A Note on "Optimum Sets of Interference-Free Sequences With Zero Autocorrelation Zone"

TL;DR

This paper introduces a simple, optimal construction method for interference-free zero correlation zone (IF-ZCZ) sequences using Zak transform tessellation, simplifying generation and reducing computational complexity.

Contribution

The paper presents a new simple construction for optimal IF-ZCZ sequences, proves its equivalence to existing methods, and analyzes spectral properties to enhance implementation efficiency.

Findings

The construction is optimal with respect to the Tang-Fan-Matsufuji bound.

The simple construction is equivalent to Popovic's general method.

Zak spectra of the sequences are sparse and highly structured.

Abstract

In this paper, a simple construction of interference-free zero correlation zone (IF-ZCZ) sequence sets is proposed by well designed finite Zak transform lattice tessellation. Each set is characterized by the period of sequences $KM$, the set size $K$ and the length of zero correlation zone $M-1$, which is optimal with respect to the Tang-Fan-Matsufuji bound. Secondly, the transformations that keep the properties of the optimal IF-ZCZ sequence set unchanged are given, and the equivalent relation of the optimal IF-ZCZ sequence set is defined based on these transformations. Then, it is proved that the general construction of the optimal IF-ZCZ sequence set proposed by Popovic is equivalent to the simple construction of the optimal IF-ZCZ sequence set, which indicates that the generation of the optimal IF-ZCZ sequence set can be simplified. Moreover, it is pointed out that the alphabet size for the special case of the simple construction of the optimal IF-ZCZ sequence set can be a factor of the period. Finally, both the simple construction of the optimal IF-ZCZ sequence set and its special case have sparse and highly structured Zak spectra, which can greatly reduce the computational complexity of implementing matched filter banks.