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# Analytic Properties of Random Processes

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Properties  ::  Continuity  ::  Differentiation  ::  Integration

## Differentiation

Recall that is differentiable at if

exists.

Similarly, a r.p. is mean-square differentiable at if

exists in the mean-square sense, that is,

If is mean-square differentiable at every , then defines another random process on the underlying probability space .

Suppose is a second-order random process, and (m.s.). Then:

1. ;
2. and if then

## Properties of the derivative

Suppose is mean-square differentiable with mean-square derivative . Suppose that is second order. Then:

1. is also second order. (This follows from the first fact above.)
2. exists, and equals .

3. exists and is equal to for all .

4. exists and is equal to .
5. Suppose and is also W.S.S. Then is also W.S.S. Also, and are jointly W.S.S. Also,
1. .
2. .

## On the existence of the mean-square derivative

It can be shown that a sufficient condition for the existence of is the existence of at . A necessary condition is the existence and equality of the mixed partials

If is W.S.S., then these two conditions are the same. So exists if and only if

By a result we will show later, we will find that

So the existence of for a WSS process is equivalent to the condition that the second moment of the PSD of is finite.

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, June 07). Analytic Properties 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/lecture7_3.htm. This work is licensed under a Creative Commons License