Homework Solutions
Utah State University
ECE 6010
Stochastic Processes
Homework # 8 Solutions

Suppose
is a Wiener process. Define a
process
by
for a
fixed positive number
.

Find the mean and autocorrelation functions of
.
Mean :
Autocorrelation :
So . 
Show that
is a stationary and find its
spectrum.
is a constant and W.S.S.Since Gaussian too Strictly Stationary.

Find the mean and autocorrelation functions of
.

Suppose
and
are zero mean and
individually and jointly W.S.S. Show that the meansquare error
associated with the noncausal Wiener filter for estimation of
from
is

Suppose
for
, where
and
are zeromean, W.S.S., and orthogonal. Suppose
that we wish to estimate
, with an estimate of the form
, where
and
are impulse responses of linear timeinvariant systems. show
that

Consider the situation of the previous problem with
,

Find the noncausal Wiener filter for estimating
from
. Find the corresponding meansquare
error.
When ,

Find the causal Wiener filter for estimating
from
. Consider
and
.
We have
picture(4149,1632)(1939,2731) (3751,2686)(0,0)[b] % (2326,1636)(0,0)[b] % (4351,1936)(0,0)[lb] % (3301,1936)(0,0)[rb] % (1951,2686)(0,0)[rb] %Let

Find the noncausal Wiener filter for estimating
from
. Find the corresponding meansquare
error.