Editing Coherence (physics) (section)

== Mathematical definition ==
{{Further|Degree of coherence}}

The ''coherence function'' between two signals <math> x(t) </math> and <math> y(t) </math> is defined as<ref>Shin. K, Hammond. J.'' Fundamentals of signal processing for sound and vibration engineers''. John Wiley & Sons, 2008.</ref>

:<math> \gamma_{xy}^{2}(f)=\frac{|S_{xy}(f)|^2}{S_{xx}(f)S_{yy}(f)}</math>

where <math> S_{xy}(f) </math> is the [[cross-spectral density]] of the signal and <math> S_{xx}(f) </math> and <math> S_{yy}(f) </math> are the power [[spectral density]] functions of <math> x(t) </math> and <math> y(t) </math>, respectively. The cross-spectral density and the power spectral density are defined as the [[Fourier transforms]] of the [[cross-correlation]] and the [[autocorrelation]] signals, respectively. For instance, if the signals are functions of time, the cross-correlation is a measure of the similarity of the two signals as a function of the time lag relative to each other and the autocorrelation is a measure of the similarity of each signal with itself in different instants of time. In this case the coherence is a function of frequency. Analogously, if <math> x(t) </math> and <math> y(t) </math> are functions of space, the cross-correlation measures the similarity of two signals in different points in space and the autocorrelations the similarity of the signal relative to itself for a certain separation distance. In that case, coherence is a function of [[wavenumber]] (spatial frequency).

The coherence varies in the interval <math> 0 \leq \gamma_{xy}^{2}(f) \leq 1 </math>. If <math> \gamma_{xy}^{2}(f)=1 </math> it means that the signals are perfectly correlated or linearly related and if <math> \gamma_{xy}^{2}(f)=0 </math> they are totally uncorrelated. If a linear system is being measured, <math> x(t) </math> being the input and <math> y(t) </math> the output, the coherence function will be unitary all over the spectrum. However, if non-linearities are present in the system the coherence will vary in the limit given above.