Editing Autocorrelation (section)

=== Autocorrelation of discrete-time signal ===
The discrete autocorrelation <math>R</math> at lag <math>\ell</math> for a discrete-time signal <math>y(n)</math> is

{{Equation box 1
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|equation = <math>R_{yy}(\ell) = \sum_{n \in Z} y(n)\,\overline{y(n-\ell)}</math>
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The above definitions work for signals that are square integrable, or square summable, that is, of finite energy. Signals that "last forever" are treated instead as random processes, in which case different definitions are needed, based on expected values. For [[wide-sense-stationary random process]]es, the autocorrelations are defined as

<math display=block>\begin{align}
R_{ff}(\tau) &= \operatorname{E}\left[f(t)\overline{f(t-\tau)}\right] \\
R_{yy}(\ell) &= \operatorname{E}\left[y(n)\,\overline{y(n-\ell)}\right] .
\end{align}</math>

For processes that are not [[Stationary process|stationary]], these will also be functions of <math>t</math>, or <math>n</math>.

For processes that are also [[Ergodic process|ergodic]], the expectation can be replaced by the limit of a time average. The autocorrelation of an ergodic process is sometimes defined as or equated to<ref name="dunn"/>

<math display=block>\begin{align}
R_{ff}(\tau) &= \lim_{T \rightarrow \infty} \frac 1 T \int_0^T f(t+\tau)\overline{f(t)}\, {\rm d}t \\
R_{yy}(\ell) &= \lim_{N \rightarrow \infty} \frac 1 N \sum_{n=0}^{N-1} y(n)\,\overline{y(n-\ell)} .
\end{align}</math>

These definitions have the advantage that they give sensible well-defined single-parameter results for periodic functions, even when those functions are not the output of stationary ergodic processes.

Alternatively, signals that ''last forever'' can be treated by a short-time autocorrelation function analysis, using finite time integrals. (See [[short-time Fourier transform]] for a related process.)