Consider a real-valued sequence x[n] with rational z-transform X(z). (a) From the

Chapter 10, Problem 10.51

(choose chapter or problem)

Consider a real-valued sequence x[n] with rational z-transform X(z). (a) From the definition of the z-transform, show that X(z) = X*(z*). (b) From your result in part (a), show that if a pole (zero) of X(z) occurs at z = z0 , then a pole (zero) must also occur at z = z~. (c) Verify the result in part (b) for each of the following sequences: (1) x[n] = (4)nu[n] (2) x[n] = o[n] - 4o[n- 1] + io[n- 2] (d) By combining your results in part (b) with the result of 10.43(b), show that for a real, even sequence, if there is a pole (zero) of H (z) at z = peF1, then there is also a pole (zero) of H(z) at z = (11 p)ei8 and at z = (11 p)e-i 8 .