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# Random Processes Through Linear Systems

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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:

In this case, we can do analysis using Fourier transforms:

Suppose the input function is a random process instead of a deterministic signal. How can we characterize the output function?

Let

The integral is to be interpreted in in the mean-square 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 tine-invariant. Then is also W.S.S., and and are jointly W.S.S. and the following hold:
1. .
2. , where .
3. . 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. This work is licensed under a Creative Commons License