Unbiased Cross-Correlation

Recall that the cross-correlation operator is cyclic (circular) since is interpreted modulo . 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 cross-correlation

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

where we choose $L\ll N$ (e.g., $L\approx\sqrt{N}$ ) in order to have enough lagged products $\overline{x(n)} y(n+l)$ at the highest lag

so that a reasonably accurate average is obtained. Note that the summation stops at

to avoid cyclic wrap-around of

modulo

. The term ``unbiased'' refers to the fact that the expected value^8.6[31] of ${\hat r}^u_{xy}(l)$ is the true cross-correlation $r_{xy}(l)$ of

and

(assumed to be samples from stationary stochastic processes).

An unbiased acyclic cross-correlation may be computed faster via DFT (FFT) methods using zero padding:

$\displaystyle \zbox {{\hat r}^u_{xy}(l) = \frac{1}{N-l}\hbox{\sc IDFT}_l(\overline{X}\cdot Y), \quad l = 0,1,2,\ldots,L-1}$

where

$\begin{eqnarray*} X &=& \hbox{\sc DFT}[\hbox{\sc CausalZeroPad}_{N+L-1}(x)]\\ Y &=& \hbox{\sc DFT}[\hbox{\sc CausalZeroPad}_{N+L-1}(y)]. \end{eqnarray*}$

Note that and belong to ${\bf C}^N$ while and belong to ${\bf C}^{N+L-1}$ . The zero-padding may be causal (as defined in §7.2.8) because the signals are assumed to be be stationary, in which case all signal statistics are time-invariant. As usual when embedding acyclic correlation (or convolution) within the cyclic variant given by the DFT, sufficient zero-padding is provided so that only zeros are ``time aliased'' (wrapped around in time) by modulo indexing.

Cross-correlation is used extensively in audio signal processing for applications such as time scale modification, pitch shifting, click removal, and many others.

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written by 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.

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See Also

Mathematics of the DFT
    Example Applications of the DFT
       Correlation Analysis
          Unbiased Cross-Correlation