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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 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 estimator8.4 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.

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).

Recall that the cross-correlation operator is cyclic (circular) since $ n+l$ is interpreted modulo $ N$. In practice, we are normally interested in estimating the acyclic cross-correlation between two signals. For this (more realistic) case, we may define instead the unbiased sample cross-correlation

$\displaystyle \zbox {{\hat r}^u_{xy}(l) \isdef \frac{1}{N-l}\sum_{n=0}^{N-1-l} \overline{x(n)} y(n+l),\;
l = 0,1,2,\ldots,L-1}

where we choose $ L\ll N$ (e.g., $ L\approx\sqrt{N}$) in order to have enough lagged products at the highest lag that a reasonably accurate average is obtained. The term ``unbiased'' refers to the fact that the expected value8.5[32] of $ {\hat r}^u_{xy}(l)$ is the true cross-correlation $ r_{xy}(l)$ of $ x$ and $ y$ (assumed to be samples from stationary stochastic processes).

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[How to cite this work] [Order a printed hardcopy]

``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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