End-to-end Throughput Maximization for Underlay Multi-hop Cognitive Radio Networks with RF Energy Harvesting

I. INTRODUCTION

• The paper formulates end-to-end throughput maximization under energy-causality and primary-receiver interference constraints, transforms the nonconvex problem into an equivalent convex one, and proposes JOTPA.JOTPA iteratively solves feasibility problems for a given throughput using dual decomposition.Source: §I · p. 3
• The paradigm treats the PT as both a spectrum and RF-energy source for SUs, pursuing spectrum efficiency and green energy utilization together.RF harvesting is framed as sustainable because ambient transmitter signals are consistently available.Source: §I · pp. 2–3
• The authors state that optimal resource allocation for end-to-end throughput maximization in underlay multi-hop EN-CRNs had not previously been studied.Source: §I · p. 3
• Across all considered scenarios, JOTPA achieves larger end-to-end throughput and higher green energy utilization than OTEPA and ETOPA.The evaluation compares the algorithms across three typical deployment scenarios.Source: §I · pp. 3–4

II. RELATED WORK

• The related-work discussion presents underlay EH-CRNs as allowing harvested energy to support self-sustainability while treating PUs as a stable energy source.It contrasts this with energy spent on sensing in interweave systems or cooperation with PUs in overlay systems.Source: §II · p. 5

III. SYSTEM MODEL

• The system has battery-free SUs harvesting RF energy from an always-active PT and using synchronized TDMA with decode-and-forward relaying to lower transmit powers.The model considers one data flow from the source SU to the destination SU.Source: §I · p. 2, §III · p. 6
• Each SU’s available energy depends on harvesting before its transmission turn, because residual or later-harvested energy cannot be carried into the next frame.Later SUs on the multi-hop path have more harvesting time than earlier SUs.Source: §I · p. 3, §III · p. 7
• The model assumes quasi-static block fading and that SUs can perfectly evaluate channel state information at the beginning of each block.Source: §III · p. 8

• The section investigates end-to-end throughput maximization in an underlay multi-hop EH-CRN.Source: §IV · p. 9
• The stated objective is to maximize throughput across the multi-hop path.Source: §IV · p. 9
• The section considers optimal resource allocation for the underlay EH-CRN.Source: §IV · p. 9

A. Problem Formulation

• Each hop’s achievable throughput depends on its harvested energy and allocated transmission time, with the primary transmitter’s signal treated as energy rather than interference.The passage also identifies σ2 as the signal-to-noise ratio at the receiving SU.Source: A. Problem Formulation · p. 9
• End-to-end throughput is the minimum achievable throughput across the hops, so maximizing it targets the path’s bottleneck.The formulation optimizes this quantity over time and power allocations.Source: A. Problem Formulation · p. 9
• The allocation chooses time and power to maximize end-to-end throughput while respecting per-SU energy causality, primary-receiver interference, and total and per-SU time constraints.The energy constraint limits allocated transmit energy to what was harvested; the interference constraint limits peak interference at the primary receiver.Source: A. Problem Formulation · pp. 9–10

B. Problem Transformation

• Introducing e_k = P_kτ_k makes allocated transmission energy the decision variable and converts the energy-causality constraint from a product of optimization variables into an affine constraint.This reparameterization replaces the original time-and-power variables with time and allocated energy.Source: B. Problem Transformation · p. 10
• The transformed end-to-end throughput R(τ, e) is jointly concave in time and allocated energy.The paper states this as Proposition 1.Source: B. Problem Transformation · p. 10
• Together, the concave objective and affine energy constraint yield an equivalent convex optimization problem in time and energy.The transformed problem retains the other constraints and can be solved with convex optimization techniques.Source: B. Problem Transformation · p. 10
• Introducing R reformulates the objective so that each hop’s throughput must be at least R under the feasible time and energy allocations.Maximizing R then maximizes the end-to-end throughput.Source: B. Problem Transformation · p. 11

C. Joint Optimal Time and Power Allocation Algorithm

• The maximum end-to-end throughput is attained only when all frame time is allocated and every hop achieves the same throughput.This characterizes the optimal allocation for the multi-hop max-min objective.Source: C. Joint Optimal Time and Power… · p. 11
• The algorithm raises or lowers R according to feasibility and repeats the feasibility test until R reaches the maximum within error threshold δ.A bisection search can be used to update R.Source: C. Joint Optimal Time and Power… · pp. 11, 13, 14
• For a fixed R > 0, the allocation problem is feasible if and only if some λ ≥ 0 gives G(λ) ≤ 0.The dual function therefore supplies a feasibility test for a candidate throughput.Source: C. Joint Optimal Time and Power… · p. 12
• For each candidate R, the method uses dual decomposition to handle coupled variables and constraints, then updates the allocation variables and dual variables to solve the associated problem.The passage describes alternating updates of time and energy for a given μ, followed by a subgradient update of μ.Source: C. Joint Optimal Time and Power… · pp. 12–13
• The paper states that the allocation and dual-variable updates converge for fixed R, and that bisection convergence makes the resulting solution optimal.Its convergence argument relies on the convexity of the relevant problems and the ellipsoid method for updating λ and μ.Source: C. Joint Optimal Time and Power… · p. 14

V. SIMULATION RESULTS

• The evaluation uses three typical scenarios created by moving the EH-CRN around the primary users to examine deployment effects.The passage describes the evaluation design, not numerical outcomes for those scenarios.Source: §V · p. 14
• JOTPA is compared with OTEPA and ETOPA to assess its performance across the considered scenarios.The passage identifies the comparison objective but does not report comparative result values.Source: §V · p. 14
• For a fair comparison, OTEPA uses JOTPA’s time-allocation method with equal power, while ETOPA uses equal time and optimizes power subject to interference and maximum-power limits.All algorithms account for the effect of time allocation on available harvested energy.Source: §V · p. 14

Simulation parameters

• The normalized frame lasts T = 1, with ξ = 0.8, σ² = 1, Pt = 40 dB, α = 2, and independent Rayleigh block-fading channels.The source-to-destination and PT-to-PR distances are each 20 m, with relay SUs equally spaced along the linear route.Source: A. Simulation Setup · pp. 14–15

JOTPA procedure

• JOTPA takes K, T, Pt, Ip, ξ, σ², δ, and channel gains as inputs, initializes sufficiently small Rmin and sufficiently large Rmax, calculates e by (17), and updates μ by the ellipsoid method with (18) until μ converges to μ*.Source: A. Simulation Setup · p. 15

Deployment scenarios

• The three deployments place the former SUs nearer to PUs, distribute SUs symmetrically around PUs, or place the latter SUs nearer to PUs.The scenarios use SU1 and SUK+1 endpoints (0, 0) and (20, 0), (-10, 0) and (10, 0), or (-20, 0) and (0, 0), respectively.Source: A. Simulation Setup · p. 16

Deployment and hop count

• For each scenario, maximum end-to-end throughput R* increases as hop count K increases; at the same K, Scenario 1 or 2 achieves larger R* than Scenario 3, consistent with Scenario 1’s more balanced energy distribution.For the six-hop EH-CRN, Scenario 3’s harvested energy increases monotonically across SUs, while Scenario 1’s allocated transmission time is approximately equal from SU1 to SU6.Source: B. Comparisons of Different Deployment… · p. 17

Strict interference constraint

• At Ip = 0 dB, Scenario 1 has the largest throughput and Scenario 3 the smallest, as the strict interference constraint leaves much harvested energy unallocated and subject to leakage.The resulting ranking favors deployments with former SUs closer to the PT when the PR interference constraint is strict.Source: B. Comparisons of Different Deployment… · pp. 17–18, Figure 5 · p. 18, Figure 6 · p. 18

Interference limit

• As Ip rises, throughput increases from near zero at Ip = −30 dB, then stops improving once harvested energy becomes the limiting resource at the given Pt.At sufficiently large Ip, the EH-CRN is equivalent to a WPCN without cognitive radio.Source: B. Comparisons of Different Deployment… · pp. 18–19, Figure 7 · p. 19

Throughput comparison

• JOTPA achieves larger R∗ than OTEPA and ETOPA for the same ξ, and the throughput gaps widen as Pt increases.OTEPA does not allocate powers according to CSI and Ip, while ETOPA's unoptimized time allocation can leave some SUs short of usable energy and others with energy discharged.Source: C. Comparisons of Different Resource… · pp. 19–20, Figure 8 · p. 20

Energy utilization

• JOTPA allocates more total energy to transmission than OTEPA and ETOPA, although JOTPA and OTEPA harvest the same energy because they use the same time-allocation method.The energy-status comparison is for three-hop EH-CRNs in Scenario 2.Source: C. Comparisons of Different Resource… · p. 20, Figure 9 · p. 20

Path loss

• For any K, JOTPA obtains much larger R∗ than OTEPA and ETOPA when α is small, but its advantage decreases as path loss makes harvested energy and correctly decoded information scarce.Fig. 10 compares R∗ against α for different algorithms.Source: C. Comparisons of Different Resource… · p. 21, Figure 10 · p. 21

Hop-count trade-off

• With α = 2, JOTPA's throughput gain is 23.72% from K = 3 to K = 4 but only 10.86% from K = 4 to K = 5, because larger K shortens each hop's time allocation.The paper states that optimal K is finite and can be found by one-dimensional search.Source: C. Comparisons of Different Resource… · p. 21

VI. CONCLUSION

• JOTPA obtains larger end-to-end throughput and higher green energy utilization than ETOPA and OTEPA under all considered scenarios.The paper presents its underlay multi-hop EH-CRN paradigm and JOTPA as optimal resource allocation.Source: §VI · p. 22
• Deploying the former SUs close to PUs achieves larger end-to-end throughput and higher green energy utilization than deploying the latter SUs close to PUs.Source: §VI · p. 22
• Without energy storage and management for battery-free SUs, the paper's underlay multi-hop EH-CRN performance is a lower bound.Future work on storage and management may improve performance; advanced schemes are needed to address PU dynamics in interweave, overlay, or hybrid paradigms.Source: §VI · p. 22

APPENDIX A PROOF OF PROPOSITION 1

• The perspective of the concave function U(e_k) makes each hop throughput jointly concave in τ_k and e_k.The end-to-end throughput, as the pointwise minimum of the hop throughputs, is jointly concave in τ and e.Source: Appendix A · p. 22

APPENDIX B PROOF OF PROPOSITION 2

• Each hop throughput increases monotonically with both allocated time and energy, while energy remains constrained by the allocated times.Under C1 and C2, changing a hop's allocated time also requires adjusting its energy allocation.Source: Appendix B · p. 23
• For given time allocations, adjusting time and energy until each SU obtains the same throughput achieves maximum end-to-end throughput.Source: Appendix B · p. 23

APPENDIX C PROOF OF PROPOSITION 3

• A feasible solution makes the Lagrangian nonpositive for nonnegative λ, so its minimum G(λ) is also nonpositive.Source: Appendix C · pp. 23–24
• These arguments address the if and only if parts of Proposition 3's feasibility condition.Source: Appendix C · p. 24
• The converse argument assumes feasibility alongside G(λ) > 0, then derives a contradiction from G(λ) ≤ L(τ, e, λ) ≤ 0.Source: Appendix C · p. 24

APPENDIX D PROOF OF PROPOSITION 4

• Slater's condition gives strong duality for problem (13), so the optimal solution satisfies the KKT condition.Source: Appendix D · p. 24
• The Lambert W function is used to solve for each hop's allocated time, after which τ_0 is calculated using Proposition 2.Lambert W is the inverse of f(t) = t exp(t).Source: Appendix D · p. 24
• The interference power constraint 0 ≤ e_k ≤ I_pτ_k determines the optimal energy allocation.Source: Appendix D · p. 24