Multi-Rate Representation of Generalized Cyclotomic Sequences of Any Odd Period

Abstract

Based on a unified representation of generalized cyclotomic classes, every generalized cyclotomic sequence of order d over $Z_{p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{r}^{e_{r}}}$ is shown to be a sum of d-residue sequences over $Z_{p_{s}^{e_{s}}}$ for $s\in \{1,2,\cdots,r \}$. For d=2, by the multi-rate approach, several generalized cyclotomic sequences are explicitly expressed by Legendre sequences, and their linear complexity properties are analyzed.