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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

$\displaystyle C_{xy}(\omega) \isdef \frac{\vert R_{xy}(\omega)\vert^2}{R_x(\omega)R_y(\omega)}.
$

In practice, these quantities can be estimated by time-averaging $ \overline{X(\omega_k)}Y(\omega_k)$ , $ \left\vert X(\omega_k)\right\vert^2$ , and $ \left\vert Y(\omega_k)\right\vert^2$ over successive signal blocks. Let $ \{\cdot\}_m$ denote time averaging across frames as in Eq.$ \,$ (8.3) above. Then an estimate of the coherence, the sample coherence function $ {\hat
C}_{xy}(\omega_k)$ , may be defined by

$\displaystyle {\hat C}_{xy}(\omega_k) \isdef
\frac{\left\vert\left\{\overline{X_m(\omega_k)}Y_m(\omega_k)\right\}_m\right\vert^2}{\left\{\left\vert X_m(\omega_k)\right\vert^2\right\}_m\cdot\left\{\left\vert Y_m(\omega_k)\right\vert^2\right\}_m}.
$

Note that the averaging in the numerator occurs before the absolute value is taken.

The coherence $ C_{xy}(\omega)$ is a real function between zero and one which gives a measure of correlation between $ x$ and $ y$ at each frequency $ \omega$ . For example, imagine that $ y$ is produced from $ x$ via an LTI filtering operation:

$\displaystyle y = h\ast x \;\implies\; Y(\omega_k) = H(\omega_k)X(\omega_k)
$

Then the magnitude-normalized cross-spectrum in each frame is

\begin{eqnarray*}
{\hat A}_{x_m y_m}(\omega_k) &\isdef &
\frac{\overline{X_m(\omega_k)}Y_m(\omega_k)}{\left\vert X_m(\omega_k)\right\vert\cdot\left\vert Y_m(\omega_k)\right\vert}
= \frac{\overline{X_m(\omega_k)}H(\omega_k)X_m(\omega_k)}{\left\vert X_m(\omega_k)\right\vert\cdot\left\vert H(\omega_k)X_m(\omega_k)\right\vert}\\ [5pt]
&=& \frac{\left\vert X_m(\omega_k)\right\vert^2 H(\omega_k)}{\left\vert X_m(\omega_k)\right\vert^2\left\vert H(\omega_k)\right\vert}
= \frac{H(\omega_k)}{\left\vert H(\omega_k)\right\vert}
\end{eqnarray*}

so that the coherence function becomes

$\displaystyle \left\vert{\hat C}_{xy}(\omega_k)\right\vert^2 =
\left\vert\frac{H(\omega_k)}{\left\vert H(\omega_k)\right\vert}\right\vert^2 = 1.
$

On the other hand, when $ x$ and $ y$ are uncorrelated (e.g., $ y$ is a noise process not derived from $ x$ ), 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, $ x(n)$ might be a known signal which is input to an unknown system, such as a reverberant room, say, and $ y(n)$ is the recorded response of the room. Ideally, the coherence should be $ 1$ 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.



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``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 © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA