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Generalized Arlery-Tan-Rabaste-Levenshtein Lower Bounds on Ambiguity Function and Their Asymptotic Achievability

Lingsheng Meng, Yong Liang Guan, Yao Ge, Zilong Liu, Pingzhi Fan

TL;DR

This work addresses fundamental limits on aperiodic delay-Doppler ambiguity in unimodular sequences by deriving generalized Arlery-Tan-Rabaste-Levenshtein-type AF lower bounds within delay-Doppler low-ambiguity zones. It introduces two weight vectors to separately capture delay and Doppler contributions and formulates the bounds through Frobenius-norm constraints on weighted AF matrices, yielding explicit bounds that reduce to known correlation bounds in special cases. The authors prove asymptotic achievability of these bounds using Chu sequence sets, demonstrating order-optimal behavior (i.e., maximal AF magnitudes scaling as $\mathcal{O}(\sqrt{N})$) for certain LAZ configurations, and provide numerical evidence supporting tightness. The results offer actionable design guidelines for ISAC and high-mobility communication scenarios by quantifying the trade-offs and demonstrating constructive sequence families that approach fundamental limits.

Abstract

This paper presents generalized Arlery-Tan-Rabaste-Levenshtein lower bounds on the maximum aperiodic ambiguity function (AF) magnitude of unimodular sequences under certain delay-Doppler low ambiguity zones (LAZ). Our core idea is to explore the upper and lower bounds on the Frobenius norm of the weighted auto- and cross-AF matrices by introducing two weight vectors associated with the delay and Doppler shifts, respectively. As a second major contribution, we demonstrate that our derived lower bounds are asymptotically achievable with selected Chu sequence sets by analyzing their maximum auto- and cross- AF magnitudes within certain LAZ.

Generalized Arlery-Tan-Rabaste-Levenshtein Lower Bounds on Ambiguity Function and Their Asymptotic Achievability

TL;DR

This work addresses fundamental limits on aperiodic delay-Doppler ambiguity in unimodular sequences by deriving generalized Arlery-Tan-Rabaste-Levenshtein-type AF lower bounds within delay-Doppler low-ambiguity zones. It introduces two weight vectors to separately capture delay and Doppler contributions and formulates the bounds through Frobenius-norm constraints on weighted AF matrices, yielding explicit bounds that reduce to known correlation bounds in special cases. The authors prove asymptotic achievability of these bounds using Chu sequence sets, demonstrating order-optimal behavior (i.e., maximal AF magnitudes scaling as ) for certain LAZ configurations, and provide numerical evidence supporting tightness. The results offer actionable design guidelines for ISAC and high-mobility communication scenarios by quantifying the trade-offs and demonstrating constructive sequence families that approach fundamental limits.

Abstract

This paper presents generalized Arlery-Tan-Rabaste-Levenshtein lower bounds on the maximum aperiodic ambiguity function (AF) magnitude of unimodular sequences under certain delay-Doppler low ambiguity zones (LAZ). Our core idea is to explore the upper and lower bounds on the Frobenius norm of the weighted auto- and cross-AF matrices by introducing two weight vectors associated with the delay and Doppler shifts, respectively. As a second major contribution, we demonstrate that our derived lower bounds are asymptotically achievable with selected Chu sequence sets by analyzing their maximum auto- and cross- AF magnitudes within certain LAZ.
Paper Structure (17 sections, 15 theorems, 59 equations, 2 figures, 1 table)

This paper contains 17 sections, 15 theorems, 59 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Definitions
  4. Properties of Aperiodic AF
  5. Aperiodic AF Lower Bounds of Unimodular Sequences
  6. Aperiodic AF Lower Bounds Associated to the LAZ
  7. Weight vector A ($\mathbf w^{A,q}$)
  8. Weight vector B ($\mathbf w^{B,q}$)
  9. LAZ with $Z_{x}=N$ Case
  10. Weight vector C ($\mathbf w^C$)
  11. Asymptotically Order-Optimal Sequence Set for AF in the LAZ
  12. Discussions and Comparisons
  13. Comparisons with the Existing AF Lower Bound
  14. Asymptotic Achievability of the Proposed Bounds
  15. Conclusions
  16. ...and 2 more sections

Key Result

Lemma 1

For any unimodular sequence $\bm x$, its aperiodic AAF satisfies

Figures (2)

  • Figure 1: Comparison of the proposed aperiodic AF lower bounds and the benchmark bound.
  • Figure 2: Comparison of the maximum AAF/CAF magnitudes of a Chu sequence pair $\mathbf{s}^1$ and $\mathbf{s}^2$ ($a_1=20$, $a_2 = 19$) and the proposed aperiodic AAF/CAF lower bounds (\ref{['C3']}a) associated to the LAZ with $Z_x= \left \lfloor \frac{9N}{10|a_1|} \right \rfloor$ and $Z_y = a_2$.

Theorems & Definitions (21)

  • Lemma 1: Zero delay
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Remark 1
  • Theorem 1: Aperiodic AF lower bounds associated to an LAZ
  • Remark 2
  • Corollary 1: Aperiodic AF lower bound for global AF
  • Remark 3
  • Corollary 2
  • ...and 11 more