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An Explicit Upper Bound of Generalized Quadratic Gauss Sums and Its Applications for Asymptotically Optimal Aperiodic Polyphase Sequence Design

Huaning Liu, Zilong Liu

TL;DR

The paper tackles the open problem of designing asymptotically order-optimal aperiodic polyphase sequence sets under Welch's bound by deriving an explicit upper bound for generalized quadratic Gauss sums using Paris' asymptotic expansion and Fibonacci zeta convergence. This bound enables deterministic constructions of four order-optimal sequence families based on Chu and Alltop sequences, including the novel result that the full Alltop set is asymptotically optimal for low aperiodic sidelobes and a subset achieving joint low aperiodic correlation and ambiguity across the full time window. The Chu-based designs yield $\mathcal{C}_1$ with low aperiodic correlation and $\mathcal{C}_2$ with aperiodic LAZ properties, while the Alltop-based designs yield $\mathcal{A}_1$ with low sidelobes and $\mathcal{A}_2$ with simultaneous optimality in correlation and ambiguity. Overall, the work provides a computable performance yardstick for incomplete Gauss-sum-based analyses and delivers concrete, practically relevant sequence sets for high-mobility sensing and ISAC scenarios.

Abstract

This work is motivated by the long-standing open problem of designing asymptotically order-optimal aperiodic polyphase sequence sets with respect to the celebrated Welch bound. Attempts were made by Mow over 30 years ago, but a comprehensive understanding to this problem is lacking. Our first key contribution is an explicit upper bound of generalized quadratic Gauss sums which is obtained by recursively applying Paris' asymptotic expansion and then bounding it by leveraging the fast convergence property of the Fibonacci zeta function. Building upon this major finding, our second key contribution includes four systematic constructions of order-optimal sequence sets with low aperiodic correlation and/or ambiguity properties via carefully selected Chu sequences and Alltop sequences. For the first time in the literature, we reveal that the full Alltop sequence set is asymptotically optimal for its low aperiodic correlation sidelobes. Besides, we introduce a novel subset of Alltop sequences possessing both order-optimal aperiodic correlation and ambiguity properties for the entire time-shift window.

An Explicit Upper Bound of Generalized Quadratic Gauss Sums and Its Applications for Asymptotically Optimal Aperiodic Polyphase Sequence Design

TL;DR

The paper tackles the open problem of designing asymptotically order-optimal aperiodic polyphase sequence sets under Welch's bound by deriving an explicit upper bound for generalized quadratic Gauss sums using Paris' asymptotic expansion and Fibonacci zeta convergence. This bound enables deterministic constructions of four order-optimal sequence families based on Chu and Alltop sequences, including the novel result that the full Alltop set is asymptotically optimal for low aperiodic sidelobes and a subset achieving joint low aperiodic correlation and ambiguity across the full time window. The Chu-based designs yield with low aperiodic correlation and with aperiodic LAZ properties, while the Alltop-based designs yield with low sidelobes and with simultaneous optimality in correlation and ambiguity. Overall, the work provides a computable performance yardstick for incomplete Gauss-sum-based analyses and delivers concrete, practically relevant sequence sets for high-mobility sensing and ISAC scenarios.

Abstract

This work is motivated by the long-standing open problem of designing asymptotically order-optimal aperiodic polyphase sequence sets with respect to the celebrated Welch bound. Attempts were made by Mow over 30 years ago, but a comprehensive understanding to this problem is lacking. Our first key contribution is an explicit upper bound of generalized quadratic Gauss sums which is obtained by recursively applying Paris' asymptotic expansion and then bounding it by leveraging the fast convergence property of the Fibonacci zeta function. Building upon this major finding, our second key contribution includes four systematic constructions of order-optimal sequence sets with low aperiodic correlation and/or ambiguity properties via carefully selected Chu sequences and Alltop sequences. For the first time in the literature, we reveal that the full Alltop sequence set is asymptotically optimal for its low aperiodic correlation sidelobes. Besides, we introduce a novel subset of Alltop sequences possessing both order-optimal aperiodic correlation and ambiguity properties for the entire time-shift window.
Paper Structure (17 sections, 8 theorems, 81 equations)

This paper contains 17 sections, 8 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Motivations and Contributions
  4. PRELIMINARY
  5. Chu and Alltop sequences
  6. Existing Results on Incomplete Exponential Sum
  7. A Refined Bound on Paris' Asymptotic Expansion Errors
  8. The Fibonacci zeta function
  9. Proposed explicit bound for generalized quadratic Gauss sums
  10. Proposed Order-Optimal Sequence Design from Chu Sequences
  11. Order-Optimal Sequence Set $\mathcal{C}_1$ with Low Aperiodic Correlation
  12. Order-Optimal Sequence Set $\mathcal{C}_2$ with Aperiodic LAZ Properties
  13. Proposed Order-Optimal Sequence Design from Alltop Sequences
  14. Order-Optimal Sequence Set $\mathcal{A}_1$ with Low Aperiodic Correlation
  15. Order-Optimal Sequence Set $\mathcal{A}_2$ with both Low Aperiodic Correlation and Ambiguity Properties
  16. ...and 2 more sections

Key Result

Lemma 1

For $r\geq 2$ we have where $bN\equiv \pm 1 \ (\bmod \ r)$, $1\leq b\leq \lfloor\frac{r}{2}\rfloor$, $s_0=\sqrt{\frac{z_0r}{\pi b}}$ and $z_0=1.1655$.

Theorems & Definitions (19)

  • Lemma 1
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • ...and 9 more