Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

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:

(7.203)

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.