A function related to cross-correlation is the coherence function, defined in terms of power spectral densities and the cross-spectral density by
The coherence is a real function between zero and one which gives a measure of correlation between and at each frequency . For example, imagine that is produced from via an LTI filtering operation:
so that the coherence function becomes
A common use for the coherence function is in the validation of input/output data collected in an acoustics experiment for purposes of system identification. For example, might be a known signal which is input to an unknown system, such as a reverberant room, say, and is the recorded response of the room. Ideally, the coherence should be at all frequencies. However, if the microphone is situated at a null in the room response for some frequency, it may record mostly noise at that frequency. This is indicated in the measured coherence by a significant dip below 1.