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1) Let h[n] be a finite length sequence of length N, where N is an odd integer, and symmetric around its mid-point. Let h[n] be the unit sample response of an LTI system. a) Prove that if h[n] is non-causal extending from n = -(N-1)/2 to n = (N-1)/2 then arg{|H1(ω)}, the phase spectrum of H(ω), is zero. What is the time delay through this system? b) Let h1[n] = h[n–(N-1)/2], or the causal version of h[n]. Show that |H1(ω)| = |H(ω)| and arg{|H1(ω = –αω, i.e., linear. Write the expression and plot that phase spectrum. What is the time delay through this system?
2) An N-point rectangular window is the sequence wR[n] defined as: ≤− ≤ = elsewhere Nn nw R 0, 011, [] Derive its Fourier transform, WR(ω). Write the expression for the magnitude and phase spectra. Compute (a) the DC gain, (b) the main lobe width, (c) the side lobe widths and (d) the phase spectrum, all as functions of the window size, N. Provide results and time and frequency domain plots for N = 32.
3) An N-point Hamming sequence wH[n] is defined as: ( )[ ] ≤− −≤− = elsewhere NnNn nw H 0, 011,/2cos46.054.0 [] π
Express wH[n] as a function of wR[n], and apply Fourier transform properties to derive the expression WH(ω) as function of WR(ω). Compute (a) the d.c. gain, (b) the main lobe width, (c) the side lobe widths and (d) the phase spectrum, all as functions of the window size, N. Provide results and time and frequency domain plots for N = 32.
Contrast the properties of WR(ω) and WH(ω).
4) For the 250-point audio segment (fs = 10 kHz) is provided in file ex2M1.mat compute its magnitude spectrum using both a rectangular window and a Hamming window. (a) Use windows of the same length as the audio segment. Plot the magnitude in linear and dB magnitude scales (and linear phase scale) and phase spectra. Highlight the differences between the two versions of the signal spectrum. (b) Repeat the analysis using windows with length twice that of the provided data, by appending zeros at end of audio data. (b) Repeat the analysis using windows with length twice that of the provided data, by appending zeros at beginning and end of audio data.