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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
over successive signal
averaging across frames as in Eq.
) 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.
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
are uncorrelated (e.g.
process not derived from
), the sample coherence converges to
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  for the case of a measured
A more elementary example is given in the next section.
Previous: Power Spectral Density EstimationNext: Coherence Function in Matlab
About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA)
, teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/