If one is modeling a signal x(n) whose transform, X(z), contains zeros, then an all-pole model may be used to effectively model a zero with an infinite number of poles. In this problem we look at how a zero is modeled with the autocorrelation method. Let x(n) = δ(n) - aδ(n - 1) where and a is real.
(a) Determine the pth-order all-pole model Ap(z) for x(n) where pis an arbitrary positive integer, and find the value of the squared error εp.
(b) For the all-pole model determined in part (a), what is the limit of Ap(z) as p → ∞? What does εp converge to as p → ∞? Justify your answers.
(c) Repeat parts (a) and (b) for