## Definition and basic concepts of random process

## Stationary and non-stationary

## Mean, correlation, and covariance function functions, the mean-square value and variance

The mean of a strictly stationary process is a constant.

first-order moment:

second-order moment:

Example 1.2 Sinusoidal Wave with Random Phase

Example 1.3 Random Binary Wave

The concept of ergodic process

## Transmission of a random process through a linear time-invariant filter

We see that if the input to a stable linear time-invariant filter is a **stationary process**, then the output of the filter is also a **stationary process**.

## Power spectral density

Einstein-Wiener-Khintchine relations.

Property 1: The zero-frequency value of the power spectral density of a stationary process equals the total area under the graph of the auto correlation function.

Property 2: The mean-square value of a stationary process equals the total area under the graph of the power spectral density.

Property 3: The power spectral density of a stationary process is always nonnegative. (Because the mean-square value must always be nonnegative.)

Property 4: The power spectral density of a real-valued random process is an even function of frequency.

Property 5: The power spectral density, appropriately normalized, has the properties usually associated with a probability density function.

## Gaussian process

## Noise

###### White Noise

## Uncorrelated

Covariance is 0.

Uncorrelated, then they are statistically independent, which, in turn, means that the joint probability density function of this set of random variables can be expressed as the product of the probability density functions of the individual random variables in the set.

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