Autocorrelation - Properties

Properties

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, which is easy to prove from the definition. In the continuous case,

the autocorrelation is an even function

when is a real function,

and the autocorrelation is a Hermitian function

when is a complex function.

• The continuous autocorrelation function reaches its peak at the origin, where it takes a real value, i.e. for any delay, . This is a consequence of the Cauchy–Schwarz inequality. The same result holds in the discrete case.

• The autocorrelation of a periodic function is, itself, periodic with the same period.

• The autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all ) is the sum of the autocorrelations of each function separately.

• Since autocorrelation is a specific type of cross-correlation, it maintains all the properties of cross-correlation.

• The autocorrelation of a continuous-time white noise signal will have a strong peak (represented by a Dirac delta function) at and will be absolutely 0 for all other .

• The Wiener–Khinchin theorem relates the autocorrelation function to the power spectral density via the Fourier transform:

• For real-valued functions, the symmetric autocorrelation function has a real symmetric transform, so the Wiener–Khinchin theorem can be re-expressed in terms of real cosines only: