Random Processes: Filtering, Estimation, and Detection

4 Random Processes

4.1 DEFINITION OF A RANDOM PROCESS

From Chapters 2 and 3 we learned that an experiment is specified by the three tuple (S, BF, P(·)), where S is a countable or noncountable set representing the outcomes of the experiment, BF is a Borel field specifying the set of events for which consistent probabilities exist, and P(·) is a probability measure that allows the calculation of the probability of all the events in the Borel field. A real random variable X(ξ) was defined as a real-valued function of S subject to (a) the event specified by {ξ : X(ξ) ≤ x} is a member of the BF for all x, thus guaranteeing the existence of the cumulative distribution, and (b) P{ξ : X(ξ) = ∞} = 0 or P{ξ : x(ξ) = −∞} = 0, or both.

If, instead of assigning a real value to each ξ, a time function X(t, ξ) is defined for each ξ, we say a random process is specified. Roughly speaking, a random process is a family of time functions together with a probability measure. For a finite sample space S we can visualize the random process as in Figure 4.1, that of a mapping from the sample space to a space of time waveforms.

A random process can also be viewed as shown in Figure 4.2. For a particular outcome ξ_i, with probability P_i, the time waveform shown X(t, ξ_i) occurs. The n time signals represent an ensemble of time waveforms.

If we evaluate these time waveforms at t₀, the values on the second column from the right are obtained. Coming down that column of values, we have a mapping from the outcomes ...