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: ${\displaystyle R_{xy}(\tau )}$ ${\displaystyle x(t)}$ ${\displaystyle y(t)}$ ${\displaystyle \tau }$ ${\displaystyle x\neq y}$

${\displaystyle R_{xy}(t_{1},t_{2})=E\{{\textbf {X}}(t_{1})\cdot {\textbf {Y}}(t_{2})\}}$

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. ${\displaystyle t_{1}}$ ${\displaystyle t_{2}}$ ${\displaystyle \tau =t_{2}-t_{1}}$

definition

The following applies to energy signals :

${\displaystyle R_{xy}(\tau )=(x\star y)(\tau )=(x^{*}(-t)*y(t))(\tau )=\int _{-\infty }^{\infty }x^{*}(t)\,y(t+\tau )\,\mathrm {d} t}$

and for power signals :

${\displaystyle R_{xy}(\tau )=(x\star y)(\tau )=(x^{*}(-t)*y(t))(\tau )=\lim _{T\to \infty }{\frac {1}{2T}}\int _{-T}^{T}x^{*}(t)\,y(t+\tau )\,\mathrm {d} t}$

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 . ${\displaystyle x^{*}}$ ${\displaystyle x}$ ${\displaystyle \star }$ ${\displaystyle *}$

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: ${\displaystyle [m]}$ ${\displaystyle n}$

${\displaystyle R_{xy}[n]}$= (Energy signals) ${\displaystyle (x\star y)[n]=\sum _{m=-\infty }^{\infty }x^{*}[m]\ y[m+n]}$

${\displaystyle R_{xy}[n]}$= (Power signals) ${\displaystyle (x\star y)[n]=\lim _{M\to \infty }{\frac {1}{2M+1}}\sum _{m=-M}^{M}x^{*}[m]\ y[m+n]}$

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:

${\displaystyle R_{xy}[m]:={\begin{cases}\ \;\,{\frac {1}{N-|m|}}\sum _{n=0}^{N-m-n}x[n]y[n+m]&{\text{für }}m\geq 0\\\ \;\,{\frac {1}{N-|m|}}\sum _{n=-m}^{N-1}x[n]y[n+m]&{\text{für }}m<0\end{cases}}}$ (Opening version)

${\displaystyle R_{xy}[m]:={\begin{cases}\ \;\,{\frac {1}{N}}\sum _{n=0}^{N-m-n}x[n]y[n+m]&{\text{für }}m\geq 0\\\ \;\,{\frac {1}{N}}\sum _{n=-m}^{N-1}x[n]y[n+m]&{\text{für }}m<0\end{cases}}}$ (untensioned version)

Cross -correlation is closely related to cross- covariance .

properties

Relationship between convolution, cross-correlation and autocorrelation.

For all true ${\displaystyle \tau }$

${\displaystyle R_{xy}(\tau )=R_{yx}(-\tau )}$

such as

${\displaystyle \left|R_{xy}(\tau )\right|\leq {\sqrt {R_{xx}(0)R_{yy}(0)}}\leq {\frac {1}{2}}(R_{xx}(0)+R_{yy}(0))}$

and

${\displaystyle \lim \limits _{\tau \to \pm \infty }R_{xy}(\tau )=0}$

with the autocorrelation functions and . ${\displaystyle R_{xx}(\tau )}$ ${\displaystyle R_{yy}(\tau )}$

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. ${\displaystyle x(t)}$ ${\displaystyle y(t)}$

In terms of computation, the cross-correlation function is usually determined using the inverse Fourier transformation of the cross-power spectrum : ${\displaystyle S_{XY}(f)}$

${\displaystyle R_{xy}(\tau )=\int _{-\infty }^{\infty }S_{XY}(f)\,e^{\mathrm {i} 2\pi f\tau }\,\mathrm {d} f}$

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. ${\displaystyle x(t)}$ ${\displaystyle y(t)}$ ${\displaystyle ({\bar {x}}(t)=0}$ ${\displaystyle {\bar {y}}(t)=0)}$

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

• Autocorrelation

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)