Autocorrelation is a mathematical tool used frequently in signal processing for analysing series of values, such as time domain signals.
Formally, the autocorrelation R at distance j for signal x(i) is
- R(j) = E{(x(n)-m)(x(n-j)-m)]},
where the expected value operator E{} is taken over n, and m is the average value (expected value) of x(i). Quite frequently, autocorrelations are calculated for zero-centered signals, that is, for signals with zero mean. The autocorrelation definition then becomes
- R(j) = E[x(n)x(n-j)],
which is the definition of autocovariance.
Multi-dimensional autocorrelation is defined similarly, that is, for example in three dimensions
- R(j,k,l) = E{[x(n,m,p)-m][x(n-j,m-k,p-l)-m]}.
In the following, we will describe properties of one-dimensional autocorrelations only, since most properties are easily transferred from the one-dimensional case to the multi-dimensional cases.
A fundamental property of the autocorrelation is symmetry, R(i) = R(-i), which is easy to prove from the definition.
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