NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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 .
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