Autocorrelation

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Let {at} be a sequence of periodn (so at = at+n for all values of t) with symbols being the integers mod q (see modular arithmetic). The periodic auto-correlation of the sequence {at} at shift τ is defined as $$A(\tau) = \sum_{t=0}^{n-1} \omega^{a_{t+\tau}-a_t},$$ where ω is a complex qth root of unity.

In most applications one considers binary sequences when q=2 and ω=-1. Then the autocorrelation at shift τ equals the number of agreements minus the number of disagreements between the sequence {at} and its cyclic shift {at+τ}. Note that in most applications one wants the autocorrelation for all nonzero shifts τ ≠ 0 (mod n) (the out-of-phase autocorrelation) to be low in absolute value. For example, this property of a sequence is extremely useful for synchronization purposes.