Assume that s[n] is a finite-length (windowed) sequence that is zero outside the interva...

Assume that s[n] is a finite-length (windowed) sequence that is zero outside the interval 0 ≤ n ≤ M −1. The p^th-order backward linear prediction error sequence for this signal is defined as

That is, s[n] is “predicted” from the p samples that follow sample n. The mean-squared backward prediction error is defined as

where the infinite limits indicate that the sum is over all nonzero values of as in the autocorrelation method used in “forward prediction.”

(a) The prediction error sequence is zero outside a finite interval N₁ ≤ n ≤ N₂. Determine N₁ and N₂.

(b) Following the approach used in this chapter to derive the forward linear predictor, derive the set of normal equations that are satisfied by the β_ks that minimize the mean-squared prediction error Give your final answer in a concise, well-defined form in terms of autocorrelation values.

(c) Based on the result in part (b), describe how the backward predictor coefficients {β_k} related to the forward predictor coefficients {α_k}?