10.1.5 Gaussian Random Processes
Here, we will briefly introduce normal (Gaussian) random processes. We will discuss some examples of Gaussian processes in more detail later on. Many important practical random processes are subclasses of normal random processes.
First, let us remember a few facts about Gaussian random vectors. As we saw before, random variables , ,..., are said to be jointly normal if, for all ,,..., , the random variable
is a normal random variable. Also, a random vector
is said to be normal or Gaussian if the random variables , ,..., are jointly normal. An important property of jointly normal random variables is that their joint PDF is completely determined by their mean and covariance matrices. More specifically, for a normal random vector X with mean and covariance matrix C, the PDF is given by
Now, let us define Gaussian random processes.
A random process is said to be a Gaussian (normal) random process if, for all
the random variables , ,..., are jointly normal.
Example
Let be a zero-mean WSS Gaussian process with , for all .
- Find .
- Find .
- Solution
-
- is a normal random variable with mean and variance
Thus,
- Let . Then, is a normal random variable. We have
Note thatTherefore,We conclude . Thus,
- is a normal random variable with mean and variance
-
An important property of normal random processes is that wide-sense stationarity and strict-sense stationarity are equivalent for these processes. More specifically, we can state the following theorem.
Theorem Consider the Gaussian random processes . If is WSS, then is a stationary process.
- Proof
-
We need to show that, for all and all , the joint CDF of
is the same as the joint CDF ofSince these random variables are jointly Gaussian, it suffices to show that the mean vectors and the covariance matrices are the same. To see this, note that is a WSS process, so
and From the above, we conclude that the mean vector and the covariance matrix ofis the same as the mean vector and the covariance matrix of
-
We need to show that, for all and all , the joint CDF of
Similarly, we can define jointly Gaussian random processes.
Two random processes and are said to be jointly Gaussian (normal), if for all
Note that from the properties of jointly normal random variables, we can conclude that if two jointly Gaussian random processes and are uncorrelated, i.e.,
the random variables
are jointly normal.
then and are two independent random processes.