Let x(n) be a zero-mean stationary process with variance x 2 and auto correlation x( l

Let x(n) be a zero-mean stationary process with variance σ_x² and auto correlation γx(l).

(a) Show that the variance σ²_d of the first-order prediction error d(n) = x(n) – ax(n – 1) is given σ²_d = σ²_x [1 + a² – 2ap_x(1)] where p_x(1) = γ_x(1)/γ_x(0) is the normalized autocorrelation sequence.

(b) Show that σ²_d attains its minimum value σ²_d = σ²_x [1 – p²_x(1)] for a = γ_x(1)/γ_x(0) = p_x(1).

(c) Under what conditions is σ²_d < σ²_x?

(d) Repeat steps (a) to (c) for the second-order prediction error d(n) = x(n) – a₁x(n – 1) – a₂x(n – 2)