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arxiv: 2605.19759 · v1 · pith:S5IA2R2Knew · submitted 2026-05-19 · 📡 eess.SP

DAFT-s-AFDM Enabled ISAC Systems: Ambiguity Function Analysis and Waveform Design

Shiqi Cui , Tianqi Mao , Fan Zhang , Zeping Sui , Christos Masouros , Zhaocheng Wang This is my paper

Pith reviewed 2026-05-20 02:09 UTC · model grok-4.3

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keywords DAFT-s-AFDMISACambiguity functionprobabilistic constellation shapingwaveform designPareto optimizationbit error rate
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The pith

A probabilistic constellation shaping framework applied to DAFT-s-AFDM waveforms jointly optimizes communication throughput and sensing ambiguity function characteristics in ISAC systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper first derives a closed-form expression for the magnitude expectation of the ambiguity function of DAFT-s-AFDM and uses it to reveal how waveform parameters affect sensing and communication performance. These insights then motivate a probabilistic constellation shaping approach that solves a multi-objective optimization problem to balance the two functions. An efficient algorithm relying on a closed-form bit error rate expression finds the Pareto-optimal operating points. Simulations confirm that the resulting waveforms outperform conventional designs with a more flexible performance tradeoff.

Core claim

By analyzing the ambiguity function of DAFT-s-AFDM and obtaining its closed-form magnitude expectation, the authors identify parameter choices that improve integrated sensing and communication. They then introduce probabilistic constellation shaping to adjust symbol probabilities, formulating and solving a multi-objective problem whose solutions, obtained via a closed-form bit error rate expression, produce Pareto-optimal waveforms that deliver superior and controllable dual-functional performance.

What carries the argument

The probabilistic constellation shaping framework on DAFT-s-AFDM, which reweights symbol probabilities to jointly control communication bit error rate and sensing ambiguity function sidelobe behavior.

If this is right

  • Parameter selection guided by the ambiguity function expression directly improves the sensing resolution and communication reliability of DAFT-s-AFDM.
  • The multi-objective formulation yields a family of waveforms whose communication-sensing tradeoff can be tuned by the designer.
  • The closed-form bit error rate enables low-complexity computation of operating points without exhaustive search.
  • Reduced multiuser interference and Doppler robustness are preserved while the peak-to-average power ratio remains low.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same shaping technique could be applied to other affine or chirp-based waveforms to extend the performance gains beyond the specific DAFT-s-AFDM case.
  • Real-time adaptation of the constellation probabilities might allow the system to respond to changing channel or target conditions.
  • The approach suggests a general method for embedding sensing constraints into constellation design for any multicarrier ISAC waveform.

Load-bearing premise

The closed-form bit error rate expression used to obtain Pareto-optimal solutions accurately captures the system behavior under the DAFT-s-AFDM model.

What would settle it

An experiment or simulation in which the measured bit error rate under the optimized DAFT-s-AFDM constellations deviates substantially from the closed-form expression, producing a different set of Pareto points than predicted.

Figures

Figures reproduced from arXiv: 2605.19759 by Christos Masouros, Fan Zhang, Shiqi Cui, Tianqi Mao, Zeping Sui, Zhaocheng Wang.

Figure 1
Figure 1. Figure 1: Transceiver diagram of the DAFT-s-AFDM-enabled monostatic ISAC system. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The DPAF for different subcarrier configurations. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: T3 of the AF for different parameter settings. OFDM and discrete Fourier transform-spread OFDM (DFT-s￾OFDM), the chirped subcarriers in DAFT-s-AFDM under L￾FDMA occupy the full signal bandwidth, rendering the delay resolution largely independent of M. This yields superior de￾lay resolution given the subcarrier occupancy M, constituting a key advantage. Fig. 2c illustrates the AF under configuration (c) of … view at source ↗
Figure 5
Figure 5. Figure 5: The zero-delay slice. responses from localized targets, consistent with the behavior observed in AFDM [13]. Notably, DFT-s-OFDM with c1 = 0 exhibits the same property. We then proceed to compute and plot the zero-delay slice for four distinct waveforms: OFDM, AFDM, DFT-s-OFDM, and DAFT-s-AFDM, all employing QPSK modulation with identical subcarrier mapping. As shown in [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 6
Figure 6. Figure 6: CA-CFAR detection results with different [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: DAFT-s-AFDM BER Comarison with PCS. algorithm achieves a runtime approximately one order of magnitude lower than that of the MBA algorithm [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: Optimal PCS constellations at three representative [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Illustration of (a) MUSIC Spectrum at SNR = 20dB [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
read the original abstract

Discrete affine Fourier transform spread affine frequency division multiplexing (DAFT-s-AFDM) is a promising waveform for integrated sensing and communication (ISAC) due to its low peak-to-average power ratio, robustness to Doppler shifts, and reduced multiuser interference in the uplink transmission. This paper presents a comprehensive ambiguity function (AF) analysis of DAFT-s-AFDM and derives the closed-form expression for the AF magnitude expectation. Several key insights into the impact of DAFT-s-AFDM parameters on ISAC performance are revealed, thus providing concrete guidance for the subsequent waveform design. Building on these insights, a novel probabilistic constellation shaping (PCS) framework is proposed for ISAC waveform enhancement, where the communication throughput and the sensing AF characteristics are jointly optimized by addressing a multi-objective problem. An efficient algorithm based on a closed-form bit error rate expression is developed to obtain the Pareto-optimal solutions. Extensive simulations validate the theoretical results and that the proposed PCS-enhanced DAFT-s-AFDM can significantly outperform the classical counterparts, achieving a superior and highly controllable tradeoff between the dual-functional performances.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes the ambiguity function (AF) of discrete affine Fourier transform spread affine frequency division multiplexing (DAFT-s-AFDM) for integrated sensing and communication (ISAC), derives a closed-form expression for the AF magnitude expectation, extracts parameter insights for ISAC performance, and proposes a probabilistic constellation shaping (PCS) framework. This framework solves a multi-objective optimization problem using a closed-form bit error rate (BER) expression to locate Pareto-optimal points that balance communication throughput against sensing AF characteristics, with simulations claiming superior and controllable dual-functional tradeoffs over classical waveforms.

Significance. If the closed-form AF expectation and BER expressions are accurate and the PCS optimization translates to realized performance, the work supplies concrete design guidelines for Doppler-robust, low-PAPR ISAC waveforms with reduced multiuser interference. The explicit closed-form derivations and simulation validation of the Pareto frontier constitute a strength that could inform practical waveform selection in high-mobility scenarios.

major comments (2)
  1. [Waveform design / multi-objective optimization section (around the closed-form BER and Pareto solver)] The central optimization in the PCS framework relies on a closed-form BER expression to identify Pareto-optimal solutions. This expression must be shown to incorporate DAFT-s-AFDM-specific effects (affine Fourier spreading, residual affine-frequency interference, and reduced MUI) when symbol probabilities become non-uniform under PCS; otherwise the claimed controllable tradeoff cannot be guaranteed. The manuscript should explicitly state the assumptions (e.g., perfect channel estimation, uniform vs. shaped constellations) used in the BER derivation and verify them against the full DAFT-s-AFDM signal model.
  2. [Ambiguity function analysis section (closed-form AF magnitude expectation)] The AF magnitude expectation derivation is presented as closed-form and used to guide parameter selection. It is unclear whether the expectation accounts for the statistical properties introduced by PCS (non-uniform symbol amplitudes) or whether it remains valid only under uniform constellations. A direct comparison or extension of the AF expression under PCS is needed to support the subsequent claim that PCS-enhanced DAFT-s-AFDM achieves a superior ISAC tradeoff.
minor comments (2)
  1. [Simulation results section] Simulation parameters (e.g., number of Monte Carlo trials, exact Doppler ranges, MUI levels, and error-bar reporting) should be stated more explicitly so that the reported performance gains can be reproduced.
  2. [System model / PCS framework] Notation for the DAFT spreading matrix and the PCS probability distribution should be introduced consistently before their first use in the optimization formulation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We have carefully reviewed the concerns regarding the closed-form expressions in the PCS framework and AF analysis. Below we address each major comment point by point, indicating the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: The central optimization in the PCS framework relies on a closed-form BER expression to identify Pareto-optimal solutions. This expression must be shown to incorporate DAFT-s-AFDM-specific effects (affine Fourier spreading, residual affine-frequency interference, and reduced MUI) when symbol probabilities become non-uniform under PCS; otherwise the claimed controllable tradeoff cannot be guaranteed. The manuscript should explicitly state the assumptions (e.g., perfect channel estimation, uniform vs. shaped constellations) used in the BER derivation and verify them against the full DAFT-s-AFDM signal model.

    Authors: We agree that the current closed-form BER expression was derived under the assumption of uniform symbol probabilities and requires extension to fully capture non-uniform probabilities under PCS along with DAFT-s-AFDM-specific effects such as affine Fourier spreading and residual interference. In the revised manuscript we will provide an updated BER derivation that incorporates these effects, explicitly list all assumptions including perfect channel estimation and shaped constellations, and add a verification step comparing the expression to the complete DAFT-s-AFDM signal model. This will ensure the Pareto-optimal solutions remain valid for the proposed PCS-enhanced waveform. revision: yes

  2. Referee: The AF magnitude expectation derivation is presented as closed-form and used to guide parameter selection. It is unclear whether the expectation accounts for the statistical properties introduced by PCS (non-uniform symbol amplitudes) or whether it remains valid only under uniform constellations. A direct comparison or extension of the AF expression under PCS is needed to support the subsequent claim that PCS-enhanced DAFT-s-AFDM achieves a superior ISAC tradeoff.

    Authors: The original closed-form AF magnitude expectation was obtained assuming equiprobable symbols. To address the concern, the revised manuscript will include an explicit extension of the AF analysis for non-uniform symbol probabilities under PCS. We will derive the adjusted expectation, provide a direct comparison of AF characteristics between uniform and shaped cases, and demonstrate that the key parameter insights and the claimed superior ISAC tradeoff continue to hold (or are appropriately adjusted) when PCS is applied. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper derives a closed-form expression for the AF magnitude expectation directly from the DAFT-s-AFDM waveform model and uses it to extract parameter insights. It then introduces a PCS framework whose multi-objective optimization is solved via a separate closed-form BER expression developed for the same model. These steps constitute independent analytical content rather than re-labeling of fitted inputs or self-referential definitions. Simulations are invoked to validate the theoretical expressions, confirming that the central tradeoff claims rest on model-derived formulas rather than tautological reductions. No load-bearing self-citation chains or ansatz smuggling are present in the provided derivation outline.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no specific free parameters, axioms, or invented entities can be extracted or audited from the provided text.

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discussion (0)

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