Sign in

Not a member? | Forgot your Password?

Search Online Books



Search tips

Free Online Books

Chapters

FIR Filter Design Software

See Also

Embedded SystemsFPGA
Chapter Contents:

Search Mathematics of the DFT

  

Book Index | Global Index


Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

Cross-Correlation



Definition: The circular cross-correlation of two signals $ x$ and $ y$ in $ {\bf C}^N$ 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.}
$

(Note that the ``lag'' $ l$ is an integer variable, not the constant $ 1$.) The DFT correlation operator `$ \star$' was first defined in §7.2.5.

The term ``cross-correlation'' comes from statistics, and what we have defined here is more properly called a ``sample cross-correlation.'' That is, $ {\hat r}_{xy}(l)$ is an estimator8.8 of the true cross-correlation $ r_{xy}(l)$ 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}
$

The last equality above follows from the correlation theorem7.4.7).


Previous: Correlation Analysis
Next: Unbiased Cross-Correlation

Order a Hardcopy of Mathematics of the DFT


About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


Comments


No comments yet for this page


Add a Comment
You need to login before you can post a comment (best way to prevent spam). ( Not a member? )