Summary Stationary and Wide Sense Stationary Random Process

PROBABILITY AND STOCHASTICS THEORY (21MAB203T) – RANDOM AND STATIONERY
PROCESS

Stationary and Wide Sense Stationary Random Process

A random process is said to be stationary if its statistical properties do not change over
time. There are two types of stationary random processes:

1. Strictly Stationary Random Process

2. Wide Sense Stationary Random Process

In this note, we will focus on the Wide Sense Stationary Random Process.

Mean and Autocorrelation Function

The mean function of a random process is defined as the expected value of the process
at any point in time. It is denoted by mx(t) and is given by:

mx(t)=E[x(t)]
where x(t) is the random process and E[⋅] denotes the expected value.

The autocorrelation function of a random process is a measure of the similarity
between the process at different points in time. It is defined as the expected value of
the product of the process at two different points in time. It is denoted by Rx(t1,t2)
and is given by:


Rx(t1,t2)=E[x(t1)x(t2)]where x(t1)and x(t2)are the values of the random process at
times $t_1$ and $t_2$, respectively.

A Wide Sense Stationary Random Process has the following properties:

1. The mean function is constant, i.e.,
mx(t)=mx for all t.

2. The autocorrelation function depends only on the difference between the two time
points, i.e., Rx(t1,t2)=Rx(t2−t1).

Example: Uniformly Distributed Random Variable

Let x(t) be a uniformly distributed random variable over the interval [-a, a]. Its mean
function is given by:




Its autocorrelation function is given by:

Rx(t1,t2)=E[x(t1)x(t2)]