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Basic Definitions in Stochastic Processes

Module by: Don Johnson. E-mail the author

A random or stochastic process is the assignment of a function of a real variable to each sample point ωω in a sample space. Thus, the process Xωt X ω t can be considered a function of two variables. For each ωω, the time function must be well-behaved and may or may not look random to the eye. Each time function of the process is called a sample function and must be defined over the entire domain of interest. For each tt, we have a function of ωω, which is precisely the definition of a random variable. Hence the amplitude of a random process is a random variable. The amplitude distribution of a process refers to the probability density function of the amplitude: p Xt x p X t x . By examining the process's amplitude at several instants, the joint amplitude distribution can also be defined. For the purposes of this module, a process is said to be stationary when the joint amplitude distribution depends on the differences between the selected time instants.

The expected value or mean of a process is the expected value of the amplitude at each tt. EXt= m X t=xp Xt xdx X t m X t x x p X t x For the most part, we take the mean to be zero. The correlation function is the first-order joint moment between the process's amplitudes at two times. R X t 1 t 2 = x 1 x 2 p X t 1 X t 2 x 1 x 2 d x 1 d x 2 R X t 1 t 2 x 2 x 1 x 1 x 2 p X t 1 X t 2 x 1 x 2 Since the joint distribution for stationary processes depends only on the time difference, correlation functions of stationary processes depend only on | t 1 t 2 | t 1 t 2 . In this case, correlation functions are really functions of a single variable (the time difference) and are usually written as R X τ R X τ where τ= t 1 t 2 τ t 1 t 2 . Related to the correlation function is the covariance function K X τ K X τ , which equals the correlation function minus the square of the mean. K X τ= R X τ m X 2 K X τ R X τ m X 2 The variance of the process equals the covariance function evaluated as the origin. The power spectrum of a stationary process is the Fourier Transform of the correlation function. 𝒮 X ω= R X τe(iωτ)dτ 𝒮 X ω τ R X τ ω τ A particularly important example of a random process is white noise. The process Xt X t is said to be white if it has zero mean and a correlation function proportional to an impulse. EXt=0 X t 0 R X τ= N 0 2δτ R X τ N 0 2 δ τ The power spectrum of white noise is constant for all frequencies, equaling N 0 2 N 0 2 which is known as the spectral height. 1

When a stationary process Xt X t is passed through a stable linear, time-invariant filter, the resulting output Yt Y t is also a stationary process having power density spectrum 𝒮 Y ω=|Hiω|2 𝒮 X ω 𝒮 Y ω H ω 2 𝒮 X ω where Hiω H ω is the filter's transfer function.

Footnotes

  1. The curious reader can track down why the spectral height of white noise has the fraction one-half in it. This definition is the convention.

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