 0, ..., (M/2 - 1) 0, otherwise r_2[n] = r_1[n - 1] Fourier transform of rectangular window is since function thus obtain the transform of r_1[n] and r_2[n] represented by W_R1 (e^i omega) and W_R2 (e^i omega) W_R1 (e^i omega) = squareroot 2/M sin(omega M/4)/sin(omega/2) e^-i omega (M/omega - 1/2) \r\n W_R2 (e^i omega) = squareroot 2/M sin(omega M/4)/sin(omega/2) e^-i omega (M/4 - 1/2 + 1) = squareroot 2/M sin(omega M/4)/sin(omega/2) e^-i omega (M/4 + 1/2) Convolution in time domain is multiplication in frequency domain Thus, W_B(e^i omega) = W_R1(e^i omega) W_R2(e^i omega) = [squareroot 2/M sin(omega M/4)/sin(omega/2)] e^-i omega(M/4 - 1/2 + M/4 + 1/2) = 2/M(sin(omega M/4)/sin(omega/2))^2 e^-(i omega M/2) To obtain (M + 1) point Bartlett window for M odd, convolve the following rectangle windows r_3[n] = {squareroot 2/M, n = 0, ...m M - 1/2 0, otherwise