Consider a causal linear time-invariant system with system function where ? is real.

(a) Write the difference equation that relates the input and the output of this system.

(b) For what range of values of ? is the system stable?

(c) For ? = ?, plot the pole-zero diagram and shade the region of convergence.

(d) Find the impulse response h[n] for the system.

(e) Show that the system is an all-pass system, i.e., that the magnitude of the frequency response is a constant. Also, specify the value of the constant.

1-a-- H(z) = %3! 1- az-!