Autocorrelation

Let be a periodic sequence, then the autocorrelation of the sequence, sometimes called the periodic autocorrelation (Zwillinger 1995, p. 223), is the sequence

(1)

where denotes the complex conjugate and the final subscript is understood to be taken modulo .

Similarly, for a periodic array with and , the autocorrelation is the -dimensional matrix given by

(2)

where the final subscripts are understood to be taken modulo and , respectively.

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.

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