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GeezerVison is showing mathworld.wolfram.com — Cross-Correlation -- from Wolfram MathWorld

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Cross-Correlation

The cross-correlation of two complex functions f and g of a real variable t, denoted f*g, is defined by

f*g=f^~*g,
(1)

where * denotes convolution, f^_(t) is the complex conjugate of f(t), and f^~(t)=f^_(-t). Since convolution is defined by

[f*g](t)=int_(-infty)^inftyf(tau)g(t-tau)dtau,
(2)

it follows that

[f*g](t)=int_(-infty)^inftyf^_(-tau)g(t-tau)dtau.
(3)

Letting tau^'=-tau, dtau^'=-dtau, so (3) is equivalent to

[f*g](t)=int_infty^(-infty)f^_(tau^')g(t+tau^')(-dtau^')
(4)
=int_(-infty)^inftyf^_(tau)g(t+tau)dtau.
(5)

The cross-correlation satisfies the identity

(g*h)*(g*h)=(g*g)*(h*h).
(6)

If f is even, then

f*g=f^_*g,
(7)

where * again denotes convolution.

See also

Autocorrelation, Convolution, Cross-Correlation Theorem, Fourier Transform

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References

Bracewell, R. "Pentagram Notation for Cross Correlation." The Fourier Transform and Its Applications. New York: McGraw-Hill, pp. 46 and 243, 1965.Papoulis, A. The Fourier Integral and Its Applications. New York: McGraw-Hill, pp. 244-245 and 252-253, 1962.

Referenced on Wolfram|Alpha

Cross-Correlation

Cite this as:

Weisstein, Eric W. "Cross-Correlation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Cross-Correlation.html

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