Let
denote the cross-correlation of functions
and
.
Then
(1)
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(2)
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(3)
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(4)
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(5)
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(6)
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(7)
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where
denotes the Fourier transform,
is the complex conjugate,
and
(8)
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(9)
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Applying a Fourier transform on each side gives the cross-correlation theorem,
(10)
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If ,
then the cross-correlation theorem reduces to the Wiener-Khinchin
theorem.