Coherence

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Coherence

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Coherence

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 averaging$\, $\ and $\over successive signal blocks. Let $\ denote time averaging. Then an estimate of the coherence, the sample coherence function${\, may be defined by
\

The magnitude-squared coherence $\ 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:

\

Then the coherence function is
\

and the magnitude-squared coherence function is simply
\

On the other hand, if $x$ and $y$ are uncorrelated noise processes, the coherence converges to zero.

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 nullin 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 $1$.

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