**Search Mathematics of the DFT**

Book Index | Global Index

**Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?**

## Coherence Function

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 [69] for the case of a measured
guitar-bridge admittance.
A more elementary example is given in the next section.

**Subsections**

**Previous:** Power Spectral Density Estimation**Next:** 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 (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/ for details.