where is the lag time, or the amount of time by which the signal has been shifted.
The autocovariance function of a WSS process is therefore given by:[2]: p. 517
(Eq.2)
which is equivalent to
.
Normalizationedit
It is common practice in some disciplines (e.g. statistics and time series analysis) to normalize the autocovariance function to get a time-dependent Pearson correlation coefficient. However in other disciplines (e.g. engineering) the normalization is usually dropped and the terms "autocorrelation" and "autocovariance" are used interchangeably.
The definition of the normalized auto-correlation of a stochastic process is
.
If the function is well-defined, its value must lie in the range , with 1 indicating perfect correlation and −1 indicating perfect anti-correlation.
Autocovariance can be used to calculate turbulent diffusivity.[4] Turbulence in a flow can cause the fluctuation of velocity in space and time. Thus, we are able to identify turbulence through the statistics of those fluctuations[citation needed].
Reynolds decomposition is used to define the velocity fluctuations (assume we are now working with 1D problem and is the velocity along direction):
where is the true velocity, and is the expected value of velocity. If we choose a correct , all of the stochastic components of the turbulent velocity will be included in . To determine , a set of velocity measurements that are assembled from points in space, moments in time or repeated experiments is required.
If we assume the turbulent flux (, and c is the concentration term) can be caused by a random walk, we can use Fick's laws of diffusion to express the turbulent flux term:
The velocity autocovariance is defined as
or
where is the lag time, and is the lag distance.
The turbulent diffusivity can be calculated using the following 3 methods:
^ abHsu, Hwei (1997). Probability, random variables, and random processes. McGraw-Hill. ISBN 978-0-07-030644-8.
^Lapidoth, Amos (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 978-0-521-19395-5.
^ abKun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3
^Taylor, G. I. (1922-01-01). "Diffusion by Continuous Movements" (PDF). Proceedings of the London Mathematical Society. s2-20 (1): 196–212. doi:10.1112/plms/s2-20.1.196. ISSN 1460-244X.
Further readingedit
Hoel, P. G. (1984). Mathematical Statistics (Fifth ed.). New York: Wiley. ISBN 978-0-471-89045-4.