A digital filter is characterized by the following properties:

(1) It is high pass and has one pole and one zero.

(2) The pole is at a distance r = 0.9 from the origin of the z-plane.

(3) Constant signals do not pass through the system.

(a) Plot the pole-zero pattern of the filter and determine its system function H (z).

(b) Compute the magnitude response |H (ω)| and the phase response < H (ω) of the filter.

(c) Normalize the frequency response H (ω) so that |H (π) = 1.

(d) Determine the input-output relation (difference equation) of the filter in the time domain.

(e) Compute the output of the system if the input is x (n) = 2 cos (π/6n + 45) -∞ < n < ∞

(You can use either algebraic or geometrical arguments.)

(1) xt")=cos 10" + 3snGn +TO)