Momentum-Accelerated Phase Synchronization for UAV Swarm Collaborative Beamforming

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors' team presents a decentralised phase-synchronisation algorithm (with a momentum-accelerated update and position-based delay compensation) that enables a UAV swarm to quickly align carrier phases to perform coherent collaborative beamforming/jamming toward a target.

 

The topic concerns distributed jamming and coherent swarming radios, using a combination of Kuramoto-style coupling and Metropolis-Hastings weights. This combination creates interesting insight.

 

The research problem around sparse connectivity and delay bias is stated correctly. The author's effort is to connect the synchronisation error to beamforming efficiency through a Ruze-style analysis and volumetric field simulations.

 

The introduction gives a general overview of counter-UAS motivations, distributed beamforming, and the closed-loop versus open-loop synchronisation landscape. It could be written in a way that better positions the proposed algorithm relative to the most directly comparable open-loop carrier synchronisation and distributed phased-array demonstrations, explaining what is truly new beyond adding momentum to a standard consensus update. Some of the ideas presented in the manuscript, i.e., about practical jamming deployments, resilience in GNSS-denied environments, and rapid EW response, need better referencing. Also, a clearer difference between what the simulation presents and what is assumed would make the introduction more effective.

 

The research design follows the development of an algorithm, and the analysis of its stability in a small-signal regime is validated with Monte Carlo simulations on random geometric graphs. However, some of the physical assumptions, such as pure free-space LOS propagation, perfect beam pointing, and the absence of mobility-induced Doppler or time-varying topology during synchronisation, are "optimistic". There is also an inconsistency in the evaluation setup: early sections use a uniform circular array as a standardised baseline, while later convergence experiments use random 3D node placements and sparse graphs, and the transition between these two cases is not always explained. Also, there is some modelling concern - the mix of frequencies and error budgets. The antenna example is at 150 MHz, while the delay discussion references X/Ku bands; the ranging-error tolerances are interpreted generically without showing how the same centimetre error maps to phase error across bands.

 

Methods are described with a solid level of mathematical detail, including notation, update laws, and the definition of the synchronisation error metric used for stopping. The parameter selection for ε and β, along with a consolidated simulation-parameter table (node density, communication radius, iteration time step, carrier frequency, and noise levels), should be included to make the method section more self-explanatory. Also, authors should more explicitly acknowledge that global convergence on S¹ can still depend on initialisation and graph structure. The propagation-delay model and the idea of compensating phase lag using relative ranging are the strengths of the manuscript. The derivation of how distance noise translates into a phase residual is straightforward and useful for the reader.

 

Results are presented clearly and backed by phase-trajectory plots, error curves, and box plots. The beam-spot evolution figure relates algorithm behaviour to physical focusing.

 

Some figures need polishing and would benefit from consistent axis scaling and a clearer separation of 'single run' illustrations from statistical summaries. Also, the iteration counts are reported without mapping to wall-clock time or communication load. For this reason, it is hard to tell whether the acceleration translates into a meaningful real-time advantage given typical UAV link rates and update periods.

 

Simulation results support the conclusions of faster convergence and the necessity of delay compensation. Some of the claims presented in the manuscript, like those about field deployability, scalability in contested environments, and near-ideal coherent combining, should be softened or paired with at least a hardware-in-the-loop plan and a more conservative discussion of oscillator noise, mobility, and synchronisation messaging overhead.

 

The English language and style are correct.

 

There are minor style issues (occasional repetition, abrupt line breaks, inconsistent symbol formatting, and a few typos) that should be fixed.

 

The reference list is largely relevant to distributed beamforming, consensus/synchronisation, and time transfer.

 

From the formal point of view, the manuscript is well written, divided in logical order, the chapters are of adequate length, and it is self-explanatory and readable with ordinary effort. It also fits the journal scope. However, it needs tighter alignment between the physical scenario, the simulation assumptions, and the claims made in the abstract and conclusion.

 

Suggestions for authors:

1. Align the carrier-frequency assumptions and error budgets (ranging, oscillator phase noise, propagation delay) and extend the sensitivity analysis so that the 'centimetre-level is enough' statement from the manuscript is demonstrated at the intended operating bands.
2. Adding a discussion on how the algorithm behaves under time-varying graphs and delayed/asynchronous updates would help readers judge robustness in real swarms.
3. Add more competitive baselines (e.g., other accelerated consensus schemes or recent decentralised carrier-sync methods), report communication/compute costs, and expand the limitations section to cover mobility, Doppler, and non-LOS channels.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors This paper proposes a momentum-accelerated synchronization strategy with a position-based pre-compensation mechanism for UAV swarm collaborative beamforming. Generally, the research framework of this paper is complete, but some comments need to be addressed.

1. The parameter optimization process and the impact of different parameter combinations on algorithm performance need to be further supplemented.
2. The applicable range and approximation error of the small-signal approximation used in the algorithm stability analysis are not explained.
3. The authors only consider simple Gaussian ranging noises and random initial phase distribution, which may be inconsistent with actual engineering scenarios.
4. It would be better if the authors provide the computational complexity analysis of the proposed algorithm and the comparison of computational overhead with traditional algorithms.
5. There are some minor grammatical errors and redundant expressions.
6. The reference format needs to be checked and standardized in accordance with journal requirements.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The paper presents a physics-aware momentum-accelerated synchronization strategy aimed at reducing convergence time and ensuring phase synchronization in UAV swarm collaborative beamforming, especially for electronic jamming tasks. However, there are some issues in the current version, to be considered for potential publication, the authors should address the questions raised below.

1. Although the paper compares the traditional Metropolis-Hastings (MH) method and the fixed-weight method, it does not sufficiently compare with other state-of-the-art synchronization methods. It is recommended to expand the comparison with existing methods, especially those focused on synchronization for UAV swarms, in order to better highlight the advantages of the proposed approach.
2. It is recommended to further investigate the computational and communication complexity of the algorithm in large-scale UAV swarms, especially regarding its feasibility in practical hardware implementations.
3. The position-based phase compensation mechanism proposed in the paper is effective, but there is insufficient discussion on handling positioning errors in extreme cases. It would be beneficial to analyze the impact of positioning errors on synchronization accuracy, especially in large-scale dynamic environments with larger errors.
4. The paper outlines future directions but lacks a detailed discussion on the challenges of practical deployment. Future work should address the integration with existing UAV communication systems, especially in complex urban environments.
5. The effect of increasing swarm sizeon convergence performance is outlined, but a deeper explanation of how node connectivity, represented through the algebraic connectivity, directly correlates with synchronization speed would be helpful. Can this relationship be quantified more clearly with additional examples?
6. The introduction could be strengthened by briefly addressing the core challenges faced in the engineering implementation of multi-node beamforming systems, such as random hovering-induced position disturbance and real-time high-precision positioning under angle of arrival (AoA) measurement noise. For instance, the paper “Quantum-position-locked loop: New concept for collaborative beam forming for UAV swarm” discusses the position disturbance and real-time beam-reforming challenges that arise during the engineering implementation of collaborative beamforming systems—issues directly relevant to multi-node beamforming in this study.Similarly, “AAV air-to-air channel: statistical properties and experimental verification” quantifies the effects of airframe occlusion, rooftop specular reflection, and UAV mobility/attitude on factors like channel capacity, outage probability, and root mean square delay spread. These channel characteristics form a critical theoretical basis and provide situational context for designing synchronization algorithms and evaluating beamforming performance in this research. Incorporating such context would clarify the motivation behind this research and strengthen its practical relevance.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Dear authors, thank you for providing me with the updated version of your manuscript and also for the cover letter. After careful proofreading, I can assume that the concerning parts were updated.

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have addressed all my questions.

Comments on the Quality of English Language

should be improved.

Reviewer 3 Report

Comments and Suggestions for Authors

The author has addressed all my concerns, and I have no further questions.