FIR system modeling Consider an "unknown" FIR system with impulse response h(n),...

FIR system modeling Consider an "unknown" FIR system with impulse response h(n), 0 ≤ n ≤ 11, given by

A potential user has access to the input and output of the system but does not have any information about its impulse response other than that it is FIR. In an effort to determine the impulse response of the system, the user excites it with a zero-mean, random sequence x(n) uniformly distributed in the range [-0.5, 0.5], and records the signal x(n) and the corresponding output y(n) for 0 ≤ n ≤ 199.

(a) By using the available information that the unknown system is FIR, the user employs the method of least squares to obtain an FIR model h(n), 0 ≤ n ≤ M - 1. Set up the system of linear equations, specifying the parameters h(0), h(1), . . . , h(M -1). Specify formulas we should use to determine the necessary autocorrelation and crosscorrelation values.

(b) Since the order of the system is unknown, the user decides to try models of different orders and check the corresponding total squared error. Clearly, this error will be zero (or very close to it if the order of the model becomes equal to the order of the system). Compute the FIR models 0 ≤ n ≤ M - 1. for M = 8, 9, 10, 11, 12, 13, 14 as well as the corresponding total squared errors M = 8, 9, ... ,14. What do you observe?

(c) Determine and plot the frequency response of the system and the models for M =11,12, 13. Comment on the results.

(d) Suppose now that the output of the system is corrupted by additive noise, so instead of the signal y(n) , 0 ≤ n ≤ 199, we have available the signal

where ω(n) is a Gaussian random sequence with zero mean and variance

Repeat part (b) of Problem 10.27 by using v(n) instead of y(n) and comment on the results. The quality of the model can be also determined by the quantity