Problem 1: The Window Method of Filter Design is based on choosing h[n] = v[n]w[n] where v[n] is impulse response of an ideal filter and w[n] is a window function.
To get the desired frequency response, one typically adjusts the cutoff frequency of the ideal filter, the window type, and the window length. (a) Use this method to design a low-pass filter with pass band 0 ≤ |Ω| < 1.3, pass-band ripple < 0.5 dB, stop band π/2 ≤ |Ω| ≤ π, and 50 dB stop-band rejection.
Mathematically, this means
20 log10 |H(e jΩ)| < 0.5 for 0 ≤ |Ω| ≤ 1.3
20 log10 |H(e jΩ)| < −50 for π/2 ≤ |Ω| ≤ π.
Design the filter (with fdatool or filterDesigner) plot the magnitude response, and report the final specifications you achieve (i.e., pass-band ripple and stop-band rejection with the filter transfer function plot).
(b) Load and listen to ”singing8.wav”. Now, try decimating by 2 and listening with soundsc(x(1:2:end),Fs/2). What do you hear?
(c) Can you use the filter from (a) to allow down-sampling by 2 without audible distortion? Explain.