Aperiodic Crosscorrelation of Sequences Derived from Characters

TL;DR

This paper investigates the aperiodic crosscorrelation properties of sequences derived from characters, showing how specific sequence constructions can significantly reduce crosscorrelation, with theoretical formulas and empirical data supporting these findings.

Contribution

It provides new constructions and asymptotic formulas for low aperiodic crosscorrelation of character-based sequences, advancing sequence design for communication systems.

Findings

Sequences with reversed or shifted halves have lower crosscorrelation.

Constructed sequences can achieve near-optimal crosscorrelation performance.

Empirical data confirms asymptotic formulas for modest sequence lengths.

Abstract

It is shown that pairs of maximal linear recursive sequences (m-sequences) typically have mean square aperiodic crosscorrelation on par with that of random sequences, but that if one takes a pair of m-sequences where one is the reverse of the other, and shifts them appropriately, one can get significantly lower mean square aperiodic crosscorrelation. Sequence pairs with even lower mean square aperiodic crosscorrelation are constructed by taking a Legendre sequence, cyclically shifting it, and then cutting it (approximately) in half and using the halves as the sequences of the pair. In some of these constructions, the mean square aperiodic crosscorrelation can be lowered further if one truncates or periodically extends (appends) the sequences. Exact asymptotic formulae for mean squared aperiodic crosscorrelation are proved for sequences derived from additive characters (including m-sequences and modified versions thereof) and multiplicative characters (including Legendre sequences and their relatives). Data is presented that shows that sequences of modest length have performance that closely approximates the asymptotic formulae.