Source-linked AI summary
Robust Beamforming for Wireless Information and Power Transmission
Zhengzheng Xiang, Meixia Tao
TL;DR
The paper addresses worst-case robust beamforming for simultaneous information transmission and energy harvesting with imperfect transmitter CSI. It transforms the semi-infinite nonconvex design into a relaxed SDP and proves the relaxation is tight because an optimal solution is rank-one. Simulations show small performance loss for modest CSI errors, while robust beamforming avoids rate-target outages observed with nonrobust design.
Problem
The paper seeks to maximize worst-case harvested energy while maintaining an information-rate threshold for every possible channel realization under bounded CSI uncertainty.
Method
The semi-infinite nonconvex problem is transformed into a finite formulation and then relaxed into an efficiently solvable semidefinite program.
Results
The SDP relaxation is tight because its optimal solution is always rank-one, yielding an optimal solution to the original problem; simulations show small loss for modest CSI errors and no robust-design outage.
Takeaways & Limitations
Worst-case robust beamforming can optimize harvested energy under bounded CSI uncertainty while satisfying the information-rate target across possible channel realizations.
Abstract
from arXiv · showhide
In this letter, we study the robust beamforming problem for the multi-antenna wireless broadcasting system with simultaneous information and power transmission, under the assumption of imperfect channel state information (CSI) at the transmitter. Following the worst-case deterministic model, our objective is to maximize the worst-case harvested energy for the energy receiver while guaranteeing that the rate for the information receiver is above a threshold for all possible channel realizations. Such problem is nonconvex with infinite number of constraints. Using certain transformation techniques, we convert this problem into a relaxed semidefinite programming problem (SDP) which can be solved efficiently. We further show that the solution of the relaxed SDP problem is always rank-one. This indicates that the relaxation is tight and we can get the optimal solution for the original problem. Simulation results are presented to validate the effectiveness of the proposed algorithm.
I. INTRODUCTION
The paper motivates robust beamforming for simultaneous wireless information transmission and energy harvesting when transmitter CSI is imperfect. It adopts a worst-case uncertainty model and transforms the resulting semi-infinite nonconvex problem into an efficiently solvable SDP with a tight relaxation.
- Energy harvesting supports energy-constrained wireless networks, while beamforming exploits transmitter CSI for information transmission and simultaneous information-power systems.
- Perfect-CSI beamforming characterizes rate-energy tradeoffs, but practical CSI is impaired by channel estimation errors, quantization, and feedback delay.
- The deterministic model bounds CSI errors and seeks the required quality of service for every possible error, providing absolute robustness when feasible.
- The paper maximizes worst-case harvested energy while guaranteeing a minimum information rate under imperfect CSI.
- The channel-uncertainty problem is converted into a finite nonconvex problem, then relaxed to an efficiently solvable SDP whose optimal solution is always rank-one.
II. SYSTEM MODEL AND PROBLEM FORMULATION
The system is a three-node MISO link with one multi-antenna transmitter and single-antenna information and energy receivers. The formulation maximizes harvested energy subject to a rate target under bounded, imperfect CSI for both receivers.
- The three-node MISO system has an N-antenna transmitter and single-antenna information and energy receivers.
- The harvesting model assumes harvested RF-band power is proportional to received baseband-signal power, with conversion efficiency η assumed to equal 1.
- The beamforming design maximizes harvested energy while ensuring the information receiver’s rate exceeds a target under a transmitter power constraint.
- The transmitter has imperfect CSI for both receivers, modeled using estimated channels and bounded error vectors.
- The worst-case objective maximizes harvested energy for the worst channel realization while maintaining the rate threshold for all possible channel realizations.
III. SEMIDEFINITE PROGRAMMING SOLUTION
The robust beamforming problem is transformed from a semi-infinite nonconvex QCQP into a convex SDP through reformulation and semidefinite relaxation. The relaxation is tight because the relaxed problem always has a rank-one optimal solution, enabling recovery of the original optimum.
- Problem structure: Channel uncertainties and nonconvex constraints make P1 a semi-infinite nonconvex QCQP that is generally NP-hard.The paper exploits the special structure of P1 to obtain an optimal convex reformulation.
- Problem reformulation: The objective and uncertainty constraints are reformulated into a finite-constraint problem that remains nonconvex.The derivation uses triangle and Cauchy-Schwarz inequalities and assumes sufficiently small channel errors.
- Assumption: Large channel estimation errors can cause large beamforming errors, so the robustness guarantee relies on sufficiently small uncertainty.The paper characterizes this as a practical assumption in the derivation.
- Semidefinite relaxation: Semidefinite relaxation drops the rank-one constraint, producing P2, a standard convex SDP that can be solved efficiently.The matrix substitutions bG = bgbgH and bH = bhbhH enable the SDP formulation.
- Tightness and recovery: The optimal solution W of P2 is rank-one, so eigenvalue decomposition can extract an optimal beamformer for P1.The rank-one result establishes that the relaxed SDP does not lose optimality.
IV. SIMULATION RESULTS
Simulations evaluate robust beamforming under bounded CSI uncertainty. The robust design maintains the information-rate target, while uncertainty affects harvested energy and nonrobust designs cause outages.
- Simulation setup: The simulations use a four-antenna MISO system with P = 10, noise covariance σ2 = 1, normalized Rayleigh fading, and 100 uncertainty samples per realization.The information-rate target is chosen below log(1 + 10∥h∥2) to ensure feasibility.
- Robust performance: Small CSI errors cause only small harvested-energy losses, while the performance gap grows as the information-rate target increases.Fig. 2 compares average harvested energy across information-rate targets and bounded channel-uncertainty levels, including perfect CSI.
- Outage comparison: Channel uncertainty ignored during design leads to frequent information-rate violations in the nonrobust beamforming design.The nonrobust design assumes perfect CSI even though the actual CSI is uncertain.
- Outage comparison: The proposed worst-case robust beamforming algorithm always satisfies the information-rate target, resulting in no outage.Outage is defined as failure to satisfy the information-rate target at the information receiver.
V. CONCLUSION
The paper develops worst-case robust beamforming for simultaneous information transmission and energy harvesting with imperfect CSI. Semidefinite relaxation yields a tight SDP formulation whose solution is always optimal for the original problem, while extensions to multiple receivers remain future work.
- Conclusion: The proposed design addresses simultaneous information transmission and energy harvesting when the transmitter has imperfect CSI.The system includes both information and energy receivers.
- Conclusion: Semidefinite relaxation transforms the original robust design problem into an SDP problem.The transformation targets the original robust beamforming formulation.
- Conclusion: The relaxation is tight, so the SDP solution always provides an optimal solution to the original problem.The conclusion states that the proposed algorithm's performance is demonstrated by simulations.
- Future work: Robust beamforming for systems with multiple information receivers and multiple energy receivers is identified as a future research direction.
APPENDIX A PROOF OF THEOREM 1
The appendix derives the rank-one property of the relaxed SDP solution through its Lagrangian dual structure. This establishes equivalence between the relaxed problem and the original beamforming problem.
- Dual formulation: The Lagrangian of P2 is formed using dual variables λ and µ.The dual function is then defined from this Lagrangian.
- Dual formulation: Because P2 is convex with strong duality, it can be solved through its dual problem.The optimal dual solution determines the matrix W⋆ that maximizes the Lagrangian.
- Rank-one proof: Boundedness of the dual problem requires µ⋆I − λ⋆bH to be positive definite.Otherwise, a rank-one choice W = twwH can make the objective unbounded as t approaches infinity.
- Rank-one proof: The proof assumes a non-rank-one optimal solution can have rank k with 2 ≤ k ≤ N and decomposes it to derive a contradiction.
- Rank-one proof: Theorem 1 concludes that the optimal solution W⋆ of P2 is rank-one.The rank-one result follows from the contradiction argument in the appendix.