a) What is the system function H(z) of this discrete-time system? b) Find by any means the response to the unit step, x[n] = urn]. a) Prove the following theorems for the (unilateral) Z-transform...

a) What is the system function H(z) of this discrete-time system?

b) Find by any means the response to the unit step, x[n] = urn].

a) Prove the following theorems for the (unilateral) Z-transform

IMTIAL-AND FINAL-VALUE THEOREMS:

(i) should give no trouble. (ii) follows by an argument that is essentially the discrete analog of the Final-Value Theorem for the L-transformation. Argue first that the Z-transform of x{n + 1]- x[n] is

Taking the limit as z ..... 1 on both sides and assuming orders of passing to the limit on the left may be interchanged, obtain

By writing out a few terms of the sum on the right, convince yourself that the limit (if it exists) is x[oo]- x[O], which gives the desired result.)

b) Test these theorems by applying them to all of the Z-transform pairs in the table in the appendix to this chapter.