question a define z-transform of sequence xn b a

Question

QUESTION

(a) Define z-transform of sequence x[n].

(b) A causal LTI system has impulse response h[n], for which the z-transform is

                    H(z) = (1+z^-1)/(1-1/2z^-1)(1+1/4z^-1)

(i) What is the Region of Convergence of H(z)? Plot the poles and zeros in the z-plane.

(ii) Find the impulse response h[n] of the system.

(c) A causal LTI system has impulse response h[n] by:

                    h[n] = [(0.3)ⁿ+(-0.5)ⁿ] u[n]

(i) Find the z-transform of h[n].

(ii) Plot the poles and zeros in the z-plane and the shade the Region of convergence.