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

Ahmet M. Elbir, Kumar Vijay Mishra, Sergiy A. Vorobyov, Robert W. Heath

arXiv:2211.02165v3eess.SPcs.IT

TL;DR

Beamforming research must handle increasingly complex arrays, propagation conditions, and uncertain environments across radar and communications. This article surveys 25 years of progress from convex and nonconvex optimization to learning-based approaches, covering architectures, paths, applications, and emerging challenges. It highlights robust optimization and machine learning as important approaches for contemporary beamformer design, while noting practical limitations in learning and hardware-intensive settings.

  • Problem

    Beamforming must address complex nonlinear designs, model mismatch, large arrays, high frequencies, and emerging propagation and hardware settings across many applications.

  • Method

    The article provides a 25-year overview organized around the shifts from convex to nonconvex optimization and from optimization to learning-based beamforming.

  • Results

    The survey covers beamforming across convex and nonconvex optimization, machine learning, transmit-receive architectures, propagation paths, and conventional and emerging applications.

  • Takeaways & Limitations

    Machine learning offers lower post-training complexity, expedited design, and robustness to imperfections, while modern beamforming spans diverse architectures and applications.

Abstract

from arXiv · show

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.

I. INTRODUCTION

Beamforming has evolved from early array-processing demonstrations into a broad research area spanning robust optimization, nonconvex design, large-array communications, and learning-based methods. This article surveys the major breakthroughs and applications of that evolution over 25 years.

  • Beamforming supports applications including radar, sonar, acoustics, astronomy, seismology, ultrasound, and communications.
  • Recent mobile communications, large arrays, high-frequency sensors, near-field recovery, and smart radio environments create demands for robust, flexible, and low-complexity algorithms.
  • Robust beamforming became necessary because small steering-vector or look-direction mismatches can severely degrade SINR.
  • Convex optimization recast robust beamformer design as efficiently solvable second-order cone programs, often using worst-case mismatch models.
  • Nonconvex beamforming gained prominence for robust adaptive, multicast, and hybrid analog/digital problems, with SDR, compressed sensing, and alternating optimization used for tractability.
  • Machine learning is attractive for nonlinear beamformer design because it can reduce post-training computation, accelerate design, and process massive antenna-array data.
  • The article organizes the field around two major shifts: convex-to-nonconvex optimization and optimization-to-learning-based beamforming.

II. CONVEX OPTIMIZATION FOR BEAMFORMING

This section develops beamforming from array observations and steering vectors to Capon optimization, showing how covariance-based weights and linear constraints produce directional responses.

  • Convex optimization transforms difficult beamformer designs into computationally attractive problems with exact or approximate solutions.
  • An antenna array models a narrowband source through a steering vector a(θ), with element spacing and wavelength determining its array response.
  • Beamforming multiplies received signals by complex weights and combines them into a scalar output for signal recovery.
  • Figure 1 classifies beamformers by transmission range, transceiver architecture, propagation path, and application.
  • The Capon beamformer uses the sample covariance matrix and steering vector to obtain an optimized minimum-variance response.
  • Capon performance depends on accurate knowledge of the steering vector and sample covariance matrix.
  • Additional linear constraints can stabilize the mainbeam or jointly specify a desired response and a null at selected directions.

B. Loaded SMI beamformer

Loaded SMI and robust beamforming address degradation caused by limited samples, steering mismatch, non-Gaussian noise, and uncertainty in array statistics.

  • Small training sample sizes can significantly deteriorate beamforming performance even when the source direction is known.
  • Loaded SMI mitigates sample deficiency by adding a regularization term to the beamforming objective.
  • Robust Capon beamforming addresses inaccurate source directions and steering vectors through uncertainty-aware optimization.
  • Worst-case formulations model steering distortions with bounded spherical or ellipsoidal uncertainty sets.
  • Ellipsoidal uncertainty can describe mismatch more accurately when information beyond a common uncertainty radius is available, but that assumption can conflict with robustness.
  • For non-Gaussian noise, the beamforming problem becomes nonconvex and can be solved with iterative reweighted MVDR techniques.

E. Beamforming for general-rank source

General-rank beamforming replaces a single steering-vector constraint with a source covariance model, then addresses covariance uncertainty and positive-semidefiniteness through tractable reformulations.

  • Incoherent scattering makes the source covariance matrix general-rank, so a single steering-vector constraint is no longer sufficient.
  • The resulting MVDR-type formulation uses the source covariance matrix, with its optimal solution obtained through a principal-eigenvector operation.
  • General-rank solutions require knowledge of the source covariance matrix, which may not be available in practice.
  • Covariance mismatch is modeled through a factorization constrained to preserve positive semidefiniteness.
  • The resulting nonconvex problem is solved efficiently using a polynomial-time difference-of-convex functions algorithm.

B. Norm-constrained beamforming based on steering vector estimation

Norm-constrained beamforming extends robust steering-vector estimation by adding constraints on mismatch or beamformer norms, while nonconvex formulations can be addressed through tractable reformulations and iterative methods.

  • B. Norm-constrained beamforming based on steering vector estimation: Additional norm constraints generalize robust Capon beamforming beyond its uncertainty constraint.The formulation is described as a more general setting with an added norm constraint for beamformer weights.
  • B. Norm-constrained beamforming based on steering vector estimation: Doubly-constrained robust Capon beamforming is iteratively solved as a covariance-fitting problem to robustly estimate the array steering vector.The approach estimates the difference between actual and presumed steering vectors without assuming the mismatch vector’s norm or probability distribution.
  • B. Norm-constrained beamforming based on steering vector estimation: A new constraint prevents the estimated source steering vector from converging to interference steering vectors or their linear combinations.This constraint addresses source-steering-vector estimation within the norm-constrained robust beamforming formulation.
  • B. Norm-constrained beamforming based on steering vector estimation: Generalized similarity constraints describe imperfect desired steering-vector knowledge through a convex, specifically ellipsoidal, uncertainty set when V has full row rank.The formulation uses selected values ∆1, η1, and η2 together with the generalized similarity condition.
  • B. Norm-constrained beamforming based on steering vector estimation: These nonconvex problems can often be solved exactly using SDR, iterative SOC programs, QMI, and BLMI approaches.The listed approaches recast or solve the nonconvex formulations through tractable optimization procedures.

C. Chance-constrained beamforming

Chance-constrained beamforming replaces a deterministic distortionless requirement with a probability constraint and extends robust design to uncertainty in both covariance matrices and steering vectors.

  • C. Chance-constrained beamforming: Chance-constrained robust adaptive beamforming requires the distortionless response to hold with at least a selected probability p.The equivalent outage probability is pout = 1 − p for violating |w^Hã| ≥ 1.
  • C. Chance-constrained beamforming: The chance constraint minimizes beamformer output power while enforcing a probabilistic signal distortionless-response condition.This formulation interprets the requirement as a non-outage probability constraint.
  • C. Chance-constrained beamforming: The nonconvex formulation models both the interference-plus-noise covariance matrix and the true steering vector as uncertain random variables.This setting is presented as more practical than assuming either quantity is precisely known.
  • C. Chance-constrained beamforming: The chance-constrained beamformer achieves higher output SINR than the compared convex LSMI and nonconvex worst-case optimization beamformers.This comparison is reported for the practical setting in which both the covariance matrix and true steering vector are imprecisely known.
  • C. Chance-constrained beamforming: Distributionally robust beamforming accounts for distributional uncertainty by considering probability distributions sharing specified first- and second-order moments.The steering-vector distribution has known mean a0 and covariance Σ ≻ 0, while the covariance-matrix distribution is represented through probability measures and an empirical mean.

D. Multicast transmit beamforming

Multicast and hybrid beamforming formulate multiuser transmission and large-array hardware design as nonconvex optimization problems, with SDR, alternating optimization, and manifold methods providing solution routes under practical architecture constraints.

  • D. Multicast transmit beamforming: Multicast beamforming broadcasts a data stream from an N-element array to U single-antenna users over channels h_u.The received signal is modeled as y_u(t_i) = h_u^H x(t_i) + e_u(t_i).
  • D. Multicast transmit beamforming: The multicast optimization is a QCQP with nonconvex constraints, which SDR addresses by lifting the rank-1 matrix M = ww^H and removing the rank constraint.The beamformer weight can then be obtained through eigenvalue decomposition of M.
  • D. Multicast transmit beamforming: Alternating optimization of the two factors w_1 and w_2 provides a more accurate solution after rewriting M = w_1w_1^H + w_2w_2^H.The factors are updated iteratively until convergence.
  • D. Multicast transmit beamforming: Hybrid beamforming can provide multiple beams with fewer RF chains than digital beamforming while reducing hardware cost and retaining satisfactory spectral efficiency.This makes hybrid architectures attractive for massive arrays such as those used in 5G communications.
  • D. Multicast transmit beamforming: Hybrid beamforming uses constant-modulus analog weights and a product F_RF F_BB, making its optimization nonconvex and nonlinear.The transmitted signal is x = F_RF F_BB s, and the objective is to maximize mutual information.
  • D. Multicast transmit beamforming: Alternating optimization of analog and digital beamformers often achieves spectral efficiency close to the digital-only solution F_C.For wideband systems, digital beamformers vary by subcarrier while the analog beamformer is shared across subcarriers.
  • D. Multicast transmit beamforming: Analog beamformers can be designed with OMP or manifold optimization under constant-modulus constraints.Manifold optimization treats the search space as a Riemannian submanifold and alternately optimizes analog and digital beamformers.
  • D. Multicast transmit beamforming: Hybrid implementations are constrained by limited phase shifters and ADCs, while lens-based beamformers reduce complexity but primarily generate directional beams.Lens-based architectures may not realize sophisticated patterns needed for spatial multiplexing or interference cancellation.

Low-resolution ADCs:

Low-resolution ADCs reduce power consumption and hardware cost in digital beamforming, with the quantized received signal then used for receiver design.

  • Low-resolution ADCs:: Low-resolution 1–3-bit ADCs reduce overall power consumption and hardware cost in digital beamformers.One-bit ADCs can eliminate automatic gain control and linear amplifiers, enabling a lower-cost RF chain.
  • Low-resolution ADCs:: The quantized received signal r_q = Q_b(W_RF^H r) is used to design the receiver with zero-forcing or maximum-rate combining.Q_b(·) denotes the quantization operator with b-bit resolution.

Finite resolution phase shifters:

Finite-resolution phase shifters reduce implementation cost by restricting beamformer weights to a discrete set, while hybrid designs can retain performance close to fully digital beamforming. Learning-based beamforming further offers data-driven mappings and reduced post-training computation for complex designs.

  • Finite resolution phase shifters:: Finite-resolution phase shifters select beamformer weights from a discrete set determined by the number of quantization bits.The continuous phase constraint is replaced by membership in W, and infinite-resolution designs can be followed by phase quantization.
  • Finite resolution phase shifters:: The MO-based hybrid architecture performs very close to fully digital beamforming, while 5-bit OMP is closest to infinite-resolution phase shifters.The performance gap from fully digital beamforming is larger for OMP-based techniques than for MO-based beamforming.
  • Learning-based beamforming: Learning-based hybrid beamforming constructs a nonlinear mapping from inputs such as channel matrices or array outputs to beamformers.The approach treats beamformer design from a model-free viewpoint and can support analog and digital beamformer feature extraction.
  • Learning-based beamforming: Parallel processing enables machine learning to reduce computational times by approximately 10-fold compared with model-based techniques.Parallel implementation of conventional convex or nonconvex optimization-based beamforming is described as not straightforward.
  • Learning-based beamforming: Beamforming can be formulated as regression with beamformer weights as outputs or classification with an index from a predefined beamformer set.Supervised learning uses labeled data, whereas unsupervised learning clusters unlabeled data through latent features or patterns.

B. Reinforcement learning

Reinforcement learning updates beamforming models through interaction with channel conditions and reward signals, while online learning adapts models when received data change. Federated and centralized learning address multi-user settings with different dataset-access trade-offs.

  • B. Reinforcement learning: Reinforcement learning updates beamforming parameters autonomously from channel conditions and average-rate rewards.The model takes previous analog and baseband beamformers as input and learns through trial-and-error interaction with the environment.
  • B. Reinforcement learning: Reinforcement learning remains harder to apply to real-world beamforming because unlabeled data require longer training, especially in dynamic short-coherence-time scenarios.The passage identifies the absence of labels and wireless-channel feature learning as practical difficulties.
  • Online learning: Online learning updates model parameters after significant changes in received inputs, such as user motion away from the base station in direction-of-arrival.The changing received array data can differ substantially from offline training data and degrade network performance.
  • D. Federated learning: Centralized learning outperforms federated learning with the same network structure because it accesses the whole dataset at once.Federated learning is suited to downlink edge deployment, while centralized learning uses decentralized data less directly.
  • D. Federated learning: Both centralized and federated learning outperform OMP, but their performance gap increases as local datasets become more non-uniform.The comparison is reported using spectral efficiency against OMP and fully digital beamforming.

V. EMERGING APPLICATIONS

Emerging beamforming applications span model-driven and convolutional designs, joint radar-communications spectrum sharing, and THz systems. These settings introduce hardware, propagation, and beam-squint constraints that require specialized beamformer designs.

  • V. EMERGING APPLICATIONS: Model-driven networks and deep unfolding aim to bound beamforming algorithm complexity while retaining performance.Convolutional beamformers are also gaining attention in acoustics and ultrasound for combining multiple, usually nonlinear, operations.
  • A. Joint radar-communications: Joint radar-communications design addresses the need to share spectrum previously separated between sensing and communications systems.An unconstrained JRC beamformer combines radar and communications beamformers using ζ to trade off their performance.
  • A. Joint radar-communications: The JRC hybrid beamformer uses an auxiliary unitary matrix to reconcile differing beamformer dimensions without distorting the radar beampattern.The auxiliary matrix P accounts for the dimensions of FRFFBB and FR.
  • B. THz communications: THz systems use ultra-massive arrays and a shared analog beamformer across subcarriers for hardware-efficient, computationally inexpensive wideband beamforming.A single analog beamformer causes lower and higher subcarriers to point in different directions, producing beam squint.
  • B. THz communications: Beam-squint angular deviation is approximately 6° for 0.3 THz with 30 GHz bandwidth and 0.4° for 60 GHz with 1 GHz bandwidth.Time-delayer networks and beam-squint-aware digital beamformers are described as approaches to address the effect.

C. Intelligent reflecting surfaces

Beamforming design spans active/passive intelligent reflecting surfaces and near-field propagation, alongside convex, nonconvex, and learning-based methods. The review presents these algorithms as a toolkit whose suitability depends on application-specific constraints.

  • C. Intelligent reflecting surfaces: IRS-assisted beamforming combines base-station beamforming with phase shifts from passive metamaterial elements to reach distant or blocked users and targets with low power consumption.The design jointly optimizes the active base-station beamformer and passive IRS phase shifts.
  • C. Intelligent reflecting surfaces: In an IRS-assisted system, the IRS phase matrix, base-station beamformer, channels, and additive noise determine the received user signal.The passage defines the user-IRS, user-base-station, and base-station-IRS channels and the diagonal IRS phase matrix.
  • Near-field beamforming: Near-field propagation produces a spherical wavefront and a range-dependent beampattern, unlike the plane-wave far-field model.The near-field region is defined relative to the Fraunhofer distance.
  • Near-field beamforming: For a uniform linear array in the near field, the array response depends on both direction θ and range r.The range-dependent parameter describes the distance between the receiver and each transmit antenna.
  • VI. SUMMARY: The review frames beamforming algorithms and variants as a toolkit for selecting techniques appropriate to specific applications.It focuses primarily on radar and communications while also covering ultrasound, acoustics, synthetic apertures, and optics.
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