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Beamforming Optimization for Intelligent Reflecting Surface with Discrete Phase Shifts
Qingqing Wu, Rui Zhang
TL;DR
The paper addresses transmit-power minimization for an IRS-assisted AP-to-user link when IRS phase shifters have finite discrete resolution. It jointly optimizes continuous AP beamforming and discrete IRS phases with a low-complexity alternating-optimization method. The analysis shows that discrete-phase IRSs retain asymptotic squared power gain, while their loss relative to continuous phases depends only on phase-shifter resolution.
Problem
Prior IRS beamforming studies mainly assume infinite-resolution phase shifters, whereas practical systems require finite discrete phase shifts.
Method
The paper jointly optimizes AP transmit beamforming and IRS discrete phase shifts under an SNR constraint using a suboptimal alternating-optimization algorithm.
Results
The IRS with discrete phase shifts achieves the same asymptotic squared power gain O(N 2) as continuous phase shifts, with power loss determined only by phase-shifter resolution.
Takeaways & Limitations
Discrete phase shifters can simplify IRS hardware and control without compromising large-N asymptotic performance.
Takeaways & Limitations
The analysis uses unit reflection amplitude and simulations assume specific channel, noise, SNR, and antenna settings.
Abstract
from arXiv · showhide
Intelligent reflecting surface (IRS) is a promising technology for achieving high spectrum efficiency in future wireless networks by leveraging massive low-cost reflecting elements with each reflecting the incident signal with a proper phase shift. However, prior works on IRS are mainly based on the optimization of infinite-resolution phase shifters which are practically infeasible due to hardware imperfections. In contrast, we study in this paper an IRS-aided wireless network, where an IRS with only finite-resolution phase shifter available at each element is deployed to assist in the communication from a multi-antenna access point (AP) to a single-antenna user. We aim to minimize the transmit power at the AP by jointly optimizing the transmit beamforming at the AP and reflect beamforming at the IRS, subject to the signal-to-noise ratio (SNR) constraint and practical discrete phase shift constraints. We first propose a suboptimal but low-complexity algorithm by exploiting the alternating optimization technique. Then, we reveal that as in the case with continuous phase shifts, the IRS with discrete phase shifts also achieves the squared power gain for asymptotically large number of reflecting elements, despite suffering a performance loss that depends only on the resolution of phase shifters.
1. INTRODUCTION
IRS is presented as a low-cost way to improve wireless performance using many passive reflecting elements with adjustable phases. This paper studies discrete phase shifts instead of the continuous shifts assumed in prior work, while retaining joint AP and IRS beamforming.
- Motivation: IRS uses many low-cost passive elements to reflect incident signals with adjustable phase shifts.The reflected signal can enhance received power or suppress co-channel interference.
- Prior work: Prior single-user IRS work reported asymptotic received-power gain of O(N 2) as the number of reflecting elements N grows.This squared gain exceeds the O(N) gain associated with massive MIMO.
- This paper: The paper considers a multi-antenna AP serving a single-antenna user through an IRS with finite-resolution discrete phase shifts.The objective is to minimize AP transmit power under a user SNR constraint by jointly optimizing active and passive beamforming.
- This paper: A low-complexity alternating-optimization algorithm determines discrete phase shifts iteratively while fixing the others.The paper also analyzes how discrete shifts affect performance as N increases.
2. SYSTEM MODEL
The system is a downlink MISO link in which an IRS with N reflecting elements assists an M-antenna AP and single-antenna user. Continuous AP beamforming is combined with discrete IRS reflection under a finite phase-shift alphabet.
- System configuration: The model contains an AP with M antennas, an IRS with N reflecting elements, and one single-antenna user.The IRS controller coordinates with the AP over a separate wireless link to exchange channel knowledge and adjust phases.
- Beamforming: The AP uses continuous transmit beamforming, while the IRS applies a diagonal phase-shift matrix to the reflected signal.The direct AP-user and IRS-reflected signals are combined at the receiver.
- Discrete phase shifts: Each IRS phase shift is selected from K=2^b equally spaced values in [0, 2π), with unit reflection amplitude β=1.The b-bit phase resolution determines the discrete alphabet.
- Signal model: The transmitted symbols are modeled as i.i.d. zero-mean unit-variance variables, and receiver noise is i.i.d. AWGN with variance σ2.The total AP transmit power is represented by the squared Euclidean norm of the beamforming vector.
3. PROBLEM FORMULATION
The paper formulates joint AP transmit-beamforming and IRS phase-shift design as transmit-power minimization under an SNR target and discrete phase constraints. The resulting problem is non-convex and difficult to solve exhaustively at large IRS sizes.
- Optimization objective: The optimization minimizes AP transmit power by jointly selecting beamforming w and discrete IRS phases θ subject to a user SNR requirement γ.Both the SNR constraint and the discrete phase-shift constraints are included in the formulation.
- Problem structure: The problem is non-convex because the SNR expression is not jointly concave in w and θ, while θn is restricted to discrete values.These coupled design variables prevent direct use of standard efficient convex optimization methods.
- Computational challenge: Exhaustively searching all discrete phase combinations requires complexity O(2bN), which is prohibitive for practical systems with large N.The exponential dependence arises from b-bit choices across N reflecting elements.
4. PROPOSED ALGORITHM
The paper develops a low-complexity alternating-optimization algorithm for jointly selecting discrete IRS phase shifts and AP transmit beamforming. Coordinate-wise phase updates produce a convergent suboptimal solution, with initialization obtained by quantizing continuous-phase shifts.
- The algorithm alternately optimizes one IRS phase shift while fixing the other N −1 shifts until convergence.
- For fixed phase shifts, maximum-ratio transmission provides the optimal AP transmit beamforming solution.
- Minimizing AP transmit power is equivalent to maximizing the combined AP-user channel power gain.
- With all other phases fixed, the objective is linear in e^jθ_n, enabling a closed-form update for the nth phase shift.
- Repeated coordinate updates make the objective non-decreasing and guarantee convergence because the objective is upper-bounded.
- Initialization quantizes phase shifts obtained from the continuous-phase problem to their nearest discrete values.
5. PERFORMANCE ANALYSIS
The performance analysis derives the asymptotic received-power behavior of discrete-phase IRSs under a simplified single-antenna AP setting. It shows that phase resolution determines a fixed loss relative to continuous shifts, while the squared power scaling with N remains achievable.
- The analysis characterizes average received power as the number of reflecting elements N approaches infinity, assuming M = 1 and negligible direct AP-user reception.
- Quantization errors are independently uniform in [−π/2^b, π/2^b), where b is the number of phase-shifter bits.
- The discrete-to-continuous received-power ratio increases monotonically with b and approaches 1 as b approaches infinity.
- As N approaches infinity, the power ratio depends only on phase-shifter resolution and remains independent of N, while discrete phase shifts retain O(N^2) power gain.
- The reported ratios are η(1) = 0.4053, η(2) = 0.8106, and η(3) = 0.9496.
- Three-bit phase shifters are described as practically sufficient for close-to-optimal performance, with a trade-off between reflecting-element count and phase resolution.
6. SIMULATION RESULTS
Simulations evaluate AP transmit power against user distance and IRS element count under discrete phase shifts, showing practical power savings and resolution-dependent asymptotic loss.
- The AP transmit-power evaluation uses Rayleigh fading, pathloss exponents 2.2, 2.8, and 3.4, noise power −80 dBm, SNR target 20 dB, and M = 5.The AP uses a ULA and the IRS uses a URA.
- The simulations compare lower-bound, exhaustive-search, alternating-optimization, initialization, and no-IRS schemes for 1-bit phase shifts.The setup varies AP-user horizontal distance and evaluates the minimum power needed for the target SNR.
- 1-bit phase shifters require significantly less AP transmit power than no IRS when the user is near the IRS.This demonstrates a signal hotspot even with coarse, low-cost phase shifters.
- As N increases, the proposed schemes' performance gap from the continuous-phase lower bound first increases and then approaches a constant determined by η(b).The comparison considers b = 1 and b = 2 at d = 50 m.
- η(1) = −3.9224 dB and η(2) = −0.9224 dB quantify the asymptotic performance loss for 1-bit and 2-bit phase shifts.
7. CONCLUSION
The paper optimizes AP and IRS beamforming under discrete phase shifts using an efficient alternating-optimization method. It reports squared power gain even with 1-bit phase shifters, while direct quantization already performs well and additional alternating optimization adds only marginal gain.
- The study jointly optimizes continuous AP transmit beamforming and discrete IRS reflect beamforming to minimize AP transmit power under a user SNR target.
- An efficient alternating-optimization algorithm solves the discrete-phase beamforming problem and achieves near-optimal performance.
- Even 1-bit phase shifters achieve the squared power gain obtained with continuous phase shifts.
- Directly quantizing the continuous-phase solution achieves quite good performance, while additional alternating optimization provides only marginal gain.