The aperiodic autocorrelation function for a real-valued stable sequence x[n] is defined...

The aperiodic autocorrelation function for a real-valued stable sequence x[n] is defined as

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

C _xx (z) = X(z)X(z⁻¹).

Determine the ROC for C_xx(z).

(b) Suppose that x[n] = aⁿu[n]. Sketch the pole–zero plot for C_xx(z), including the ROC. 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).