NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University
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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 averaging, and over successive signal blocks. Let denote time averaging. Then an estimate of the coherence, the sample coherence function, may be defined by
The magnitude-squared coherence is a real function between and which gives a measure of correlation between and at each frequency (DFT bin number ). For example, imagine that is produced from via an LTI filtering operation:
Then the coherence function is
and the magnitude-squared coherence function is simply
On the other hand, if and are uncorrelated noise processes, the coherence converges to zero.
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 nullin the room response for some frequency, it may record mostly noise at that frequency. This will be indicated in the measured coherence by a significant dip below .