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# Basic Concepts of Random Processes

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Definitions  ::  Ergodicity  ::  Autocorrelations  ::  Sinusoidal  ::  Poisson  ::  Gaussian  ::  Properties  ::  Spectra  ::  Cases  ::  Random  ::  Independent

## Means and Autocorrelations

For a second order r.p., and for all .

## Properties of Autocorrelation functions

1. . (This is the second moment)
2. (Schwartz inequality)
3. (symmetric)

## Wide-sense stationarity

Since depends only on , we write (by an abuse of notation'')

Thus,

If a random process is second order and strictly stationary, it must also be WSS. On the other hand, if a process is WSS, it is not necessarily strictly stationary.

We say a process is covariant stationary if depends only on , or, equivalently, depends only on .

## Properties of for WSS r.p.s

1. (independent of )
2. .
3. (even function)
4. A defining property of these functions:

for all and all and for all .

Any function with this property is a nonnegative definite function.

Before proceeding with more properties, a few examples.
Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, June 07). Basic Concepts of Random Processes. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lecture6_3.htm. This work is licensed under a Creative Commons License