Source-linked AI summary
MISO Wireless Communication Systems via Intelligent Reflecting Surfaces
Xianghao Yu, Dongfang Xu, Robert Schober
TL;DR
The paper addresses joint beamformer and IRS phase-shift design for spectral- and energy-efficient IRS-assisted wireless communication. It proposes fixed point iteration and manifold optimization, reporting improved spectral efficiency and lower complexity than the state-of-the-art approach.
Problem
Future wireless networks need spectral- and energy-efficient designs, while the optimal joint design of AP beamforming and IRS phase shifts remains insufficiently understood.
Method
The paper jointly optimizes the AP beamformer and IRS phase shifts using fixed point iteration and manifold optimization for a point-to-point IRS-assisted MISO system.
Results
The proposed algorithms achieve higher spectral efficiency and lower computational complexity than the state-of-the-art SDR method, while guaranteeing locally optimal solutions.
Takeaways & Limitations
Increasing IRS reflecting elements is more efficient than enlarging the AP transmit antenna array for improving spectral efficiency, with lower energy consumption from passive IRS deployment.
Abstract
from arXiv · showhide
Intelligent reflecting surfaces (IRSs) have received considerable attention from the wireless communications research community recently. In particular, as low-cost passive devices, IRSs enable the control of the wireless propagation environment, which is not possible in conventional wireless networks. To take full advantage of such IRS-assisted communication systems, both the beamformer at the access point (AP) and the phase shifts at the IRS need to be optimally designed. However, thus far, the optimal design is not well understood. In this paper, a point-to-point IRS-assisted multiple-input single-output (MISO) communication system is investigated. The beamformer at the AP and the IRS phase shifts are jointly optimized to maximize the spectral efficiency. Two efficient algorithms exploiting fixed point iteration and manifold optimization techniques, respectively, are developed for solving the resulting non-convex optimization problem. The proposed algorithms not only achieve a higher spectral efficiency but also lead to a lower computational complexity than the state-of-the-art approach. Simulation results reveal that deploying large-scale IRSs in wireless systems is more efficient than increasing the antenna array size at the AP for enhancing both the spectral and the energy efficiency.
I. INTRODUCTION
The paper motivates IRS-assisted wireless communication as a low-cost way to control propagation while avoiding additional energy-hungry RF chains. It studies joint AP beamformer and IRS phase-shift design for spectral efficiency under a non-convex optimization problem.
- Motivation: Growing multimedia demand requires wireless networks with higher capacity and motivates spectral- and energy-efficient designs.Conventional capacity-enhancement approaches add antenna elements, APs, or high-frequency RF chains, increasing cost and power consumption.
- IRS opportunity: IRSs use passive metasurface devices to manipulate electromagnetic waves and change signal transmission direction.Their passive implementation can control propagation without deploying additional energy-hungry RF chains.
- System and design problem: Effective IRS-assisted communication requires jointly designing AP transmit beamforming and IRS phase shifts.The paper focuses on a point-to-point MISO system with programmable phase shifters, quasi-static flat fading, and perfect CSI assumptions.
- Open challenge: Optimizing IRS phase shifts is difficult because the associated unit modulus constraints make the objective non-convex.The authors state that globally optimal solutions are generally intractable and therefore target locally optimal solutions.
- Proposed approach: Two algorithms based on fixed point iteration and manifold optimization jointly optimize the AP beamformer and IRS phase shifts.The proposed methods are reported to achieve locally optimal solutions and outperform the state-of-the-art SDR method in spectral efficiency and computational complexity.
A. Problem Formulation
The problem formulation optimizes the AP beamforming vector and IRS phase-shift matrix to maximize achievable spectral efficiency. Reformulation produces a QCQP with non-convex unit modulus constraints, motivating alternatives to SDR.
- Reformulation: The reformulated problem introduces optimization variable v, composed of an auxiliary variable t and the IRS phase shifts.The first M elements encode the phase-shift variables, while the corresponding beamformer is obtained afterward.
- Beamformer recovery: The beamforming vector is optimally recovered from the phase-shift solution using maximum ratio transmission.This separates beamformer recovery from solving the reformulated phase-shift problem.
- Optimization structure: P1 is an NP-hard QCQP because its unit modulus constraints are intrinsically non-convex.The stated problem size is M + 1.
- Baseline: SDR relaxes the rank-one constraint and uses Gaussian randomization, yielding an approximate solution with asymptotic objective value at least π/4 of optimum.The paper uses SDR as the state-of-the-art baseline and notes its computational expense for large IRS sizes.
B. Fixed Point Iteration
The fixed point iteration method solves the reformulated unit-modulus problem by iteratively improving its objective. Its limit points are locally optimal, and the resulting algorithm also recovers the IRS phase shifts and AP beamformer.
- Fixed Point Iteration: A limit point of the fixed point iteration is a locally optimal solution for P1.The result follows because the iteration uses a non-negative real-valued quantity and satisfies the cited local-optimality condition.
- Convergence: The iteration monotonically increases the objective value and converges because that value is upper bounded.The proof uses the Hermitian structure of R and the unit-modulus constraints.
- Implementation: Algorithm 1 initializes v, repeatedly applies the fixed point update, and stops when the objective increment falls below a small threshold ϵ.After convergence, it takes the first M elements for the IRS phase shifts and designs the AP beamformer according to (6).
C. Manifold Optimization
Manifold optimization treats the unit-modulus phase-shift constraints as a Riemannian manifold and applies gradient-based search with transport and retraction operations. The resulting conjugate-gradient algorithm is guaranteed to converge to a critical point.
- Manifold formulation: The unit-modulus constraints define a complex circle manifold, making the feasible search space a Riemannian submanifold of C^M.
- Manifold formulation: The Riemannian gradient is the Euclidean gradient projected orthogonally onto the tangent space at the current point.
- Conjugate-gradient method: Vector transport maps search directions between successive tangent spaces so conjugate-gradient updates remain well-defined.
- Algorithm and guarantee: Algorithm 2 iteratively updates the Riemannian gradient, transported search direction, and manifold point until convergence, then designs the IRS phase shifts and AP beamformer.
- Conjugate-gradient method: Retraction maps a tangent-space search direction back onto the manifold to produce the next iterate.
- Algorithm and guarantee: The manifold algorithm is guaranteed to converge to a critical point of P2 where the Riemannian gradient is zero.
D. Discussion
The proposed methods use heuristic initialization and have lower stated computational complexity than the SDR benchmark. The SDR approach is described as prohibitively expensive relative to the proposed algorithms.
- Initialization: Both proposed algorithms require initialization near the optimum because they produce locally optimal solutions.
- Initialization: The initialization relaxes the unit-modulus constraints to a norm constraint, solves an eigenvalue problem, and extracts phases for the starting point.
- Computational complexity: The fixed-point method has a closed-form solution in each iteration, while the manifold conjugate-gradient method has stated worst-case complexity O(...).
- Computational complexity: The SDR benchmark has complexity O(...), described as prohibitively high compared with the proposed algorithms.
IV. SIMULATION RESULTS
The simulations evaluate the proposed algorithms against SDR under independent Rayleigh fading, averaging results over 1000 channel realizations with a specified convergence threshold.
- Performance is benchmarked against SDR under independent Rayleigh fading with path-loss exponent 3 and reference distance 10 m.
- The simulations use P = 5 dBm transmit power, σ2 = −80 dBm user noise power, and average results over 1000 channel realizations.
- Both proposed algorithms stop when the objective-function increment is less than ϵ = 10^-6.
A. Average Spectral Efficiency vs. AP-User Distance
The proposed algorithms achieve nearly identical spectral efficiency and significantly outperform SDR across AP-user distances in the stated setup. The performance gap is smaller near the AP or IRS, but joint design matters more when the user is far from both.
- Algorithm comparison: Under rAI = 50 m, rAu + rIu = 70 m, Nt = 8, and M = 10, manifold optimization and fixed-point iteration achieve almost the same spectral efficiency.
- Algorithm comparison: Both proposed algorithms significantly outperform SDR, which yields only an approximate solution.
- User position: Near the AP or IRS, strong direct or reflecting channels reduce the performance gap between the proposed algorithms and SDR.
- User position: At rAu = 40 m, joint beamformer and IRS phase-shift design is needed for satisfactory communication performance.
B. Computational Complexity
The proposed algorithms achieve locally optimal solutions, with manifold optimization offering slightly higher spectral efficiency for large IRSs and substantially lower runtime than SDR.
- Both proposed algorithms lead to locally optimal solutions for the joint beamformer and IRS phase-shift design.
- For large IRS sizes, manifold optimization achieves slightly higher spectral efficiency than fixed point iteration because it is more likely to escape saddle points.
- The proposed algorithms require much less runtime than the SDR method, confirming their computational efficiency.
C. Beyond Massive MIMO
IRS-assisted systems significantly outperform conventional MRT without IRSs, and increasing IRS reflecting elements is more efficient than enlarging the AP antenna array for spectral and energy efficiency.
- IRS-assisted systems significantly outperform the MRT strategy without IRSs in spectral efficiency.The comparison uses optimal MRT beamforming for the conventional system.
- Increasing the number of IRS reflecting elements is more efficient than enlarging the transmit antenna array size in terms of spectral efficiency.The performance gain increases with the number of elements.
- IRS-assisted wireless systems are more spectral- and energy-efficient than conventional wireless systems.Increasing antenna elements requires additional RF chains and power amplifiers, whereas large-scale IRSs are passive.
V. CONCLUSIONS
The paper jointly designs the AP beamformer and IRS phase shifts using fixed point iteration and manifold optimization to address unit modulus constraints. Simulations show substantial potential for high-speed green communication networks.
- The paper investigates joint design of the AP beamformer and IRS phase shifts in an IRS-assisted wireless communication system.
- Fixed point iteration and manifold optimization effectively tackle the unit modulus constraints that obstruct IRS phase-shift optimization.
- Identifying the manifold structure of IRS phase shifts enables the application of manifold optimization techniques.
- Simulation results reveal substantial potential for IRSs to support high-speed green communication networks.