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Intelligent Reflecting Surface Aided MIMO Broadcasting for Simultaneous Wireless Information and Power Transfer

Cunhua Pan, Hong Ren, Kezhi Wang, Maged Elkashlan, Arumugam Nallanathan, Jiangzhou Wang, Lajos Hanzo

arXiv:1908.04863v4eess.SP

TL;DR

IRS-assisted SWIPT addresses a research area with few existing contributions while targeting WSR maximization in energy-harvesting systems. The paper develops a BCD-based solution with convergent iterative subproblem algorithms, reports KKT convergence and rapid convergence, and finds that IRS enhances SWIPT performance.

  • Problem

    IRS-assisted SWIPT has few existing contributions, despite SWIPT's appeal for future energy-hungry Internet-of-Things applications.

  • Method

    The paper uses a block coordinate descent algorithm and low-complexity iterative algorithms for its subproblems, with bisection enabled by a monotonically decreasing function.

  • Results

    The proposed algorithms converge to KKT points, while simulations show that IRS enhances SWIPT performance and the algorithm converges rapidly.

  • Takeaways & Limitations

    Invoking an IRS in SWIPT enhances system performance, and the rapidly convergent algorithm is appealing for practical applications.

  • Takeaways & Limitations

    The channel information is assumed imperfectly known, and discrete phase-shift design remains an open issue.

Abstract

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An intelligent reflecting surface (IRS) is invoked for enhancing the energy harvesting performance of a simultaneous wireless information and power transfer (SWIPT) aided system. Specifically, an IRS-assisted SWIPT system is considered, where a multi-antenna aided base station (BS) communicates with several multi-antenna assisted information receivers (IRs), while guaranteeing the energy harvesting requirement of the energy receivers (ERs). To maximize the weighted sum rate (WSR) of IRs, the transmit precoding (TPC) matrices of the BS and passive phase shift matrix of the IRS should be jointly optimized. To tackle this challenging optimization problem, we first adopt the classic block coordinate descent (BCD) algorithm for decoupling the original optimization problem into several subproblems and alternatively optimize the TPC matrices and the phase shift matrix. For each subproblem, we provide a low-complexity iterative algorithm, which is guaranteed to converge to the Karush-Kuhn-Tucker (KKT) point of each subproblem. The BCD algorithm is rigorously proved to converge to the KKT point of the original problem. We also conceive a feasibility checking method to study its feasibility. Our extensive simulation results confirm that employing IRSs in SWIPT beneficially enhances the system performance and the proposed BCD algorithm converges rapidly, which is appealing for practical applications.

I. INTRODUCTION

IRS-assisted SWIPT uses passive reflecting components alongside BS precoding to improve energy harvesting and information transmission, while the coupled design creates a difficult WSR optimization problem. The paper formulates this problem and develops convergent algorithms for its subproblems, feasibility checking, and overall solution.

  • IRS-assisted wireless communication: IRSs are passive, low-power reflecting devices that can reconfigure wireless propagation and integrate into existing networks without changing physical-layer standards.They reflect incident signals without decoding or amplification and can be fabricated in small, lightweight forms for building surfaces.
  • Optimization challenge: Jointly designing BS active beamforming and IRS passive beamforming is difficult because their optimization variables are coupled.This coupling leads to a complex optimization problem.
  • SWIPT motivation: SWIPT supports energy-hungry IoT networks, but severe channel attenuation weakens ER received power and limits ER link distance.The paper addresses this limitation by placing an IRS near ERs to provide additional transmission links and enhance harvested power.
  • Problem formulation: The paper formulates WSR maximization for IRS-assisted SWIPT MIMO by jointly optimizing BS TPC matrices and IRS phase shifts while satisfying ER energy-harvesting requirements.The paper identifies this as the first treatment of WSR maximization in IRS-assisted SWIPT MIMO systems.
  • Proposed algorithms: BCD alternately optimizes TPC matrices and IRS phase shifts, with iterative subproblem algorithms guaranteed to converge to their respective KKT points.The TPC and phase-shift procedures are developed separately, including an SCA-based approach and a non-convex QCQP formulation with an energy-harvesting constraint.
  • Analysis and results: The overall BCD algorithm converges to a KKT point of the original problem, while simulations show enhanced system performance, rapid convergence, expanded ER operating range, and sensitivity to IRS location and path-loss exponent.The feasibility issue is also studied through an alternative optimization problem and iterative algorithm.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system models an IRS-assisted multiuser MIMO SWIPT downlink in which the IRS supports both information reception and energy harvesting. It specifies the channel, signal, rate, harvesting, and CSI assumptions used for the formulation.

  • A. System Model: The BS communicates with multiple multi-antenna IRs and ERs over the same frequency band in an IRS-aided multiuser MIMO downlink.The model includes K_I information receivers, K_E energy receivers, and an IRS with M reflective elements.
  • A. System Model: The IRS is placed near energy receivers to extend sensor operational range and enhance harvested power under severe channel attenuation.Low-power sensors require operational energy, but deployment close to the BS limits implementation; the IRS provides additional transmission links.
  • A. System Model: Each IR receives BS transmission through direct and IRS-reflected channels, with d data streams and a linear TPC matrix F_k.The channel model includes BS–IRS, BS–IR, BS–ER, IRS–IR, and IRS–ER links, while multiple reflections are ignored.
  • A. System Model: The BS is assumed to know all relevant CSIs and computes the IRS phases, so the resulting performance represents an upper bound.The phase shifts are fed to the IRS controller through dedicated feedback channels.
  • A. System Model: The IR performance is measured by achievable data rates based on equivalent channels and interference-plus-noise covariance matrices.The equivalent IR channel combines the direct BS–IR path with the IRS-assisted path, and the rate is expressed in nat/s/Hz.
  • A. System Model: The ER constraint requires the weighted harvested power across all ERs to exceed a predefined threshold.The weights α_l prioritize ERs, while Q̄ specifies the minimum weighted harvested power.

B. Problem Formulation

The problem jointly optimizes BS transmit precoding and IRS phase shifts to maximize IR weighted sum rate while satisfying power, unit-modulus, and energy-harvesting constraints. The resulting problem is difficult because its variables are coupled, its EH constraint is non-convex, and feasibility is not guaranteed.

  • B. Problem Formulation: The objective is to jointly optimize TPC matrices F and IRS phase matrix Φ for maximum IR weighted sum rate.The formulation includes scheduling weights ω_k for IR priorities.
  • B. Problem Formulation: The optimization imposes a BS power limit, unit-norm phase-shifter constraints, and a weighted harvested-power requirement.The IRS is passive, and the phase shifts are computed at the BS using CSI and related parameters.
  • B. Problem Formulation: The TPC matrices and IRS phase shifts are coupled, making the joint optimization difficult to solve.The phase variables are represented through φ_m = e^jθ_m.
  • B. Problem Formulation: The EH constraint is non-convex, so algorithms developed for the unconstrained WSR problem cannot be directly applied.Removing the EH constraint recovers a recently studied WSR maximization problem.
  • B. Problem Formulation: The problem may be infeasible because the power and harvested-energy constraints can conflict.The paper first develops an algorithm assuming feasibility and then studies feasibility separately.

III. LOW-COMPLEXITY ALGORITHM DEVELOPMENT

The original WSR problem is reformulated into an equivalent WMMSE-based form that enables block coordinate descent. Auxiliary decoding and weighting matrices are optimized alongside the TPC and IRS variables.

  • III. LOW-COMPLEXITY ALGORITHM DEVELOPMENT: The paper transforms the original problem into a tractable formulation that decouples TPC matrices from IRS phase shifts.The classic BCD algorithm then alternately optimizes variable blocks.
  • III. LOW-COMPLEXITY ALGORITHM DEVELOPMENT: A WMMSE reformulation replaces the complex rate objective with an equivalent objective involving auxiliary matrices W and decoding matrices U.The reformulated objective uses h_k(W,U,F,Φ) = log|W_k| − Tr(W_kE_k) + d.
  • III. LOW-COMPLEXITY ALGORITHM DEVELOPMENT: The BCD procedure updates one variable set at a time while holding the remaining matrices fixed.The reformulated objective is easier to handle despite introducing more optimization variables.
  • III. LOW-COMPLEXITY ALGORITHM DEVELOPMENT: For fixed Φ, W, and F, the optimal decoding and auxiliary matrices U and W are obtained from first-order conditions.The decoding matrix estimates each IR’s signal vector, and the resulting MSE matrix determines the auxiliary update.
  • III. LOW-COMPLEXITY ALGORITHM DEVELOPMENT: After updating U and W, the remaining BCD subproblem optimizes the TPC matrices F and phase shifts Φ.The constraints retain the original power, energy-harvesting, unit-modulus, and reformulated-problem conditions.

B. Optimizing the Precoding Matrices F

The precoding subproblem is addressed through convex optimization, dual decomposition, bisection, and successive convex approximation. The resulting iterative algorithm is designed for lower complexity and is proved to converge to a KKT point.

  • B. Optimizing the Precoding Matrices F: With W, U, and Φ fixed, the TPC subproblem is convex and can be solved by standard convex solvers, but their complexity is high.Convexity follows from the objective and relevant constraints being convex in F.
  • B. Optimizing the Precoding Matrices F: The TPC update is obtained from first-order conditions, with dual variables selected to satisfy complementary slackness for the power-related constraints.The solution uses matrix pseudoinverses and dual variables λ and μ.
  • B. Optimizing the Precoding Matrices F: The non-convex EH-constrained subproblem is handled by successive convex approximation.The method iteratively replaces the relevant expression with a first-order Taylor approximation and solves the resulting convex subproblem.
  • B. Optimizing the Precoding Matrices F: A low-complexity nearly optimal TPC solution is derived using Lagrangian dual decomposition.Slater’s condition gives zero duality gap for the convex subproblem under the stated feasibility assumption.
  • B. Optimizing the Precoding Matrices F: The total power P(λ) decreases monotonically with λ, enabling bisection search for the dual variable.This monotonicity is stated as Lemma 1 and is used in Algorithm 1.
  • B. Optimizing the Precoding Matrices F: SVD reduces repeated matrix-inverse calculations by expressing (A + λI)^−1 through diagonal operations and matrix products.The resulting per-iteration computation has lower complexity than directly calculating an inverse of the same dimension.
  • B. Optimizing the Precoding Matrices F: Algorithm 2 generates precoding sequences that converge to the KKT optimum point of the TPC subproblem.The convergence claim is given in Theorem 1 for the sequence produced by the SCA algorithm.
  • B. Optimizing the Precoding Matrices F: The complexity analysis identifies the bisection-based TPC update as the main contribution to each SCA iteration.The analysis assumes N_B ≥ N_I ≥ d and derives the total complexity from the inner algorithm’s iterations.

C. Optimizing the Phase Shift Matrix

The phase-shift subproblem is reformulated using successive convex approximation and majorization-minimization to handle its non-convex constraints. The resulting algorithm is globally optimal for an intermediate problem and converges to a KKT point for the phase-shift problem.

  • The phase-shift matrix Φ is optimized while the transmit precoding matrix is fixed.
  • The quadratic phase-shift objective is handled with SCA by replacing a convex term with its first-order Taylor lower bound.This converts the relevant constraint into a linear constraint.
  • MM constructs tractable upper bounds for the objective and solves successive approximate subproblems.The surrogate must satisfy three stated conditions before each update.
  • A price mechanism with bisection search solves the non-convex phase-shift subproblem despite its unit-modulus constraint.The search exploits the monotonicity of J(p).
  • Algorithm 3 finds the globally optimal solution of Problems (49) and (48).
  • Algorithm 4 produces a convergent objective sequence whose final phase-shift solution satisfies the KKT conditions of Problem (31).Its dominant eigenvalue calculation has complexity O(M^3).

D. Overall Algorithm to Solve Problem (8)

The overall BCD algorithm alternates optimization of the BS precoders, IRS phase shifts, decoding matrices, and auxiliary matrices. Its objective sequence is guaranteed to converge to a KKT solution of the original problem, with rapid convergence observed in simulations.

  • Algorithm 5 alternates optimal updates of precoding matrices, phase shifts, decoding matrices, and auxiliary matrices.
  • The BCD objective-value sequence is guaranteed to converge, and the final solution satisfies the KKT conditions of Problem (8).
  • Simulation results show that Algorithm 5 converges rapidly, demonstrating its low complexity.

IV. FEASIBILITY CHECK FOR PROBLEM (8)

The EH and transmit-power constraints can make Problem (8) infeasible, so feasibility must be checked before optimizing the WSR. The proposed check compares the maximum achievable harvested power with the required threshold.

  • Problem (8) may be infeasible because energy-harvesting and limited-transmit-power constraints conflict.
  • The feasibility method constructs an optimization problem that maximizes the harvested-power objective.
  • Problem (8) is feasible when the optimized objective exceeds the harvested-power requirement Q̄; otherwise, it is infeasible.
  • Because precoding matrices and phase shifts are coupled, global optimization is difficult, so alternating optimization obtains a suboptimal solution.
  • For fixed phase shifts, the precoding subproblem uses an eigenvalue and eigenvector of G in its solution.
  • For fixed precoding, the phase-shift objective is treated as a difference-of-convex program and solved using SCA.

V. SIMULATION RESULTS

Simulations evaluate harvested power, convergence, and WSR under varying IRS size, energy requirements, path loss, and receiver locations. IRS assistance improves harvesting and WSR, while the BCD method converges rapidly and generally outperforms the fixed-phase and No-IRS schemes.

  • Simulation setup: The simulations average results over 100 random locations and channel generations in an IRS-aided SWIPT MIMO setting.The scenario contains four ERs and two IRs, with Rayleigh and Rician fading assigned according to link geometry.
  • Harvested power: Harvested power decreases as ERs move farther from the BS, while IRS assistance provides more harvested power than No-IRS, especially with larger M.The reflected IRS link supplies an additional strong link and expands the ER operational range.
  • Harvested power: At Q̄ = 2 × 10^-4 W, No-IRS supports only 5.5 m, whereas M = 40 phase shifters support distances up to 9 m.
  • Convergence: The WSR increases monotonically with BCD iterations, and a few iterations achieve a large portion of the converged value.
  • WSR performance: As M increases, BCD significantly outperforms fixed phase and No-IRS; at M = 60, its gain over No-IRS reaches up to 10 bit/s/Hz.
  • Propagation and location: Increasing αIRS sharply reduces IRS-aided WSR; at αIRS = 3, the gain over No-IRS is only 7 bit/s/Hz.The paper attributes this to weaker reflected signal power and recommends carefully selecting an obstacle-free, low-loss location.

VI. CONCLUSIONS

The paper formulates IRS-assisted SWIPT MIMO design as WSR maximization under ER energy-harvesting requirements, jointly optimizing BS precoding and IRS phase shifts. Its BCD-based algorithms are shown to converge appropriately, while simulations indicate performance enhancement and rapid convergence.

  • The system maximizes IR weighted sum rate while guaranteeing ER energy-harvesting requirements under non-convex unit-modulus constraints.
  • The proposed BCD algorithm alternately optimizes BS transmit-precoding matrices and the IRS phase-shift matrix.
  • Each BCD subproblem uses a low-complexity iterative algorithm whose generated solutions are guaranteed to converge to a KKT point.
  • Simulation results demonstrate that IRS deployment enhances SWIPT system performance.
  • The proposed BCD algorithm converges rapidly and is described as suitable for practical implementations.
  • Perfect CSI is assumed at the BS; robust design with imperfect CSI and discrete phase-shift design remain future work.

APPENDIX A PROOF OF LEMMA 1

The appendix establishes convergence and optimality properties for the phase-shift iterative procedure. It shows feasibility, monotonic objective behavior, convergence of the objective sequence, and satisfaction of the KKT conditions.

  • The appendix also proves that the parameter-dependent optimal quantity P(λ) is non-increasing in λ.
  • For the other case, contradiction arguments establish that the solution produced by Algorithm 3 is globally optimal for the equivalent reformulated problem.
  • For one case, the algorithm directly attains the optimal solution when the relevant constraint is not tight at optimum.
  • The proof establishes that the phase-shift solution sequence remains feasible for the problem's unit-modulus and energy-harvesting constraints.
  • The objective-function sequence is monotonically decreasing and therefore converges because the objective has a lower bound.
  • The converged phase-shift solution satisfies the KKT conditions of the target problem.

APPENDIX D PROOF OF THEOREM 4

The appendix proves convergence properties for the full alternating algorithm. The generated solution sequence remains feasible, and the converged solution satisfies the KKT conditions of the original problem.

  • Algorithm 5 generates a solution sequence that is always feasible for Problem (8).
  • The appendix invokes a monotonicity argument to establish the convergence behavior of Algorithm 5.
  • The converged solution is represented by {W⋆, U⋆, F⋆, Φ⋆} and is analyzed through the KKT conditions of the constituent problem.
  • The listed stationarity and constraint equations together constitute exactly the KKT conditions of Problem (8).
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