{"id":"36155355-1990-4d92-93db-23438331d60d","arxiv_id":"2506.11851","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A robust statistical-CSI beamforming design for satellite-to-terrestrial interference mitigation in integrated terrestrial-satellite networks, with a position-aided approximation of the interference integral.","lead":"This paper designs satellite transmit beamforming to limit interference to terrestrial users in shared-spectrum networks, using only statistical channel information and base station positions instead of real-time channel sharing. It presents iterative and closed-form beamforming schemes plus an approximation that avoids computing a complex interference integral.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3's claimed monotone decrease of I_sg with the penalty factor ς is false in a valid two-antenna instance, so Algorithm 3's bisection is not guaranteed to enforce the interference threshold.","rationale":"The reader's stated weakest assumption is the accuracy of the user-distribution model and the position-aided approximation. That is a legitimate practical limitation, but it is deployment-dependent and could in principle be mitigated by more robust modeling or estimation. The stronger, internal issue is Proposition 3: Algorithm 3's guarantee depends exactly on I_sg(ς) being monotone non-increasing. The appendix proof is invalid, and a valid two-antenna UPA instance of the paper's own statistical model shows the interference increasing with ς. This does not overturn the WQTIA or WWEIA schemes, which optimize with the constraint directly, but it invalidates the advertised closed-form MMSE solution and the claim that the interference threshold can be enforced by bisection. Since no code or data are supplied, the numerical simulations cannot compensate for the false monotonicity claim. The verdict should therefore move from CONDITIONAL to REJECT unless the monotonicity is re-proven under additional stated constraints that exclude the constructed instance.","tokens_in":21267,"tokens_out":18105,"duration_ms":178035,"concrete_test":"Evaluate I_sg(ς)=Tr(P(ς)^H Υsg P(ς))/K_G from (33)-(34) for the constructed two-antenna instance: arΥss=diag(1,100), Υss=diag(2,200), Υsg=diag(1,2), ε=1, at ς=0, 10, and 50. If the values increase rather than decrease, Proposition 3 is false and the bisection in Algorithm 3 is not guaranteed to satisfy the interference threshold. Independently re-derive ∇_ς I_sg from (33)-(34) without the trace inequality used in Appendix C; the sign of the derivative is not determined by Tr{MN}≤Tr{M}Tr{N} when M and N are indefinite.","verdict_should_be":"REJECT","load_bearing_attack":"The low-complexity MMSEIA scheme meets I_thr only if I_sg(ς) is monotone non-increasing, as asserted in Proposition 3 and used by the bisection in Algorithm 3. The proof in Appendix C is not valid: the matrices called Hermitian are not Hermitian in general, and the trace inequalities do not bound the sign of ∇_ς I_sg. More importantly, the statement is false. Take M_S=2 with orthonormal UPA/ULA steering vectors v1=[1,1]^T/√2 (ϑ=0) and v2=[1,-1]^T/√2 (ϑ=1), K_S=2 satellite UTs along v1,v2, and two terrestrial interfered UTs along the same directions. In the {v1,v2} basis, with Rician factor q=1 and ε=K_Sσ_s²/P_T=1, set arΥss=diag(1,100), Υss=diag(2,200), and Υsg=diag(1,2). These are of the required statistical form. Using (33)-(34), I_sg(0)/P_T ≈ 1.02, I_sg(10)/P_T ≈ 1.26, and I_sg(50)/P_T ≈ 1.76: the average interference rises as ς grows, opposite to Proposition 3. Thus the closed-form bisection claim can fail in a valid instance, and this is a load-bearing flaw in the central claim, not a mere proof gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies downlink transmit beamforming from a LEO satellite to satellite user terminals in a spectrum-sharing integrated terrestrial and satellite network, using only statistical CSI and without sharing terrestrial CSI. It models the satellite-to-terrestrial interference as an integral over the terrestrial user distribution, formulates a weighted-sum-rate maximization with an interference threshold and power budget, solves it by a multidimensional complex quadratic transform and by a WMMSE-based iteration, and derives a closed-form MMSE beamformer whose penalty coefficient is tuned by bisection. It also proposes a terrestrial-base-station-position-aided approximation of the integral interference term and analyzes the approximation error for a single BS at the sub-satellite point. Numerical simulations compare the proposed schemes with conventional beamformers.","tokens_in":21537,"tokens_out":13210,"duration_ms":131221,"significance":"The problem is timely, and the sCSI-only interference integral formulation is a useful starting point for interference management without cross-system CSI. The paper is not circular: the penalty coefficient is chosen algorithmically, and no parameters are fitted to results. The WSR/WMMSE iterations and the position-aided approximation are potentially valuable, and the complexity tables make the computational claims concrete. However, the closed-form MMSEIA scheme currently rests on a false monotonicity proposition, so the paper's most distinctive practical claim is not established.","major_comments":[{"comment":"The proof that ∇ς I_sg(ς) ≤ 0 is invalid, and the statement is false. The matrices M and N defined after Eq. (51) are not Hermitian in general, so the von Neumann trace inequality cannot be applied; the subsequent inequality Tr{MN} ≤ Tr{M}Tr{N} is not valid for Hermitian matrices with indefinite eigenvalues. More importantly, the claimed monotonicity fails in a concrete instance of the model: with M_S=2, orthonormal steering vectors v1=[1,1]^T/√2 and v2=[1,-1]^T/√2, K_S=2, Rician factor κ=1, K_Sσ_s²/P_T=1, and in the {v1,v2} basis Υ_ss=diag(2,200), arΥ_ss=diag(1,100), Υ_sg=diag(1,2), the updates (33)–(34) give I_sg(0)/P_T≈1.02, I_sg(10)/P_T≈1.26, and I_sg(50)/P_T≈1.76. Thus I_sg increases with ς, opposite to Proposition 3.","section":"Appendix C / Proposition 3 / Eq. (35)"},{"comment":"Because Proposition 3 is false, the bisection method in Algorithm 3 is not guaranteed to find a penalty coefficient that satisfies I_sg(ς) ≤ I_thr. In the counterexample above, bisection would be applied to a non-monotone function and can converge to a value with interference above the threshold. The paper should either replace the bisection with a provably correct procedure, such as a constrained convex solve of the MMSE problem, or add explicit conditions on Υ_ss, arΥ_ss, and Υ_sg under which monotonicity holds and can be verified.","section":"Algorithm 3 / Section IV-B"},{"comment":"The proof establishes only a stationary point of the Lagrangian of problem (32), not global optimality. Problem (32) is not jointly convex in (P,β) because of the 1/β² and 1/β terms, and the reduction to a scalar minimization in ζ does not show convexity of f(ζ). The statement that the given solution 'achieves the optimum' is therefore an overclaim; at most a locally optimal or heuristic closed-form solution is obtained. The authors should either prove global optimality under explicit conditions or reword the claim.","section":"Appendix B / Proposition 2 / Eqs. (32)–(34)"},{"comment":"The error analysis covers only a single terrestrial BS located at the sub-satellite point under uniform user density. For the general multi-BS geometry used in the Section VI simulations, no expression or bound for the approximation error is provided, so the claim that the PA schemes nearly meet the interference threshold is supported only by simulation. Please either extend the analysis to the general geometry or clearly state this as a limitation of the theoretical guarantees.","section":"Section V / Proposition 4 / Eq. (39)"}],"minor_comments":[{"comment":"The symbol 'Rbs' appears as a subscript in several places; use R_bs consistently with the equation formatting.","section":"Section II, Eq. (10)"},{"comment":"Step 2 should specify the initial interval for ς, the bisection tolerance, and what to do if no feasible ς exists; currently the procedure is described only in words.","section":"Algorithm 3"},{"comment":"The caption contains the typo 'convergernce'; it should be 'convergence'.","section":"Section VI, Fig. 7"},{"comment":"The rows 'Noise Figure F 9 dB' and 'Noise Temperature T 290 K' are ambiguous; clarify whether T is the system noise temperature or the reference temperature in the SNR formula, and how F and T enter the noise power calculation.","section":"Table I"},{"comment":"The sentence describing the complexity of Step 6 is grammatically ambiguous and should be split; also, define N_r and N_ϕ before first use.","section":"Section III-B"}],"recommendation":"major_revision","confidential_remarks":"The main concern is whether the authors can repair Proposition 3 and Algorithm 3. The counterexample indicates a genuine correctness failure in the closed-form MMSEIA scheme, not merely a missing proof step. If the monotonicity cannot be restored under useful conditions, the MMSEIA scheme should be presented as a heuristic without a threshold guarantee, or replaced. The WSR/WMMSE iterative schemes and the position-aided approximation may still be salvageable, but the manuscript needs substantial revision before it can be considered for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the genuinely new piece is the CSI-free integral-form interference model (Eq. 13) and the terrestrial BS position-aided approximation with its error analysis. That is worth reading. The WSR-MCQT and WMMSE iterative designs are standard machinery applied carefully to that model, and the simulations suggest they behave as advertised. The closed-form MMSE scheme, however, has a load-bearing flaw.\n\nProposition 3 claims I_sg is non-increasing in the penalty factor ς, and Algorithm 3 bisects on that monotonicity. The proof in Appendix C does not work: the matrices called Hermitian are not Hermitian in general, and the trace inequalities do not give the claimed sign. This is not just a cosmetic proof gap. I checked the stress-test counterexample: with M_S=2, K_S=2, K_G=2, orthonormal steering directions, \\barΥss=diag(1,100), Υss=diag(2,200), Υsg=diag(1,2), and ε=1, Eqs. (33)-(34) give I_sg/P_T ≈ 1.02 at ς=0, ≈ 1.26 at ς=10, and ≈ 1.76 at ς=50. The average interference increases as the penalty grows, opposite to Proposition 3. So the bisection in Algorithm 3 can converge to the wrong ς or miss the threshold entirely. The paper's simulations meet the threshold for their chosen parameters, but the claimed guarantee is false.\n\nSecondary issues: the model switches between f(r,ϕ) as a probability density and as a user density, which leaves a normalization ambiguity in Eq. (10). The position-aided approximation is analyzed only for one BS at the sub-satellite point with uniform user density, so the evidence for realistic cell layouts is thinner than the prose suggests. No code or data is provided. The citation pattern looks normal for this subfield.\n\nWhat survives: Proposition 1 and the derivation of the closed-form MMSE solution are fine; the WQTIA and WWEIA schemes do not depend on the false monotonicity claim; and the integral model plus PA approximation are a sensible way to avoid shared CSI. The paper should be revised to remove or repair the MMSEIA bisection claim, or to add a feasibility check before declaring the threshold satisfied.\n\nI would send this to peer review, because the core modeling idea is useful and the flaw is specific and fixable. I would not cite or build on the MMSEIA bisection result until it is corrected.","headline":"Useful CSI-free interference model and two sound iterative schemes, but Proposition 3 is false, so the closed-form MMSE bisection is not guaranteed to enforce the interference threshold.","tokens_in":22112,"tokens_out":4689,"would_cite":false,"duration_ms":45093,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Robust satellite beamforming protects terrestrial users without shared CSI.","keywords":["integrated terrestrial and satellite networks","spectrum sharing","transmit beamforming","statistical CSI","interference mitigation","MMSE beamforming","weighted sum rate maximization","position-aided approximation"],"falsifier":"Run a Monte Carlo simulation of a multi-cell deployment with base stations away from the sub-satellite point and non-uniform user density, compute $\\Upsilon^{\\mathrm{int}}_{sg}$ from the true user positions, then evaluate $\\frac{1}{K_G}\\mathrm{Tr}(P^H \\Upsilon^{\\mathrm{int}}_{sg} P)$ for the beamformer produced with the position-aided $\\tilde{\\Upsilon}_{sg}$; if this interference exceeds $I_{\\mathrm{thr}}$ by more than a small numerical tolerance in a realistic geometry, the paper's approximation guarantee fails.","tokens_in":21035,"feed_emoji":"📡","tokens_out":5937,"duration_ms":54523,"temperature":0.7,"pith_summary":"Satellite links sharing sub-6 GHz spectrum with terrestrial networks can be shaped so that they respect a terrestrial interference limit using no instantaneous channel information from the terrestrial side. The paper builds this on a statistical interference model: the satellite-to-terrestrial channel covariance is written as an integral over the terrestrial user distribution and base-station geometry, so only slowly varying statistics and known locations are needed. Three beamformers are derived: an iterative weighted-sum-rate design, a lower-complexity equivalent WMMSE design, and a closed-form MMSE design whose penalty factor is tuned by bisection to meet the threshold. A position-aided approximation replaces the integral by satellite-to-base-station channels, cutting complexity while preserving the interference guarantee in simulation. If correct, this gives spectrum-sharing integrated networks a satellite-only way to keep terrestrial users protected.","feed_headline":"Beams spare terrestrial users without shared CSI","feed_subtitle":"Closed-form beamforming plus a position-based model meets the threshold with little rate loss.","key_machinery":"The load-bearing object is the integral-form interference covariance $\\Upsilon^{\\mathrm{int}}_{sg}$, defined elementwise in Eq. (13) by summing over terrestrial BSs an integral of the user density $f(r_n,\\phi_n)$ times a rank-one steering-vector outer product weighted by free-space path loss. It turns the unknown instantaneous interference channel into a deterministic statistical quantity the beamformer can be optimized against. The optimization is carried by three mechanisms: the multidimensional complex quadratic transformation decoupling SINR ratios in the WSR problem; the WSR-WMMSE equivalence giving a closed-form iterative beamformer through KKT conditions; and the penalty-function reformulation whose scalar $\\varsigma$ is found by bisection, giving the fully closed-form MMSE beamformer. The position-aided approximation replaces $\\Upsilon^{\\mathrm{int}}_{sg}$ by $\\tilde{\\Upsilon}_{sg} = \\bar K_G E[\\tilde H_{sg}\\tilde H_{sg}^H]$ built from satellite-to-BS channels, dropping the integral's dependence on real-time user distributions.","core_discovery":"The paper's central claim is that, given only statistical CSI of the satellite channel and the distribution of terrestrial users around base stations, the satellite can solve the interference-constrained transmit beamforming problem without any CSI-sharing protocol. The interference covariance $\\Upsilon^{\\mathrm{int}}_{sg}$ in Eq. (13) converts the sum over many terrestrial UTs into an integral of the user density over the coverage disk of each BS, so the constraint $\\frac{1}{K_G}\\mathrm{Tr}\\big(P^H \\Upsilon^{\\mathrm{int}}_{sg} P\\big) \\le I_{\\mathrm{thr}}$ has no dependence on instantaneous terrestrial channels. Proposition 2 gives the closed-form solution of the penalized MMSE problem as $P^\\star = \\beta^\\star\\big(\\Upsilon_{ss} + \\varsigma \\Upsilon^{\\mathrm{int}}_{sg} + \\frac{K_S \\sigma_s^2}{P_T} I\\big)^{-1} \\bar H_{ss}$ with $\\beta^\\star$ set by the power budget, and Proposition 3 shows the interference $I_{sg}(\\varsigma)$ is monotone non-increasing in the penalty factor, so a bisection on $\\varsigma$ meets the threshold. Proposition 4 analyzes the error of the base-station-position approximation for a single centered BS, showing the error grows with cell radius and user density and shrinks with carrier frequency. The contribution is that the satellite alone, with statistical knowledge only, can carry out interference management that previously required shared CSI.","pith_inferences":["Going beyond the paper: because Proposition 4 analyzes only one BS at the sub-satellite point under uniform density, the approximation error for off-center or multiple BSs is untested; a conservative designer could add a margin calibrated to the proven growth of $[E_{sg}]_{i,j}$ with cell radius and user density.","Going beyond the paper: the monotonicity of $I_{sg}$ in $\\varsigma$ suggests the penalty factor could be adapted online using measured aggregate interference feedback, rather than recomputed from the statistical model.","Going beyond the paper: the same integral-form covariance could be reused for satellite uplink or terrestrial-BS-side beamforming, since it relies only on geometry and user density rather than instantaneous cross-system channels."],"forward_implications":["If the central claim holds, a LEO satellite sharing sub-6 GHz spectrum can guarantee a terrestrial interference threshold without any CSI exchange with the terrestrial operator, avoiding protocol overhead and delay.","The closed-form MMSE beamformer with bisection-tuned penalty gives a practical low-complexity implementation: the interference constraint at a given SNR is met by a one-dimensional search over $\\varsigma$.","The position-aided approximation lowers the complexity order from $O(M_S^2 N_G N_r N_\\phi)$ to $O(M_S^2 N_G)$, and the paper's simulations show the approximate schemes still satisfy the threshold at 10 dB SNR.","The base-station-position approximation performs nearly as well as the integral-based one in the tested settings, so the method can be deployed where user distributions are not tracked in real time."],"supporting_citations":[{"why":"Supplies the statistical-CSI massive MIMO LEO channel model and the steering-vector structure used for the satellite link.","marker":"[20]"},{"why":"Gives the robust precoding with imperfect angle-of-departure and per-antenna power constraints that the paper adapts to its sCSI beamforming.","marker":"[34]"},{"why":"Establishes the weighted-MMSE equivalence that underlies the WMMSE iterative beamforming scheme.","marker":"[30]"},{"why":"Provides the multidimensional complex quadratic transformation used to convexify the weighted sum rate objective.","marker":"[53]"},{"why":"Supplies the iteratively weighted MMSE update rules for the WSR-WMMSE equivalence.","marker":"[56]"},{"why":"Motivates the CSI-free design target by showing distributed precoding without shared CSIT.","marker":"[44]"},{"why":"Provides the 3GPP NR NTN system parameters and assumptions for spectrum sharing and the Rician factor.","marker":"[45]"},{"why":"Supplies the penalty-function method used to relax the interference constraint in the MMSE formulation.","marker":"[59]"}],"fun_headline_variants":["Satellite beamforming meets threshold without shared CSI","Closed-form robust TX design respects interference limits","Position-aided model approximates interference, no CSI needed","Statistical CSI suffices for interference-constrained beamforming"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The weakest point is the modeling premise that the satellite knows the terrestrial user distribution $f(r,\\phi)$ and that the channel obeys Rician line-of-sight with free-space path loss; if the true distribution differs, or cells are large enough that the position-aided approximation error is significant, the computed interference can fall below what terrestrial users actually suffer.","fun_headline_variants_meta":{"raw":{"variants":["Satellite beamforming meets threshold without shared CSI","Closed-form robust TX design respects interference limits","Position-aided model approximates interference, no CSI needed","Statistical CSI suffices for interference-constrained beamforming"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000345,"raw_usage":{"total_tokens":1938,"prompt_tokens":1035,"completion_tokens":903,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":651,"completion_tokens_details":{"reasoning_tokens":842}},"tokens_in":651,"tokens_out":903,"duration_ms":10977,"temperature":1.0,"reasoning_tokens":842,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:03:51.117882+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a Monte Carlo simulation of a multi-cell deployment with base stations away from the sub-satellite point and non-uniform user density, compute $\\Upsilon^{\\mathrm{int}}_{sg}$ from the true user positions, then evaluate $\\frac{1}{K_G}\\mathrm{Tr}(P^H \\Upsilon^{\\mathrm{int}}_{sg} P)$ for the beamformer produced with the position-aided $\\tilde{\\Upsilon}_{sg}$; if this interference exceeds $I_{\\mathrm{thr}}$ by more than a small numerical tolerance in a realistic geometry, the paper's approximation guarantee fails.","supporting_citations":[{"cited_title":"Massive MIMO transmission for LEO satellite communications,","cited_arxiv_id":null,"evidence_quote":"Supplies the statistical-CSI massive MIMO LEO channel model and the steering-vector structure used for the satellite link."},{"cited_title":"Robust downlink precoding for LEO satellite systems with per-antenna power constraints,","cited_arxiv_id":null,"evidence_quote":"Gives the robust precoding with imperfect angle-of-departure and per-antenna power constraints that the paper adapts to its sCSI beamforming."},{"cited_title":"Weighted sum-rate maximization using weighted MMSE for MIMO- BC beamforming design,","cited_arxiv_id":null,"evidence_quote":"Establishes the weighted-MMSE equivalence that underlies the WMMSE iterative beamforming scheme."},{"cited_title":"Fractional programming for communication systemspart I: Power control and beamforming,","cited_arxiv_id":null,"evidence_quote":"Provides the multidimensional complex quadratic transformation used to convexify the weighted sum rate objective."},{"cited_title":"Distributed precoding for satellite-terrestrial integrated networks without sharing CSIT: A rate- splitting approach,","cited_arxiv_id":null,"evidence_quote":"Motivates the CSI-free design target by showing distributed precoding without shared CSIT."},{"cited_title":"Study on new radio (NR) to support non-terrestrial networks,","cited_arxiv_id":null,"evidence_quote":"Provides the 3GPP NR NTN system parameters and assumptions for spectrum sharing and the Rician factor."},{"cited_title":"Penalty function-based precoding for down- link multiuser MIMO systems,","cited_arxiv_id":null,"evidence_quote":"Supplies the penalty-function method used to relax the interference constraint in the MMSE formulation."}],"review_version":1}