Connexions

You are here: Home » Content » Statistical Signal Processing » Basic Definitions in Stochastic Processes

Recently Viewed

This feature requires Javascript to be enabled.

Inside Collection (Course):

Course by: Don Johnson. E-mail the author

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.

Content actions

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks