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 and is assumed when defining . 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 square integrable 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 when the limit exists (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 inverse Fourier transform of the absolute square of is given by

(6)

For the square integrable complex function , Schwarz's inequality gives

(7)

Thus its complex modulus is maximum at the origin. For a square integrable real function , itself is maximum at the origin; in other words,

(8)

To see this, let be a real number. Then

(9)

(10)

(11)

Define

			(12)
			(13)

Then the preceding inequality is for every real number . If , the result is immediate. If , the discriminant satisfies , so and hence . It follows that

(14)

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