### 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:

Note that and belong to while and belong to . 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.

**Next Section:**

Autocorrelation

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Cross-Correlation