### 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*

*e.g.*, ) in order to have enough lagged products 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

^{8.9}[33] of is the true cross-correlation 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:

*time scale modification*,

*pitch shifting*,

*click removal*, and many others.

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Autocorrelation

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Cross-Correlation