November 2012
Intermediate to advanced
794 pages
22h 9m
English
book
Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
by John J. Shynk
Content preview from Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
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7.8 DIFFERENCE EQUATIONS
Similar results can be derived for a random sequence Y[k] that is processed by a discrete-time system represented by the following linear difference equation:
yielding the output X[k]. Taking expectations gives the following equation for the mean:
(7.204) 
and it is straightforward to generate similar equations for the various correlation functions:
(7.205) 
(7.206) 
where the random sequences are not necessarily wide-sense stationary. The first equation is derived by replacing k with k1 in (7.203), which is then multiplied by X[k2] and the expectation is taken. The second equation is obtained in the same way except that (7.203) is multiplied by Y[k2]. These difference equations can be analyzed using methods similar to those of the previous example for DEs except that the z-transform is used instead of the Laplace transform. We defer that discussion to Chapter 8 where we focus on wide-sense stationary signals, though a first-order example is considered in Problem 7.24.
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