Multiple Wide Sense Stationary Random Processes

Abstract

In Chapters 7 and 12 we defined multiple random variables X and Y as a mapping from the sample space S of the experiment to a point (x, y) in the x-y plane. We now extend that definition to be a mapping from S to a point in the x-y plane that evolves with time, and denote that point as (x[n], y[n]) for –∞ < n < ∞. The mapping, denoted either by (X[n], Y[n]) or equivalently by [X[n] Y[n]T, is called a jointly distributed random process. An example is the mapping from a point at some geographical location, where the possible choices for the location constitute S, to the daily temperature and pressure at that point or (T[n], P[n]). Instead of treating the random processes, which describe temperature and pressure, separately, it makes more sense to analyze them jointly. This is especially true if the random processes are correlated. For example, a drop in barometric pressure usually indicates the onset of a storm, which in turn will cause a drop in the temperature.