Consider an LTI system with system function H(z) = h0 + h1z−1. The signal y[n] is the ou...

Consider an LTI system with system function H(z) = h₀ + h₁z⁻¹. The signal y[n] is the output of this system due to an input that is white noise with zero mean and unit variance.

(a) What is the autocorrelation function r_yy[m] of the output signal y[n]?

(b) The 2nd-order forward prediction error is defined as

Without using the Yule–Walker equations directly, find a₁ and a₂, such that the variance of e[n] is minimized.

(c) The backward prediction filter for y[n] is defined as

Find b₁ and b₂ such that the variance of is minimized. Compare these coefficients to those determined in part (b).