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 (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 value8.9 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.