System identification Consider an unknown LTI system and an FIR system model as show...

System identification Consider an unknown LTI system and an FIR system model as shown in Fig. P10.26. Both systems are excited by the same input sequence {x(n)}. The problem is to determine the coefficients {h(n), 0 ≤ n ≤ M —1} of the FIR model of the system to minimize the average squared error between the outputs of the two systems.

(a) Determine the equation for the FIR filter coefficients {h(n), 0 ≤ n ≤ M — 1} that minimize the least-squares error

where

and N >> M

(b) Repeat part (a) if the output of the unknown system is corrupted by an additive white noise sequence with variance