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Doppler Resilient Complementary Sequences: Tighter Aperiodic Ambiguity Function Bound and Optimal Constructions

Zheng Wang, Yang Yang, Zhengchun Zhou, Avik Ranjan Adhikary, Pingzhi Fan

TL;DR

This work addresses the design of Doppler-resilient complementary sequence sets by deriving a new weight-vector–based lower bound for the aperiodic ambiguity function of unimodular DRCSs, generalizing and tightening existing bounds. The approach leverages a Frobenius-norm framework with weight vectors and analyzes two main bounds that recover classic results as special cases. It then introduces an asymptotically optimal DRCS construction based on quasi-Florentine rectangles and Butson-type Hadamard matrices, yielding small-alphabet DRCS sets with zero auto-AF and maximal cross-AF magnitude for distinct codes, and demonstrates asymptotic achievability of the new bound. Overall, the paper provides both a tighter theoretical bound and practical, flexible constructions that improve Doppler-resilient waveform design for joint sensing and communication in mobile environments.

Abstract

Doppler-resilient complementary sequence sets (DRCSs) are crucial in modern communication and sensing systems in mobile environments. In this paper, we propose a new lower bound for the aperiodic ambiguity function (AF) of unimodular DRCSs based on weight vectors, which generalizes the existing bound as a special case. The proposed lower bound is tighter than the Shen-Yang-Zhou-Liu-Fan bound. Finally, we propose a novel class of aperiodic DRCSs with small alphabets based on quasi-Florentine rectangles and Butson-type Hadamard matrices. Interestingly, the proposed DRCSs asymptotically satisfy the proposed bound.

Doppler Resilient Complementary Sequences: Tighter Aperiodic Ambiguity Function Bound and Optimal Constructions

TL;DR

This work addresses the design of Doppler-resilient complementary sequence sets by deriving a new weight-vector–based lower bound for the aperiodic ambiguity function of unimodular DRCSs, generalizing and tightening existing bounds. The approach leverages a Frobenius-norm framework with weight vectors and analyzes two main bounds that recover classic results as special cases. It then introduces an asymptotically optimal DRCS construction based on quasi-Florentine rectangles and Butson-type Hadamard matrices, yielding small-alphabet DRCS sets with zero auto-AF and maximal cross-AF magnitude for distinct codes, and demonstrates asymptotic achievability of the new bound. Overall, the paper provides both a tighter theoretical bound and practical, flexible constructions that improve Doppler-resilient waveform design for joint sensing and communication in mobile environments.

Abstract

Doppler-resilient complementary sequence sets (DRCSs) are crucial in modern communication and sensing systems in mobile environments. In this paper, we propose a new lower bound for the aperiodic ambiguity function (AF) of unimodular DRCSs based on weight vectors, which generalizes the existing bound as a special case. The proposed lower bound is tighter than the Shen-Yang-Zhou-Liu-Fan bound. Finally, we propose a novel class of aperiodic DRCSs with small alphabets based on quasi-Florentine rectangles and Butson-type Hadamard matrices. Interestingly, the proposed DRCSs asymptotically satisfy the proposed bound.
Paper Structure (11 sections, 19 theorems, 69 equations, 3 figures, 5 tables)

This paper contains 11 sections, 19 theorems, 69 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ambiguity Function
  4. Low Ambiguity Zone and Existing Aperiodic DRCS Set Bound
  5. Quasi-Florentine Rectangle
  6. Butson-type Hadamard Matrix
  7. Proposed Lower Bounds of DRCSs
  8. Discussions on the Proposed Lower Bounds for DRCSs
  9. Asymptotically Optimal Construction of DRCSs
  10. Comparisons with the Existing Aperiodic AF Lower Bound
  11. Concluding Remarks

Key Result

Lemma 1

For a $\left(K, M, N, \theta_{\max }, \Pi\right)$-DRCS set, where $\Pi=\left(-Z_x, Z_x\right) \times\left(-Z_y, Z_y\right), 1 \leq Z_x, Z_y \leq N$, the lower bound of the aperiodic AF is given by

Figures (3)

  • Figure 1: A glimpse of the aperiodic auto-AF and cross-AF of the sequence set $\mathcal{C}$ in Example \ref{['ex1']}.
  • Figure 2: A glimpse of the aperiodic auto-AF and cross-AF of the sequence set $\mathcal{C}$ in Example \ref{['ex2']}.
  • Figure 3: Comparison between the optimality factors (\ref{['rho1']}) and shen2024 with respect to $(p^n, p^n, p^n-1, p^n, \Pi)$-DRCS for $3 \leq p < 120,$$n=2$ and $\Pi=(-p^n+1,p^n-1)\times (-p^n+1,p^n-1)$.

Theorems & Definitions (45)

  • Lemma 1: shen2024
  • Definition 1: Optimality Factor
  • Definition 2: Avik2024
  • Theorem 1: Avik2024
  • Corollary 1: Avik2024
  • Lemma 2: Avik2024
  • Definition 3: had1973complex
  • Remark 1
  • Lemma 3: ye22
  • Lemma 4
  • ...and 35 more