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.