Random Processes Through Linear Systems
Continuous :: Discrete :: White Noise
Continuous time systems
Recall: A signal through a linear system produces an output

The system is
causal
if
for
. In this
case,

The system is
time invariant
if
for all
and
. In this case,
is obtained by
convolution:
Let
The integral is to be interpreted in in the meansquare sense. Properties:

exists
.

.
That is, the mean of the output is the response of the mean of the
input:
 . That is, the correlation between the input and the output is the response to the autocorrelation with respect to the 1st input.
 .

If
is W.S.S. and
is tineinvariant. Then
is also W.S.S., and
and
are jointly
W.S.S. and the following hold:
 .
 , where .
 . The quantity is sometimes called the power transfer function.
Copyright 2008,
Todd Moon.
Cite/attribute Resource
.
admin. (2006, June 07). Random Processes Through Linear Systems. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lecture8_1.htm.
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