The a periodic auto correlation function for a real-valued stable sequence x[n] is defined as

(a) Show that the z-transform of c_x[n] is?

C_xx(z) = X(z) X(z?1).

?Determine the region of convergence for C_xx(z).

(b) Suppose that x[n] = aⁿ u[n]. Sketch the pole-zero plot for C_xx(z), including the region of convergence. Also, find c_xx[n] by evaluating the inverse z-transform of C_xx(z).

(c) Specify another sequence, x₁[n], that is not equal to x[n] in part (b), but that has the same autocorrelation function, c_xx[n], as x[n] in part (b).

(d) Specify a third sequence, x₂[n], that is not equal to x[n] or x₁[n], but that has the same autocorrelation function as x[n] in part (b)

Cxa(n] = E *[k]x[n + k]. %3D k-00