Cross-Correlation

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

$$f\star g \equiv f^*(-t)*g(t),$$ (1)

where $*$ denotes Convolution and $f^*(t)$ is the Complex Conjugate of $f(t)$. The Convolution is defined by

$$f(t)*g(t)=\int_{-\infty}^\infty f(\tau)g(t-\tau)d\tau,$$ (2)

therefore

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

Let $\tau'\equiv -\tau$, so $d\tau'=-d\tau$ and

$$f\star g = \int_\infty^{-\infty} f^*(\tau')g(t+\tau')(-d\tau')$$

$$= \int_{-\infty}^\infty f^*(\tau)g(t+\tau)\,d\tau.$$ (4)

The cross-correlation satisfies the identity

$$(g\star h)\star(g\star h) = (g\star g)\star(h\star h).$$ (5)

If $f$ or $g$ is Even, then

$$f\star g = f*g,$$ (6)

where $*$ denotes Convolution.

See also Autocorrelation, Convolution, Cross-Correlation Theorem