In statistics, given a real stochastic process X(t), the autocovariance is the covariance of the variable against a time-shifted version of itself. If the process has the mean , then the autocovariance is given by
where E is the expectation operator.
If X(t) is stationary process, then the following are true:
- for all t, s
is the lag time, or the amount of time by which the signal has been shifted.
As a result, the autocovariance becomes
where RXX represents the autocorrelation in the signal processing sense.
However, often the autocovariance is called autocorrelation even if this normalization has not been performed.
The autocovariance can be thought of as a measure of how similar a signal is to a time-shifted version of itself with an autocovariance of σ2 indicating perfect correlation at that lag. The normalisation with the variance will put this into the range [−1, 1].
The autocovariance of a linearly filtered process
See also 
- P. G. Hoel, Mathematical Statistics, Wiley, New York, 1984.
- Lecture notes on autocovariance from WHOI