At the beginning of Chapter 7, we pointed out that the concept of random variable was needed to describe with a model the variability of certain phenomena said to be random. Speech signal observed at a microphone’s output is an example. There is no use to try and describe it with a deterministic expression such as *x*(*t*) = *A* cos(2π*f*_{0}*t*), which is relevant however when describing electrical voltage, hence the idea of using random variables for describing the phenomenon at every instant. This leads us to the following definition.

**Definition 8.1** *A random process is a set of time-indexed random variables X*(*t*) *defined in the same probability space. If the possible values for t belong to* , *the process is called a continuous-time random process. If the possible values fort belong to* , *then we are dealing with a discrete-time random process*^{1}.

The definition implies that a random process associates a real value called a realization with every *instant t* and every *outcome ω.* A *random process* can therefore be interpreted as two different perspectives (Figure 8.1):

- either as a set of functions of time, also called
*trajectories,*each one associated with an outcome; - or as a set of
*random variables,*each one associated with a given time.

In MATLAB^{®}, the ...

Start Free Trial

No credit card required