Cross-Correlation

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Related Concepts

Autocorrelation; Cross-Correlation; Maximal-Length Linear Sequences; Modular Arithmetic; Sequences; Stream Cipher

Definition

Let {at} and {bt} be two sequences of period n (so at = at + n and bt = bt + n for all values of t) over an alphabet being the integers mod q (Modular Arithmetic). The cross-correlation between the sequences {at} and {bt} at shift τ is defined as $$ C(\tau ) ={ \sum \limits _{t=0}^{n-1}}{\omega }^{{a}_{t+\tau }-{b}_{t} } $$ where \(\omega \) is a complex q-th root of unity. Note that in the special case of binary sequences, then q = 2 and \(\omega = -1\).

In the special case when the two sequences are the same, the cross-correlation is the same as the autocorrelation.

Applications

Many applications in stream ciphers and communication systems require large families of cyclically distinct sequences with a low maximal nontrivial value of the auto- and cross-correlation between any two sequences in the family.

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