Dictionary:Crosscorrelation

{{#category_index:C|crosscorrelation}} A measure of the similarity of two waveforms, of the degree of linear relationship between them, or of the extent to which one is a linear function of the other. For two waveforms $G(t)$ and $H(t)$, the normalized crosscorrelation function $\phi _{GH}(\tau )$ is given as a function of the time shift $\tau$; between the functions by

$\phi _{GH}(\tau )={\frac {\displaystyle \int _{-\infty }^{+\infty }G(t)H(t+\tau )dt}{\displaystyle \int _{-\infty }^{+\infty }G(t)dt\int _{-\infty }^{+\infty }H(t)dt}}.$

For digital data this becomes

$\phi _{GH}(\tau )={\frac {\displaystyle \sum _{k=-\infty }^{+\infty }G_{k}H_{k+\tau }}{\displaystyle \sum _{k=-\infty }^{+\infty }G_{k}\displaystyle \sum _{k=-\infty }^{+\infty }H_{k}}}.$

The denominator in the above two expressions is the normalizing factor and is often omitted (as in Wiener filtering). When normalized, a crosscorrelation of 1 indicates a perfect match, values near zero indicate very little correlation, and negative values indicate that one of the wavelets is inverted. Normalized crosscorrelation is also called correlation coefficient. See also autocorrelation. Nonnormalized crosscorrelation can be accomplished by reversing one function in time and convolving:

$\phi _{ab}(\tau )=a(t)^{*}b(-t).$

The equivalent operation in the frequency domain involves multiplying the amplitudes of common frequencies and subtracting phase-response curves. See Sheriff and Geldart (1995, 287–288 and 541–543).