Cross Correlation

The cross correlation between two signals x(t) and y(t) is defined as :

Note that the cross-correlation is the convolution of .

If then using the fact that the auto-correlation integral peaks at 0 , the cross correlation peaks at .

It may be said that cross-correlation function gives a measure of resemblance between the shifted versions of signal x(t) and y(t). Hence it is used to in Radar and Sonar applications to measure distances . In these systems, a transmitter transmits signals which on reflection from a target are received by a receiver. Thus the received signal is a time shifted version of the transmitted signal . By seeing where the cross-correlation of these two signals peaks, one can determine the time shift and hence the distance of the target.

The Fourier transform of is of-course