Formally, the autocorrelation R at distance j for signal x(i) is R(j) = E{[x(n)m]*[x(nj)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 zerocentered signals, that is, for signals with zero mean. The autocorrelation definition then becomes R(j) = E[x(n)*x(nj)], which is the definition of autocovariance[?].
Multidimensional autocorrelation is defined similarly, that is, for example in three dimensions R(j,k,l) = E{[x(n,m,p)m]*[x(nj,mk,pl)m]}. In the following, we will describe properties of onedimensional autocorrelations only, since most properties are easily transferred from the onedimensional case to the multidimensional 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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