Cross-Correlation
The cross-correlation of two complex functions f(t) and g(t) of a real variable t, denoted f*g is defined by
| f*g=f^_(-t)*g(t), |
(1)
|
where * denotes convolution and f^_(t) is the complex conjugate of f(t). Since convolution is defined by
it follows that
=int_(-infty)^inftyf^_(-tau)g(t-tau)dtau.](/index.cgi/speech/https://mathworld.wolfram.com/images/equations/Cross-Correlation/NumberedEquation3.svg){kind=link}
Letting tau^'=-tau, dtau^'=-dtau, so (3) is equivalent to
The cross-correlation satisfies the identity
| (g*h)*(g*h)=(g*g)*(h*h). |
(6)
|
If f or g is even, then
| f*g=f*g, |
(7)
|
where * again denotes convolution.
See also
Autocorrelation, Convolution, Cross-Correlation Theorem, Fourier TransformExplore with Wolfram|Alpha
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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-CorrelationCite this as:
Weisstein, Eric W. "Cross-Correlation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Cross-Correlation.html