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Transmission Mask Analysis for Range-Doppler Sensing in Half-Duplex ISAC

Dikai Liu, Yifeng Xiong, Marco Lops, Fan Liu, Jianhua Zhang

TL;DR

This work analyzes masked modulation for half-duplex ISAC to achieve range–Doppler sensing. It derives a closed-form $E_r(k,l,\nu)$ and shows the range-sidelobe response ($k\neq l$) is Doppler-invariant, while the range-mainlobe ($k=l$) exhibits sparse Doppler sidelobes under periodic masking. In the moderately dynamic regime, Singer CDSs are minimax-optimal, balancing low maximal Doppler sidelobes with reduced mainlobe fluctuation; in the highly dynamic regime, a concave dependence of Doppler energy on the mask autocorrelation implies an inevitable tradeoff, with no mask being optimal for all criteria. Numerical results validate the theory and demonstrate practical mask-design guidance, highlighting when CDS-type masks are ideal and when tradeoffs must be accepted. These insights inform high-throughput ISAC waveform design under half-duplex constraints and dynamic targets.

Abstract

In this paper, we analyze the periodic transmission masks for MASked Modulation (MASM) in half-duplex integrated sensing and communication (ISAC), and derive their closed-form expected range-Doppler response $\mathbb{E}\{r(k,l,ν)\}$. We show that range sidelobes ($k\neq l$) are Doppler-invariant, extending the range-sidelobe optimality to the 2-D setting. For the range mainlobe ($k=l$), periodic masking yields sparse Doppler sidelobes: Cyclic difference sets (CDSs) (in particular Singer CDSs) are minimax-optimal in a moderately dynamic regime, while in a highly dynamic regime the Doppler-sidelobe energy is a concave function of the mask autocorrelation, revealing an inevitable tradeoff with mainlobe fluctuation.

Transmission Mask Analysis for Range-Doppler Sensing in Half-Duplex ISAC

TL;DR

This work analyzes masked modulation for half-duplex ISAC to achieve range–Doppler sensing. It derives a closed-form and shows the range-sidelobe response () is Doppler-invariant, while the range-mainlobe () exhibits sparse Doppler sidelobes under periodic masking. In the moderately dynamic regime, Singer CDSs are minimax-optimal, balancing low maximal Doppler sidelobes with reduced mainlobe fluctuation; in the highly dynamic regime, a concave dependence of Doppler energy on the mask autocorrelation implies an inevitable tradeoff, with no mask being optimal for all criteria. Numerical results validate the theory and demonstrate practical mask-design guidance, highlighting when CDS-type masks are ideal and when tradeoffs must be accepted. These insights inform high-throughput ISAC waveform design under half-duplex constraints and dynamic targets.

Abstract

In this paper, we analyze the periodic transmission masks for MASked Modulation (MASM) in half-duplex integrated sensing and communication (ISAC), and derive their closed-form expected range-Doppler response . We show that range sidelobes () are Doppler-invariant, extending the range-sidelobe optimality to the 2-D setting. For the range mainlobe (), periodic masking yields sparse Doppler sidelobes: Cyclic difference sets (CDSs) (in particular Singer CDSs) are minimax-optimal in a moderately dynamic regime, while in a highly dynamic regime the Doppler-sidelobe energy is a concave function of the mask autocorrelation, revealing an inevitable tradeoff with mainlobe fluctuation.
Paper Structure (16 sections, 5 theorems, 35 equations, 3 figures)

This paper contains 16 sections, 5 theorems, 35 equations, 3 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Transmission and Sensing Reception Model
  4. Performance Metrics
  5. Analysis of the Range-Doppler Response
  6. The range sidelobes $k\neq l$
  7. The range mainlobes $k=l$, Moderately Dynamic Regime
  8. The range mainlobes $k=l$, Highly Dynamic Regime
  9. Numerical Results
  10. Range mainlobe ($k=l$), global Doppler behavior (highly dynamic regime)
  11. Range mainlobe ($k=l$), zoomed in (moderately dynamic regime)
  12. Range-Doppler response at a fixed $k$
  13. Conclusions
  14. Proof of Proposition \ref{['prop:doppler_sidelobe_range_main']}
  15. Proof of Proposition \ref{['prop:eisl_max']}
  16. ...and 1 more sections

Key Result

Lemma 1

For $\nu\in\{0,1,\dotsc,MN-1\}$, we have $S_{k,MN}(\nu)=0$ for $\nu\not\equiv 0\pmod{M}$.

Figures (3)

  • Figure 1: The range-Doppler responses of different masks at range mainlobes ($k=l$), full Doppler span $\nu\in\{0,\ldots,MN-1\}$.
  • Figure 2: The range-Doppler responses of different masks at range mainlobes ($k=l$), local Doppler window $\nu\in\{0,\ldots,M-1\}$ (zoom-in of Fig. \ref{['fig:doppler_global']}).
  • Figure 3: The range-Doppler response of the Singer cds and a random mask, at $k=20$.

Theorems & Definitions (6)

  • Lemma 1
  • Remark 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Corollary 1