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Let $\star$ denote cross-correlation. Then the cross-correlation of two functions $f(t)$ and $g(t)$ of a real variable $t$ is defined by

f\star g \equiv f^*(-t)*g(t),
\end{displaymath} (1)

where $*$ denotes Convolution and $f^*(t)$ is the Complex Conjugate of $f(t)$. The Convolution is defined by
f(t)*g(t)=\int_{-\infty}^\infty f(\tau)g(t-\tau)d\tau,
\end{displaymath} (2)

f\star g \equiv \int_{-\infty}^\infty f^*(-\tau)g(t-\tau)\,d\tau.
\end{displaymath} (3)

Let $\tau'\equiv -\tau$, so $d\tau'=-d\tau$ and
$\displaystyle f\star g$ $\textstyle =$ $\displaystyle \int_\infty^{-\infty} f^*(\tau')g(t+\tau')(-d\tau')$  
  $\textstyle =$ $\displaystyle \int_{-\infty}^\infty f^*(\tau)g(t+\tau)\,d\tau.$ (4)

The cross-correlation satisfies the identity
(g\star h)\star(g\star h) = (g\star g)\star(h\star h).
\end{displaymath} (5)

If $f$ or $g$ is Even, then
f\star g = f*g,
\end{displaymath} (6)

where $*$ denotes Convolution.

See also Autocorrelation, Convolution, Cross-Correlation Theorem

© 1996-9 Eric W. Weisstein