Let be a periodic sequence, then the autocorrelation of the sequence, sometimes called the periodic autocorrelation (Zwillinger 1995, p. 223), is the sequence
(1)
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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)
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where the final subscripts are understood to be taken modulo and , respectively.
For a complex function , the autocorrelation is defined by
(3)
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(4)
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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)
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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)
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is maximum at the origin; in other words,
(7)
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To see this, let be a real number. Then
(8)
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(9)
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(10)
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Define
(11)
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(12)
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Then plugging into above, we have . This quadratic equation does not have any real root, so , i.e., . It follows that
(13)
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with the equality at . This proves that is maximum at the origin.