ISAC with Affine Frequency Division Multiplexing: An FMCW-Based Signal Processing Perspective

Cong Yi; Fan Liu; Haoran Yin; Huseyin Arslan; Jiajun Zhu; Yanqun Tang; Yuanhan Ni; Zhiqiang Wei

arxiv: 2511.12308 · v3 · pith:YADMRX74new · submitted 2025-11-15 · 📡 eess.SP

ISAC with Affine Frequency Division Multiplexing: An FMCW-Based Signal Processing Perspective

Jiajun Zhu , Yanqun Tang , Cong Yi , Haoran Yin , Yuanhan Ni , Fan Liu , Zhiqiang Wei , Huseyin Arslan This is my paper

Pith reviewed 2026-05-17 22:07 UTC · model grok-4.3

classification 📡 eess.SP

keywords AFDMISACFMCWdelay-DopplerDAFTsensinghigh-mobilitymatched filtering

0 comments

The pith

A specific parameter selection makes AFDM subcarriers mathematically equivalent to Nyquist-sampled FMCW waveforms.

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

The paper establishes that a targeted parameter selection criterion creates an exact mathematical equivalence between AFDM subcarriers and Nyquist-sampled FMCW radar signals. This equivalence supplies a physical picture of AFDM's sensing behavior in high-mobility settings and produces a direct mapping from the discrete affine Fourier transform index to the delay and Doppler parameters of the wireless channel. From this foundation the authors derive an input-output model in the delay-Doppler-parameterized DAFT domain that isolates the coupling effect caused by chirp-channel interaction, then construct two matched-filter sensing procedures, one low-complexity in the time-frequency domain and one precise in the DD-DAFT domain. The work shows that these procedures support pilot-free sensing while exposing concrete trade-offs among sensing accuracy, communication overhead, and computation. A reader would care because the equivalence turns an abstract multicarrier waveform into something that can reuse established FMCW radar intuition for integrated sensing and communication.

Core claim

The central claim is that an innovative parameter selection criterion establishes a precise mathematical equivalence between AFDM subcarriers and Nyquist-sampled FMCW. This connection supplies a clear physical insight into AFDM's sensing mechanism and enables a direct mapping from the DAFT index to delay-Doppler parameters of wireless channels. Building on the equivalence, the paper develops a novel input-output model in a DD-parameterized DAFT domain that explicitly reveals the inherent DD coupling effect arising from chirp-channel interaction, then designs two matched-filtering sensing algorithms whose performance is validated through simulation.

What carries the argument

The parameter selection criterion that produces exact equivalence between AFDM subcarriers and Nyquist-sampled FMCW, which in turn supports the DD-DAFT domain model and the direct DAFT-to-DD mapping.

If this is right

Pilot-free sensing becomes feasible by treating AFDM as an FMCW-like waveform.
The DD-DAFT domain model isolates the coupling effect so that the second algorithm can resolve it exactly.
Sensing performance, communication overhead, and computational complexity trade off against one another in a quantifiable way.
The proposed AFDM variant outperforms classical AFDM and other waveform variants in most simulated scenarios.

Where Pith is reading between the lines

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

The equivalence may allow reuse of existing FMCW radar processing pipelines inside AFDM transceivers.
Accounting for the DD coupling appears necessary for high-resolution sensing; ignoring it would limit the first algorithm's accuracy.
The mapping from DAFT index to delay-Doppler parameters could simplify joint waveform design in other multicarrier ISAC systems.

Load-bearing premise

The wireless channel must exhibit the exact delay-Doppler coupling that arises from chirp-channel interaction and the high-mobility regime must allow the DAFT-domain representation to hold without unmodeled distortions.

What would settle it

A numerical or experimental test in which the chosen parameters are applied to an AFDM waveform and the resulting range-Doppler map deviates measurably from the map produced by a reference Nyquist-sampled FMCW waveform under the same channel conditions.

Figures

Figures reproduced from arXiv: 2511.12308 by Cong Yi, Fan Liu, Haoran Yin, Huseyin Arslan, Jiajun Zhu, Yanqun Tang, Yuanhan Ni, Zhiqiang Wei.

**Figure 1.** Figure 1: The AFDM-based ISAC scenario. that explicitly decouples these effects. Both methods are applicable to pilot-free AFDM. The remainder of this paper is organized as follows. Section II introduces the system model. In Section III, we establish the relationship between any AFDM subcarrier and FMCW signal. Based on this, we introduce a new AFDM input-output relationship with DD parameters in Section IV. And in … view at source ↗

**Figure 2.** Figure 2: AFDM modulation block diagram. where n = 0, . . . , Nc −1 and Nc is the number of subcarriers. The primary distinction from OFDM is the inclusion of two additional chirp parameters, c1 and c2. These parameters define the chirp characteristics of the subcarriers. Specifically, the term e j2πc1n 2 functions as a time-domain chirp window, while e j2πc2m2 acts as a chirp filter in the frequency domain. When c1… view at source ↗

**Figure 3.** Figure 3: Time domain waveforms and TF distributions of the [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: DDMs of TF and DD MF algorithms for PO = 0 and PO = 1 cases. 0 5 10 13 15 20 25 30 SNR in dB 100 101 PSLR in dB dechirp, PO=0 TFMF, PO=0 DDMF, PO=0 dechirp, PO=0.5 TFMF, PO=0.5 DDMF, PO=0.5 dechirp, PO=1 TFMF, PO=1 DDMF, PO=1 13 15 (a) PSLR vs SNR 0 5 10 15 20 23 25 27 30 SNR in dB 6 8 10 12 14 20 18 16 ImageSNR in dB dechirp, PO=0 TFMF, PO=0 DDMF, PO=0 dechirp, PO=0.5 TFMF, PO=0.5 DDMF, PO=0.5 dechirp, PO… view at source ↗

**Figure 5.** Figure 5: PSLR, Image SNR, and Pd of the proposed AFDM under different PO and algorithms. [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

**Figure 6.** Figure 6: The Image SNR, BER, PSLR, and Pd performance comparison of waveforms under TF and DD algorithms. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

read the original abstract

This paper investigates the sensing potential of affine frequency division multiplexing (AFDM) in high-mobility integrated sensing and communication (ISAC) from the perspective of radar waveforms. We introduce an innovative parameter selection criterion that establishes a precise mathematical equivalence between AFDM subcarriers and Nyquist-sampled frequency-modulated continuous-wave (FMCW). This connection not only provides a clear physical insight into AFDM's sensing mechanism but also enables a direct mapping from the DAFT index to delay-Doppler (DD) parameters of wireless channels. Building on this, we develop a novel input-output model in a DD-parameterized DAFT (DD-DAFT) domain for AFDM, which explicitly reveals the inherent DD coupling effect arising from the chirp-channel interaction. Subsequently, we design two matched-filtering sensing algorithms. The first is performed in the time-frequency domain with low complexity, while the second is operated in the DD-DAFT domain to precisely resolve the DD coupling. Simulations show that our algorithms achieve effective pilot-free sensing and demonstrate a fundamental trade-off between sensing performance, communication overhead, and computational complexity. The proposed AFDM outperforms classical AFDM and other variants in most scenarios.

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.

Desk Editor's Note private letter to a colleague

AFDM can be made equivalent to FMCW for ISAC sensing via parameter choice, yielding a new DD model and detectors, but check the assumptions on distortions.

read the letter

The main thing to know is that this work establishes a parameter selection rule making AFDM subcarriers equivalent to Nyquist-sampled FMCW waveforms. That equivalence lets them map DAFT indices directly to delay-Doppler parameters and creates a DD-DAFT input-output model that isolates the coupling effect from the chirp interacting with the channel. They then propose two matched-filter sensing algorithms based on that model. One runs in the time-frequency domain for lower complexity, and the other works in the DD-DAFT domain to handle the coupling more accurately. Simulations indicate these achieve solid pilot-free sensing and reveal trade-offs with communication overhead and computation. The proposed setup beats classical AFDM and other variants in most tested scenarios. What the paper does well is provide a concrete physical interpretation of how AFDM senses in high-mobility settings by linking it to familiar FMCW processing. This could help designers adapt existing radar techniques to AFDM-based ISAC systems. The soft spot is around the assumptions needed for the exact equivalence. The model assumes the DAFT-domain response matches the FMCW form without extra distortions from things like finite sequence length, Doppler spread beyond the range, or non-ideal sampling. If real channels introduce residual terms, the mapping and algorithms may not perform as expected. The paper derives everything from the parameter criterion, but more analysis or simulations with impairments would help confirm robustness. This paper is for specialists in integrated sensing and communications working on high-mobility links, such as in vehicular or aerial networks. A reader looking for waveform innovations in that area would find the mapping and algorithms worth examining. I think it deserves to go to peer review. The connection is specific and the algorithms are tailored, so referees can check the derivations and simulation details.

Referee Report

3 major / 2 minor

Summary. The manuscript claims that a novel parameter selection criterion establishes a precise mathematical equivalence between AFDM subcarriers and Nyquist-sampled FMCW waveforms. This equivalence yields a DD-parameterized DAFT (DD-DAFT) input-output model that explicitly captures the DD coupling effect from chirp-channel interaction, enabling two matched-filtering sensing algorithms (one low-complexity in the time-frequency domain and one in the DD-DAFT domain) for pilot-free ISAC in high-mobility settings. Simulations are reported to demonstrate effective sensing performance, fundamental trade-offs with communication overhead and complexity, and outperformance relative to classical AFDM and other variants.

Significance. If the equivalence and resulting model hold under the stated conditions, the work supplies a concrete signal-processing bridge between AFDM and established FMCW radar techniques, furnishing both physical insight into AFDM sensing and directly mappable DD parameters. The proposed algorithms and observed trade-offs could inform practical ISAC waveform design for high-mobility channels, particularly if the derivations prove reproducible and the simulations are fully specified.

major comments (3)

[Section 3] Parameter selection criterion (Section 3): The central claim of 'precise mathematical equivalence' between AFDM subcarriers and Nyquist-sampled FMCW is load-bearing for both the physical insight and the direct DAFT-to-DD mapping. The derivation must explicitly demonstrate that the chosen parameters eliminate residual phase/amplitude terms, aliasing, or other distortions arising from finite-length effects and high-mobility chirp-channel interaction beyond the modeled DD coupling; otherwise the equivalence is conditional rather than general.
[Section 4] DD-DAFT input-output model (Section 4): The assertion that the model 'explicitly reveals the inherent DD coupling effect' requires the explicit equation (or set of equations) showing how the coupling term emerges from the chirp-channel interaction and matches the FMCW-derived form exactly. If any approximation or unmodeled term remains, both the novelty of the model and the justification for the DD-DAFT-domain matched filter are weakened.
[Section 6] Simulation results (Section 6): The reported performance advantages, trade-offs, and outperformance claims rest on simulations whose channel generation procedure, exact metric definitions (e.g., sensing MSE or detection probability), and handling of realistic impairments are not fully specified. This prevents independent verification that the equivalence survives the impairments highlighted in the stress-test note and therefore undermines the empirical support for the algorithms.

minor comments (2)

[Abstract] Abstract: The statement that 'simulations show' effective sensing would benefit from one or two quantitative highlights (e.g., achieved range or velocity resolution) to give readers an immediate sense of the gains.
Notation: Ensure consistent distinction between the standard DAFT domain and the proposed DD-DAFT domain throughout; occasional slippage between the two terms can confuse readers unfamiliar with the new model.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough and constructive review. The comments highlight important points on explicitness of derivations and reproducibility of simulations. We will revise the manuscript accordingly to strengthen these aspects while preserving the core contributions. Point-by-point responses follow.

read point-by-point responses

Referee: [Section 3] Parameter selection criterion (Section 3): The central claim of 'precise mathematical equivalence' between AFDM subcarriers and Nyquist-sampled FMCW is load-bearing for both the physical insight and the direct DAFT-to-DD mapping. The derivation must explicitly demonstrate that the chosen parameters eliminate residual phase/amplitude terms, aliasing, or other distortions arising from finite-length effects and high-mobility chirp-channel interaction beyond the modeled DD coupling; otherwise the equivalence is conditional rather than general.

Authors: We agree that explicit verification of the equivalence is essential. In the revised manuscript, Section 3 will be expanded with a complete derivation that substitutes the selected parameters (chirp rate matching the FMCW sweep and subcarrier spacing aligned to Nyquist sampling) into the AFDM waveform expression. This will show term-by-term cancellation of residual phase and amplitude factors, as well as the absence of aliasing under the finite-length and high-mobility conditions considered. The equivalence is thereby established precisely within the stated modeling assumptions rather than conditionally. revision: yes
Referee: [Section 4] DD-DAFT input-output model (Section 4): The assertion that the model 'explicitly reveals the inherent DD coupling effect' requires the explicit equation (or set of equations) showing how the coupling term emerges from the chirp-channel interaction and matches the FMCW-derived form exactly. If any approximation or unmodeled term remains, both the novelty of the model and the justification for the DD-DAFT-domain matched filter are weakened.

Authors: The DD-DAFT model is obtained by applying the parameter equivalence to the received AFDM signal and re-expressing the result in the DD-parameterized DAFT domain. In the revision, we will insert the intermediate steps that isolate the coupling term arising from the product of the linear chirp phase and the time-varying channel response. The resulting expression will be shown to match the FMCW-derived DD coupling form exactly, with no residual approximations beyond the standard narrowband and finite-duration assumptions already stated in the paper. This will directly support the subsequent matched-filter designs. revision: yes
Referee: [Section 6] Simulation results (Section 6): The reported performance advantages, trade-offs, and outperformance claims rest on simulations whose channel generation procedure, exact metric definitions (e.g., sensing MSE or detection probability), and handling of realistic impairments are not fully specified. This prevents independent verification that the equivalence survives the impairments highlighted in the stress-test note and therefore undermines the empirical support for the algorithms.

Authors: We acknowledge the need for full reproducibility. The revised Section 6 will specify the exact DD channel generation procedure (including delay and Doppler spreads, number of paths, and Jakes' spectrum implementation), provide closed-form definitions for all reported metrics (sensing MSE, detection probability, and communication BER), and detail the incorporation of realistic impairments such as additive noise, phase noise, and residual synchronization errors. These additions will enable independent verification of the claimed performance and trade-offs. revision: yes

Circularity Check

0 steps flagged

Parameter selection criterion for AFDM-FMCW equivalence is a constructive design choice, not a circular reduction.

full rationale

The paper's central step introduces a parameter selection criterion to achieve exact mathematical equivalence between AFDM subcarriers and Nyquist-sampled FMCW, enabling the DD-DAFT domain model and DD coupling revelation. This is presented as an innovative criterion rather than a fitted prediction or self-definition; the equivalence holds under the stated assumptions on channel DD coupling and high-mobility regime. No load-bearing self-citation, uniqueness theorem from prior work, or renaming of known results is used for the core claims. The matched-filtering algorithms and simulations follow directly from the constructed model without reducing to the inputs by construction. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard properties of the discrete affine Fourier transform and linear time-varying channel models; no new free parameters beyond the selection criterion itself or invented physical entities are introduced in the abstract.

axioms (2)

standard math The discrete affine Fourier transform (DAFT) provides an orthonormal basis for representing chirp-modulated signals in high-mobility channels.
Invoked when mapping DAFT indices to delay-Doppler parameters.
domain assumption Wireless channels in high-mobility scenarios exhibit delay-Doppler coupling that interacts with the chirp structure of AFDM.
Used to justify the DD-DAFT input-output model.

pith-pipeline@v0.9.0 · 5534 in / 1579 out tokens · 28322 ms · 2026-05-17T22:07:18.925026+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

precise mathematical equivalence between AFDM subcarriers and Nyquist-sampled frequency-modulated continuous-wave (FMCW)... DD coupling effect arising from the chirp-channel interaction
IndisputableMonolith/Foundation/DimensionForcing.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

K=8 chirp periods... period Np... floor(k/K) coupling term

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Towards Standardizing Affine Frequency Division Multiplexing (AFDM) for Future Wireless Networks
eess.SP 2026-06 unverdicted novelty 3.0

AFDM is analyzed for standardization potential, showing backwards compatibility with 4G/5G, radar, and LoRa systems while highlighting robustness in delay-Doppler channels for NTN, ISAC, V2X, and UWA applications.