A function related to cross-correlation is the coherence function, defined in terms of power spectral densities and the cross-spectral density by
In practice, these quantities can be estimated by time-averaging , , and over successive signal blocks. Let denote time averaging across frames as in Eq. (8.3) above. Then an estimate of the coherence, the sample coherence function , may be defined by
Note that the averaging in the numerator occurs before the absolute value is taken.
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:
Then the magnitude-normalized cross-spectrum in each frame is
so that the coherence function becomes
On the other hand, when and are uncorrelated (e.g., is a noise process not derived from ), the sample coherence converges to zero at all frequencies, as the number of blocks in the average goes to infinity.
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. An example is shown in Book III [71] for the case of a measured guitar-bridge admittance. A more elementary example is given in the next section.