Engineering
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.
