Correlation Analysis
Correlation analysis applies only to stationary stochastic processes (§C.1.5).
Cross-Correlation
Definition: The cross-correlation of two signals
and
may be defined by
![]() |
(C.24) |
I.e., it is the expected value (§C.1.6) of the lagged products in random signals
![$ x$](http://www.dsprelated.com/josimages_new/sasp2/img38.png)
![$ y$](http://www.dsprelated.com/josimages_new/sasp2/img165.png)
Cross-Power Spectral Density
The DTFT of the cross-correlation is called the cross-power spectral density, or ``cross-spectral density,'' ``cross-power spectrum,'' or even simply ``cross-spectrum.''
Autocorrelation
The cross-correlation of a signal with itself gives the autocorrelation function of that signal:
![]() |
(C.25) |
Note that the autocorrelation function is Hermitian:
![$\displaystyle r_x(-l) = \overline{r_x(l)}
$](http://www.dsprelated.com/josimages_new/sasp2/img2674.png)
When
![$ x$](http://www.dsprelated.com/josimages_new/sasp2/img38.png)
Sample Autocorrelation
See §6.4.
Power Spectral Density
The Fourier transform of the autocorrelation function
is
called the power spectral density (PSD), or power
spectrum, and may be denoted
![$\displaystyle S_x(\omega) \isdef \hbox{\sc DTFT}_\omega(r_x).
$](http://www.dsprelated.com/josimages_new/sasp2/img2675.png)
When the signal
![$ x$](http://www.dsprelated.com/josimages_new/sasp2/img38.png)
Sample Power Spectral Density
See §6.5.
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