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## Homework Statement

Let e(t), for t = 0,Â±1,Â±2, . . ., be a realization of an IID sequence of zero-mean random

variables with variance Ïƒe^2, and let v(t) be the output of the filter:

v(t) = (B(q)/A(q) )e(t)

where B(q) and A(q) are polynomials in the forward shift operator q (i.e. qv(t) = v(t + 1)),

given by:

B(q) = 1 +Æ©(from k=1 to m) bk*q^-k ---- bk â†’b subscript k

A(q) = 1 +Æ©(from k=1 to n ) ak*q^-k -----ak â†’a subscript k

and the solutions to A(z) = 0 and B(z) are all inside the unit circle.

(a) Let x(t) be the output of the following filter:

x(t) = ((A(q) âˆ’ B(q))/B(q)) v(t)

(b) Write the relationship between x(t) and v(t) as a difference equation. What is the

coefficient of v(t) (i.e. at zero delay)? Is this filter stable?

(c) Compute the expected value E[x(t âˆ’ k)e(t âˆ’ k)], for k = 0, 1, 2, . . ..

(d) Let y(t) be defined as:

y(t) = (B(q)/A(q))x(t)

Can this signal be used as a â€œone-stepâ€ ahead prediction of v(t)? Justify your answer.

(e) Compute the expected value and autocorrelation of v(t) âˆ’ y(t).

## Homework Equations

**3. The Attempt at a Solution [/b**

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