In the mathematical sciences, a stationary process (or strict(ly) stationary process) is a stochastic process in which the probability density function of some random variable X does not change over time or position. As a result, parameters such as the mean and variance also do not change over time or position. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In the mathematics of probability, a stochastic process is a random function. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ... In statistics, mean has two related meanings: the average in ordinary English, which is more correctly called the arithmetic mean, to distinguish it from geometric mean or harmonic mean. ... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ...

As an example, the measurement of white noise is stationary. Alternatively, the measurement of a cymbal clashing is not stationary. Although a cymbal clash is basically white noise, the measurement of that noise varies over time: Before the clash, there is silence, and after the clash, the noise gradually diminishes. Four thousandths of a second of white noise White noise ( ▶ (help· info)) is a random signal (or process) with a flat power spectral density. ... Sabian Paragon cymbals Cymbals (Fr. ... Four thousandths of a second of white noise White noise ( ▶ (help· info)) is a random signal (or process) with a flat power spectral density. ...

Stationarity is used as a tool in time series analysis, where the raw data are often transformed to become stationary, for example, economic data are often seasonal and/or dependent on the price level. Processes are described as trend stationary if they are a linear combination of a stationary process and one or more processes exhibiting a trend. Transforming this data to leave a stationary data set for analysis is referred to as de-trending. In statistics and signal processing, a time series is a sequence of data points, measured typically at successive times, spaced apart at uniform time intervals. ... Economics (from the Greek [oikos], house, and [nomos], rule, hence household management) is a social science that studies the production, distribution, trade and consumption of goods and services. ... A series of measurements of a process may be treated as a time series, and then trend estimation is the application of statistical techniques to make and justify statements about trends in the data. ...

A discrete-time stationary process where the sample space is also discrete (so that the random variable may take one of N possible values) is known as a Bernoulli scheme. When N=2, the process is called a Bernoulli process. Discrete time is non-continuous time. ... In mathematics, the Bernoulli scheme is a generalization of the Bernoulli process to more than two possible outcomes. ... In probability and statistics, a Bernoulli process is a discrete_time stochastic process consisting of finite or infinite sequence of independent random variables X1, X2, X3,..., such that For each i, the value of Xi is either 0 or 1; For all values of i, the probability that Xi = 1 is...

Weak or wide-sense stationarity

A weaker form of stationarity commonly employed in signal processing is known as weak-sense, wide-sense stationarity (WSS), second-order stationarity or covariance stationarity. WSS random processes only require that 1st and 2nd moments do not vary with respect to time. Signal processing is the processing, amplification and interpretation of signals and deals with the analysis and manipulation of signals. ... This article is in need of attention from an expert on the subject. ...

So, a continuous-time random process x(t) which is WSS has the following restrictions on its mean function In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In the mathematics of probability, a stochastic process can be thought of as a random function. ... In statistics, mean has two related meanings: the average in ordinary English, which is more correctly called the arithmetic mean, to distinguish it from geometric mean or harmonic mean. ...

1.

and correlation function In probability theory and statistics, correlation, also called correlation coefficient, is a numeric measure of the strength of linear relationship between two random variables. ...

2.

The first property implies that the mean function m_x(t) must be constant. The second property implies that the correlation function depends only on the difference between t₁ and t₂ and only needs to be indexed by one variable rather than two variables. Thus, instead of writing,

we usually abbreviate the notation and write

When processing WSS random signals with linear, time-invariant (LTI) filters, it is helpful to think of the correlation function as a linear operator. Since it is a circulant operator (depends only on the difference between the two arguments), its eigenfunctions are the Fourier complex exponentials. Additionally, since the eigenfunctions of LTI operators are also complex exponentials, LTI processing of WSS random signals is highly tractable --- all computations can be performed in the frequency domain. Thus, the WSS assumption is widely employed in signal processing algorithms. The word linear comes from the Latin word linearis, which means created by lines. ... A time-invariant system is one whose output does not depend explicitly on time. ... In electrical engineering, specifically in signal processing and control theory, LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. ... In electronics and signal processing, a filter is a device or process that modifies a signal. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In linear algebra, a circulant matrix is a special kind of Toeplitz matrix where each row vector is shifted one element to the right relative to the preceding row vector. ... In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. ... Fourier (SAMPA: [fVri:eI]) can mean: Jean Baptiste Joseph Fourier, a French mathematician and physicist. ... The exponential function is one of the most important functions in mathematics. ... In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. ... The exponential function is one of the most important functions in mathematics. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ... Signal processing is the processing, amplification and interpretation of signals and deals with the analysis and manipulation of signals. ...