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# Random Processes Through Linear Systems

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Continuous  ::  Discrete  ::  White Noise

## Discrete-time filters

Let . The output of a discrete-time system is

Causal: for . Time-invariant: for all . Then

We can transform in the time-invariant case using a Z-transform,

or

We also deal with the discrete-time 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:

1. exists .
2. .
3. If is W.S.S. and is time-invariant, then is W.S.S. and , are jointly W.S.S. Then:
1. , where .
These properties can be expressed in the Z-transform domain:
1. .
2. .
These can be further expressed on the unit circle as:
1. .
2. .

Copyright 2008, by the Contributing Authors. 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_2.htm. This work is licensed under a Creative Commons License