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Random Processes

Module by: Nick Kingsbury

Summary: This module introduces random processes.

We discussed Random Signals briefly and now we return to consider them in detail. We shall assume that they evolve continuously with time tt, although they may equally well evolve with distance (e.g. a random texture in image processing) or some other parameter.
We can imagine a generalization of our previous ideas about random experiments so that the outcome of an experiment can be a 'Random Object', an example of which is a signal waveform chosen at random from a set of possible signal waveforms, which we term an Ensemble. This ensemble of random signals is known as a Random Process.
Figure 1: Ensemble representation of a random process.
An example of a Random Process Xtα X t α is shown in Figure 1, where tt is time and αα is an index to the various members of the ensemble.
  • tt is assumed to belong to some set (the time axis).
  • αα is assumed to belong to some set (the sample space).
  • If is a continuous set, such as ℝ or 0∞ 0 , then the process is termed a Continuous Time random process.
  • If is a discrete set of time values, such as the integers ℤ, the process is termed a Discrete Time Process or Time Series.
  • The members of the ensemble can be the result of different random events, such as different instances of the sound 'ah' during the course of this lecture. In this case αα is discrete.
  • Alternatively the ensemble members are often just different portions of a single random signal. If the signal is a continuous waveform, then αα may also be a continuous variable, indicating the starting point of each ensemble waveform.
We will often drop the explicit dependence on αα for notational convenience, referring simply to random process Xt X t .
If we consider the process Xt X t at one particular time t=t1 t t1 , then we have a random variable Xt1 X t1 .
If we consider the process Xt X t at NN time instants t1t2…tN t1 t2 … tN , then we have a random vector: X=Xt1Xt2…XtNT X X t1 X t2 … X tN We can study the properties of a random process by considering the behavior of random variables and random vectors extracted from the process, using the probability theory derived earlier in this course.

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