[Paper Review] Twenty-Five Years of Advances in Beamforming: From Convex and Nonconvex Optimization to Learning Techniques

Ahmet M. Elbir, Kumar Vijay Mishra+2 more|arXiv (Cornell University)|Nov 3, 2022

Antenna Design and Optimization4 citations

TL;DR

This paper provides a comprehensive review of beamforming advancements over the past 25 years, tracing the evolution from convex and nonconvex optimization techniques to modern learning-based approaches. It details how these methods address challenges in robustness, array mismatch, and complex propagation environments across radar, wireless communications, and emerging applications like intelligent reflecting surfaces and near-field beamforming.

ABSTRACT

Beamforming is a signal processing technique to steer, shape, and focus an electromagnetic wave using an array of sensors toward a desired direction. It has been used in several engineering applications such as radar, sonar, acoustics, astronomy, seismology, medical imaging, and communications. With the advances in multi-antenna technologies largely for radar and communications, there has been a great interest on beamformer design mostly relying on convex/nonconvex optimization. Recently, machine learning is being leveraged for obtaining attractive solutions to more complex beamforming problems. This article captures the evolution of beamforming in the last twenty-five years from convex-to-nonconvex optimization and optimization-to-learning approaches. It provides a glimpse of this important signal processing technique into a variety of transmit-receive architectures, propagation zones, paths, and conventional/emerging applications.

Motivation & Objective

• To survey the evolution of beamforming techniques from convex optimization to nonconvex and learning-based methods over the past 25 years.
• To identify key algorithmic developments that enhance robustness against steering vector mismatches, limited snapshots, and array manifold errors.
• To analyze the role of machine learning in solving complex beamforming problems beyond traditional optimization.
• To examine emerging beamforming architectures such as hybrid beamforming, intelligent reflecting surfaces (IRS), and near-field beamforming.
• To provide a unified overview of beamforming across diverse applications including radar, wireless communications, ultrasound, and optics.

Proposed method

• Systematically categorizes beamforming methods by transmission range (far- vs. near-field), transceiver architecture (analog, digital, hybrid), and propagation paths (LoS, NLoS).
• Reviews convex optimization-based beamformers such as worst-case robust beamforming, SMI, and ellipsoidal uncertainty modeling to improve robustness.
• Analyzes nonconvex beamforming techniques using semi-definite relaxation (SDR), compressed sensing (CS), and alternating optimization for hybrid beamforming and multicast beamforming.
• Introduces learning-based beamforming, including supervised (SL), unsupervised (UL), reinforcement learning (RL), federated learning (FL), and online learning (OL), with application-specific advantages.
• Presents joint active/passive beamforming design for IRS-assisted systems using a joint optimization framework with power and phase constraints.
• Derives a range-dependent array response model for near-field beamforming, where the response depends on both direction and distance, requiring spherical wavefront modeling.

Experimental results

Research questions

• RQ1 How have convex optimization techniques improved robustness in beamforming under steering vector and direction mismatches?
• RQ2 What are the key challenges and solutions in nonconvex beamforming problems involving constraints on signal covariance, constant-modulus, and SNR?
• RQ3 How do different machine learning paradigms (SL, RL, FL, OL) enhance beamforming performance in dynamic or data-limited environments?
• RQ4 What role does intelligent reflecting surface (IRS) technology play in extending coverage and reducing power consumption in beamforming systems?
• RQ5 How does near-field beamforming differ from far-field beamforming, and what modeling changes are required for accurate beamformer design?

Key findings

• Convex optimization-based beamformers, such as worst-case robust beamforming and SMI, significantly improve performance under steering vector uncertainty and limited snapshots.
• Nonconvex beamforming techniques using SDR and compressed sensing enable efficient solutions for hybrid beamforming and multicast beamforming with constant-modulus constraints.
• Reinforcement learning outperforms supervised and unsupervised learning in beamforming due to its reward-based optimization of long-term performance.
• Federated learning enables distributed beamforming in multi-user scenarios with privacy preservation and reduced feedback overhead.
• IRS-assisted beamforming achieves significant performance gains by jointly optimizing beamformers and phase shifts, enabling coverage extension with low power.
• Near-field beamforming requires a range-dependent array response model, where the beam pattern varies with distance, necessitating spherical wavefront modeling instead of plane wave approximations.