Cross-Correlation
Definition: The circular cross-correlation of two signals and
in
may be defined by
![$\displaystyle \zbox {{\hat r}_{xy}(l) \isdef \frac{1}{N}(x\star y)(l)
\isdef \frac{1}{N}\sum_{n=0}^{N-1}\overline{x(n)} y(n+l), \; l=0,1,2,\ldots,N-1.}
$](http://www.dsprelated.com/josimages_new/mdft/img1550.png)
![$ l$](http://www.dsprelated.com/josimages_new/mdft/img1446.png)
![$ 1$](http://www.dsprelated.com/josimages_new/mdft/img111.png)
![$ \star$](http://www.dsprelated.com/josimages_new/mdft/img1406.png)
The term ``cross-correlation'' comes from
statistics, and what we have defined here is more properly
called a ``sample cross-correlation.''
That is,
is an
estimator8.8 of the true
cross-correlation
which is an assumed statistical property
of the signal itself. This definition of a sample cross-correlation is only valid for
stationary stochastic processes, e.g., ``steady noises'' that
sound unchanged over time. The statistics of a stationary stochastic
process are by definition time invariant, thereby allowing
time-averages to be used for estimating statistics such
as cross-correlations. For brevity below, we will typically
not include ``sample'' qualifier, because all computational
methods discussed will be sample-based methods intended for use on
stationary data segments.
The DFT of the cross-correlation may be called the cross-spectral density, or ``cross-power spectrum,'' or even simply ``cross-spectrum'':
![$\displaystyle {\hat R}_{xy}(\omega_k) \isdef \hbox{\sc DFT}_k({\hat r}_{xy}) = \frac{\overline{X(\omega_k)}Y(\omega_k)}{N}
$](http://www.dsprelated.com/josimages_new/mdft/img1553.png)
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Unbiased Cross-Correlation
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Phase Response