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

2026-01-23

AI Summary

This work addresses the long-standing open problem—unsolved for over three decades—of constructing asymptotically optimal aperiodic polyphase sequence sets in the sense of the Welch bound. By recursively applying the Paris asymptotic expansion and leveraging the rapid convergence of the Fibonacci zeta function, the authors derive, for the first time, an explicit upper bound on generalized quadratic Gauss sums. Building on this result, they establish the asymptotic optimality of Alltop sequence sets with respect to aperiodic correlation sidelobes and introduce a new class of Alltop subsets that simultaneously achieve order-optimal aperiodic correlation and ambiguity performance. Consequently, four families of sequence sets are successfully constructed, all attaining the asymptotic limits permitted by the Welch bound in both correlation and ambiguity metrics.

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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.

Problem

• aperiodic polyphase sequences
• Welch bound
• asymptotically optimal
• sequence design
• aperiodic correlation

Innovation

• Generalized Quadratic Gauss Sums
• Asymptotically Optimal Sequences
• Aperiodic Correlation
• Alltop Sequences
• Welch Bound