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)),
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).
3. The Attempt at a Solution [/b