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Robust and Secure Sum-Rate Maximization for Multiuser MISO Downlink Systems with Self-sustainable IRS
Shaokang Hu, Zhiqiang Wei, Yuanxin Cai, Chang Liu, Derrick Wing Kwan Ng, Jinhong Yuan
TL;DR
The paper addresses robust and secure sum-rate maximization for multiuser MISO downlink systems using a self-sustainable IRS under secure-communication and imperfect-CSI requirements. It jointly designs AP beamforming, AN, discrete IRS phases, and harvesting schedules through an iterative optimization approach. Simulations show trade-offs between sum-rate and self-sustainability, saturation with many harvesting elements, and strong performance from low-resolution phase shifters.
Problem
The central problem is designing secure IRS-assisted multiuser MISO downlinks when IRS energy consumption, nonlinear harvesting, and imperfect CSI cannot be ignored.
Method
The paper jointly optimizes beamformers, AN covariance, discrete IRS phase shifts, and harvesting schedules using an iterative algorithm with S-procedure, SCA, and SDR.
Results
Simulations reveal a trade-off between sum-rate and IRS self-sustainability, saturation with many harvesting elements, and considerable performance from small-bit discrete phase shifters.
Takeaways & Limitations
Self-sustainable IRSs can enhance physical-layer security while remaining effective with limited-resolution phase shifters.
Abstract
from arXiv · showhide
This paper investigates robust and secure multiuser multiple-input single-output (MISO) downlink communications assisted by a self-sustainable intelligent reflection surface (IRS), which can simultaneously reflect and harvest energy from the received signals. We study the joint design of beamformers at an access point (AP) and the phase shifts as well as the energy harvesting schedule at the IRS for maximizing the system sum-rate. The design is formulated as a non-convex optimization problem taking into account the wireless energy harvesting capability of IRS elements, secure communications, and the robustness against the impact of channel state information (CSI) imperfection. Subsequently, we propose a computationally-efficient iterative algorithm to obtain a suboptimal solution to the design problem. In each iteration, S-procedure and the successive convex approximation are adopted to handle the intermediate optimization problem. Our simulation results unveil that: 1) there is a non-trivial trade-off between the system sum-rate and the self-sustainability of the IRS; 2) the performance gain achieved by the proposed scheme is saturated with a large number of energy harvesting IRS elements; 3) an IRS equipped with small bit-resolution discrete phase shifters is sufficient to achieve a considerable system sum-rate of the ideal case with continuous phase shifts.
I. INTRODUCTION
The paper targets sustainable, secure, and robust 6G-compatible IRS-assisted downlink communication while accounting for IRS energy consumption and imperfect CSI. It jointly optimizes transmission, IRS operation, and harvesting resources to maximize sum-rate.
- Future wireless networks must balance capacity growth with communication cost and energy efficiency.
- IRSs use controllable phase shifts to coherently combine reflected signals and provide additional wireless-channel degrees of freedom.
- A hundred-element IRS with 3-bit phase shifters consumes 0.15 W, making IRS operational power comparable to radiated communication power.
- RF energy harvesting offers a controllable alternative to grid or battery powering for uninterrupted IRS operation.
- Perfect eavesdropper CSI is difficult to obtain, motivating robust beamforming against CSI imperfections.
- The proposed design jointly optimizes information and AN beamformers, discrete IRS phase shifts, and the harvesting schedule under security and imperfect-CSI constraints.
- The resulting non-convex mixed-integer problem is addressed with an iterative suboptimal algorithm using transformations, SCA, and SDR.
B. IRS Model
The self-sustainable IRS model switches individual elements between reflection and energy-harvesting modes, using discrete phase shifts and a nonlinear RF-to-DC conversion model.
- Each IRS element is selected for either reflection or energy harvesting, so the schedule balances reflected communication signals against harvested power.
- Reflection-mode elements reflect all incident signals, whereas harvesting-mode elements absorb received signal energy and do not reflect.
- 1) Phase Shift Model:: The IRS uses uniformly quantized phase shifts with B = 2^b realizable levels and fixes each reflection amplitude coefficient to 1.
- Harvested RF power passes through matching, voltage-multiplication, and energy-storage stages before delivery to the IRS load.
- The IRS includes a controller that switches element operation between reflection and power-harvesting modes.
- 2) Non-linear Energy Harvesting Model:: The adopted nonlinear harvesting model uses a sigmoidal function of received RF power and has a maximum harvestable power parameter M_P.
C. Channel State Information
The paper models imperfect CSI with bounded uncertainty sets for cascaded, AP–IRS, direct user, and eavesdropper channels, then jointly designs secure, self-sustainable IRS-assisted transmission.
- Channel uncertainty: Bounded CSI error models cover cascaded AP–IRS–user, AP–IRS, direct AP–user, cascaded AP–IRS–eavesdropper, and direct AP–eavesdropper channels.The model guarantees performance for errors within specified norm bounds.
- Channel uncertainty: The AP–IRS channel is treated as quasi-static and estimated over a large timescale because AP and IRS positions are usually fixed.The paper cites dual-link pilot transmission and coordinate-descent-based estimation as examples.
- Optimization formulation: The optimization maximizes the worst-case system sum-rate while jointly selecting AP precoding, artificial-noise covariance, IRS modes, and discrete phase shifts.The formulation also enforces AP power, discrete-phase, harvested-power, and information-leakage constraints.
- Secure transmission: Information-leakage limits τ_k,j control the maximum tolerable leakage from legitimate user k to potential eavesdropper j under CSI uncertainty.These constraints provide a lower bound on the system secrecy rate and allow per-eavesdropper secrecy control.
IV. SOLUTION OF THE OPTIMIZATION PROBLEM
The solution procedure reformulates the coupled mixed-discrete robust problem using augmented mode variables, matrix lifting, lower bounds, and slack-variable transformations before solving the resulting constrained problem.
- Problem reformulation: The original problem is non-convex because beamformers, artificial noise, IRS phases, and binary modes are coupled, while CSI uncertainty creates infinitely many constraints.Exhaustive search is computationally intractable even for moderate system sizes.
- Iterative solution: SCA handles difference-of-convex functions after coupling decomposition, while the S-procedure addresses infinitely many constraints caused by imperfect CSI.The resulting iterative method jointly optimizes the variables in each iteration.
- Problem reformulation: Augmented mode-selection variables combine IRS operating modes and discrete phase shifts into a generalized finite set.The zero value denotes harvesting, while nonzero discrete complex values denote reflection modes.
- Matrix lifting: Matrix lifting rewrites SINR, leakage, power, and related constraints using W = ww^H and V = vv^H representations.The lifted formulation preserves the relationships among beamforming, IRS selection, and phase variables for subsequent convexification.
- Mode selection: Binary mode-selection variables indicate which of B+1 modes is selected for each IRS element, and continuous constraints replace the binary constraint.The first row of the selection matrix represents energy-harvesting activity.
- Convexification: A logarithmic lower bound, auxiliary slack variables, and equivalent constraint transformations produce a tractable intermediate optimization problem.The reformulation replaces the objective with a lower bound and separates selected non-convex constraints.
B. Optimization Variables Decoupling
The decoupling step targets three variable couplings—fractional SINR terms, binary–continuous products, and matrix products—using equivalent reformulations and auxiliary variables.
- Coupling structure: The formulation contains fractional SINR variables, binary–continuous products, and products among W_k, Z, and V.These couplings are responsible for much of the non-convexity in the objective and constraints.
- SINR reformulation: The SINR is rewritten in difference-of-concave form to address its fractional coupling.The resulting concave components depend on auxiliary variables ξ_k and ι_k.
- Mode coupling: Big-M reformulation converts logical products involving mode-selection variables into equivalent constraints with auxiliary variables.The constant MBig is chosen sufficiently large, and channel-error vectors are explicitly represented.
- Element-wise structure: Element-selection matrices isolate each IRS element’s contribution to received power and channel-error terms.T_n depends only on the n-th IRS element, exploiting independent AP–IRS channel uncertainty across elements.
- Matrix-product decoupling: A transformation decomposes products involving {W_k, V} and {Z, V} into separated variables for the objective and robust constraints.The paper illustrates this decomposition using constraint C9 and applies the same approach to C5 and C10.
C. S-Procedure
The S-procedure converts robust constraints over continuous CSI-error sets into finitely many linear matrix inequalities or equivalent finite constraint systems.
- Robust reformulation: CSI uncertainty creates infinitely many possibilities for constraints C3f, C5, C9, and C10, motivating the use of the S-procedure.The method converts robust conditions into finite-dimensional matrix constraints.
- S-procedure: The standard S-procedure replaces an implication between two quadratic inequalities with a matrix condition involving a nonnegative scalar.The equivalence requires the stated feasibility condition on the quadratic functions.
- Application to C3f: Constraint C3f is rewritten under the channel-uncertainty model and transformed into finitely many LMIs using the S-procedure.The resulting formulation introduces nonnegative variables associated with the uncertainty conditions.
- Application to C5, C9, and C10: For constraints C5, C9, and C10, slack variables are introduced to manage infinitely many quadratic matrix inequalities caused by CSI uncertainty.The same robust-conversion strategy is then applied to obtain finite constraint representations.
- Generalized S-procedure: The generalized S-procedure converts the remaining infinite inequalities in C5b, C12, and C13 into equivalent finite constraints.This extension applies to matrix-valued quadratic forms under positive-semidefinite uncertainty conditions.
D. SCA- and SDR-based Iterative Algorithm
The algorithm combines SCA and SDR to address the non-convex resource-allocation problem, iteratively solving convex approximations while retaining polynomial-time complexity. The SDR relaxation is tight, and joint optimization can outperform alternating optimization.
- SCA-based iterative optimization: SCA replaces non-convex constraints and objective terms with first-order Taylor-based lower bounds at feasible iterative points.The procedure targets constraints C4d, C5a, C9, C10, C11b, and the objective function.
- SCA-based iterative optimization: The proposed algorithm repeatedly solves the convexified problem, updates variables, and stops upon convergence or reaching tmax.Algorithm 1 initializes tmax, sets t = 0, and updates the iteration variables in the main SCA loop.
- SDR relaxation: SDR relaxes the rank constraint in C8 so the resulting problem can be solved with a standard convex-programming solver.Theorem 1 states that a rank-one solution can always be obtained when Pmax > 0 and the relaxed problem is feasible.
- SCA-based iterative optimization: The SCA procedure provides a lower bound for the original problem, which is tightened by iteratively updating feasible solutions.The resulting algorithm converges to a suboptimal solution according to the cited convergence proof.
- Complexity and comparison: Both the proposed algorithm and AO have polynomial-time complexity, while the proposed method jointly optimizes all variables in each iteration.The paper contrasts this with AO, which separately and alternately optimizes variables, and reports higher performance for the proposed algorithm.
A. Simulation Setup
The simulations place users and potential eavesdroppers around concentric circles and evaluate the proposed design against upper bounds, baselines, and AO. Channel uncertainty and eavesdropper capacity constraints are included in the default setup.
- Network geometry: Users and most potential eavesdroppers are randomly distributed on circles with radii 1 m and 20 m, while another eavesdropper lies on an 80 m circle centered at the AP.The AP and the users’ circle center are separated by d0 = 60 m.
- Channel and security settings: The default simulations use normalized channel-estimation mean square error κ^2 = 0.1 and cap potential-eavesdropper channel capacity at τ = 1.5 bits/s/Hz.The same normalized uncertainty parameter κ is applied across the listed channel links unless otherwise specified.
- Comparison schemes: The comparison includes continuous-phase and no-eavesdropper upper bounds, non-robust no-IRS and self-sustainable-IRS baselines, imperfect-CSI and random-phase variants, and AO.The baselines isolate robustness, security, IRS deployment, phase randomness, and energy-consumption effects.
B. Average System Sum-Rate versus the Distance Between AP and IRS
The proposed IRS-assisted scheme improves average system sum-rate across AP–IRS distances by jointly optimizing beamforming and IRS phase shifts. Its advantage reflects additional IRS path gain, security-aware artificial-noise design, and joint optimization beyond fixed, random, or alternating baselines.
- The proposed scheme achieves substantially higher average system sum-rate than the baseline without an IRS across AP–IRS distances.
- The IRS adds an optimizable reflected path gain, improving the end-to-end channel and system performance.
- Joint optimization of transmit beamforming and IRS phase shifts provides a considerable gain over the fixed-beamforming baseline.
- Random IRS phase shifts prevent reflected information beams from consistently aligning with desired channels, limiting the usable path gain.
- The proposed scheme outperforms alternating optimization because it jointly optimizes all design variables in each iteration rather than splitting them into two subproblems.
APPENDIX A PROOF OF THEOREM 1
The proof transforms a rank-relaxed problem into an equivalent convex formulation, then uses strong duality and KKT conditions to characterize the optimal beamforming matrix.
- Problem transformation: Three slack variables and constraints transform the rank-relaxed problem into an equivalent formulation.The introduced variables are CC5a,k,j, CC9,k, and CC10,i,k; the added constraints are C14, C15, and C16.
- Duality analysis: The transformed problem is jointly convex and satisfies Slater’s constraint qualification, establishing strong duality.Its Lagrangian is then formed with respect to W_k.
- KKT analysis: KKT conditions are examined using scalar and matrix Lagrange multipliers associated with the transformed constraints.The multipliers correspond to constraints C1, C3f, C5a, C7, C9, C10, C14, C15, and C16.
- Rank characterization: The optimal beamforming matrix W_k lies in the null space of the matrix M*_C7,k.The proof investigates the structure of M*_C7,k to reveal the rank of W*_k.
- Rank characterization: The proof uses a unit-norm eigenvector spanning the relevant null space to characterize W*_k, with ν enforcing the transmitter-power constraint.The matrix C7,k is stated to be positive definite and full rank; violating C1 would make the dual problem unbounded.
- Problem transformation: The transformed formulation includes semidefinite constraints linking CC9,k and CC10,i,k to W_k and their associated matrices.These are stated as constraints C15 and C16 for all relevant users and indices.