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In the Mathematical Sciences , a stationary process (or '''strict(ly) stationary process''') is a Stochastic Process whose Probability Distribution at a fixed time or position is the same for all times or positions. As a result, parameters such as the Mean and Variance , if they exist, also do not change over time or position.

As an example, White Noise is stationary. However, the sound of a Cymbal crashing is not stationary because the acoustic power of the crash (and hence its variance) diminishes with time.

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.

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 .


WEAK OR WIDE-SENSE STATIONARITY


A weaker form of stationarity commonly employed in Signal Processing is known as weak-sense stationarity, '''wide-sense stationarity''' (WSS), or '''covariance stationarity'''. WSS random processes only require that 1st and 2nd Moments do not vary with respect to time. Any strictly stationary process which has a Mean and a Covariance is also WSS.

So, a Continuous -time Random Process ''x''(''t'') which is WSS has the following restrictions on its mean function

:\mathbb{E}\{x(t)\} = m_x(t) = m_x(t + au) \,\, orall \, au \in \mathbb{R}

and Correlation function

:\mathbb{E}\{x(t_1)x(t_2)\} = R_x(t_1, t_2) = R_x(t_1 + au, t_2 + au) = R_x(t_1 - t_2, 0) \,\, orall \, au \in \mathbb{R}.

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 t_1 and t_2 and only needs to be indexed by one variable rather than two variables. Thus, instead of writing,

:\,\!R_x(t_1 - t_2, 0)\,

we usually abbreviate the notation and write

:R_x( au) \,\! \mbox{ where } au = t_1 - t_2.

When processing WSS random signals with Linear , Time-invariant ( LTI ) Filter s, 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 Eigenfunction s are the Fourier Complex Exponential s. Additionally, since the Eigenfunction s of LTI operators are also Complex Exponential s, 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.


SEE ALSO