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*

where we choose (

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

where

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.

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