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On the Construction and Correlation Properties of Permutation-Interleaved Zadoff-Chu Sequences

Qin Yuan, Chunlei Li, Xiangyong Zeng

TL;DR

This work advances CAZAC sequence design by interleaving Zadoff-Chu sequences with permutation polynomials over $\mathbb{Z}_N$, constructing new CAZAC (and ZAC) sequences via two PP-based interleaving schemes and their inverses. The authors prove that these interleaved sequences achieve ZAC properties under precise modulus factorizations and polynomial forms, and they establish the sufficiency of Berggren–Popović's conjecture in this setting while proving inequivalence to both ZC sequences and QPP-interleaved ZC sequences. They also analyze aperiodic autocorrelation for $N=2^n$ using Weyl-sum techniques, deriving bounds of the form $|\tilde{\theta}(d)| \le C(\varepsilon,\varepsilon_1) N^{1-(\varepsilon_1-\varepsilon)/4}$. Collectively, the results expand the catalog of CAZAC sequence constructions, provide rigorous inequivalence results, and offer practical insights into the correlation properties of interleaved ZC sequences for radar and communications, while pointing to open problems in permutation-polynomial design over integer rings.

Abstract

Constant amplitude zero auto-correlation (CAZAC) sequences are widely applied in waveforms for radar and communication systems. Motivated by a recent work [Berggren and Popović, IEEE Trans. Inf. Theory 70(8), 6068-6075 (2024)], this paper further investigates the approach to generating CAZAC sequences by interleaving Zadoff-Chu (ZC) sequences with permutation polynomials (PPs). We propose one class of high-degree PPs over the integer ring Z N , and utilize them and their inverses to interleave ZC sequences for constructing CAZAC sequences. It is known that a CAZAC sequence can be extended to an equivalence class by five basic opertations. We further show that the obtained CAZAC sequences are not covered by the equivalence classes of ZC sequences and interleaved ZC sequences by quadratic PPs and their inverses, and prove the sufficiency of the conjecture by Berggren and Popović in the aforementioned work. In addition, we also evaluate the aperiodic auto-correlation of certain ZC sequences from quadratic PPs.

On the Construction and Correlation Properties of Permutation-Interleaved Zadoff-Chu Sequences

TL;DR

This work advances CAZAC sequence design by interleaving Zadoff-Chu sequences with permutation polynomials over , constructing new CAZAC (and ZAC) sequences via two PP-based interleaving schemes and their inverses. The authors prove that these interleaved sequences achieve ZAC properties under precise modulus factorizations and polynomial forms, and they establish the sufficiency of Berggren–Popović's conjecture in this setting while proving inequivalence to both ZC sequences and QPP-interleaved ZC sequences. They also analyze aperiodic autocorrelation for using Weyl-sum techniques, deriving bounds of the form . Collectively, the results expand the catalog of CAZAC sequence constructions, provide rigorous inequivalence results, and offer practical insights into the correlation properties of interleaved ZC sequences for radar and communications, while pointing to open problems in permutation-polynomial design over integer rings.

Abstract

Constant amplitude zero auto-correlation (CAZAC) sequences are widely applied in waveforms for radar and communication systems. Motivated by a recent work [Berggren and Popović, IEEE Trans. Inf. Theory 70(8), 6068-6075 (2024)], this paper further investigates the approach to generating CAZAC sequences by interleaving Zadoff-Chu (ZC) sequences with permutation polynomials (PPs). We propose one class of high-degree PPs over the integer ring Z N , and utilize them and their inverses to interleave ZC sequences for constructing CAZAC sequences. It is known that a CAZAC sequence can be extended to an equivalence class by five basic opertations. We further show that the obtained CAZAC sequences are not covered by the equivalence classes of ZC sequences and interleaved ZC sequences by quadratic PPs and their inverses, and prove the sufficiency of the conjecture by Berggren and Popović in the aforementioned work. In addition, we also evaluate the aperiodic auto-correlation of certain ZC sequences from quadratic PPs.
Paper Structure (13 sections, 18 theorems, 144 equations, 3 tables)

This paper contains 13 sections, 18 theorems, 144 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Permutation polynomials over $\mathbb{Z}_{N}$
  4. CAZAC sequences and their equivalence
  5. Permutation-interleaved Zadoff-Chu sequences
  6. Auxiliary lemmas
  7. Permutation-interleaved ZC sequences
  8. Permutation-interleaved ZC sequences with the ZAC property
  9. Proofs of Theorems \ref{['mainresult']} (i) and \ref{['mainresult2']} (i)
  10. Proofs of Theorems \ref{['mainresult']} (ii) and \ref{['mainresult2']} (ii)
  11. Inequivalent permutation-interleaved ZC sequences
  12. Aperiodic auto-correlation of interleaved ZC sequences
  13. Conclusion

Key Result

Lemma 1

( Rivest) Let $N=2^n$ with $n\geq 2$ and $\pi(x)=a_mx^m + a_{m-1}x^{m-1} + \dots + a_2x^2 + a_1x + a_0\in \mathbb{Z}_{N}[x]$. Then the polynomial $\pi$ is a PP over $\mathbb{Z}_{2^n}$ if and only if

Theorems & Definitions (45)

  • Lemma 1
  • Lemma 2
  • Definition 1: CAZAC sequences
  • Definition 2: Equivalence Classes
  • Definition 3
  • Remark 1
  • Conjecture 1
  • Lemma 3
  • Lemma 4
  • Proposition 1
  • ...and 35 more