Autocorrelation

Autocorrelation of a real-valued stochastic process measures the correlation of values separated by a given lag. For a weakly stationary process with mean , the autocovariance and autocorrelation functions are

			(1)
			(2)

where denotes the expectation value of . They depend only on the lag , not on the time . Their sample counterpart is the sample autocorrelation.

For a finite sequence whose indices are taken modulo its length, the corresponding notion is periodic autocorrelation. The non-wrapping counterpart is aperiodic autocorrelation.

For a complex function , the autocorrelation is defined by

			(3)
			(4)

where denotes cross-correlation and is the complex conjugate (Bracewell 1965, pp. 40-41).

Note that the notation is sometimes used for and that the quantity

(5)

is sometimes also known as the autocorrelation of a continuous real function (Papoulis 1962, p. 241).

The autocorrelation discards phase information, returning only the power, and is therefore an irreversible operation.

There is also a somewhat surprising and extremely important relationship between the autocorrelation and the Fourier transform known as the Wiener-Khinchin theorem. Let , and denote the complex conjugate of , then the Fourier transform of the absolute square of is given by

(6)

is maximum at the origin; in other words,

(7)

To see this, let be a real number. Then

(8)

(9)

(10)

Define

			(11)
			(12)

Then plugging into above, we have . This quadratic equation does not have any real root, so , i.e., . It follows that

(13)

with the equality at . This proves that is maximum at the origin.