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stochastic process
Communication Principles Chapter 3 Random Process Mind Map, a random process is a set of all sample functions, and a random process is regarded as a set of random variables at different moments in the time process.
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stochastic process
- bacteria
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- Outline
stochastic process
basic concept
mean (mathematical expectation)
is a deterministic function of time a(t)
The swing center of the n sample function curve representing the random process
variance
Difference between mean square value and mean square
Represents the degree of deviation of the random process at time t relative to the mean a(t)
Correlation function R(t1,t2) or covariance function B(t1,t2)
Since R(t1, t2) measures the degree of correlation of the same process, it is called an autocorrelation function.
B(t1,t2)=R(t1,t2)-a(t1)a(t2)
Measures the degree of correlation between two processes, called the cross-correlation function
stationary random process
definition
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 mean value has nothing to do with t and is a constant a
The autocorrelation function is only related to the time interval
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.
ergodicity
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.
autocorrelation function
R(τ)=E[ε(t)ε(tτ)]
nature
R(0)=E[ε²(t)], indicating the average power of ε(t)
R(τ)=R(-τ), indicating the even function of τ
|R()τ|≤R(0), indicating the upper bound of R(τ)
R(∞)=E²[ε(t)]=a², indicating the DC power of ε(t)
R(0)-R(∞)=σ², σ² is the variance, which represents the AC power of the stationary process
Power Spectral Density
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
Gaussian random process
definition
If any n-dimensional distribution of a random process obeys the normal distribution, it is called a normal process or Gaussian process.
important properties
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.
Gaussian random variable
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)
Stationary stochastic processes through linear systems
Output process mean
The mean is a constant
Autocorrelation function of the output process
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
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
Probability distribution of the output process
If the input process of a linear system is Gaussian, the output process of the system is also Gaussian.
narrowband random process
Most communication systems are of the narrowband bandpass type
The signal or noise passing through the narrowband system must be a narrowband random process
Expressions for narrowband processes
envelope phase
In phase
Sine wave plus narrowband Gaussian noise
The probability density function of the envelope is the generalized Rayleigh distribution, also known as the Rician distribution
When the signal is very small, the Rician distribution degenerates into the Rayleigh distribution.
When the signal-to-noise ratio is very large, the Rician distribution is approximated to a Gaussian distribution.
Gaussian white noise and band-limited white noise
White Noise
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.
low pass 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
band 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.