Under the constraint that the filter has only one coefficient co, no that in output simplifies to:= c‘,X,„ find the coefficient co that minimizes the mean-squared error MSE =E((S5— Se).

Infinite sequence

Let {5,} be an infinite sequence of i.i.d.Al(1, 1) random variables.

Define a new random uquence X, by subtracting 1 from the product of three conucutive Sk values, according to:

X, = S„,S,S,_, —1.

Let  =   be an estimate of &arrived at by passing X, through en LTI filter:

  S,         Al(1, 1)                     X, = Se ,S,S,_,— 1

Under the constraint that the filter has only one coefficient co, no that in output simplifies to:= c‘,X,„ find the coefficient co that minimizes the mean-squared error MSE =E((S5— Se).

Under the constraint that the filter has only two coefficients co and q, so that its output is:

Sk= c„.X, + c,X,_ „

find the coefficients ca, c, that minimize the mean-squared error MSE = Eqk — Se).

In the unconstrained case when there are an infinite number of filter coefficients without any causality comb-ail., no that the filter output is:

the frequency response C(e0) = E,-,o of the filter that minimize MSE =               — V)

can be written as:

A+ Been(e) + Ccos (20)

D + E cos (I)) + Fens (28)

Find numeric values for the constants A, B, C, D, E, and F.