Physics-Inspired Discrete-Phase Optimization for 3D Beamforming with PIN-Diode Extra-Large Antenna Arrays

TL;DR

This paper addresses the challenge of beamforming with PIN-diode arrays that support only discrete phase shifts, a problem that is NP-hard when optimizing across large arrays. The authors formulate the objective as a Rayleigh quotient and reformulate it into a sequence of QUBO subproblems via a bisection method, solving them with simulated annealing. Key contributions include extending 3D beamforming to extra-large arrays (up to $10^4$ elements), providing an exact QUBO reformulation within the bisection framework, and introducing an early-stop SA scheme that reduces compute time significantly without sacrificing beam quality. The results show discrete 2- and 4-phase configurations can closely match continuous-beam performance in large arrays, enabling scalable, CSI-free beamforming for applications like wireless power transmission and sensing.

Abstract

Large antenna arrays can steer narrow beams towards a target area, and thus improve the communications capacity of wireless channels and the fidelity of radio sensing. Hardware that is capable of continuously-variable phase shifts is expensive, presenting scaling challenges. PIN diodes that apply only discrete phase shifts are promising and cost-effective; however, unlike continuous phase shifters, finding the best phase configuration across elements is an NP-hard optimization problem. Thus, the complexity of optimization becomes a new bottleneck for large-antenna arrays. To address this challenge, this paper suggests a procedure for converting the optimization objective function from a ratio of quadratic functions to a sequence of more easily solvable quadratic unconstrained binary optimization (QUBO) sub-problems. This conversion is an exact equivalence, and the resulting QUBO forms are standard input formats for various physics-inspired optimization methods. We demonstrate that a simulated annealing approach is very effective for solving these sub-problems, and we give performance metrics for several large array types optimized by this technique. Through numerical experiments, we report 3D beamforming performance for extra-large arrays with up to 10,000 elements.

Figures (11)

• Figure 1: Beamforming with PIN-diode-based low-resolution discrete (2- or 4-) phase shifters, aiming to maximize the power radiated on the target area out of the total transmitted power.
• Figure 2: Comparison of the QUBO structure and the QUBO solver's goal between common work and this work.
• Figure 3: High-level SA system architecture, presenting typical optimization processing in SA.
• Figure 4: Maximum batch iteration ($G$) for a stuck configuration ($x$) depending on the recent best min $\mathcal{H}(x)$.
• Figure 5: Proof of concept (the figure directly from the early version of this work stockley2023optimizing). Beamforming using a 7$\times$7 array of 4-phase elements, with backplane, azimuth=2 ($\phi$ = 114.64 degrees), polar elevation angle=0.25 ($\theta$ = 14.3 degrees). The phases are $\{1,j,-1,-j\}$ or equivalently 0, 90, 180, 270 degrees. The problem is solved by SA (left), while the same problem is solved by QA on the D-Wave quantum annealer (right). The small differences are due to the heuristic nature of the annealing process, meaning that slightly different (but still close to optimal) weights are found.