Source-linked AI summary
Beamforming Optimization for Intelligent Reflecting Surface Assisted MIMO: A Sum-Path-Gain Maximization Approach
Boyu Ning, Zhi Chen, Wenjie Chen, Jun Fang
TL;DR
The paper addresses spectral-efficiency maximization in IRS-assisted point-to-point MIMO by jointly designing source precoding and IRS phase shifters. It replaces the difficult optimization with SPGM-based phase-shifter design solved by ADMM, followed by SVD-based precoding, and reports near-optimal performance with moderate computational complexity.
Problem
Joint optimization of source precoding and IRS phase shifters for IRS-assisted MIMO spectral efficiency is difficult because the resulting problem is intractable and non-convex.
Method
The method designs IRS phase shifters using an SPGM criterion and low-complexity ADMM, then derives source precoding through SVD and water-filling.
Results
The proposed scheme achieves near-optimal performance with moderate computational complexity.
Takeaways & Limitations
SPGM provides an efficient phase-shifter design that exploits both conventional beamforming gain and IRS aperture gain.
Abstract
from arXiv · showhide
Recently, intelligent reflecting surface (IRS) has emerged as an appealing technique that enables wireless communications with low hardware cost and low power consumption. In this letter, we consider an IRS-assisted point-to-point multi-input multi-output (MIMO) system, where a source communicates with its destination with the help of an IRS. Our goal is to maximize the spectral efficiency of this system by jointly optimizing the (active) precoding at the source and the (passive) phase shifters (PSs) at the IRS. However, this turns out to be an intractable mixed integer non-convex optimization problem. To circumvent the intractability, we propose a new sum-path-gain maximization (SPGM) criterion to obtain a high-quality and efficient suboptimal solution to this problem. Specifically, the PSs are first designed based on a simplified optimization problem, which aims to maximize the sum-gains of the spatial paths between the source and the destination. Then, a low-complexity alternating direction method of multipliers (ADMM) algorithm is utilized to solve this simplified problem. Finally, with the above obtained PSs, the source precoding is derived by performing the singular value decomposition (SVD) on the effective channel between the source and the destination. Numerical results demonstrate that the proposed scheme can achieve near-optimal performance.
I. INTRODUCTION
The introduction motivates IRS-assisted MIMO as a low-cost approach to improving spectral efficiency, then presents an efficient suboptimal design based on SPGM, ADMM, and SVD precoding.
- Motivation: IRS uses controllable passive phase shifts to improve spectrum efficiency through reflected-signal combining.Unlike fixed passive reflectors, IRS elements independently adjust phase shifts; reflected and direct-link signals can combine constructively at the receiver.
- Motivation: IRS-assisted MIMO has received limited attention because its system setup is more challenging than MISO communication.Prior work considered beamforming and IRS phase-shifter design mainly in IRS-assisted MISO systems, while MIMO work included a sophisticated BCD algorithm.
- Contribution: The paper jointly optimizes source precoding and IRS phase shifters to maximize spectral efficiency in point-to-point MIMO.The proposed approach targets the IRS-assisted link between Alice and Bob.
- Contribution: SPGM replaces the difficult direct optimization with a simplified problem that maximizes the sum-gains of spatial paths between Alice and Bob.The phase-shifter solution is obtained using a low-complexity ADMM algorithm.
- Contribution: With the phase shifters fixed, Alice’s precoding matrix is obtained by applying SVD to the effective Alice-Bob channel.This provides the active-precoding stage after passive phase-shifter design.
II. SYSTEM MODEL AND PROBLEM FORMULATION
The system model describes an IRS-assisted point-to-point MIMO link with direct and cascaded paths, then formulates spectral-efficiency maximization under phase-shifter constraints.
- System model: Alice transmits Ns data streams through a linear precoder to Bob using an IRS with Nr passive elements.Alice has Nt antennas and Bob has Nb antennas; Ns is selected as rank(Heff) to exploit spatial multiplexing.
- System model: The IRS phase-shifter vector contains unit-modulus entries parameterized by element phases, with reflection amplitude β.The reflection matrix is represented using the phase-shifter vector and conjugation operation.
- System model: The received signal combines the direct Alice-Bob link with the cascaded Alice-IRS-Bob link, while signals reflected two or more times are ignored.The IRS applies a diagonal phase-shift reflection matrix before forwarding the signal toward Bob.
- Problem formulation: The design problem maximizes spectral efficiency by jointly choosing the precoder and phase shifters under uni-modular constraints.The effective Alice-Bob channel is used in the formulation, with Ns = rank(Heff).
- Problem formulation: The optimization is difficult because the objective is non-convex in the phase-shifter vector and includes equality constraints.The model assumes global channel state information is available at a centralized controller.
III. JOINT PS AND PRECODING DESIGN
The joint design first selects IRS phase shifters using SPGM and then optimizes source precoding through SVD and water-filling power allocation.
- PS and precoding design: SPGM provides an efficient criterion for designing the IRS phase-shifter vector.The criterion is used to address the difficulty of directly solving the phase-shifter optimization problem.
- PS and precoding design: After phase-shifter design, the precoding matrix is optimized using SVD and water-filling power allocation.This separates passive phase-shifter design from active source precoding.
A. PS Design under the SPGM Criterion
The paper replaces the difficult optimal phase-shifter design with the SPGM criterion, which maximizes the sum-gains of spatial paths and can be solved using ADMM under unit-modulus constraints. The resulting objective is bounded relative to the original problem, with tighter practical behavior for small and moderate numbers of data streams.
- The original phase-shifter optimization is difficult because the variables are implicitly coupled and subject to unit-modulus constraints.
- SPGM maximizes the sum-gains of the spatial paths between Alice and Bob to enhance the effective channel quality.
- The SPGM problem provides a lower bound on the optimal value of the original optimization problem.
- In MISO systems, the SPGM-based rate equals the rate of the original formulation.
- The bounds ˆR ≤ ˜R ≤ Ns ˆR become looser as Ns increases, while SPGM remains effective for small and moderate Ns.
- ADMM reformulates the unit-modulus problem into an equivalent constrained problem with closed-form phase updates and an unconstrained least-squares update.
- The proposed ADMM process for solving the reformulated problem has a convergence guarantee.
B. Precoding Design with the Obtained PSs
After obtaining the phase shifters, the method optimizes transmit power allocation and derives the source precoder using the effective channel. These steps complete the joint PS and precoding design summarized in Algorithm 1.
- Given the obtained phase shifters, the optimal power allocation is found using water-filling.
- Algorithm 1 summarizes the proposed joint phase-shifter and precoding design procedure.
- The optimal precoding matrix is then obtained from the effective channel decomposition and the power-allocation matrix.
IV. NUMERICAL RESULTS
Numerical experiments compare the proposed SPGM-based ADMM scheme with exhaustive search, SDR, random phase-shifter designs, and a no-IRS baseline. The proposed scheme achieves comparable performance to exhaustive search and SDR at lower complexity, while outperforming random and no-IRS designs; spectral efficiency also grows quadratically with the number of IRS elements.
- Experimental setup: The numerical evaluation averages results over 1,000 random channel realizations under the stated Rician fading and array-model assumptions.The setup uses uniform linear arrays and independently generated complex Gaussian non-line-of-sight channel entries.
- Experimental setup: The proposed SPGM-based ADMM algorithm is compared with SDR, random phase shifters, exhaustive search over 500,000 random phase-shifter configurations, and optimal precoding without IRS.The experiments evaluate spectral efficiency versus transmit power and IRS element count.
- Spectral-efficiency comparison: The proposed scheme achieves comparable performance to exhaustive search and SDR-based phase-shifter design with considerably lower computational complexity.The SDR solution may require eigen-decomposition and projection or Gaussian randomization when it is not rank one.
- Spectral-efficiency comparison: The proposed scheme significantly outperforms random phase-shifter designs and optimal precoding without IRS.This indicates that the scheme exploits both beamforming gain and IRS aperture gain to improve the Alice–Bob channel quality.
- IRS-size scaling: Under a line-of-sight-only component, spectral efficiency grows at the same rate as the quadratic SNR growth with the number of IRS elements N_r.The comparison uses N_t = 16, N_b = 4, and P = 10 dB.
V. CONCLUSIONS
The paper studies joint phase-shifter and source-precoding design for an IRS-assisted point-to-point MIMO system. It proposes SPGM for efficient IRS phase-shifter design, derives source precoding by SVD with water-filling, and reports near-optimal performance with moderate complexity by exploiting beamforming and aperture gains.
- The paper considers joint phase-shifter and source-precoding design in an IRS-assisted point-to-point MIMO system.
- The proposed SPGM criterion efficiently designs the IRS phase shifters.
- The source precoding matrix is derived using SVD with water-filling allocations.
- The proposed scheme achieves near-optimal performance with moderate computational complexity by exploiting conventional beamforming gain and IRS aperture gain.