# Dictionary:Crosscorrelation theorem

Other languages:
English • ‎español

The Fourier transform of the crosscorrelation of g1(t) and g2(t) is

${\displaystyle \phi _{12}(\tau )\leftrightarrow {\overline {G_{1}(f)}}G_{2}(f)=\Phi _{12}(f)}$,

where ${\displaystyle G_{1}(f),G_{2}(f),\Phi _{12}(f)}$ are the Fourier transforms of ${\displaystyle g_{1}(t),g_{2}(t),\phi _{12}(\tau )}$, and the superscribed bar indicates a complex conjugate. Here ${\displaystyle \Phi _{12}(f)}$ is called the cross-energy spectrum. See Figure F-22 and Sheriff and Geldart (1995, 285, 538, and 541-542) [1].

Figure F-22. Fourier transform theorems.

## References

1. Sheriff, R. E. and Geldart, L. P., 1995, Exploration Seismology, 2nd Ed., Cambridge Univ. Press.