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Beamforming for MISO Interference Channels with QoS and RF Energy Transfer
Stelios Timotheou, Ioannis Krikidis, Gan Zheng, Björn Ottersten
TL;DR
The paper studies protocols and transmission techniques for wireless networks with RF-EH capabilities, under QoS and EH constraints. It formulates the minimum required energy and develops optimization approaches, finding that MRT-ZF provides the best trade-off between computational complexity and optimality.
Problem
The paper addresses protocols and transmission techniques for wireless networks with RF-EH capabilities, including QoS and EH constraints.
Method
The paper formulates the minimum required energy and uses SDP with rank relaxation, alongside closed-form and SOCP-based beamforming solutions.
Results
MRT-ZF provides the best trade-off between computational complexity and optimality.
Takeaways & Limitations
ZF has a closed-form solution, while the proposed formulations address variable-weight beamforming and optimization across beamforming schemes.
Takeaways & Limitations
Applying the investigated schemes to scenarios with limited channel information remains future work.
Abstract
from arXiv · showhide
We consider a multiuser multiple-input single-output interference channel where the receivers are characterized by both quality-of-service (QoS) and radio-frequency (RF) energy harvesting (EH) constraints. We consider the power splitting RF-EH technique where each receiver divides the received signal into two parts a) for information decoding and b) for battery charging. The minimum required power that supports both the QoS and the RF-EH constraints is formulated as an optimization problem that incorporates the transmitted power and the beamforming design at each transmitter as well as the power splitting ratio at each receiver. We consider both the cases of fixed beamforming and when the beamforming design is incorporated into the optimization problem. For fixed beamforming we study three standard beamforming schemes, the zero-forcing (ZF), the regularized zero-forcing (RZF) and the maximum ratio transmission (MRT); a hybrid scheme, MRT-ZF, comprised of a linear combination of MRT and ZF beamforming is also examined. The optimal solution for ZF beamforming is derived in closed-form, while optimization algorithms based on second-order cone programming are developed for MRT, RZF and MRT-ZF beamforming to solve the problem. In addition, the joint-optimization of beamforming and power allocation is studied using semidefinite programming (SDP) with the aid of rank relaxation.
I. INTRODUCTION
RF energy transfer offers a controllable harvesting technology for communication systems with critical QoS constraints, but applying RF-EH to complex network configurations remains open. This paper studies a MISO interference channel with simultaneous QoS and EH requirements and develops fixed- and variable-beamforming solutions.
- Motivation: RF energy transfer can be controlled and is suitable for applications with critical QoS constraints.Ambient RF radiation is captured by receiver antennas and converted into DC voltage through rectennas.
- Related work: Protocols and transmission techniques for wireless networks with RF-EH capabilities remain a relatively new research direction.Prior work includes simultaneous information and energy transfer, time switching, power splitting, and multiple-receiver approaches.
- Problem: Most prior work focuses on simple network topologies with single transmitters, leaving RF-EH in more complex configurations open.The paper addresses this gap with a MISO interference channel whose receivers have both QoS and EH constraints.
- Contributions: For fixed beamforming, the paper minimizes total transmitted power by jointly selecting source powers and receiver power-splitting ratios under known CSI.It studies MRT, ZF, RZF, and the hybrid MRT-ZF scheme; ZF has a closed-form solution, while general schemes use optimization methods.
- Fixed beamforming results: ZF always yields feasible solutions under the stated antenna condition but requires significantly more transmit power than MRT.The antenna condition is that each transmitter has at least as many antennas as there are receivers.
- Joint optimization and comparison: MRT-ZF provides significantly better solution quality than standard beamforming approaches with only a small increase in computational complexity.The joint beamforming, power-allocation, and power-splitting problem is also approximated using SDP with rank relaxation; the SDP gives optimal solutions in the examined instances but becomes prohibitive as problem size grows.
II. SYSTEM MODEL AND PROBLEM FORMULATION
The paper models a K-user MISO interference channel with single-antenna receivers using power splitting to support information decoding and RF energy harvesting. Transmit powers, unit-norm beamforming vectors, channel fading, noise, splitting ratios, and harvesting efficiency define the received SINR and stored electrical power.
- System model: The system has K multiple-antenna sources and K single-antenna receivers, with each transmitter using a unit-norm beamforming vector.The considered symmetric topology uses the minimum required number of transmit antennas for zero-forcing cross-interference.
- System model: Independent frequency-nonselective Rayleigh block fading and AWGN with variance σ2 characterize all wireless links.Channel coefficients remain constant during one slot and change independently between slots.
- Power splitting: Each receiver splits received RF power between baseband information detection and RF-EH circuitry for DC conversion and energy storage.The splitting ratio ρ_i assigns 100ρ_i% of received power to data detection and the remainder to harvesting.
- Receiver constraints: The receiver model includes additional baseband circuit noise, while harvested electrical power depends on RF input and conversion efficiency ζ_i.The conversion efficiency satisfies ζ_i ∈(0, 1].
- Receiver constraints: The resulting SINR describes data detection, while the RF-EH model measures stored electrical power from the harvesting branch.The paper focuses on rectenna input requirements and does not model energy-storage efficiency.
A. Optimization problem
The optimization minimizes total transmitted power while satisfying strict per-receiver QoS and RF-EH requirements through beamforming, power allocation, and power splitting. Introducing variable splitting ratios makes the resulting problem non-convex and difficult to solve.
- Problem formulation: The energy threshold λ_i is intended to trigger efficient rectenna operation during each transmission interval.The associated harvested energy can support equipment involved in signal generation, decoding, amplification, and radio transmission.
- Problem formulation: The objective is to minimize total transmitted power subject to QoS and energy-harvesting constraints involving beamforming, power allocation, and splitting.The QoS constraint requires SINR at least γ_i, while the energy constraint requires RF circuitry input above λ_i.
- Non-convexity: The formulation is non-convex because power splitting parameters destroy the second-order-cone structure of SINR constraints.Even fixed splitting ratios leave RF-EH constraints as sums of concave terms.
- Non-convexity: With fixed splitting parameters, the problem remains a non-convex QCQP; variable splitting parameters further increase its difficulty.This non-convexity motivates specialized optimization procedures for the beamforming schemes studied in the paper.
III. OPTIMAL BEAMFORMING
The paper jointly optimizes beamforming vectors, transmit powers, and power-splitting parameters using an SDP obtained by lifting beamforming variables and relaxing rank constraints. The relaxed problem is convex, but exactness depends on the resulting matrix ranks.
- Joint optimization: Joint optimization of beamforming, power, and splitting provides optimal performance but has relatively high computational complexity.The authors position it mainly for small problems or as a performance benchmark.
- SDP formulation: Beamforming vectors are lifted into matrix variables W_i = v_i v_i^H, converting quadratic expressions into semidefinite constraints.The original rank-one constraints are then relaxed to make the optimization tractable.
- SDP formulation: After rank relaxation, the resulting problem is a convex SDP with a linear objective, positive semidefinite cones, and linear matrix inequalities.Its numerical solution can be obtained with SDP solvers.
- Rank relaxation: If every optimized W_i has rank 1, the relaxed solution is exactly optimal for the original beamforming problem.Otherwise, the relaxed objective supplies a lower bound on the minimum required power, and a feasible solution must be extracted.
- Extensions: Transmit-power constraints can be added as linear constraints without losing convexity or significantly increasing complexity.The SDP framework therefore accommodates additional per-transmitter power limits.
B. Rank Issue and The Proposed Algorithm
The paper addresses possible higher-rank SDP solutions with eigenvector extraction and randomization, while establishing exact rank-one optimality for K = 2 and K = 3. Simulations found rank-one solutions more generally, but that property remains unproven.
- Empirical rank behavior: In simulations, the relaxed SDP always produced rank-one solutions that were also optimal for the original problem, although this property lacks a proof.The authors leave a general guarantee for future work.
- Proposed algorithm: The proposed procedure solves the relaxed SDP, extracts principal eigenvectors for rank-one matrices, and then solves fixed-weight power allocation.Higher-rank cases can use randomization to obtain feasible beamforming vectors.
- Proposed algorithm: The algorithm returns the resulting beamforming vectors after combining extracted beamforming directions with optimized power allocation.This provides a heuristic solution when the relaxation returns higher-rank matrices.
- Rank results: For general K, the SDP is not known to guarantee rank-one solutions.The paper therefore distinguishes proved small-K exactness from the broader empirical observation.
- Rank results: For K = 2 and K = 3, rank-one solutions exist that optimally solve both the relaxed SDP and the original formulation.This theorem provides exactness for these two network sizes.
IV. CONVENTIONAL MISO BEAMFORMING SCHEMES
The section compares conventional fixed beamformers for balancing QoS/SINR and RF-EH requirements. ZF, MRT, RZF, and hybrid MRT-ZF represent different trade-offs between interference suppression and energy transfer.
- ZF beamforming: ZF cancels co-channel interference, favoring SINR constraints but placing less emphasis on RF-EH requirements.
- MRT beamforming: MRT maximizes each receiver's SNR using direct-link channel knowledge, but ignores simultaneous transmissions and can create strong cross-interference.Its low complexity makes it suitable for practical applications with strict computational or time constraints.
- MRT beamforming: The resulting interference can benefit RF-EH scenarios, so MRT targets energy transfer differently from conventional interference-limited designs.
- Design trade-offs: ZF can be problematic for ill-conditioned channels, while optimizing RZF regularization is nontrivial because it depends on channel and design factors.
- RZF beamforming: RZF retains controlled interference and accounts for noise variance, providing a better useful-signal/interference trade-off than ZF for RF-EH constraints.Its regularization parameter controls the amount of interference that remains.
- MRT-ZF beamforming: MRT-ZF linearly combines MRT and ZF to seek a favorable trade-off between SINR and RF-EH constraints.The combination weights determine the contribution of the two beamformers; setting the ZF contribution to zero yields MRT.
V. PROBLEM SOLUTION FOR FIXED BEAMFORMING
For fixed beamforming, the paper derives a closed-form ZF solution and formulates the MRT and RZF cases as SOCP problems. The resulting formulation jointly optimizes transmit powers and power-splitting parameters for fixed weights.
- MRT and RZF beamforming: MRT and RZF share a common problem structure that can be transformed into a convex SOCP formulation.
- SOCP formulation: The SOCP formulation provides optimal, robust, and fast solutions using off-the-shelf optimization solvers.
- ZF beamforming: ZF admits an optimal closed-form solution, and a feasible solution always exists despite QoS and RF-EH constraints.The stated feasibility holds regardless of how demanding those constraints are.
- Fixed-weight optimization: Solving the fixed-weight formulation yields optimal power allocation and power-splitting parameters for any fixed beamforming vectors.The formulation can also be extended with transmit-power constraints.
C. Problem solution for MRT-ZF beamforming
The MRT-ZF solution constructs a hybrid beamformer through variable contributions from MRT and ZF, then uses convexification and fixed-weight optimization to obtain a valid solution.
- Hybrid formulation: MRT-ZF link gains can be represented using variables associated with the MRT, ZF, and cross terms.
- Convexification: The resulting nonconvex problem is relaxed and convexified into a second-order-cone formulation.
- Validity of the relaxation: Because the relaxed problem gives a lower-bound approximation, its solution may not be valid for the original optimization problem.
- Feasible solution construction: The decision variables from the relaxed solution construct normalized MRT-ZF weights, after which the fixed-weight problem is solved to obtain a valid solution.If that solution is infeasible, the procedure employs ZF beamforming.
D. Implementation issues
The evaluation studies centralized optimization under randomly generated channel instances and compares feasibility and optimality across beamforming schemes. Results show distinct feasibility, optimality, and complexity trade-offs.
- Centralized implementation: Most schemes, except ZF, involve global optimization requiring centralized knowledge of all downlink channels.A centralized controller would communicate beamforming vectors, transmit powers, and power-splitting factors, creating complexity and signaling overhead.
- Implementation scope: The proposed network structure targets optimal total energy consumption, while practical distributed implementations remain outside the work's scope.The optimization problems provide theoretical bounds and guidelines for evaluating distributed implementations.
- Evaluation setup: The simulations use randomly generated channel instances with equal QoS and EH thresholds across users and unit conversion efficiency.
- Feasibility: ZF, MRT-ZF, and optimal beamforming produced no infeasible instances, whereas MRT, LPρ, and RZF had many infeasible cases, especially at γ = 20 dB.
- Optimality: MRT required 50%-100% more power when splitting parameters were fixed, and generally outperformed RZF and ZF among feasible fixed-weight cases.The reported advantage over ZF was attributed to MRT's strong cross-interference facilitating EH constraints.
- MRT-ZF performance: MRT-ZF always produced feasible solutions and showed excellent performance, particularly for small numbers of users; its relative performance worsened at higher SINR thresholds and for K = 8.
- Optimal beamforming: The SDP relaxation produced rank-1 solutions in all simulations, indicating optimal solutions for all considered instances.
- Channel conditions: Required transmission power decreases as the ratio of direct to indirect channel variance increases.
B. Parameter effect
The study examines how QoS, RF-EH, channel, and system parameters affect required transmit power, and how interference exploitation changes the ZF-versus-optimal comparison.
- Smaller detection and energy-harvesting thresholds reduce transmit-power requirements because the constraints become easier to satisfy.
- Required power remains nearly flat across some λ_i and γ_i values, indicating threshold ranges with limited additional power impact.
- A QoS threshold of 20 dB requires enough transmitted power to harvest at least −22 dBm.
- Increasing the direct-to-indirect channel-variance ratio δ decreases required transmission power because SINR constraints become easier to satisfy.
- Average transmitted power increases as the number of users K increases.
- Interference exploitation can reduce total transmitted power by up to 45 times at low SINR thresholds in the considered scenarios.Low SINR permits additional interference that benefits RF-EH without violating QoS constraints; high SINR instead requires near-total interference cancellation.
- The ZF/optimal power ratio increases with the RF-EH threshold because the optimal scheme manages interference to support energy harvesting.
D. Trade-off between beamforming schemes
The paper compares fixed and jointly optimized beamforming under QoS and RF-EH constraints, balancing solution quality against computational cost as problem size grows.
- The SDP beamforming scheme provides optimal solutions for all examined problem instances, making it a benchmark for solution quality.
- ZF and MRT-ZF require less than 0.025s and 0.08s, respectively, across considered K, whereas SDP requires about 190s for K = 24.
- Optimal beamforming becomes prohibitive especially for K ≥12, while MRT-ZF remains substantially faster as K increases.
- For K = 24, MRT-ZF takes only 3.3 times ZF's execution time, whereas optimal beamforming exceeds MRT-ZF by more than 2500 times.
- MRT-ZF offers the best trade-off between optimality, feasibility, and execution time, particularly for large K.
- The study evaluates ZF, MRT, RZF, MRT-ZF, and variable-weight beamforming, with SOCP algorithms for fixed designs and SDP for joint optimization.
- The proposed approximate SDP formulation is theoretically optimal for two and three users and empirically achieved optimal performance.
APPENDIX A PROOF OF THEOREM 2
The appendix proves the closed-form optimality structure for ZF by reducing the problem into independent per-user subproblems and identifying the smallest feasible scalar solution.
- Under ZF, the constraint sets separate, allowing the overall problem to decompose into K independent subproblems.
- Feasibility requires x_i > max(λ_i, α_i + β_i), with ρ_i constrained to the interval (0, 1).
- At least one constraint must bind at the optimum; binding either candidate equality yields the same solution.
- Because x_i increases monotonically with P_i, the optimal solution is the smallest feasible x_i satisfying the derived quadratic condition.
- The optimal ZF power is obtained by summing the optimal solutions of the independent per-user subproblems.