Question

Problem 3. Let y[nl-xInj-2xIn-11+xIn-2], find H(z), H(jo), and h[n]. Sketch the magnitude of H(jo). What kind of filter does this operation represent?

Answer 1

Let y (n) = x (n) - 2x (n - 1) + x (n - 2) Apply transformation, if f (t) rightarrow^F (W) f (t) rightarrow^Z f (t) rightarrow^L F (8) then f (t - t_0) rightarrow^e^- jw t_0 F (w) f (t - t_0) rightarrow^Z Z^- t_0 F (Z) f (t - t_0) rightarrow^L e^- s t_0 F (s) Now apply Fourier transform, rightarrow Y (w) = X (w) - 2 e^- jw X (w) + e^- j w 2 X (w) rightarrow Y (w)/X (w) = H (w) = 1 - 2 e^- jw + e^- j 2w Apply Laplace transform, Y (s) = X (s) - 2 e^- s X (s) + e^- 2s X (s) rightarrow Y (s)/X (s) = H (s) = 1 - 2 e^- s + e^- 2s Apply Z-transform, Y (Z) = X (Z) - 2 Z^- 1 X (Z) + Z^- 2 X (Z) rightarrow Y (Z)/X (Z) = H (Z) = 1 - 2 Z^- 1 + Z^- 2 \r\n we have H (Z) = 1 - 2 Z^- 1 + Z^- 2 rightarrow h (n) = {1, - 2, 1} co-efficient in the polynomial