Random Processes Through Linear Systems
Continuous :: Discrete :: White Noise
Discretetime filters
Let . The output of a discretetime system is
Causal: for . Timeinvariant: for all . Then
We can transform in the timeinvariant case using a Ztransform,
or
We also deal with the discretetime Fourier transform, obtained by evaluating on the unit circle . We write
(This is an abuse of notation, but makes the notation consistent with continuous time.)
For a random process, we still have
but we interpret this in a m.s. sense:
Properties:
 exists .
 .

If
is W.S.S. and
is timeinvariant, then
is W.S.S. and
,
are jointly W.S.S.
Then:
 , where .
 .
 .
 .
 .