A digital filter is characterized by the following properties: 1. It is highpass and h...

A digital filter is characterized by the following properties:

1. It is highpass 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 and the phase response of the filter.

(c) Normalize the frequency response

(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

(You can use either algebraic or geometrical arguments.)