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== Background ==
Analysis of the loads on a wind turbine are usually carried out through use of power spectra. A power spectrum is defined as the power spectral density function of a signal plotted against frequency. The power spectral density function of a plot is defined as the Fourier transform of the covariance function.<ref>Remote sensing: models and methods for image processing, R. a. Schowengerdt</ref><ref>Remote Sensing: Models and Methods for Image Processing, Robert A. Schowengerd</ref> Regarding analysis of loads, the analysis involves time series, in which case the covariance function becomes the [[Autocovariance|autocovariance]] function. In the signal processing sense, the autocovariance can be related to the [[Autocorrelation|autocorrelation]] function.
In [[statistics]], given a real [[stochastic process]] ''Z''(''t''), the '''autocovariance''' is the [[covariance]] of the variable against a time-shifted version of itself. If the process has the [[mean]] <math>E[Z_t] = \mu_t</math>, then the autocovariance is given by
:<math>C_{ZZ}(t,s) = E[(Z_t - \mu_t)(Z_s - \mu_s)] = E[Z_t Z_s] - \mu_t \mu_s.\,</math>
where ''E'' is the [[expected value|expectation]] operator.
=== Stationarity ===
If ''Z''(''t'') is [[stationary process]], then the following are true:
:<math>\mu_t = \mu_s = \mu \,</math> for all ''t'', ''s''
and
:<math>C_{ZZ}(t,s) = C_{ZZ}(s-t) = C_{ZZ}(\tau)\,</math>
where
:<math>\tau = s - t\,</math>
is the lag time, or the amount of time by which the signal has been shifted.
As a result, the autocovariance becomes
:<math>C_{ZZ}(\tau) = E[(Z(t) - \mu)(Z(t+\tau) - \mu)]\,</math>
::::<math> = E[Z(t) Z(t+\tau)] - \mu^2\,</math>
::::<math> = R_{ZZ}(\tau) - \mu^2,\,</math>
where ''R''<sub>ZZ</sub> represents the [[autocorrelation]] in the signal processing sense.
== Deterministic processes ==
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