Linear Model-Based Beamforming

• The framework reformulates classical beamforming as a constrained regularized linear optimization problem, replacing direct covariance inversion with robust least-squares estimation.
• It leverages Tikhonov regularization and subspace methods to enhance stability and performance across applications like MIMO, broadband imaging, and distributed array processing.
• Practical implementations utilize efficient closed-form solvers and iterative algorithms, achieving near-oracle performance under conditions of low SNR, model mismatch, and limited data.

A linear model-based beamforming framework comprises a class of algorithms that rigorously formulate sensor-array beamforming (for tasks such as signal estimation, spatial filtering, communications, and acoustic imaging) as the solution to problems in linear algebra or statistical estimation, with explicit modeling of array responses, spatial covariance, and system uncertainties. This approach covers a spectrum of methodologies: from robust adaptive beamforming with regularized least squares, through time-domain and subspace-structured models with spatiotemporal priors, to advanced frameworks for distributed, nonlinear, and continuous-domain arrays. Its core unifying principle is the explicit recasting of array processing as constrained or regularized optimization over a (possibly high-dimensional) linear operator mapping between physical sources or signals and array outputs.

1. Reformulation of Classical Beamforming in Linear Model Terms

Classical beamforming, typified by the Capon (MVDR) and LCMV designs, starts by seeking a linear combination of array sensor outputs that passes a signal from a desired direction while suppressing noise and interferers. Traditionally, the optimum weight vector is computed by inverting the spatial covariance matrix; however, this step is acutely sensitive to ill-conditioning and model mismatch.

The robust regularized least-squares (RLS) framework re-expresses the MVDR solution using the unique Hermitian square-root of the covariance matrix $R$, writing the beamformer output as a scaled inner product of two vectors $v = R^{1/2} a$ and $u = R^{1/2} x$, where $a$ is the presumed steering vector and $x$ the current array snapshot. Both $v$ and $u$ are then estimated via Tikhonov-regularized least-squares problems: $$v^* = \arg\min_v \| R^{1/2} a - v \|_2^2 + \eta_v \| v \|_2^2; \qquad u^* = \arg\min_u \| R^{1/2} x - u \|_2^2 + \eta_u \| u \|_2^2$$ with the final output $y = \alpha^* (v^*)^H u^*$ and normalization $\alpha^* = 1 / \left((v^*)^H v^*\right)$. This decoupling avoids direct inversion of $R$ and allows the regularization parameters $\eta_v, \eta_u$ to control robustness to steering vector mismatch and covariance estimation errors. The approach is empirically shown to achieve SINR performance matching or exceeding competing robust algorithms in the presence of both model errors and limited snapshots, especially at low SNR or high-mismatch regimes (Suliman et al., 2016).

2. Regularization and Robustness in the Linear Model Perspective

The introduction of regularized least-squares estimation into the linear model-based framework systematically stabilizes solutions that would otherwise suffer from poor conditioning or model mismatch. Regularization—typically via Tikhonov (ridge) terms or more general convex constraints—yields closed-form or efficiently computable solutions, e.g., $(I + \eta I)^{-1} b$ for identity regularization.

This paradigm generalizes beyond classical spatial arrays. In distributionally robust receive combining, minimax optimality is achieved by formulating the receive beamforming as a regularized risk minimization over all second-order distributions within a moment uncertainty set: $$\min_W \max_{R_x \in \mathcal{U}} \; \mathrm{Tr}\left[ W R_x W^H - W R_{xs} - R_{xs}^H W^H + R_s \right]$$ For diagonal loading sets, the solution becomes the classical ridge estimator, providing a direct interpretation of diagonal loading and kernel methods as distributionally robust minimax solutions (Wang et al., 22 Jan 2024). Under such linear model regularization, one avoids explicit channel estimation: only the relevant sample or empirical covariance moments are required.

3. Advanced Linear Model-Based Beamforming Architectures

Significant generalizations of the linear model framework have been developed to handle broadband signals, spatiotemporal regularization, multiuser or distributed arrays, and convolutional or nonlinear array geometries.

• Time-domain linear model-based passive acoustic mapping (TD-LM-PAM): Here, the received signals are modeled as a linear function of a discretized spatiotemporal map of acoustic activity, with the forward model operator explicitly constructed to account for time-of-flight delays dictated by array geometry. Inversion is performed via convex regularization—$\ell_1$, sparsity+TV, or learned denoising priors—and solved with first-order methods (FISTA, ADMM). This yields substantial data efficiency compared to traditional frequency-domain cross-spectral methods, with superior axial resolution and SNR even at 20% of the normal data volume (Gelvez-Barrera et al., 25 Nov 2025).
• Broadband beamforming via linear embedding: For wideband arrays, observed signals are projected onto a Slepian subspace, enabling array data to be acquired and reconstructed from a compressed set of linear measurements. Reconstruction and subsequent beamforming can be performed with provably small residual error and substantial savings in hardware complexity. The recovery algorithm leverages closed-form MMSE or penalized least-squares in the subspace, and performance bounds relate directly to the eigenvalue decay of the underlying covariance, quantifying signal recoverability as a function of embedding dimension (DeLude et al., 2022).
• Low-complexity online convolutional beamforming: Integrating the multichannel linear prediction (MCLP) dereverberation model into the beamforming framework, a Kalman filter-derived affine projection algorithm enables real-time joint dereverberation and beamforming. This configuration reduces algorithmic complexity from $O(Q^2)$ for full-matrix RLS to $O(Q)$ per time step—a crucial benefit for arrays with large mic counts or long filter lengths. Empirical evaluation shows competitive or superior speech enhancement performance at a fraction of the computational cost (Braun et al., 2021).

4. Applications to Spatially Diverse Array Platforms and Problem Classes

The linear model-based framework is inherently adaptable to various spatial array configurations and problem classes:

• MIMO and Multiuser Transceiver Design: The unified quadratic matrix programming (QMP) approach treats both transmit and receive beamforming as matrix-variable quadratic programs with quadratic constraints, subsuming multi-user downlink, coordinated multicell, and cognitive radio scenarios. Closed-form or efficiently computable solutions are available under single-constraint (Lagrangian with 1-D search) or multi-constraint (SDP relaxation) settings. This enables algorithmic recipes for a broad spectrum of wireless settings with rigorous MSE minimization goals (Xing et al., 2012).
• Rank-one optimization for transmit beamforming: Beamforming problems formulated as quadratically constrained quadratic programs (QCQPs) admit a semidefinite relaxation (SDR) that is provably tight; all optimal solutions to the relaxed problem are automatically rank-one, and the original beamforming vector can be extracted exactly. The framework elegantly accommodates perfect CSI, norm-bounded uncertainty, chance-constraints, and RIS-aided designs within a unified mix of LMI and SOC constraints (Le et al., 2023).
• Continuous-aperture and spherical harmonics domain beamforming: For next-generation holographic and spherical arrays, the linear model extends to the continuum—aperture responses are projected onto a basis (e.g., spherical harmonics), and beamformers are derived as finite-dimensional convex combinations of these spatial modes. This yields analytical closed-forms for MRC, ZF, and MMSE beamformers (e.g., $$w_{\text{MMSE},k}(\mathbf r) = \sum_j[\left(I + P R\right)^{-1}]_{j,k} h_j(\mathbf r)$$), and demonstrates empirical performance exceeding conventional discrete arrays, especially under ZF and MMSE (Ouyang et al., 10 Nov 2024, Rafaely et al., 2023).

5. Theoretical Properties and Performance Guarantees

The linear model-based approach, by grounding estimation in well-posed regularized problems, enables concrete theoretical analysis of robustness, consistency, and optimality:

• Avoidance of ill-conditioning: Replacing covariance matrix inversion with regularized regression—applied to operators such as $R^{1/2}$—systematically bounds sensitivity to small eigenvalues, a fundamental cause of instability in classical algorithms.
• Robustness to model mismatch: By not enforcing perfect data-model fit but trading off data fidelity against norm constraints, the framework inherently limits performance degradation from steering vector or noise model errors. This implicit robustness generalizes diagonal loading and eigenvalue thresholding as special cases, and experimental benchmarks demonstrate greater resilience in low-SNR, low-snapshot, or high-mismatch regimes (Suliman et al., 2016).
• Distributional minimax optimality: When the uncertainty set over moment matrices is convex and monotonic, the linear model-based solution is the global minimax optimal estimator against all distributions in the set. This holds for both classical and kernelized regression, as proven in (Wang et al., 22 Jan 2024).
• Degrees-of-freedom and interference management: In MIMO cellular networks, structured linear beamforming with the notion of packing ratios achieves the piecewise-linear optimal spatially-normalized DoF in two-cell, few-user regimes, and unstructured (random-equation) alignment achieves optimality in broader settings where linear beamforming dominates asymptotic interference alignment (Sridharan et al., 2013).

6. Algorithmic Implementation, Limitations, and Extensions

Implementation of linear model-based frameworks commonly proceeds via:

• Covariance or forward-operator estimation and SVD or EVD preprocessing.
• Efficient closed-form or first-order solver steps (conjugate gradient, FISTA, ADMM) for large-scale or high-dimensional systems.
• Hyperparameter selection via cross-validation, L-curve, GCV, or problem-specific analytic criteria (e.g., MVDR-COPRA).
• For time-domain applications, precomputing and storing the (often sparse) linear operator mapping, and leveraging hardware- or streaming-friendly algorithms (e.g., fast Slepian transforms, block Kalman updates).

Limitations noted include assumptions of stationary, homogeneous, or linear propagation models, practical constraints on memory or computational resources (especially with high-dimensional or 3D forward operators), and sensitivity to mis-specified priors or regularization parameters when the physical process is not well-captured by the chosen model (Gelvez-Barrera et al., 25 Nov 2025). Extension opportunities are abundant: modeling non-stationary transients, hybrid time-frequency methods, operator compression, and robustification under further classes of uncertainty remain active research fronts.

7. Summary and Outlook

Linear model-based beamforming frameworks generalize classical array-processing methods by explicit, regularized modeling of the measurement process, physical array geometry, and uncertainties in system parameters. They enable robust, theoretically principled, and often efficiently computable solutions to an encompassing range of beamforming problems: from adaptive spatial filtering, passive imaging, and dereverberation, to MIMO wireless transmission and large-scale holographic arrays. Empirical evidence shows near-oracle performance under realistic mismatch and data scarcity conditions, systematic robustness to ill-conditioning, and extensibility to nonlinear, distributed, and data-driven approaches. This framework provides a unifying technical foundation for the design and analysis of advanced array processing algorithms across acoustics, communications, radar, and beyond (Suliman et al., 2016, Gelvez-Barrera et al., 25 Nov 2025, DeLude et al., 2022, Wang et al., 22 Jan 2024, Xing et al., 2012, Rafaely et al., 2023, Sridharan et al., 2013).