Consider an all-pass analog filter

(a) Use MATLAB functions to plot the magnitude and phase responses of G(s). Indicate whether the phase is linear.

(b) A discrete filter H(z) is obtained from G(s) by the bilinear transformation. By trial and error, find the value of K in the bilinear transformation so that the poles and zeros of H(z) are on the imaginary axis of the Z-plane. Use MATLAB functions, to do the bilinear transformation, and to plot the magnitude and unwrapped phase of H(z) and its poles. Is it an all-pass filter? If so, why?

(c) Let the input to the filter H(z) be x[n] = sin(0.2Ï€n), 0 ‰¤ n < 100, and the corresponding output be y[n]. Use MATLAB functions to compute and plot y[n]. From these results would you say that the phase of H(z) is approximately linear? Why or why not?

3 s* – 4s° + 8s² – 8s + 4 G(s) : s* + 4s + 8s“ + 8s + 4 3