{"id":"21491703-e2fc-47d6-9317-c919e369c2e6","arxiv_id":"1908.09270","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a multi-hop decode-and-forward relay network with power-splitting SWIPT, the paper derives closed-form optimal power-splitting ratios for source transmit power minimization and minimum-rate maximization.","lead":"This paper derives simple formulas for how much transmit power a source needs and what data rate a multi-hop wireless network can achieve when relays harvest energy from the signal they receive. The formulas tell each relay how to split the incoming signal between energy harvesting and decoding, which could help battery-free IoT sensor networks relay data longer.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The optimal PS-ratio formulas in Theorems 1 and 2 omit the noise variances sigma_k^2; for unequal sigma_k^2 the stated rho_k^star violates the SNR constraints and is not even feasible.","rationale":"The reader's weakest assumption concerns physical idealization of energy harvesting; my concern is instead an internal correctness issue in the theorem statements. I independently re-derived the optimal A_k variables: the E0* and R* formulas are supported by telescoping, but the PS-ratio formulas printed in Theorem 1, Theorem 2, (28), and (39) omit the noise variances and therefore do not satisfy the constraints unless all sigma_k^2 are equal. The numerical section sets equal noise variances, so the simulations do not expose this. The correction is mechanical, and the convexity/telescoping argument for E0* and R* survives, so the appropriate disposition remains CONDITIONAL: the paper should be accepted only after the PS-ratio formulas and the distributed implementation updates are corrected. Because the reader identified a different weakest point, agreement_with_reader is disagree.","tokens_in":22512,"tokens_out":14233,"duration_ms":141788,"concrete_test":"Use K=2 with all beta_k=1, |h_k|^2=1, and gamma_k_bar=1, but set sigma^2=(1,100,1). Compute the Theorem 1 formulas: E0*=102, rho_1=1-1/3=2/3, rho_2=1-(1/(2/3))(1/3)=1/2, rho_3=0. Evaluate the k=2 constraint of problem (6): E0 Gamma_2/(sigma_2^2 beta_2) rho_1(1-rho_2) = 102*(1/100)*(2/3)*(1/2)=0.34 < gamma_2_bar=1. If the PS formulas are corrected to 1-rho_k=c_k/sum_{j=k}^{K+1} c_j with c_k=gamma_k_bar sigma_k^2 beta_k/Gamma_k, all three constraints are met with equality. This single numerical test settles whether the omitted sigma^2 factors are a typographical slip or a substantive error: in the general problem they are substantive.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The closed forms for E0* and R* are correct: telescoping the active constraints gives E0*=sum c_k with c_k=gamma_k_bar sigma_k^2 beta_k/Gamma_k, and analogously for the rate problem. The load-bearing flaw is in the companion PS-ratio formulas. At E0*=sum c_k, every hop constraint must be tight, so with A_k=prod_{j<=k} rho_j we have A_{k-1}-A_k=c_k/E0*, which forces 1-rho_k=c_k/sum_{j=k}^{K+1} c_j. The formula stated in Theorem 1 and in (28) instead uses gamma_k_bar beta_k/Gamma_k over sum_j gamma_j_bar beta_j/Gamma_j, with no sigma^2 factors. These agree only if all sigma_k^2 are equal. A concrete two-relay example shows the stated vector is infeasible: take K=2, beta_k=|h_k|^2=gamma_k_bar=1, and sigma^2=(1,100,1). Then E0*=102, but the paper's formula gives rho_1=2/3, rho_2=1/2, and the hop-2 SNR is E0 rho_1(1-rho_2)/sigma_2^2 = 102*(2/3)*(1/2)/100 = 0.34, below the required threshold gamma_2_bar=1. Thus the claimed globally optimal PS ratio solution does not even produce a feasible point for the modeled problem. The same omission appears in Theorem 2 and (39), where the needed ratio is sigma_k^2 beta_k/Gamma_k over sum_j sigma_j^2 beta_j/Gamma_j. The error is masked in the simulations because all nodes are assigned the same noise variance, and it is inherited by the distributed implementation equations (14)-(15). This is an internal inconsistency in the theorem statements, not merely a modeling idealization.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies a single-antenna multi-hop decode-and-forward relay chain in which each relay uses power splitting, harvesting energy from a fraction ρ_k of the received signal and using that energy to decode and forward, while the destination is modeled as node K+1 with ρ_{K+1}=0. The authors formulate two optimization problems: minimizing the source transmit power E0 subject to per-hop SNR thresholds γ̄_k, and maximizing the minimum achievable rate subject to a fixed source power. They claim globally optimal closed-form solutions in Theorems 1 and 2, an equivalence between the two problems when all thresholds are equal, a closed-form estimate for the maximum number of relays, and centralized and distributed methods for computing the PS ratios. The paper also reports simulations comparing the proposed scheme with a fixed PS ratio of 0.5 under perfect and imperfect channel state information.","tokens_in":22922,"tokens_out":9534,"duration_ms":93287,"significance":"For the idealized model, the closed-form expressions for E0* in Theorem 1 and R* in Theorem 2 are correct and can be verified by telescoping the active constraints; this part of the paper is a genuine low-complexity design contribution. The derivations are self-contained and contain no fitted parameters, and the formulas are falsifiable statements about the modeled system. However, the companion closed-form PS-ratio formulas are not correct as stated when the noise variances σ_k^2 differ across nodes, and the printed proof of Theorem 2 in Appendix B has internal sign and algebra errors. The numerical section uses identical noise variances for all nodes, which masks the error. With corrected PS ratios and a repaired proof, the paper would be a solid contribution to low-complexity SWIPT relaying; as stated, the central optimality claim for the PS ratios does not hold for the general problem.","major_comments":[{"comment":"The optimal PS ratio formula in Theorem 1 and Eq. (28) omits the noise variances σ_k^2. In the transformed problem (16), all constraints are active at the optimum, so A_{k-1}^* - A_k^* = (σ_k^2 β_k γ̄_k / Γ_k) / (Σ_{j=1}^{K+1} σ_j^2 β_j γ̄_j / Γ_j). Therefore the correct ratio is ρ_k^* = 1 - (1/∏_{j=1}^{k-1} ρ_j^*) (σ_k^2 β_k γ̄_k / Γ_k) / (Σ_j σ_j^2 β_j γ̄_j / Γ_j). The printed formula replaces the σ_k^2 factor in the numerator and the σ_j^2 factors in the denominator sum by 1, which is only valid when all σ_k^2 are equal. A concrete counterexample: take K=2, β_k=|h_k|^2=γ̄_k=1, and σ^2=(1,100,1). Then E0*=102, and the printed formulas give ρ_1=2/3 and ρ_2=1/2. The hop-2 SNR is E0 ρ_1 (1-ρ_2)/σ_2^2 = 102 × (2/3) × (1/2) / 100 = 0.34, which is below the required threshold γ̄_2=1. Thus the printed “globally optimal” PS vector is not even feasible for the modeled problem, contradicting the central claim of Theorem 1.","section":"Section III-A, Theorem 1, Eq. (28)"},{"comment":"The same omission of σ_k^2 appears in the rate-maximization PS ratios. With γ̂^* = E0 / (Σ_j σ_j^2 β_j / Γ_j) from Eq. (38), the tight constraint at hop k forces A_{k-1}^* - A_k^* = (σ_k^2 β_k / Γ_k) / (Σ_j σ_j^2 β_j / Γ_j), so the correct optimal ratio is ρ_k^* = 1 - (1/∏_{j=1}^{k-1} ρ_j^*) (σ_k^2 β_k / Γ_k) / (Σ_j σ_j^2 β_j / Γ_j). The printed expression in Theorem 2 and Eq. (39), which uses β_k/Γ_k over Σ_j β_j/Γ_j, is feasible only when all σ_k^2 are identical. Since the PS ratios are part of the claimed optimal solution, Theorem 2 as stated is not correct for the general model with unequal noise variances.","section":"Section III-B, Theorem 2, Eq. (39)"},{"comment":"The printed proof of Theorem 2 is internally inconsistent. The Lagrangian in Eq. (30) has a plus sign on the γ̂ constraint term, so the stationarity condition in Eq. (31) becomes 1 + λ_0 Σ_k σ_k^2 β_k / (E0 Γ_k) = 0, forcing λ_0 to be negative. This is incompatible with the λ_k ≥ 0 convention used in Appendix A and with the complementary-slackness step in Eq. (35). In addition, Eq. (36) incorrectly places γ̂ inside the denominator; solving Eq. (34) directly gives λ_0^* = -E0 / (Σ_k σ_k^2 β_k / Γ_k), with no γ̂ in the expression. These errors mean the proof as printed does not establish Theorem 2, although the final γ̂^* formula can be recovered by summing the tight constraints.","section":"Appendix B, Eqs. (30)-(36)"}],"minor_comments":[{"comment":"The text says “antenna noise variance σ_1^2 = ... = -80 dBm,” but in Section II σ_k^2 is the information-decoding circuit noise variance while δ_k^2 is the antenna noise variance; the notation should be corrected to avoid the impression that the antenna noise is being set to the ID noise value.","section":"Section IV, simulation parameters"},{"comment":"The derivation of the relay-node count approximation replaces E[1/Γ_k] by 1/E[Γ_k] without comment. In Rayleigh fading, E[1/|h|^2] is not finite, so this substitution is not a standard expectation calculation; please state explicitly that this is a heuristic mean-gain approximation and, if possible, provide a bound or simulation-based validation.","section":"Appendix C, Eq. (40)"},{"comment":"The approximation in Eq. (5) drops the antenna noise δ_k^2 in the SINR expression assuming δ_k^2 ≪ σ_k^2; this assumption should be restated in the simulation section where the noise parameter is set, so that the reader can verify consistency.","section":"Section II, Eq. (5)"},{"comment":"The verbal explanation accompanying Fig. 14 says a node's rate increases by reducing its ρ̄_k, but for intermediate relays changing ρ_k also affects the energy available to all later hops; the figure should be labeled as a schematic of the equalization argument rather than a proof of monotonicity in the multi-hop setting.","section":"Appendix B, Fig. 14"}],"recommendation":"major_revision","confidential_remarks":"The core E0* and R* formulas are correct and the flaw is localized to the PS-ratio expressions and the proof in Appendix B, so a revision can repair the paper without changing its scope. The equal-noise simulation setup hides the error; the authors should test heterogeneous σ_k^2 and update the distributed implementation equations, which currently inherit the incorrect PS-ratio formulas."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague — quick read of arXiv:1908.09270. The paper has a real result buried in it, but the headline solution as printed is not correct. The closed forms for minimum source power (Theorem 1) and maximum min rate (Theorem 2) are right: telescoping the active constraints gives E0^* = sum gamma_k_bar sigma_k^2 beta_k / Gamma_k and R^* = log2(1 + E0 / sum sigma_k^2 beta_k / Gamma_k). That part checks out and is a legitimate extension of the authors' earlier AF dual-hop work to multi-hop DF.\n\nWhere it falls down is the companion PS-ratio formulas. They omit the noise variances sigma_k^2 entirely. The correct ratios, from equality in the constraints, are rho_k = 1 - c_k / sum_{j=k}^{K+1} c_j with c_k = gamma_k_bar sigma_k^2 beta_k / Gamma_k (and analogous for the rate problem). What the paper prints instead uses gamma_k_bar beta_k / Gamma_k over sum gamma_j_bar beta_j / Gamma_j. Those agree only when all sigma_k^2 are equal. The simulations use equal noise variances, so the flaw is hidden; but for unequal variances the printed ratios are not even feasible. A two-relay example with sigma^2 = (1,100,1) makes the second hop SNR 0.34, below the threshold of 1. This is not a modeling idealization; it's an error in the theorem statements.\n\nThe appendix proofs also have sign mistakes in the Lagrangian (Equation (36) is off), and the node-count formula in Appendix C has a geometric-series indexing error and an unjustified expectation approximation. Those are secondary. The central issue is that the paper's claimed globally optimal PS ratio solution is wrong as stated.\n\nCredit where due: the E0* and R* expressions are new for this setting, the equivalence between the two problems under equal thresholds is a nice observation, and the distributed implementation idea is sensible once the ratio formula is fixed. The simulation comparisons are plausible but no code or data are shipped, so I can't verify them.\n\nWho is this for? People working on SWIPT relay optimization. If the PS ratio formulas are corrected, the paper is a useful specialized contribution. As printed, I would not rely on it. A serious referee could catch this and demand the fix; if the authors revise, the paper could be sound. I'd send it to review rather than desk reject, because the core formulas are defensible and the error is localized.","headline":"The closed forms for E0 and R telescope correctly, but the printed PS-ratio formulas drop the noise variances and are infeasible for unequal sigma^2; fixable but load-bearing.","tokens_in":23422,"tokens_out":3892,"would_cite":false,"duration_ms":35638,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A multi-hop relay network with decode-and-forward SWIPT has closed-form optimal power-splitting ratios, giving the minimum source power and maximum end-to-end rate as explicit formulas.","keywords":["SWIPT","decode-and-forward relaying","multi-hop relay network","power splitting","RF energy harvesting","source transmit power minimization","max-min rate optimization","closed-form solution"],"falsifier":"For the mathematical claim itself, run a dense grid search over $\\rho$ for a small instance, say $K=2$ with fixed positive thresholds and randomly drawn channel gains, and compare the numerical optimum of problem (6) with $E_0^\\star=\\sum_{k=1}^{K+1}\\bar\\gamma_k\\sigma_k^2\\beta_k/\\Gamma_k$; any mismatch would refute Theorem 1. To test the model's applicability, repeat the comparison after adding a fixed per-relay circuit power $P_c>0$ and a saturating energy-harvesting efficiency; a mismatch there would show the closed forms describe the idealized model rather than a real relay's energy budget.","tokens_in":22323,"feed_emoji":"📡","tokens_out":11259,"duration_ms":103326,"temperature":0.7,"pith_summary":"This paper tries to establish that for a multi-hop decode-and-forward relay chain whose relays harvest energy from a split of the received radio signal, the two main resource-allocation problems have exact closed-form solutions. The first problem asks for the smallest source transmit power that lets every hop meet its own signal-to-noise threshold; the second asks for the largest end-to-end rate a fixed source power can support. The paper proves that the answer to the first is $E_0^\\star = \\sum_{k=1}^{K+1} \\bar\\gamma_k \\sigma_k^2 \\beta_k / \\Gamma_k$ and the answer to the second is $R^\\star = \\log_2\\!\\left(1 + E_0 \\big/ \\sum_{k=1}^{K+1} \\sigma_k^2 \\beta_k / \\Gamma_k\\right)$, with each relay's optimal power-splitting ratio given by a recursive formula. A network operator would care because these formulas replace iterative optimization with arithmetic: the source can compute its power without knowing the relays' split ratios, and each relay can set its split from local information. Under equal threshold targets the two problems reduce to a single equation, which the paper also turns into a closed-form estimate of how many relay hops a given source power can support.","feed_headline":"Closed-form formulas optimize every power split in a SWIPT relay chain","feed_subtitle":"Minimum source power and maximum end-to-end rate each reduce to one equation for battery-free relays.","key_machinery":"The load-bearing device is the change of variables $A_k=\\prod_{j=1}^k \\rho_j$, with $A_0=1$ and $A_{K+1}=0$, together with $Q=1/E_0$ for the power-minimization problem and an auxiliary SNR variable $\\hat\\gamma$ for the rate-maximization problem. In these variables the non-convex product constraints become a chain of linear inequalities, $A_{k-1}-A_k \\ge Q\\sigma_k^2\\beta_k\\bar\\gamma_k/\\Gamma_k$ (or the same with $\\hat\\gamma$ in place of $\\bar\\gamma_k$), so each problem becomes convex with a single active constraint chain. Karush–Kuhn–Tucker analysis then forces all dual variables to be equal, a telescoping sum collapses to unity, and the closed-form objectives and recursive PS ratios follow. The machinery is doing the work of removing the coupling between successive hops' split ratios.","core_discovery":"The paper's central claim is that, under the modeled SWIPT system, the globally optimal power-splitting ratios and the optimal objective values are available in closed form for both problems. For a source and destination separated by $K$ decode-and-forward relays with no direct link, each relay splits its received signal with ratio $\\rho_k$: fraction $\\rho_k$ goes to energy harvesting and $1-\\rho_k$ to information decoding. The paper proves Theorem 1: the minimum source power meeting per-hop SNR constraints $\\bar\\gamma_k$ is $E_0^\\star=\\sum_{k=1}^{K+1}\\bar\\gamma_k\\sigma_k^2\\beta_k/\\Gamma_k$, where $\\Gamma_k=\\prod_{j=1}^k \\beta_j|h_j|^2$ accumulates the energy-conversion efficiencies and channel gains hop by hop. It proves Theorem 2: with fixed source power $E_0$, the maximum of the minimum hop rate is $R^\\star=\\log_2(1+\\hat\\gamma^\\star)$ with $\\hat\\gamma^\\star=E_0/\\sum_{k=1}^{K+1}\\sigma_k^2\\beta_k/\\Gamma_k$. In both cases the optimal split at relay $k$ is $\\rho_k^\\star=1-\\big(\\prod_{j=1}^{k-1}\\rho_j^\\star\\big)^{-1}\\big(\\beta_k/\\Gamma_k\\big)\\big/\\sum_{j=1}^{K+1}\\beta_j/\\Gamma_j$; in the threshold version the numerator and denominator instead carry the extra factors $\\bar\\gamma_k$ and $\\bar\\gamma_j$, and the last node sets $\\rho_{K+1}^\\star=0$. The paper also claims that with a common SNR threshold the two problems coincide, giving $E_0/\\hat\\gamma=\\sum_{k=1}^{K+1}\\sigma_k^2\\beta_k/\\Gamma_k$, and that this yields a closed-form hop-count estimate for a homogeneous network.","pith_inferences":["The formulas assume zero circuit power at each relay, so they should be read as an upper bound on what a real energy-harvesting relay chain can do; adding a fixed per-relay power draw $P_c$ would shrink the feasible region and require new optimization, though the same $A_k$ change of variables may still be useful.","The recursive form of $\\rho_k^\\star$ suggests a natural online protocol: relay $k$ only needs the scalar $\\prod_{j<k}\\rho_j^\\star$ or an equivalent accumulated term from its predecessor, so the optimal splitting can be tracked hop by hop as channels change, extending the paper's distributed implementation beyond static CSI.","A testable prediction is that under nonlinear energy-harvesting efficiency (saturation at high input power), the optimal split will direct more signal to information decoding than the linear model predicts; the closed forms provide the linear baseline for measuring that shift.","The paper explicitly frames its result for single-antenna half-duplex PS-SWIPT relays with a supercapacitor and lists AF relaying, time switching, and multi-antenna nodes as extensions, so the closed forms should not be read beyond that scope."],"forward_implications":["Minimum source power for a DF-SWIPT multi-hop link is a closed-form weighted sum of the per-hop SNR thresholds, so a source can provision power without solving an optimization problem or knowing the relays' PS ratios.","The maximum-minimum end-to-end rate under a fixed source power is known directly from $R^\\star=\\log_2(1+\\hat\\gamma^\\star)$, and the optimal operating point equalizes all hop SNRs to $\\hat\\gamma^\\star$.","When all nodes share the same SNR target, source-power minimization and rate maximization are two sides of one equation, $E_0/\\hat\\gamma=\\sum_{k=1}^{K+1}\\sigma_k^2\\beta_k/\\Gamma_k$, so either quantity can be traded against the other without new computation.","In a homogeneous network with equal distances, the same relation yields a closed-form estimate of the maximum number of relays a given source power and QoS target can support, which can be embedded in routing decisions.","The centralized and distributed PS-ratio computation schemes both use only the closed-form expression; the distributed scheme lets each relay compute the next relay's split with local CSI and two scalar values."],"supporting_citations":[{"why":"Supplies the PS-SWIPT signal model and the rate expression with negligible antenna noise that equations (4) and (5) build on.","marker":"[47]"},{"why":"Supports dropping the antenna-noise term relative to the information-decoding circuit noise in the achievable rate.","marker":"[49]"},{"why":"Provides the wireless-information-and-power-exchange model for energy-constrained links that motivates the D2D application.","marker":"[50]"},{"why":"Gives the battery-equipped cooperative SWIPT relay model that this multi-hop formulation extends.","marker":"[21]"},{"why":"Supplies the prior optimal power-splitting framework for AF multi-relay SWIPT that the DF analysis builds on.","marker":"[23]"},{"why":"Introduces the max-min SNR constraint formulation used in the rate-maximization problem.","marker":"[33]"},{"why":"Provides the earlier multi-hop energy-harvesting relay analysis whose node-count result the closed-form hop-count estimate extends.","marker":"[36]"}],"fun_headline_variants":["One equation optimizes all relay power splits in SWIPT","Battery-free relay chain: exact split ratios in closed form","Closed-form solutions for every power split in SWIPT hops","Every relay split exact optimal, in one closed form"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every relay's harvested energy is exactly a fixed fraction of the received radio-frequency energy, with no energy spent on decoding or circuit operation, so all harvested energy is available to forward the next hop; if relays need a minimum operating power, have nonlinear harvester efficiency, or consume circuit power, the closed-form formulas describe a different system.","fun_headline_variants_meta":{"raw":{"variants":["One equation optimizes all relay power splits in SWIPT","Battery-free relay chain: exact split ratios in closed form","Closed-form solutions for every power split in SWIPT hops","Every relay split exact optimal, in one closed form"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00111,"raw_usage":{"total_tokens":4742,"prompt_tokens":1177,"completion_tokens":3565,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":793,"completion_tokens_details":{"reasoning_tokens":3498}},"tokens_in":793,"tokens_out":3565,"duration_ms":24437,"temperature":1.0,"reasoning_tokens":3498,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:18:20.590337+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the mathematical claim itself, run a dense grid search over $\\rho$ for a small instance, say $K=2$ with fixed positive thresholds and randomly drawn channel gains, and compare the numerical optimum of problem (6) with $E_0^\\star=\\sum_{k=1}^{K+1}\\bar\\gamma_k\\sigma_k^2\\beta_k/\\Gamma_k$; any mismatch would refute Theorem 1. To test the model's applicability, repeat the comparison after adding a fixed per-relay circuit power $P_c>0$ and a saturating energy-harvesting efficiency; a mismatch there would show the closed forms describe the idealized model rather than a real relay's energy budget.","supporting_citations":[{"cited_title":"MIMO broadcasting for simultaneo us wire- less information and power transfer,","cited_arxiv_id":null,"evidence_quote":"Supplies the PS-SWIPT signal model and the rate expression with negligible antenna noise that equations (4) and (5) build on."},{"cited_title":"Wireless information and power transfer: A dynamic power splitting approach,","cited_arxiv_id":null,"evidence_quote":"Supports dropping the antenna-noise term relative to the information-decoding circuit noise in the achievable rate."},{"cited_title":"Wireless informat ion and power exchange for energy-constrained device-to-device c ommunica- tions,","cited_arxiv_id":null,"evidence_quote":"Provides the wireless-information-and-power-exchange model for energy-constrained links that motivates the D2D application."},{"cited_title":"Simultaneous wireless information and power transfer for cooperative relay netwo rks with battery,","cited_arxiv_id":null,"evidence_quote":"Gives the battery-equipped cooperative SWIPT relay model that this multi-hop formulation extends."},{"cited_title":"Op timal power splitting for simultaneous wireless information and power transfer in amplify-and-forward multiple-relay systems,","cited_arxiv_id":null,"evidence_quote":"Supplies the prior optimal power-splitting framework for AF multi-relay SWIPT that the DF analysis builds on."},{"cited_title":"Joint resource allocation in SWIPT-based mult iantenna decode- and-forward relay networks,","cited_arxiv_id":null,"evidence_quote":"Introduces the max-min SNR constraint formulation used in the rate-maximization problem."},{"cited_title":"Multi-hop relaying using energy harvesting,","cited_arxiv_id":null,"evidence_quote":"Provides the earlier multi-hop energy-harvesting relay analysis whose node-count result the closed-form hop-count estimate extends."}],"review_version":1}