Random processes

We discussed Random Signals briefly and now we return to consider them in detail. We shall assume that they evolve continuouslywith time $t$ , although they may equally well evolve with distance (e.g. a random texture inimage 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 bea 'Random Object', an example of which is a signal waveform chosen at random from a set of possible signal waveforms, whichwe term an Ensemble . This ensemble of random signals is known as a Random Process .

An example of a Random Process $X(t, \alpha )$ is shown in , where $t$ is time and $\alpha$ is an index to the various members of the ensemble.

• $t$ is assumed to belong to some set $$ (the time axis).
• $\alpha$ is assumed to belong to some set $$ (the sample space).
• If $$ is a continuous set, such as $\mathbb{R}$ or $\left\{0\right\}$∞ , then the process is termed a Continuous Time random process.
• If $$ is a discrete set of time values, such as the integers $\mathbb{Z}$ , 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 $\alpha$ is discrete.
• Alternatively the ensemble members are often just different portions of a single random signal. If the signal is acontinuous waveform, then $\alpha$ may also be a continuous variable, indicating the starting point of eachensemble waveform.

If we consider the process $\{X(t)\}$ at one particular time $t=t_1$ , then we have a random variable $X(t_1)$ .

If we consider the process $\{X(t)\}$ at $N$ time instants $\{t_1, t_2, \dots , t_N\}$ , then we have a random vector : $$X=\begin{pmatrix}X(t_1) & X(t_2) & \dots & X(t_N)\\ \end{pmatrix}^T$$ We can study the properties of a random process by considering the behavior of random variables and random vectors extractedfrom the process, using the probability theory derived earlier in this course.