mean (mathematical expectation)

variance

Correlation function R(t1,t2) or covariance function B(t1,t2)

If the statistical characteristics of a random process have nothing to do with the starting point of time, that is, the time translation does not affect any of its statistical characteristics, then the random process is said to be a stationary random process in the strict sense, referred to as a strictly stationary random process (a strictly stationary random process must be a generalized Stable, the opposite may not be true)

Generalized stationary random process

The one-dimensional probability density function has nothing to do with time, while the two-dimensional distribution function is only related to the time interval.

Any realization in a random process has experienced all possible states of the random process. A random process with ergodic states must be a stationary process, and the reverse is not necessarily true.

The numerical characteristics of a process with ergodic properties can be completely replaced by the time average of any realization in a random process.

If the statistical average of a stationary process is equal to the time average of any of its realizations, the stationary process is said to have ergodic properties.

R(τ)=E[ε(t)ε(tτ)]

nature

Px(f)=lim(T→∞)|XT(f)|²/T

The power spectral density of any sample function of an ergodic process is equal to the power spectral density of the process

Power spectral density is non-negative and real-even

The power spectral density of a stationary process and its autocorrelation function are a pair of Fourier transform relationships

If any n-dimensional distribution of a random process obeys the normal distribution, it is called a normal process or Gaussian process.

The n-dimensional distribution of a Gaussian process depends only on the mean, variance and normalized covariance of each random variable. Therefore, for a Gaussian process, you only need to study its numerical characteristics

A generalized stationary Gaussian process is also strictly stationary. Because, if the Gaussian process is generalized stationary, that is, the mean value of q has nothing to do with time, and the covariance function is only related to the time interval, but has nothing to do with the starting point of time, then its n-dimensional distribution has nothing to do with the starting point of time, so it is also strictly stationary. of. Therefore, if the Gaussian process is broadly stationary, it is also strictly stationary.

If the values ​​of Gaussian processes at different times are uncorrelated, then they are also statistically independent.

The process generated by a Gaussian process after linear transformation is still a Gaussian process.

The value of a Gaussian process at any time is a normally distributed random variable, also called a Gaussian random variable.

One-dimensional probability density function: f(x)=1/√2π·σ·exp(-(x-a)²/2σ²)

The mean is a constant

The autocorrelation function of the output process is only a function of the time interval

If the input system process of a linear system is stationary, then the output process is stationary.

The power spectral density of the output process is the power spectral density of the input process multiplied by the square of the system frequency response modulus

If the input process of a linear system is Gaussian, the output process of the system is also Gaussian.

envelope phase

In phase

If the power spectral density of the noise is a constant at all frequencies, the noise is called white noise, represented by n(t)

Since white noise has infinite bandwidth, its average power is infinite

If the probability distribution of white noise values ​​obeys a Gaussian distribution, it is called Gaussian white noise.

If white noise passes through an ideal rectangular low-pass filter or an ideal low-pass channel, the output noise is called low-pass white noise

If white noise passes through an ideal rectangular bandpass filter or an ideal bandpass channel, the output noise is called band-identical white noise.