Stationary Processes

Laboratory Work №2

INTODUCION TO STOCHASTIC PROCESSES

POWER SPECTRAL DENSITY,

CROSS SPECTRAL DENSITY, CORRELATION

Brief Theoretical Review

A stochastic process (random process, or random function), see Fig.1 can be defined as a family of random variables . The variables are indexed with the parameter t which belongs to the set T, the indexed set, or the parameter set. The set of possible values which random variables may assume is called the state space of the process. Therefore, a random variable is completely characterized by its probability density function (PDF).

Fig.1 Random process

The probability density function describes the general distribution of the magnitude of the random process, but it gives no information on the time or frequency content of the process.

Stationary Processes

If the distribution of is identical to the distribution of for all such that and the stochastic process is said to be stationary.

A stationary process is said to be ergodic if the ensemble average equals the time average of the sample functions, that is,

for almost all ω. The symbol E denotes mathematical expectation, that is, integration with respect o the measure P.

The widely used characteristics for a random process are mean value, m and covariance function :

mean value:

The mean value,` , is the height of the rectangular area having the same area as that under the function x(t).

Therandom processcan be characterized by the following measures: mean square value, variance, standard deviation.

Given two random variables X (t₁) and X (t₂), a measure of linear relationship between them is specified by a covariance function, namely:

- ensemble average

or

- time average

This is an autocorrelation function. The autocorrelation, or autocovariance, describes the general dependency of x(t) with its value at a short time later, x(t+t) (see Fig. below).

The covariance function of the ergodic process possesses with the following properties:

1⁰ ;

2⁰ - the covariance function is an even function;

3⁰ - the correlation value is maximum for zero shift.

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