11. Deterministic and random signals and their characteristics.

Where - boundaries of possible values;

- is a differential law of random variable distribution and is called one-dimensional probability density;

- integral distribution function of a random variable.

For practical applications, the following statistical characteristics of a random variable are important:

1) The expectation of a random variable:

, (2)

if events are equally probable, then

2) Variance of the random variable (deviation from the mean):

, (3)

if events are equally probable:

.

3) Standard Deviation (MSD):

. (four)

A stationary process is a process if its -dimensional distribution law depends on a time interval. but does not depend on the position on the number axis. For strictly stationary processes, the expectation and variance do not depend on time.

When considering random variables, time and population are taken into account, i.e. sampling. A random process is called ergodic if, in determining any statistical characteristics, averaging over the population and over the sample is equal to time averaging.

A stationary random process with a constant spectral density is called white noise . It is characterized by the fact that its two values ​​are not correlated. White noise with an unlimited frequency spectrum is called mathematical white noise. White noise with a limited frequency spectrum is called physical white noise.

Correlation - the magnitude of the similarity of the two signals. If two different signals are compared, then the cross-correlation function is a measure of their similarity. If the signal is compared to itself, then the degree of similarity is determined by the autocorrelation function.

The main characteristics of deterministic signals are its energy characteristics.

Energy characteristics of signals:

1. Instant (current) power: . (five)

2. Energy: . (6)

3. Average power on the interval:

. (7)

4. If the signal is equal to the sum of two signals:

,

,

. (eight)

The mutual energy and power of two signals characterize the degree of similarity of two signals .

5. If the signals coincide, the mutual energy increases by 4 times, and such systems are called coherent :

.

6. If the signals are opposite, then they are called orthogonal . Energy orthogonality always implies orthogonality in power, but not vice versa:

.

7. If the signals do not completely coincide, they are called partially coincident signals.