Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search


Definition: The circular cross-correlation of two signals $ x$ and $ y$ in $ \mathbb{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 the ``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).

Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2016-05-31 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University