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

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

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

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

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:

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

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

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

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