Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Permutation Polynomial Interleaved Zadoff-Chu Sequences

Fredrik Berggren, Branislav M. Popovic

TL;DR

This work demonstrates that interleaving Zadoff-Chu sequences with permutation polynomials, especially quadratic permutation polynomials and their inverses, preserves the CAZAC property and yields new, information-rich CAZAC sequences. The authors prove CAZAC preservation under key conditions (Theorems 1 and 2) and connect these interleaved sequences to GCL constructions, highlighting period and inversion considerations. They also explore the uniqueness and orthogonality of the resulting sequences, showing that a set of orthogonal interleaved ZC sequences can be formed in many cases, though a full $N$-element orthogonal set is not universal. The results provide practical avenues for expanding CAZAC sequence libraries useful for reference signals, synchronization, and preambles in modern wireless systems, with potential for larger, orthogonal CAZAC sets via permutation-polynomial interleaving.

Abstract

Constant amplitude zero autocorrelation (CAZAC) sequences have modulus one and ideal periodic autocorrelation function. Such sequences are used in cellular radio communications systems, e.g., for reference signals, synchronization signals and random access preambles. We propose a new family CAZAC sequences, which is constructed by interleaving a Zadoff-Chu sequence by a quadratic permutation polynomial (QPP), or by a permutation polynomial whose inverse is a QPP. It is demonstrated that a set of orthogonal interleaved Zadoff-Chu sequences can be constructed by proper choice of QPPs.

Permutation Polynomial Interleaved Zadoff-Chu Sequences

TL;DR

This work demonstrates that interleaving Zadoff-Chu sequences with permutation polynomials, especially quadratic permutation polynomials and their inverses, preserves the CAZAC property and yields new, information-rich CAZAC sequences. The authors prove CAZAC preservation under key conditions (Theorems 1 and 2) and connect these interleaved sequences to GCL constructions, highlighting period and inversion considerations. They also explore the uniqueness and orthogonality of the resulting sequences, showing that a set of orthogonal interleaved ZC sequences can be formed in many cases, though a full -element orthogonal set is not universal. The results provide practical avenues for expanding CAZAC sequence libraries useful for reference signals, synchronization, and preambles in modern wireless systems, with potential for larger, orthogonal CAZAC sets via permutation-polynomial interleaving.

Abstract

Constant amplitude zero autocorrelation (CAZAC) sequences have modulus one and ideal periodic autocorrelation function. Such sequences are used in cellular radio communications systems, e.g., for reference signals, synchronization signals and random access preambles. We propose a new family CAZAC sequences, which is constructed by interleaving a Zadoff-Chu sequence by a quadratic permutation polynomial (QPP), or by a permutation polynomial whose inverse is a QPP. It is demonstrated that a set of orthogonal interleaved Zadoff-Chu sequences can be constructed by proper choice of QPPs.
Paper Structure (10 sections, 51 equations, 3 figures, 1 table)

This paper contains 10 sections, 51 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definitions
  3. Zadoff-Chu Sequences Interleaved by Permutation Polynomials
  4. CAZAC Property
  5. Uniqueness of the Interleaved Zadoff-Chu Sequences
  6. Relation to GCL Sequences
  7. Set of Orthogonal Interleaved Zadoff-Chu Sequences
  8. Conclusions
  9. Lemmas for Theorem 1.
  10. Lemma for Theorem 2.

Figures (3)

  • Figure 1: Fraction of QPPs which produce an interleaved Zadoff-Chu sequence which cannot be obtained by basic mathematical operations on a non-interleaved Zadoff-Chu sequence.
  • Figure 2: Maximum number of unique sequences constructed by using different QPPs or by not using any QPPs but using different root indices $u$.
  • Figure 3: Values of the magnitude of the normalized cross-correlation $|\theta_{ij}(d)|/N$ at $d=0$ for different $N$, for the interleaved Zadoff-Chu sequences constructed by using different QPPs and a common Zadoff-Chu sequence.