Coherence

A function related to cross-correlation is the coherence function $\Gamma_{xy}(\omega)$, defined in terms of power spectral densities and the cross-spectral density by

$$\Gamma_{xy}(k) \isdef \frac{R_{xy}(k)}{\sqrt{R_x(k)R_y(k)}}$$

In practice, these quantities can be estimated by averaging $\overline{X(k)}Y(k)$, $\left\vert X(k)\right\vert^2$ and $\left\vert Y(k)\right\vert^2$ over successive signal blocks. Let $\{\cdot\}_m$ denote time averaging across frames as in Eq. (E.3) of §E.3 above Then the time average of $\left\vert X(k)\right\vert^2$, for example, is given by

$$\left\{\left\vert X_m(k)\right\vert^2\right\}_m \isdef \frac{1}{M}\sum_{m=0}^{M-1}\left\vert X_m(k)\right\vert^2$$

where $X_m(k)$ is the DFT of the $m$th block of time data $x_m$, and $M$ is the total number of time blocks available. In this notation, an estimate of the coherence, the sample coherence function ${\hat\Gamma}_{xy}(k)$, may be defined by

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

The magnitude-squared coherence $\left\vert\Gamma_{xy}(k)\right\vert^2$ is a real function between 0 and $1$ which gives a measure of correlation between $x$ and $y$ at each frequency (DFT bin number $k$). For example, imagine that $y$ is produced from $x$ via an LTI filtering operation:

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

Then the coherence function in each frame is

$${\hat \Gamma}_{x_my_m}(k) \isdef \frac{\overline{X_m(k)}Y_m(k)}{... ...ht\vert^2\left\vert H(k)\right\vert} = \frac{H(k)}{\left\vert H(k)\right\vert}$$

and the magnitude-squared coherence function is simply

$$\left\vert{\hat \Gamma}_{xy}(k)\right\vert = \left\vert\frac{H(k)}{\left\vert H(k)\right\vert}\right\vert = 1.$$

On the other hand, when $x$ and $y$ are uncorrelated (e.g., $y$ is a noise process not derived from $x$), the coherence converges to zero at all frequencies.

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 will be indicated in the measured coherence by a significant dip below $1$.