Cross correlation

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In signal analysis , the cross-correlation function is used to describe the correlation between two signals and when there are different time shifts between the two signals. Cross stands for the case of the function:

If it is a weakly stationary process , the correlation function is no longer on the choice of the times is and , but only on their difference dependent.


The following applies to energy signals :

and for power signals :

with as the complex conjugate function of , the operator symbol as the shorthand notation of the cross-correlation and as that of the convolution operation .

Analogously, the discrete cross-correlation, which plays an essential role in the field of discrete signal processing, with the sequence and a shift is defined as:

= (Energy signals)
= (Power signals)

In digital signal processing, on the other hand, finite averaging with arguments starting at index 0 is required due to the architecture of computer registers, of which there is a pre-stressed and a non-pre-stressed version:

(Opening version)
(untensioned version)

Cross -correlation is closely related to cross- covariance .


Relationship between convolution, cross-correlation and autocorrelation.

For all true

such as


with the autocorrelation functions and .

She shows z. B. peaks in time shifts that correspond to the signal transit time from the measurement location of the signal to the measurement location of the signal . Differences in transit time from a signal source to both measurement locations can also be determined in this way. The cross-correlation function is therefore particularly suitable for determining transmission paths and locating sources.

In terms of computation, the cross-correlation function is usually determined using the inverse Fourier transformation of the cross-power spectrum :

Connection with the cross covariance

Is one of the signals or zero symmetrical, i. H. their mean value over the signal is zero or , the cross-correlation is identical to the cross-covariance. Well-known representatives of the zero-symmetric functions are, for example, the sine and cosine functions.


  • Bernd Girod, Rudolf Rabenstein, Alexander Stenger: Introduction to systems theory . 4th edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0176-0 .
  • Rüdiger Hoffmann: Signal analysis and recognition . Springer, ISBN 3-540-63443-6 .

See also

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