(a) The unit-step response of a discrete system is the system response c(k) with the input r(k) = 1 for k ≥ 0 . Show that if the discrete system is stable, the unit-step response, c(k), approaches a constant as k → ∞. [Let T (z) be the closed-loop system transfer function. Assume that the poles of T (z) are distinct (no repeated poles).]

(b) Find the conditions on the closed-loop system transfer function T (z) such that the unit-step response approaches zero as k → ∞.

(c) The discrete unit-impulse response of a discrete system is the system response c(k) with the input r(k) = 1 for k = 0 and r(k) = 0 for k ≥ 1 . Show that if the discrete system is stable, the unit impulse response, c(k), approaches zero. [Let T (z) be the closed-loop system transfer function. Assume that the poles of T (z) are distinct (no repeated poles).]