Cross correlation
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.
definition
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 .
properties
For all true
such as
and
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.
literature
- 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
Web links
- mpi-magdeburg: Cross correlation and combinatorics ( Memento from July 11, 2012 in the Internet Archive ) (PDF; 201 kB)
- THE CROSS CORRELATION (accessed on July 16, 2018)
- Correlation Technique (accessed July 16, 2018)
- Feature list-based cross-correlation methods for medical image processing (accessed on July 16, 2018)
- Approaches to the data-driven formulation of structural hypotheses for dynamic systems (accessed on July 16, 2018)