Assume that s[n] is a finite-length (windowed) sequence that is zero outside the interval 0 ≤...

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

where the infinite limits indicate that the sum is over all nonzero values of (e[m])2 as in the autocorrelation method used in “forward prediction.”

(a) The prediction error sequence e[n] is zero outside a finite interval N_{1} ≤ n ≤ N_{2}. Determine N_{1} and N_{2}. (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 βks that minimize the mean-squared prediction error E. 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 {βk} related to the forward predictor coefficients {αk}?