Two-Valued Periodic Complementary Sequences

We present a novel transform for periodic complementary sets (PCSs) over two-valued alphabets from a large set of difference families. This is achieved by generalizing Golomb's idea in 1992, which was for transformed perfect sequences with zero autocorrelations only. Based on the properties of difference family, a sufficient condition for such two-valued PCSs is derived. Systematic constructions of two-valued periodic complementary pairs are presented. It is shown that many lengths for which binary PCSs do not exist become admissible for our proposed two-valued PCSs.

I. Introduction

In wireless communications and radar sensing, perfect sequences (which have zero periodic autocorrelation sidelobes) are desirable for optimal performances of a variety of applications, such as spread spectrum communications, channel estimation, synchronization, and object detection or ranging [1]. For low-complexity digital implementation, perfect sequences with small alphabet sizes are demanded, nevertheless, they are limited to some short lengths. For example, the existing known binary and quaternary perfect sequences only have lengths (periods) of 4 and 2, 4, 8, 16, respectively [2]. The existing known perfect sequences over the alphabet set have lengths of 3, 4, 6, 7, 8, 10, 12, 28 only [3] . Polyphase perfect sequences [also known as perfect roots-of-unity sequences (PRUSs)] widely exist. However, as conjectured by Mow, the minimum alphabet size of PRUSs of length ( positive integers) is for even and odd and is otherwise [4].