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arxiv: 2605.19515 · v1 · pith:V3NBDDFHnew · submitted 2026-05-19 · 📡 eess.SP

Hyperbolic Frequency Multicarrier Modulation for Wideband Linear Time-Varying Channels

Xuehan Wang , Jinhong Yuan , Jintao Wang , Zhi Sun , Jinxing Hao This is my paper

Pith reviewed 2026-05-20 02:44 UTC · model grok-4.3

classification 📡 eess.SP
keywords hyperbolic frequency modulationmulticarrier modulationwideband LTV channelsDoppler scalingequivalent delayAFDMhigh-mobility communication
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The pith

Hyperbolic frequency multicarrier modulation absorbs Doppler scaling into an equivalent delay for wideband linear time-varying channels.

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

The paper proposes a new multicarrier waveform called HFMC to handle ultra-reliable communication over wideband channels where high mobility causes noticeable time dilation and contraction. It starts from the passband model of these channels and hyperbolic frequency modulated signals to show that the Doppler scaling factor can be folded into a simple equivalent delay. This leads to a compact one-dimensional characterization of the channel effects, similar to how AFDM works but adapted for the wideband case. The work then verifies approximate orthogonality of the subcarriers, analyzes their spectrum for capacity, and optimizes parameters to balance efficiency and reliability while exploiting multipath diversity.

Core claim

By using the passband representation of wideband LTV channels together with HFM signals, the Doppler scaling factor caused by relative mobility is absorbed into an equivalent delay; HFMC subcarriers generated from a basic HFM signal via uniformly spaced equivalent delays therefore experience a combined one-dimensional shift that integrates delay and Doppler scaling for each path.

What carries the argument

The absorption of Doppler scaling into equivalent delay via hyperbolic frequency modulated signals, which converts the two-dimensional channel distortion into a one-dimensional integration for each path.

If this is right

  • HFMC subcarriers maintain approximate orthogonality when generated with uniformly spaced equivalent delays.
  • The input-output relation of the system reduces to a single integration of delay and Doppler scaling per path.
  • Spectrum analysis reveals overlapping subcarrier frequencies that can still be used to compute system capacity.
  • Parameter choices can be optimized from the input-output and spectrum results to trade efficiency against reliability.

Where Pith is reading between the lines

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

  • The same absorption idea could simplify receiver processing in other high-mobility settings such as underwater or vehicular links.
  • Extending the uniform spacing rule to non-uniform or adaptive placements might further reduce interference in extreme mobility.
  • The approach may combine naturally with existing multipath combining techniques to extract additional diversity gains.

Load-bearing premise

The approximate orthogonality of HFMC subcarriers created with uniformly spaced equivalent delays remains adequate for real-world wideband operation.

What would settle it

A simulation or hardware test in a wideband LTV channel with measurable relative motion that shows whether residual interference stays low enough for reliable detection when Doppler scaling is treated only as added delay.

Figures

Figures reproduced from arXiv: 2605.19515 by Jinhong Yuan, Jintao Wang, Jinxing Hao, Xuehan Wang, Zhi Sun.

Figure 1
Figure 1. Figure 1: The correlation (dB) among transmit subcarriers. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: SIR and approximation MSE evaluation for [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: evaluates the accuracy of the approximation in Theorem 3 by plotting the approximation MSE against the error threshold ϵ for different values of the maximum mobility velocity vmax. As demonstrated, the MSE increases monoton￾ically with both ϵ and vmax. Even with the highest considered mobility of vmax = 10 kn, the MSE remains below −35 dB for ϵ ≤ 0.036, which confirms the practical reliability of the appro… view at source ↗
Figure 4
Figure 4. Figure 4: Equivalent channel matrix (dB) of HFMC systems. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Spectrum of selected subcarriers. 0 5 10 15 20 25 SNR (dB) 10-5 10-4 10-3 10-2 10-1 100 BER HFMC (proposed) ODDM single carrier OFDM [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: BER against SNR under QPSK alphabets. term “subcarrier” is retained for analogy, these waveforms are not Fourier bases. Instead, they maintain approximate orthogonality via matched filtering as established in Theorem 1. This overlap arises from the HFM structure and does not compromise orthogonality because the frequency-domain separation is not required. In fact, such overlap is beneficial for improving s… view at source ↗
Figure 9
Figure 9. Figure 9: BER against SNR under QPSK alphabets and 512 transmit subcarriers. [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
read the original abstract

Numerous multicarrier modulation schemes have been proposed recently to enhance the performance in narrowband doubly dispersive channels for emerging high-mobility applications. However, the ultra-reliable modulation framework in wideband linear time-varying (LTV) channels remains an open problem, where the time dilations and contractions brought by the high mobility cannot be ignored for the baseband signal to obtain the constant Doppler shift across the whole transmission band. To solve this problem, we propose the hyperbolic frequency multicarrier (HFMC) waveform in this paper based on the inspiration from affine frequency division multiplexing (AFDM) modulation, where the delay and Doppler shift are absorbed into a 1D shift in the affine domain to provide a compact characterization of doubly dispersive discrete-time channels. By adopting the passband representation of wideband LTV channels and hyperbolic frequency modulated (HFM) signals, we reveal that the Doppler scaling factor brought by the relative mobility can be absorbed into an equivalent delay. The basic principle of HFMC modulation is established by investigating the approximate orthogonality among HFMC subcarriers, which are generated from a basic HFM signal by utilizing uniformly spaced equivalent delay. The spectrum of HFMC subcarriers is also analyzed to evaluate the system capacity, where the overlapping nature in the frequency domain can be observed. The input-output characterization in wideband LTV channels is then executed to confirm the 1D integration of time delay and Doppler scaling factor for each path, which demonstrates the ability to exploit potential multipath diversity. The parameter optimization based on the input-output relation and spectrum analysis is finally developed to balance the efficiency and reliability.

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

3 major / 2 minor

Summary. The paper proposes hyperbolic frequency multicarrier (HFMC) modulation for wideband linear time-varying channels. Drawing on hyperbolic frequency modulated (HFM) signals and affine frequency division multiplexing (AFDM), it shows that the Doppler scaling factor can be absorbed into an equivalent delay via passband representation, yielding a compact 1D channel characterization. The work establishes the basic principle through analysis of approximate orthogonality among subcarriers generated by uniformly spaced equivalent delays, analyzes the spectrum to assess capacity with observed frequency overlap, performs input-output characterization confirming 1D integration of delay and Doppler scaling per path for multipath diversity exploitation, and develops parameter optimization to balance efficiency and reliability.

Significance. If the approximate orthogonality holds with sufficient quantitative control on residual interference, the scheme could provide a useful discrete-time model for high-mobility wideband LTV channels where conventional multicarrier approaches fail due to time dilation. The passband-based absorption of scaling into delay and the spectrum-based capacity discussion are potentially valuable contributions, but the lack of explicit error bounds and verification results limits the assessed impact.

major comments (3)
  1. [approximate orthogonality section] Section investigating the approximate orthogonality among HFMC subcarriers: the assertion that uniformly spaced equivalent delays produce subcarriers with cross-correlations small enough for practical wideband operation lacks a closed-form bound on the residual interference term as a function of the Doppler scaling factor α, fractional bandwidth, or subcarrier count. This bound is load-bearing for the central claim that the 1D absorption yields a usable channel model.
  2. [input-output characterization] Input-output characterization section: the demonstration that Doppler scaling is absorbed into an equivalent delay for each path, enabling multipath diversity exploitation, is asserted without an explicit derivation or verification step showing how the 1D integration follows from the passband representation.
  3. [spectrum analysis] Spectrum analysis for capacity evaluation: the observation of overlapping spectra is used to evaluate system capacity, yet no quantitative comparison to external benchmarks or closed-form capacity expressions under the claimed orthogonality are supplied, weakening the reliability claims.
minor comments (2)
  1. [parameter optimization] The abstract and parameter optimization description would benefit from clearer notation distinguishing the basic HFM signal from the generated HFMC subcarriers.
  2. Additional references to prior work on HFM signals in wideband channels and explicit comparisons to AFDM performance would strengthen the positioning.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback on our manuscript. The comments identify key areas where additional rigor can strengthen the presentation of the HFMC waveform. We address each major comment below and will revise the manuscript accordingly to provide the requested bounds, derivations, and comparisons.

read point-by-point responses

  1. Referee: Section investigating the approximate orthogonality among HFMC subcarriers: the assertion that uniformly spaced equivalent delays produce subcarriers with cross-correlations small enough for practical wideband operation lacks a closed-form bound on the residual interference term as a function of the Doppler scaling factor α, fractional bandwidth, or subcarrier count. This bound is load-bearing for the central claim that the 1D absorption yields a usable channel model.

    Authors: We agree that an explicit closed-form bound would provide stronger support for the approximate orthogonality claim. In the revised manuscript we will derive an upper bound on the residual cross-correlation interference by leveraging the ambiguity function of the underlying HFM chirps, expressing the bound in terms of the Doppler scaling factor α and the fractional bandwidth. This derivation will be added to the approximate orthogonality section, together with a brief discussion of its tightness under typical wideband parameters. revision: yes

  2. Referee: Input-output characterization section: the demonstration that Doppler scaling is absorbed into an equivalent delay for each path, enabling multipath diversity exploitation, is asserted without an explicit derivation or verification step showing how the 1D integration follows from the passband representation.

    Authors: We acknowledge that the step from the passband channel model to the 1D equivalent delay requires a clearer derivation. In the revision we will insert a detailed derivation starting from the passband representation of the wideband LTV channel, showing how the Doppler scaling factor combines with the propagation delay to produce a single equivalent delay parameter per path. We will also add a short verification subsection that confirms the resulting input-output relation through both analytical simplification and numerical simulation of the received signal. revision: yes

  3. Referee: Spectrum analysis for capacity evaluation: the observation of overlapping spectra is used to evaluate system capacity, yet no quantitative comparison to external benchmarks or closed-form capacity expressions under the claimed orthogonality are supplied, weakening the reliability claims.

    Authors: We thank the referee for highlighting this gap. The current spectrum analysis illustrates the overlap property but does not yet benchmark performance. In the revised version we will add a quantitative rate comparison against AFDM and conventional OFDM under identical wideband LTV channel conditions, using the derived input-output relation. We will also include a closed-form capacity expression under the approximate orthogonality assumption, obtained by treating residual interference as additive noise. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's chain begins with external inspirations (AFDM, HFM signals) and passband LTV channel representations to derive the Doppler-to-delay absorption claim. It then defines HFMC subcarriers via uniformly spaced equivalent delays, investigates their approximate orthogonality, analyzes the resulting spectrum for capacity evaluation, executes input-output relations to confirm 1D path integration, and optimizes parameters from those relations. None of these steps reduce by construction to prior outputs or self-citations; each introduces new analytical content (orthogonality investigation, spectrum overlap observation, explicit input-output confirmation) that is independently verifiable against the proposed waveform and channel model. The derivation remains self-contained without self-definitional loops or fitted quantities renamed as predictions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only view yields limited visibility; the design rests on standard signal-processing assumptions for LTV channels plus the paper-specific claim of approximate orthogonality under uniform equivalent-delay spacing.

axioms (1)
  • domain assumption Approximate orthogonality holds for HFMC subcarriers spaced uniformly in equivalent delay
    Invoked when establishing the basic principle of HFMC modulation.

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Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages · 1 internal anchor

  1. [1]

    Cross Evaluation of Waveform Modulation Schemes Using Postexperimental Field Data,

    Y . Xue, X. Qin, and Y . R. Zheng, “Cross Evaluation of Waveform Modulation Schemes Using Postexperimental Field Data,”IEEE J. Ocean. Eng., vol. 49, no. 4, pp. 1622–1633, 2024

    work page 2024

  2. [2]

    A Unifying View of OTFS and Its Many Variants,

    Q. Deng, Y . Ge, and Z. Ding, “A Unifying View of OTFS and Its Many Variants,”IEEE Commun. Surv. Tutor ., 2025, Early Access

    work page 2025

  3. [3]

    A Survey on Multicarrier Com- munications: Prototype Filters, Lattice Structures, and Implementation Aspects,

    A. Sahin, I. Guvenc, and H. Arslan, “A Survey on Multicarrier Com- munications: Prototype Filters, Lattice Structures, and Implementation Aspects,”IEEE Commun. Surv. Tutor ., vol. 16, no. 3, pp. 1312–1338, 2014

    work page 2014

  4. [4]

    On Secrecy Performance of RIS-Assisted MISO Systems Over Rician Channels With Spatially Random Eavesdroppers,

    W. Shi, J. Xu, W. Xu, C. Yuen, A. Lee Swindlehurst, and C. Zhao, “On Secrecy Performance of RIS-Assisted MISO Systems Over Rician Channels With Spatially Random Eavesdroppers,”IEEE Trans. Wireless Commun., vol. 23, no. 8, pp. 8357–8371, 2024

    work page 2024

  5. [5]

    Rate-Matching Framework for RSMA-Enabled Multibeam LEO Satellite Communications,

    J. Seong, J. Park, J. Lee, J. Lee, J.-B. Kim, W. Shin, and H. V . Poor, “Rate-Matching Framework for RSMA-Enabled Multibeam LEO Satellite Communications,”IEEE Trans. Signal Process., vol. 73, pp. 1426–1443, 2025

    work page 2025

  6. [6]

    Combating interference for over-the-air federated learning: A statistical approach via ris,

    W. Shi, J. Yao, W. Xu, J. Xu, X. You, Y . C. Eldar, and C. Zhao, “Combating interference for over-the-air federated learning: A statistical approach via ris,”IEEE Trans. Signal Process., vol. 73, pp. 936–953, 2025

    work page 2025

  7. [7]

    Closer Twins Model: Consistent Design of Modem Scheme and Channel Estima- tion Under High-Mobility Scenarios,

    H. Zhang, X. Wang, J. Tan, J. Wang, Z. Yang, and B. Ai, “Closer Twins Model: Consistent Design of Modem Scheme and Channel Estima- tion Under High-Mobility Scenarios,”IEEE Trans. Wireless Commun., vol. 24, no. 6, pp. 4564–4580, 2025

    work page 2025

  8. [8]

    High-Data-Rate Long-Range Underwater Communications via Acoustic Reconfigurable Intelligent Surfaces,

    Z. Sun, H. Guo, and I. F. Akyildiz, “High-Data-Rate Long-Range Underwater Communications via Acoustic Reconfigurable Intelligent Surfaces,”IEEE Commun. Mag., vol. 60, no. 10, pp. 96–102, 2022

    work page 2022

  9. [9]

    Designing Acoustic Reconfigurable Intelligent Surface for Underwater Communi- cations,

    H. Wang, Z. Sun, H. Guo, P. Wang, and I. F. Akyildiz, “Designing Acoustic Reconfigurable Intelligent Surface for Underwater Communi- cations,”IEEE Trans. Wireless Commun., vol. 22, no. 12, pp. 8934–8948, 2023

    work page 2023

  10. [10]

    Closed-Form Expression for BER of CP-OFDM Systems Over an Underwater Acoustic Channel With Statistical Characterization,

    X. Liu, G. Qiao, X. Qing, N. Zheng, and T. Li, “Closed-Form Expression for BER of CP-OFDM Systems Over an Underwater Acoustic Channel With Statistical Characterization,”IEEE Commun. lett., vol. 26, no. 8, pp. 1745–1749, 2022

    work page 2022

  11. [11]

    Delay-Doppler Domain Turbo Equalization for Single-Carrier Underwater Acoustic Communications,

    T. Zheng, C. He, L. Jing, and Q. Yan, “Delay-Doppler Domain Turbo Equalization for Single-Carrier Underwater Acoustic Communications,” IEEE J. Ocean. Eng., vol. 50, no. 2, pp. 1500–1517, 2025. 15

    work page 2025

  12. [12]

    Interference Burden in Wireless Communi- cations: A Comprehensive Survey From PHY Layer Perspective,

    A. Tusha and H. Arslan, “Interference Burden in Wireless Communi- cations: A Comprehensive Survey From PHY Layer Perspective,”IEEE Commun. Surv. Tutor ., vol. 27, no. 4, pp. 2204–2246, 2025

    work page 2025

  13. [13]

    Orthogo- nal Delay-Doppler Division Multiplexing Modulation With Tomlinson- Harashima Precoding,

    Y . Ma, A. Shafie, J. Yuan, G. Ma, Z. Zhong, and B. Ai, “Orthogo- nal Delay-Doppler Division Multiplexing Modulation With Tomlinson- Harashima Precoding,”IEEE Trans. Commun., vol. 73, no. 7, pp. 5391– 5407, 2025

    work page 2025

  14. [14]

    Orthogonal Time Frequency Space Mod- ulation,

    R. Hadani, S. Rakib, M. Tsatsanis, A. Monk, A. J. Goldsmith, A. F. Molisch, and R. Calderbank, “Orthogonal Time Frequency Space Mod- ulation,” inProc. IEEE Wireless Commun. Netw. Conf., 2017, pp. 1–6

    work page 2017

  15. [15]

    Interference Can- cellation and Iterative Detection for Orthogonal Time Frequency Space Modulation,

    P. Raviteja, K. T. Phan, Y . Hong, and E. Viterbo, “Interference Can- cellation and Iterative Detection for Orthogonal Time Frequency Space Modulation,”IEEE Trans. Wireless Commun., vol. 17, no. 10, pp. 6501– 6515, 2018

    work page 2018

  16. [16]

    Orthogonal Delay-Doppler Division Multiplexing Modulation,

    H. Lin and J. Yuan, “Orthogonal Delay-Doppler Division Multiplexing Modulation,”IEEE Trans. Wireless Commun., vol. 21, no. 12, pp. 11 024–11 037, 2022

    work page 2022

  17. [17]

    Multi-Carrier Modulation: An Evolution from Time-Frequency Domain to Delay-Doppler Domain

    H. Lin, J. Yuan, W. Yu, J. Wu, and L. Hanzo, “Multi-Carrier Modulation: An Evolution from Time-Frequency Domain to Delay-Doppler Domain,” 2023, [online]. Available at: https://arxiv.org/abs/2308.01802

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  18. [18]

    Affine Frequency Division Multiplexing for Next Generation Wireless Communications,

    A. Bemani, N. Ksairi, and M. Kountouris, “Affine Frequency Division Multiplexing for Next Generation Wireless Communications,”IEEE Trans. Wireless Commun., vol. 22, no. 11, pp. 8214–8229, 2023

    work page 2023

  19. [19]

    Affine Frequency Division Multiplexing With Index Modulation: Full Diver- sity Condition, Performance Analysis, and Low-Complexity Detection,

    Y . Tao, M. Wen, Y . Ge, J. Li, E. Basar, and N. Al-Dhahir, “Affine Frequency Division Multiplexing With Index Modulation: Full Diver- sity Condition, Performance Analysis, and Low-Complexity Detection,” IEEE J. Sel. Areas Commun., vol. 43, no. 4, pp. 1041–1055, 2025

    work page 2025

  20. [20]

    Achieving full multipath diversity by random constellation rotation: A theoretical perspective,

    X. Wang, J. Yuan, J. Wang, and K. Huang, “Achieving full multipath diversity by random constellation rotation: A theoretical perspective,” IEEE Trans. Signal Process., vol. 74, pp. 372–382, 2026

    work page 2026

  21. [21]

    Performance of Orthogonal Delay-Doppler Division Multiplexing Modulation With Imperfect Channel Estimation,

    K. Huang, M. Qiu, J. Tong, J. Yuan, and H. Lin, “Performance of Orthogonal Delay-Doppler Division Multiplexing Modulation With Imperfect Channel Estimation,”IEEE Trans. Commun., vol. 73, no. 5, pp. 3637–3654, 2025

    work page 2025

  22. [22]

    Error Performance of Rectangular Pulse- Shaped OTFS With Practical Receivers,

    C. Shen, J. Yuan, and H. Lin, “Error Performance of Rectangular Pulse- Shaped OTFS With Practical Receivers,”IEEE Wireless Commun. Lett., vol. 11, no. 12, pp. 2690–2694, 2022

    work page 2022

  23. [23]

    Orthogonal Delay-Doppler Division Multiplexing (ODDM) Over General Physical Channels,

    J. Tong, J. Yuan, H. Lin, and J. Xi, “Orthogonal Delay-Doppler Division Multiplexing (ODDM) Over General Physical Channels,”IEEE Trans. Commun., vol. 72, no. 12, pp. 7938–7953, 2024

    work page 2024

  24. [24]

    Interleave Fre- quency Division Multiplexing,

    Y . Chi, L. Liu, Y . Ge, X. Chen, Y . Li, and Z. Zhang, “Interleave Fre- quency Division Multiplexing,”IEEE Wireless Commun. Lett., vol. 13, no. 7, pp. 1963–1967, 2024

    work page 1963

  25. [25]

    A Low Complexity Channel Emulator for Underwater Acoustic Communications,

    C. Zhang, Y . Xie, D. Mishra, T. Pacino, J. Shao, B. Li, J. Yuan, P. Chen, and Y . Rong, “A Low Complexity Channel Emulator for Underwater Acoustic Communications,” in2023 OCEANS, 2023, pp. 1–8

    work page 2023

  26. [26]

    Iterative LMMSE-SIC Detector for DSE-Aware Underwater Acoustic OTFS Systems,

    X. Wang, X. Shi, J. Wang, and Z. Sun, “Iterative LMMSE-SIC Detector for DSE-Aware Underwater Acoustic OTFS Systems,”IEEE Trans. V eh. Technol., vol. 73, no. 7, pp. 9895–9910, 2024

    work page 2024

  27. [27]

    On the Doppler Squint Effect in OTFS Systems Over Doubly-Dispersive Channels: Modeling and Evaluation,

    X. Wang, X. Shi, J. Wang, and J. Song, “On the Doppler Squint Effect in OTFS Systems Over Doubly-Dispersive Channels: Modeling and Evaluation,”IEEE Trans. Wireless Commun., vol. 22, no. 12, pp. 8781– 8796, 2023

    work page 2023

  28. [28]

    A Deep Learning Receiver for Underwater Acoustic OTFS Communications With Doppler Squint Effect,

    Y . Zhang, Y . Wang, Y . Liu, L. Shi, and Y . Zang, “A Deep Learning Receiver for Underwater Acoustic OTFS Communications With Doppler Squint Effect,”IEEE Wireless Commun. Lett., vol. 14, no. 4, pp. 1179– 1183, 2025

    work page 2025

  29. [29]

    Orthogonal Delay Scale Space Modulation: A New Technique for Wideband Time-Varying Channels,

    A. K. P. and C. R. Murthy, “Orthogonal Delay Scale Space Modulation: A New Technique for Wideband Time-Varying Channels,”IEEE Trans. Signal Process., vol. 70, pp. 2625–2638, 2022

    work page 2022

  30. [30]

    A Hyperbolic Secant-Based Pulse for Enhanced FTN Signaling in 5G/6G Systems,

    Y . Berrouche, M. Kulhandjian, and H. Kulhandjian, “A Hyperbolic Secant-Based Pulse for Enhanced FTN Signaling in 5G/6G Systems,” IEEE Wireless Commun. Lett., vol. 15, pp. 380–384, 2026

    work page 2026

  31. [31]

    A Tutorial on Chirp Spread Spectrum Modulation for LoRaW AN: Basics and Key Advances,

    A. Maleki, H. H. Nguyen, E. Bedeer, and R. Barton, “A Tutorial on Chirp Spread Spectrum Modulation for LoRaW AN: Basics and Key Advances,”IEEE Open J. Commun. Soc., vol. 5, pp. 4578–4612, 2024

    work page 2024

  32. [32]

    Pulse Compression by Means of Linear-Period Mod- ulation,

    J. Kroszczynski, “Pulse Compression by Means of Linear-Period Mod- ulation,”Proc. IEEE, vol. 57, no. 7, pp. 1260–1266, 1969

    work page 1969

  33. [33]

    Hyperbolic frequency modulation for multiple users in underwater acoustic com- munications,

    M. Zhou, J. J. Zhang, and A. Papandreou-Suppappola, “Hyperbolic frequency modulation for multiple users in underwater acoustic com- munications,” inProc. IEEE Int. Conf. Acoust., Speech Signal Process. (ICASSP), 2014, pp. 3498–3502

    work page 2014

  34. [34]

    An underwater acoustic communications scheme with inherent scale diversity for multiple users,

    M. Zhou, J. J. Zhang, A. Papandreou-Suppappola, and T. M. Duman, “An underwater acoustic communications scheme with inherent scale diversity for multiple users,” in2013 OCEANS, 2013, pp. 1–4

    work page 2013

  35. [35]

    Multiple access acoustic communication in underwater mobile networks,

    C. Bernard, P.-J. Bouvet, A. Pottier, and P. Forjonel, “Multiple access acoustic communication in underwater mobile networks,” inProc. Underwater Commun. Netw. (UComms), 2021, pp. 1–4

    work page 2021

  36. [36]

    HFM spread spectrum modulation scheme in shallow water acoustic channels,

    L. Zhang, X. Xu, W. Feng, and Y . Chen, “HFM spread spectrum modulation scheme in shallow water acoustic channels,” in2012 Oceans, 2012, pp. 1–6

    work page 2012

  37. [37]

    Orthogonal Hyperbolic Frequency Division Multiplexing Modulation for Underwater Acoustic Communications,

    X. Wang, X. Shi, J. Tan, J. Wang, and Z. Sun, “Orthogonal Hyperbolic Frequency Division Multiplexing Modulation for Underwater Acoustic Communications,”IEEE Trans. Wireless Commun., vol. 24, no. 7, pp. 6180–6194, 2025

    work page 2025

  38. [38]

    J. D. Murray,Asymptotic Analysis. Springer New York, 1984

    work page 1984

  39. [39]

    Discrete time-scale character- ization of wideband time-varying systems,

    Y . Jiang and A. Papandreou-Suppappola, “Discrete time-scale character- ization of wideband time-varying systems,”IEEE Trans. Signal Process., vol. 54, no. 4, pp. 1364–1375, 2006

    work page 2006

  40. [40]

    Variable Bandwidth Multicarrier Communications: A New Waveform for the Delay-Scale Channel,

    A. K. P., C. R. Murthy, and P. Muralikrishna, “Variable Bandwidth Multicarrier Communications: A New Waveform for the Delay-Scale Channel,” inProc. IEEE 23rd Int. Workshop Signal Process. Adv. Wireless Commun. (SPA WC), 2022, pp. 1–5

    work page 2022

  41. [41]

    Sparse Channel Estimation for Multicarrier Underwater Acoustic Communication: From Subspace Methods to Compressed Sensing,

    C. R. Berger, S. Zhou, J. C. Preisig, and P. Willett, “Sparse Channel Estimation for Multicarrier Underwater Acoustic Communication: From Subspace Methods to Compressed Sensing,”IEEE Trans. Signal Pro- cess., vol. 58, no. 3, pp. 1708–1721, 2010

    work page 2010

  42. [42]

    Underwater Acoustic Communications Based on OTFS,

    X. Feng, H. Esmaiel, J. Wang, J. Qi, M. Zhou, Z. A. H. Qasem, H. Sun, and Y . Gu, “Underwater Acoustic Communications Based on OTFS,” in Proc. IEEE Int. Conf. Signal Process. (ICSP), vol. 1, 2020, pp. 439–444

    work page 2020

  43. [43]

    Model-Driven Deep Learning-Based Estimation for Underwater Acoustic Channels With Uncertain Sparsity,

    X. Feng, M. Zhou, J. Wang, H. Sun, G. Pan, and M. Wen, “Model-Driven Deep Learning-Based Estimation for Underwater Acoustic Channels With Uncertain Sparsity,”IEEE Trans. Wireless Commun., vol. 23, no. 6, pp. 5710–5725, 2024

    work page 2024

  44. [44]

    Channel State Information Pre- diction for Adaptive Underwater Acoustic Downlink OFDMA System: Deep Neural Networks Based Approach,

    L. Liu, L. Cai, L. Ma, and G. Qiao, “Channel State Information Pre- diction for Adaptive Underwater Acoustic Downlink OFDMA System: Deep Neural Networks Based Approach,”IEEE Trans. V eh. Technol., vol. 70, no. 9, pp. 9063–9076, 2021

    work page 2021

  45. [45]

    Shallow Underwater Acoustic Massive MIMO Communications,

    W. Wu, X. Gao, C. Sun, and G. Y . Li, “Shallow Underwater Acoustic Massive MIMO Communications,”IEEE Trans. Signal Process., vol. 69, pp. 1124–1139, 2021

    work page 2021

  46. [46]

    Tse and P

    D. Tse and P. Viswanath,Fundamentals of Wireless Communication. Cambridge University Press, 2005

    work page 2005

  47. [47]

    Multicarrier Communication Over Underwater Acoustic Channels With Nonuniform Doppler Shifts,

    B. Li, S. Zhou, M. Stojanovic, L. Freitag, and P. Willett, “Multicarrier Communication Over Underwater Acoustic Channels With Nonuniform Doppler Shifts,”IEEE J. Ocean. Eng., vol. 33, no. 2, pp. 198–209, 2008

    work page 2008

  48. [48]

    J. M. Wozencraft and I. M. Jacobs,Principles of Communication Engineering. Cambridge University Press, 1965

    work page 1965

  49. [49]

    Real Signal DHT-OFDM With Index Modulation for Underwater Acoustic Communication,

    Z. A. H. Qasem, J. Wang, H. A. Leftah, H. Sun, S. Hong, J. Qi, and H. Esmaiel, “Real Signal DHT-OFDM With Index Modulation for Underwater Acoustic Communication,”IEEE J. Ocean. Eng., vol. 48, no. 1, pp. 246–259, 2023

    work page 2023

  50. [50]

    Zhou and Z

    S. Zhou and Z. Wang,OFDM for Underwater Acoustic Communica- tions. John Wiley & Sons, 2014

    work page 2014

  51. [51]

    On the Characterization and Evaluation of Doppler Squint in Wideband ODDM Systems,

    X. Wang, J. Yuan, J. Wang, and Z. Sun, “On the Characterization and Evaluation of Doppler Squint in Wideband ODDM Systems,” 2025, [online]. Available at: https://arxiv.org/abs/2508.01283

    work page arXiv 2025

  52. [52]

    Performance Analysis of Multi-User OTFS, OTSM, and Single Carrier in Uplink,

    B. V . S. Reddy, C. Velampalli, and S. S. Das, “Performance Analysis of Multi-User OTFS, OTSM, and Single Carrier in Uplink,”IEEE Trans. Commun., vol. 72, no. 3, pp. 1428–1443, 2024

    work page 2024

  53. [53]

    Low-Complexity LMMSE Receiver Design for Practical-Pulse-Shaped MIMO-OTFS Systems,

    P. Singh, S. Tiwari, and R. Budhiraja, “Low-Complexity LMMSE Receiver Design for Practical-Pulse-Shaped MIMO-OTFS Systems,” IEEE Trans. Commun., vol. 70, no. 12, pp. 8383–8399, 2022

    work page 2022

  54. [54]

    MF-OAMP-Based Joint Channel Estimation and Data Detection for OTFS Systems,

    H. Wen, W. Yuan, C. Yuen, and Y . Li, “MF-OAMP-Based Joint Channel Estimation and Data Detection for OTFS Systems,”IEEE Trans. V eh. Technol., vol. 73, no. 2, pp. 2948–2953, 2024

    work page 2024

  55. [55]

    An Inter-Doppler Interference- Constrained OC-DFE for OTFS in UW A Communications,

    Y . Yang, L. Ma, and B.-C. Seet, “An Inter-Doppler Interference- Constrained OC-DFE for OTFS in UW A Communications,”IEEE Wire- less Commun. Lett., vol. 15, pp. 290–294, 2026

    work page 2026