x[n] and y[n] are sample sequences of jointly wide-sense stationary, zero-mean rando...

x[n] and y[n] are sample sequences of jointly wide-sense stationary, zero-mean random processes. The following information is known about the autocorrelation function Φ_xx[m] and cross correlation Φ_yx[m]:

(a) The linear estimate of y given x is denoted It is designed to minimize

where the is formed by processing x[n] with an FIR filter whose impulse response h[n] is of length 3 and is given by

Determine h₀, h₁, and h₂ to minimize

(b) In this part, the linear estimate of y given x, is again designed to minimize in Eq. (P11.17-1), but with different assumptions on the structure of the linear filter. Here the estimate is formed by processing x[n] with an FIR filter whose impulse response g[n] is of length 2 and is given by

Determine the g₁ and g₂ to minimize .

(c) The signal, x[n] can be modeled as the output from a two-pole filter H(z) whose input is w[n], a wide-sense stationary, zero-mean, unit-variance white-noise signal.

Determine a₁ and a₂ based on the least-squares inverse model in Section 11.1.2.

(d) We want to implement the system shown in Figure P11.17 where the coefficients a_i are from all-pole modeling in part (c) and the coefficients h_i are the values of the impulse response of the linear estimator in part (a). Draw an implementation that minimizes the total cost of delays, where the cost of each individual delay is weighted linearly by its clock rate.

(e) Let be the cost in part (a) and let be the cost in part (b), where each is defined as in Eq. (P11.17-1). Is larger than, equal to, or smaller than b, or is there not enough information to compare them?

(f) Calculate and when Φ_yy[0] = 88. (Hint: The optimum FIR filters calculated in parts (a) and (b) are such that