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arxiv: 2604.21819 · v1 · submitted 2026-04-23 · 💻 cs.NI

Iterative Receiver Processing at Relays in PNC-Enabled Multi-Hop Underwater Acoustic Networks

Gewei Zhang , Deqing Wang , Lizhao You , Xiangming Cai , Liqun Fu This is my paper

Pith reviewed 2026-05-08 13:41 UTC · model grok-4.3

classification 💻 cs.NI
keywords physical layer network codingunderwater acoustic networksiterative detection and decodingOFDMmulti-hop networksrelay nodesbit error rate performance
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The pith

Iterative receiver processing at relays improves BER performance in PNC-enabled multi-hop underwater acoustic networks

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

The paper develops an iterative detection and decoding strategy for relay nodes to support physical-layer network coding in multi-hop underwater acoustic networks that use OFDM. It addresses error accumulation and channel impairments like multipath and Doppler effects that degrade reliability over multiple hops. The strategy combines an adaptive factor graph detector suited to time-varying channels, a refinement algorithm using parity checks for better soft information, and a low-complexity LMMSE detector. Simulations and real lake and sea experiments show these methods outperform baselines, reaching low bit error rates even at moderate speeds and SNRs. A reader would care because this could make multi-hop underwater communication more dependable for applications needing extended range.

Core claim

The authors introduce iterative receiver algorithms for relays in PNC-enabled multi-hop UWA networks. These integrate an adaptive channel-aware factor graph detection algorithm for time-varying channels, a parity-check-constrained soft-information refinement algorithm, and an LMMSE detection algorithm on a superimposed model. The adaptive method achieves BERs around 10^{-5} at 1.5 m/s relative velocity and 8 dB SNR in simulations, while lake experiments and sea trials in the Taiwan Strait demonstrate superior BER performance compared to baseline schemes under practical conditions.

What carries the argument

The iterative receiver processing strategy that integrates adaptive channel-aware factor graph detection, parity-check-constrained soft-information refinement, and LMMSE detection based on a superimposed model for handling PNC at relays.

If this is right

  • The adaptive detection algorithm can achieve bit error rates on the order of 10^{-5} at a relative velocity of 1.5 m/s and 8 dB SNR.
  • Error accumulation across multiple relay nodes is reduced, enabling more reliable end-to-end transmission.
  • The LMMSE-based scheme offers a low-complexity alternative while maintaining performance gains.
  • The algorithms prove robust in real-world underwater acoustic channels as shown by lake and sea trial results.

Where Pith is reading between the lines

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

  • This strategy could support longer multi-hop chains in underwater networks by further limiting error propagation.
  • Similar iterative approaches might improve performance in other high-mobility or multipath wireless systems.
  • Optimizing the algorithms for lower power consumption could enhance their use in battery-limited underwater devices.

Load-bearing premise

The channel estimation and models used accurately represent the time-varying UWA channels, and the iterative processing converges reliably under practical conditions.

What would settle it

A field experiment in the sea at higher relative velocities or lower SNRs where the BER fails to reach the order of 10^{-5} would disprove the performance claims.

Figures

Figures reproduced from arXiv: 2604.21819 by Deqing Wang, Gewei Zhang, Liqun Fu, Lizhao You, Xiangming Cai.

Figure 2
Figure 2. Figure 2: A bi-directional multi-hop network using PNC with 3 relay nodes. view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of relay reception and processing at the relay in multi-hop UWA PNC networks: the uplink and the downlink node exchange information view at source ↗
Figure 4
Figure 4. Figure 4: The structure of iterative receiver, which primarily includes three modules: signal detection, channel decoding and soft-information refinement. After view at source ↗
Figure 5
Figure 5. Figure 5: Adaptive channel-aware factor graph signal detection at the view at source ↗
Figure 6
Figure 6. Figure 6: A Tanner graph of an LDPC code, illustrating the relationships view at source ↗
Figure 7
Figure 7. Figure 7: UWA CIRs with fixed deviated speed σu which correspond to fixed Doppler factors of ap,u = 6.77 × 10−5 and 1.00 × 10−3 . (27), we define the coefficient matrix of the linear method as C = RxyR−1 yy. Step 5 Conditional Probability with Mean and Variance: To derive the soft input soft output solution, xˆ is modeled as a Gaussian random variable conditioned on the superimposed symbol L = LA + LB. Consequently,… view at source ↗
Figure 8
Figure 8. Figure 8: Performance comparison of different iterative receiver schemes with a number of outer iterations and multi-hop transmission in UWA channels. view at source ↗
Figure 9
Figure 9. Figure 9: Performance of different ICI depths in the UWA channels. To distinguish ICI depths strategies, we denote the proposed method as IACA, while the traditional fixed ICI depth approach is defined as Fixed-D. The thresholds for IACA-FGD are marked as 60%, 70%, 80% and 90%. performance, reaching a BER close to 10−5 at an SNR of 8 dB. Considering that non-iterative cases of all schemes tend to saturate around a B… view at source ↗
Figure 10
Figure 10. Figure 10: Performance comparison of soft-information refinement in the view at source ↗
Figure 11
Figure 11. Figure 11: Performance comparison of different CERs in the UWA channels. System simulation is conducted under CER view at source ↗
Figure 12
Figure 12. Figure 12: UWA channel measurement scenario and real-world deployment. view at source ↗
Figure 13
Figure 13. Figure 13: UWA channel measurements conducted in the Furong Lake and the Taiwan Strait, where each scenario includes the CIRs, Doppler spectrum, and view at source ↗
Figure 14
Figure 14. Figure 14: Experiment performance comparison in the UWA channels measured from the Furong Lake and the Taiwan Strait. view at source ↗
read the original abstract

Physical-layer network coding (PNC) can increase end-to-end throughput in bi-directional multi-hop underwater acoustic (UWA) networks. However, multipath delay spread and Doppler-induced inter-carrier interference (ICI) in UWA channels can degrade the reliability of PNC transmission in a three-node relay configuration. More critically, error accumulation across multiple relay nodes leads to a pronounced increase in the end-to-end bit error rate (BER) in multi-hop networks. To address this issue, we develop an iterative detection and decoding processing strategy for relay nodes within a PNC-enabled multi-hop UWA network based on orthogonal frequency division multiplexing (OFDM) modulation. The proposed design integrates three key algorithms: (i) an adaptive channel-aware factor graph detection algorithm that is suited for time-varying UWA channels; (ii) a parity-check-constrained soft-information refinement algorithm that improves the accuracy of the information feedback from the decoder to the detector; and (iii) a linear minimum mean square error (LMMSE) detection algorithm based on a superimposed model, which offers low computational complexity as an alternative scheme. Extensive simulation results demonstrate that the adaptive detection algorithm achieves BERs on the order of $10^{-5}$ at a relative velocity of 1.5 m/s UWA channel and a signal-to-noise (SNR) of 8~dB. Both lake experiments and sea trials in the Taiwan Strait confirm that the proposed iterative receiver algorithms outperform baseline schemes in terms of BER performance under practical UWA channel conditions, showing their robustness and applicability in real multi-hop deployments.

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 paper proposes an iterative detection and decoding strategy for relay nodes in physical-layer network coding (PNC) enabled multi-hop underwater acoustic (UWA) networks using OFDM modulation. It introduces three algorithms: (i) an adaptive channel-aware factor graph detector suited to time-varying channels, (ii) a parity-check-constrained soft-information refinement algorithm, and (iii) an LMMSE detector based on a superimposed model for lower complexity. Simulations claim BERs on the order of 10^{-5} at 1.5 m/s relative velocity and 8 dB SNR for the adaptive detector, while lake experiments and sea trials in the Taiwan Strait are reported to show outperformance over baseline schemes under practical UWA conditions, addressing error accumulation across hops.

Significance. If the central claims hold, the work addresses a practically important problem in UWA communications by mitigating error propagation in multi-hop PNC setups, which could improve end-to-end reliability and throughput in challenging environments with multipath and Doppler effects. The combination of simulation results with lake and sea trial data provides external validation beyond self-referential fitting, strengthening potential applicability, though the absence of detailed convergence and estimation-error metrics limits the strength of the robustness claims.

major comments (2)
  1. [Abstract and experimental results sections] The headline performance claims (BER ~10^{-5} at 1.5 m/s and 8 dB SNR, plus experimental outperformance) rest on the assumptions that the adaptive factor-graph detector plus LMMSE superimposed model accurately tracks Doppler/multipath-induced ICI without mismatch and that the iterative loop converges reliably to avoid error accumulation across hops. The manuscript provides end-to-end BER results but lacks quantitative checks such as convergence curves, channel-estimation MSE versus ground truth, or multi-hop error-propagation measurements in the experimental sections, which are load-bearing for confirming no model mismatch under Taiwan Strait conditions.
  2. [Algorithm description (parity-check refinement)] The description of the parity-check-constrained soft-information refinement algorithm does not include analysis or bounds on how feedback accuracy improves across iterations in the presence of residual ICI, which is central to the claim that the overall iterative receiver prevents pronounced BER increase in multi-hop settings.
minor comments (2)
  1. [Abstract] The abstract states 'multi-hop' but does not specify the number of hops used in simulations or experiments; this should be clarified for reproducibility.
  2. [LMMSE algorithm subsection] Notation for the superimposed model in the LMMSE detector could be made more explicit with a dedicated equation reference to distinguish it from the factor-graph approach.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive comments, which help clarify the validation of our iterative receiver design. We address each major comment below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Abstract and experimental results sections] The headline performance claims (BER ~10^{-5} at 1.5 m/s and 8 dB SNR, plus experimental outperformance) rest on the assumptions that the adaptive factor-graph detector plus LMMSE superimposed model accurately tracks Doppler/multipath-induced ICI without mismatch and that the iterative loop converges reliably to avoid error accumulation across hops. The manuscript provides end-to-end BER results but lacks quantitative checks such as convergence curves, channel-estimation MSE versus ground truth, or multi-hop error-propagation measurements in the experimental sections, which are load-bearing for confirming no model mismatch under Taiwan Strait conditions.

    Authors: We agree that explicit convergence curves and channel-estimation error metrics would provide stronger support for the claims regarding reliable iteration and ICI tracking. The reported end-to-end BER results in multi-hop simulations and field trials already show that the proposed algorithms prevent the expected BER increase from error propagation, with the adaptive detector maintaining performance at the stated operating points. In the revised manuscript we will add simulation convergence plots illustrating BER reduction over iterations. For the lake and sea trial sections we will expand the discussion of hop-by-hop BER stability as indirect evidence of convergence under real conditions, while noting that ground-truth channel estimates are unavailable in field deployments. We will also add a short clarification on the modeling assumptions for Doppler-induced ICI. revision: partial

  2. Referee: [Algorithm description (parity-check refinement)] The description of the parity-check-constrained soft-information refinement algorithm does not include analysis or bounds on how feedback accuracy improves across iterations in the presence of residual ICI, which is central to the claim that the overall iterative receiver prevents pronounced BER increase in multi-hop settings.

    Authors: The parity-check refinement exploits the LDPC code structure to correct inconsistent soft values before feeding them back to the factor-graph detector, as described in the algorithm section. While the manuscript does not supply theoretical bounds on accuracy gain under residual ICI, the iterative improvement is validated by the simulation BER curves and the multi-hop experimental results. We will revise the description to include a more explicit step-by-step explanation of how parity checks suppress residual interference effects, accompanied by additional simulation results showing the evolution of soft-information quality across iterations. revision: partial

standing simulated objections not resolved
  • Derivation of theoretical bounds on the improvement of feedback accuracy for the parity-check refinement in the presence of residual ICI, as providing such bounds would require a separate analytical study outside the scope of the present work.

Circularity Check

0 steps flagged

No significant circularity; performance claims rest on external simulations and trials

full rationale

The paper presents three iterative receiver algorithms (adaptive factor-graph detector, parity-check refinement, LMMSE superimposed model) as explicit design choices for handling time-varying UWA channels in PNC multi-hop OFDM setups. BER results (order 10^{-5} at 1.5 m/s and 8 dB SNR) and outperformance claims are obtained from independent Monte-Carlo simulations plus lake/sea trial measurements; these are not obtained by fitting parameters to the target metric and then renaming the fit as a prediction. No self-citation chain, uniqueness theorem, or ansatz-smuggling is invoked to justify the central claims. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the effectiveness of the three proposed algorithms within established communication frameworks, with no new physical entities or ad-hoc constants introduced beyond standard parameters like SNR and velocity.

axioms (1)
  • domain assumption OFDM modulation and PNC principles apply to UWA channels with multipath and Doppler effects
    The work assumes standard models for underwater acoustic propagation and network coding.

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