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Lagged correlation

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Convolution or cross-correlation are similar operands that can be applied to any square-integrable real functions f and g.


 \left( f \, * \, g \right)(\tau) \equiv \int_{-\infty}^{\infty} f(t) \, g(\tau-t) \,dt  .


 \left( f \, \star \, g \right)(\tau) \equiv \int_{-\infty}^{\infty} f^{*}(t) \, g(\tau+t) \,dt \equiv C_{fg}(\tau).
In particular  f \, \star \, f is designated the autocorrelation.


Failed to parse (unknown error): \widehat{C}_{fg}(\tau) \equiv \frac{\int_{-\infty}^{\infty} F^{*}(t) \, G(\tau+t) \,dt}{\int_{-\infty}^{\infty} F^{*}(t) \, F(t) \,dt \, \int_{-\infty}^{\infty} G^{*}(t) \, G(t) \,dt} = \left< \frac{F(t)}{\|F\|} ,\, \frac{G(\tau+t)}{\|G\|} \right> , where  F \equiv f - \overline{f} . The idea is to remove the average and, optionally, other linear or physical trends from  f .
In particular we have
Failed to parse (unknown error): \widehat{C}_{fg}(0) = \left< \frac{F}{\|F\|} ,\, \frac{G}{\|G\|} \right> .
\widehat{C}_{ff}(0) = 1 .


\widehat{C}_{fg}(k) \equiv \frac{1}{n-1} \sum_{i} \frac{ \left( f_i - \overline{f} \right)^{*} \left( g_{k+i} - \overline{g} \right) }{\sigma_f \sigma_g} , where \sigma_f is the variance of f.
In particular  \widehat{C}_{fg}(0) is the famous correlation coefficient.


When both f and g are even functions then the convolution and cross-correlation products are rigorously equivalent. Below, we show these products when applied to box functions (left figure) and to gaussian functions (right figure). In both cases, they are even functions. f is represented by the red line, g is represented by the blue line, the product f(t) \, g(\tau - t) is represented by the solid blue line. The solid blue area is the convolution product for lag-time \tau and the whole convolution product function (for all \tau) is represented by the green line.

The big conclusions we can make is that if f and g are the same functions, but with a lag of \epsilon between them, then the cross-correlation product is maximum when \tau = \epsilon.

Matlab functions


Calculates the cross correlation from two signals.

%Consider arrays t, y1(t), y2(t)
tlag = linspace(-t(N/2), t(N/2), N+1);
crosscorr = xcorr( y1, y2, N/2, 'coeff');
plot(tlag,crosscorr); xlabel('Phase in time units');ylabel('coefficient');


Calculates the cross power spectral density (PSD) function of signals  x(t) and  y(t).

periodogram or pwelch

Calculates the power spectral density (PSD) function of signal  x(t). The periodogram may use several window functions as option. The pwelch will use the welch window.


Calculates the coherence-squared spectrum function. If you apply an inverse Fourier transform (IFT), then you get the cross-correlation function in a very efficient way.

See also