The cross-correlation is related to the convolution by:
so that if either f or g is an even function
Also:
In analogy with the convolution theorem, the cross-correlation satisfies
where denotes the Fourier transform, and an asterisk again indicates the complex conjugate. Coupled with fast Fourier transform algorithms, this property is often exploited for the efficient numerical computation of cross-correlations.
The cross-correlation is related to the spectral density. See Wiener–Khinchin theorem
the cross correlation of a convolution of f and h with a function g is the convolution of the correlation of f and g with the kernel h:
so that if either f or g is an even function
Also:
In analogy with the convolution theorem, the cross-correlation satisfies
where denotes the Fourier transform, and an asterisk again indicates the complex conjugate. Coupled with fast Fourier transform algorithms, this property is often exploited for the efficient numerical computation of cross-correlations.
The cross-correlation is related to the spectral density. See Wiener–Khinchin theorem
the cross correlation of a convolution of f and h with a function g is the convolution of the correlation of f and g with the kernel h:
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