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Zak-Transform-Induced Optimal Sequences and Their Applications in OTFS

Xiuping Peng, Congying Wu, Zilong Liu, Chunlei Li, Jianye Zhang, Pingzhi Fan

Abstract

This paper introduces a novel finite Zak transform (FZT)-aided framework for constructing multiple zero-correlation zone (ZCZ) sequence sets with optimal correlation properties. Specifically, each sequence is perfect with zero auto-correlation sidelobes, each ZCZ sequence set meets the Tang-Fan-Matsufuji bound with equality, and the maximum inter-set cross-correlation of multiple sequence sets meets the Sarwate bound with equality. Our study shows that these sequences can be sparsely expressed in the Zak domain through properly selected index and phase matrices. Particularly, it is found that the maximum inter-set cross-correlation beats the Sarwate bound if every index matrix is a circular Florentine array. Several construction methods of multiple ZCZ sequence sets are proposed, demonstrating both the optimality and high flexibility. Additionally, it is shown that excellent synchronization performance can be achieved by the proposed sequences in orthogonal-time-frequency-space (OTFS) systems.

Zak-Transform-Induced Optimal Sequences and Their Applications in OTFS

Abstract

This paper introduces a novel finite Zak transform (FZT)-aided framework for constructing multiple zero-correlation zone (ZCZ) sequence sets with optimal correlation properties. Specifically, each sequence is perfect with zero auto-correlation sidelobes, each ZCZ sequence set meets the Tang-Fan-Matsufuji bound with equality, and the maximum inter-set cross-correlation of multiple sequence sets meets the Sarwate bound with equality. Our study shows that these sequences can be sparsely expressed in the Zak domain through properly selected index and phase matrices. Particularly, it is found that the maximum inter-set cross-correlation beats the Sarwate bound if every index matrix is a circular Florentine array. Several construction methods of multiple ZCZ sequence sets are proposed, demonstrating both the optimality and high flexibility. Additionally, it is shown that excellent synchronization performance can be achieved by the proposed sequences in orthogonal-time-frequency-space (OTFS) systems.

Paper Structure

This paper contains 15 sections, 83 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Related Works
  4. Motivations and Contributions
  5. Organization of This Work
  6. Preliminaries
  7. Perfect Sequences
  8. The Zak Transform
  9. Circular Florentine Arrays
  10. Proposed Zak-Transform-Induced Multiple ZCZ Sequence Sets
  11. Sets Of Perfect Sequences With Optimal Correlation
  12. The Proposed Constructions
  13. Comparison With The Previous Related Constructions
  14. Evaluation of the Synchronization performance in OTFS
  15. Conclusions

Figures (9)

  • Figure 1: (a) The PACF of ${\textbf{s}}_1^0$ obtained from Example 4; (b) The PCCF of ${\textbf{s}}_2^0$ and ${\textbf{s}}_3^0$.
  • Figure 2: The correlation properties of Example 4.
  • Figure 3: The correlation properties of Example 6.
  • Figure 4: The relationship of OTFS modulation and IFZT
  • Figure 5: Synchronization model of SISO-OTFS system
  • ...and 4 more figures