Near-Field Nulling Control Beam Focusing

• NCBF is a near-field beamforming method that uses LCMV beamforming to achieve focused energy at targets with imposed nulls at interfering locations.
• It leverages spherical wave models to account for both angular and range-dependent variations, enabling multi-user interference mitigation in XL-MIMO and related platforms.
• Recent advances integrate deep learning and codebook partitioning to optimize phase-magnitude weight vectors, achieving high spatial resolution and low-latency operation.

Near-Field Nulling Control Beam Focusing (NCBF) is a near-field beamforming paradigm in which an array concentrates electromagnetic energy at a designated spatial location while forcing low gain or nulls at designated interfering locations. Its defining feature is operation in the radiating near-field or Fresnel region, where spherical wavefronts replace the plane-wave assumption and the array response depends on both angle and range rather than angle alone. In extra-large MIMO (XL-MIMO), modular XL-arrays, large intelligent surfaces, and related platforms, NCBF is therefore used for multi-user interference mitigation, spatial selectivity in both distance and direction, and beam pattern control under realistic electromagnetic constraints (Zhang et al., 2022, Karimi et al., 23 Sep 2025).

1. Physical regime and beam-control principle

Near-field beam focusing differs fundamentally from far-field beam steering. In the far field, beamforming is directional and the channel response depends only on angle; in the near field, beamforming is spatially focal and the channel response depends on both angle and distance. The Fraunhofer distance is given by

$d_F = \frac{2D^2}{\lambda},$

with $D$ the array diameter and $\lambda$ the wavelength. When users lie within this regime, the standard plane-wave approximation no longer holds, and the steering or focusing vector must account for per-element path lengths to the focal point (Zhang et al., 2022, Qu et al., 2023).

This change in propagation model gives NCBF an additional spatial degree of freedom. Users at the same angle but different ranges can be spatially separated, and interference can be mitigated in the distance domain as well as in the angular domain. For modular XL-arrays, accurate characterization of amplitude and phase variations across elements requires the non-uniform spherical wave model, while sub-array based uniform spherical wave models provide tractable approximations. These models reveal that modular XL-arrays can significantly enhance the spatial resolution, but at the cost of generating undesired grating lobes; in the near-field uniform spherical wave model, however, the non-linear phase variations across the array elements provide a higher grating lobe suppression capability than the conventional far-field uniform plane wave model (Li et al., 2023).

A common misconception is that NCBF is simply conventional null steering applied to a larger array. The near-field literature does not support that interpretation: the design variable is not only direction, but location in space, and model mismatch between far-field steering and near-field focusing produces gain loss that increases as the target gets closer and as the array grows larger (Qu et al., 2023).

2. Canonical formulation and performance measures

A standard formulation of NCBF uses Linearly Constrained Minimum Variance (LCMV) beamforming to maximize gain at the desired user while imposing zero gain at interfering users. With user positions $\mathbf{p}_1,\ldots,\mathbf{p}_K$ and array responses collected in

$$\mathbf{C} = [\mathbf{h}'(\mathbf{p}_1), \ldots, \mathbf{h}'(\mathbf{p}_K)],$$

the LCMV solution is

$$\mathbf{w}_{\mathrm{lcmv}} = \mathbf{R}^{-1}\mathbf{C}\left(\mathbf{C}^\mathrm{H}\mathbf{R}^{-1}\mathbf{C}\right)^{-1}\mathbf{d},$$

where $\mathbf{R}$ is the signal covariance matrix and $\mathbf{d}$ is the desired gain vector, equal to $1$ at the target user and $0$ at interferers. In the equivalent constraint form, the beamformer satisfies

$D$0

The resulting NCBF weight vector is commonly decomposed into magnitude and phase components as

$D$1

This phase-magnitude factorization is central to recent learning-based implementations (Karimi et al., 23 Sep 2025, Karimi et al., 26 Sep 2025).

Reported performance is often summarized by an NCBF gain, defined as

$D$2

This metric directly quantifies multi-user interference suppression by comparing the gain at the desired focal point with the strongest residual gain at interfering locations (Karimi et al., 23 Sep 2025).

3. Dual-estimator deep learning architectures

A recent line of work replaces repeated optimization with supervised prediction of NCBF weights from user locations. In "DNN-Based Nulling Control Beam Focusing for Near-Field Multi-User Interference Mitigation" (Karimi et al., 23 Sep 2025), the proposed framework uses a dual-estimator architecture comprising two fully connected deep neural networks (FCDNNs): a phase estimator for $D$3 and a magnitude estimator for $D$4. The inputs are the coordinates of all users, both the desired target and interfering users, typically their angular and radial positions in $D$5D, with extension to $D$6D noted as $D$7. For an array of $D$8 elements, each network outputs an $D$9-dimensional vector.

The training pipeline normalizes angular values and distances, maps phases to $\lambda$0 with a common reference to eliminate ambiguity, and normalizes magnitudes to unit power and expresses them in dB. Hidden layers use ReLU and the output layer is linear. Best performance was achieved for architectures with $\lambda$1–$\lambda$2 hidden layers and $\lambda$3–$\lambda$4 neurons per layer. Ground-truth labels are produced by the LCMV algorithm over a dataset covering both collinear and non-collinear scenarios in the near-field region from $\lambda$5–$\lambda$6 m, with $\lambda$7 samples used for final training in the reported $\lambda$8-user case of $\lambda$9 desired user and $\mathbf{p}_1,\ldots,\mathbf{p}_K$0 interferers (Karimi et al., 23 Sep 2025).

The reported results are specific. Test Circular Mean Absolute Error for phase estimation is $\mathbf{p}_1,\ldots,\mathbf{p}_K$1 radians, and test RMSE for magnitude estimation is $\mathbf{p}_1,\ldots,\mathbf{p}_K$2 dB. Average multi-user interference suppression is $\mathbf{p}_1,\ldots,\mathbf{p}_K$3 dB, interference mitigation exceeds $\mathbf{p}_1,\ldots,\mathbf{p}_K$4 dB across all tested cases, and null placement angular deviation is less than $\mathbf{p}_1,\ldots,\mathbf{p}_K$5. Full-wave simulations in Ansys HFSS, using a $\mathbf{p}_1,\ldots,\mathbf{p}_K$6 patch antenna array with realistic inter-element coupling, non-uniform element patterns, $\mathbf{p}_1,\ldots,\mathbf{p}_K$7 cm spacing, and $\mathbf{p}_1,\ldots,\mathbf{p}_K$8 GHz operation, show close agreement between DNN-predicted and LCMV-based beam patterns. Batch prediction time is reported as under $\mathbf{p}_1,\ldots,\mathbf{p}_K$9 ms for up to $$\mathbf{C} = [\mathbf{h}'(\mathbf{p}_1), \ldots, \mathbf{h}'(\mathbf{p}_K)],$$0 antennas, whereas LCMV exhibits cubic scaling with antenna number, which is presented as the central computational motivation for the learning-based alternative (Karimi et al., 23 Sep 2025).

4. Codebook partitioning and region-specialized models

A related architecture, "A Deep Neural Network Codebook Approach for Near-Field Nulling Control Beam Focusing" (Karimi et al., 26 Sep 2025), addresses scalability by partitioning the Fresnel region into spatial subsections and assigning a lightweight DNN pair to each subsection. The partitioning is correlation-based rather than uniform in space. The radial sampling rule is given as

$$\mathbf{C} = [\mathbf{h}'(\mathbf{p}_1), \ldots, \mathbf{h}'(\mathbf{p}_K)],$$1

and the angular sampling rule as

$$\mathbf{C} = [\mathbf{h}'(\mathbf{p}_1), \ldots, \mathbf{h}'(\mathbf{p}_K)],$$2

The resulting subsections are denser closer to the array, and each subsection is served by a dedicated lightweight model (Karimi et al., 26 Sep 2025).

Each codebook region contains two DNNs, again split into phase and magnitude estimation. The shared structure is an input layer of $$\mathbf{C} = [\mathbf{h}'(\mathbf{p}_1), \ldots, \mathbf{h}'(\mathbf{p}_K)],$$3 nodes, hidden layers $$\mathbf{C} = [\mathbf{h}'(\mathbf{p}_1), \ldots, \mathbf{h}'(\mathbf{p}_K)],$$4, and an output layer of $$\mathbf{C} = [\mathbf{h}'(\mathbf{p}_1), \ldots, \mathbf{h}'(\mathbf{p}_K)],$$5 nodes. Each DNN is trained on $$\mathbf{C} = [\mathbf{h}'(\mathbf{p}_1), \ldots, \mathbf{h}'(\mathbf{p}_K)],$$6 samples with an $$\mathbf{C} = [\mathbf{h}'(\mathbf{p}_1), \ldots, \mathbf{h}'(\mathbf{p}_K)],$$7 train-test split, batch size $$\mathbf{C} = [\mathbf{h}'(\mathbf{p}_1), \ldots, \mathbf{h}'(\mathbf{p}_K)],$$8, $$\mathbf{C} = [\mathbf{h}'(\mathbf{p}_1), \ldots, \mathbf{h}'(\mathbf{p}_K)],$$9 epochs, and exponential learning-rate decay with factor $$\mathbf{w}_{\mathrm{lcmv}} = \mathbf{R}^{-1}\mathbf{C}\left(\mathbf{C}^\mathrm{H}\mathbf{R}^{-1}\mathbf{C}\right)^{-1}\mathbf{d},$$0. Inputs are normalized polar coordinates of all users, and labels are LCMV weights whose phases are taken relative to the first element and whose magnitudes are scaled to unit norm and converted to dB (Karimi et al., 26 Sep 2025).

For $$\mathbf{w}_{\mathrm{lcmv}} = \mathbf{R}^{-1}\mathbf{C}\left(\mathbf{C}^\mathrm{H}\mathbf{R}^{-1}\mathbf{C}\right)^{-1}\mathbf{d},$$1 subsections, the reported average phase and magnitude prediction errors are $$\mathbf{w}_{\mathrm{lcmv}} = \mathbf{R}^{-1}\mathbf{C}\left(\mathbf{C}^\mathrm{H}\mathbf{R}^{-1}\mathbf{C}\right)^{-1}\mathbf{d},$$2 radians and $$\mathbf{w}_{\mathrm{lcmv}} = \mathbf{R}^{-1}\mathbf{C}\left(\mathbf{C}^\mathrm{H}\mathbf{R}^{-1}\mathbf{C}\right)^{-1}\mathbf{d},$$3 dB, respectively, across the sample subsections. In full-wave Ansys HFSS simulations, the DNN codebook achieves interference suppression better than $$\mathbf{w}_{\mathrm{lcmv}} = \mathbf{R}^{-1}\mathbf{C}\left(\mathbf{C}^\mathrm{H}\mathbf{R}^{-1}\mathbf{C}\right)^{-1}\mathbf{d},$$4 dB, with a performance gap within $$\mathbf{w}_{\mathrm{lcmv}} = \mathbf{R}^{-1}\mathbf{C}\left(\mathbf{C}^\mathrm{H}\mathbf{R}^{-1}\mathbf{C}\right)^{-1}\mathbf{d},$$5 dB of the LCMV method, in both collinear and non-collinear cases. The codebook framing suggests a division of NCBF into localized inference problems, which is a plausible route for extending near-field beam control to larger Fresnel regions without a single monolithic model (Karimi et al., 26 Sep 2025).

5. Continuous-region nulling, RIS realizations, and beam-shape control

Although many NCBF formulations impose nulls at discrete user locations, related work shows that the same near-field principles extend to continuous low-exposure regions. In "Near-Field Beampointing with Low Exposure Regions: a Dominant Subspace Projection Approach" (Defraigne et al., 13 Feb 2026), the received power is written as

$$\mathbf{w}_{\mathrm{lcmv}} = \mathbf{R}^{-1}\mathbf{C}\left(\mathbf{C}^\mathrm{H}\mathbf{R}^{-1}\mathbf{C}\right)^{-1}\mathbf{d},$$6

and a discretized low-exposure region yields a large constraint matrix

$$\mathbf{w}_{\mathrm{lcmv}} = \mathbf{R}^{-1}\mathbf{C}\left(\mathbf{C}^\mathrm{H}\mathbf{R}^{-1}\mathbf{C}\right)^{-1}\mathbf{d},$$7

The proposed Dominant Subspace Projection method uses the singular value decomposition

$$\mathbf{w}_{\mathrm{lcmv}} = \mathbf{R}^{-1}\mathbf{C}\left(\mathbf{C}^\mathrm{H}\mathbf{R}^{-1}\mathbf{C}\right)^{-1}\mathbf{d},$$8

to retain only the dominant basis vectors of the constraint space. The reported outcome is nearly optimal user power, exemplified by about $$\mathbf{w}_{\mathrm{lcmv}} = \mathbf{R}^{-1}\mathbf{C}\left(\mathbf{C}^\mathrm{H}\mathbf{R}^{-1}\mathbf{C}\right)^{-1}\mathbf{d},$$9 of MRT/optimal, together with uniform power mitigation over the low exposure region and substantially reduced complexity (Defraigne et al., 13 Feb 2026).

Near-field NCBF has also been transposed to reconfigurable intelligent surfaces. "Physics-Aware RIS Codebook Compilation for Near-Field Beam Focusing under Mutual Coupling and Specular Reflections" (Papadopoulos et al., 19 Jan 2026) models the incident field on each RIS cell as the sum of direct illumination, mutual coupling, and specular reflections:

$\mathbf{R}$0

The radiated field then becomes

$\mathbf{R}$1

The MATCH algorithm compiles codebooks through geometric-optics initialization, sensitivity-based local refinement, Pareto-optimal global search, and final refinement. In the reported full-wave simulations, focus-region energy improves from $\mathbf{R}$2 under initialization to $\mathbf{R}$3 after sensitivity refinement, $\mathbf{R}$4 after global optimization, and $\mathbf{R}$5 after final refinement, while residual leakage drops to $\mathbf{R}$6 (Papadopoulos et al., 19 Jan 2026).

Near-field beam-shape control can also be interpreted as a variant of NCBF. "Quasi-Closed-Form Driven Near-Field Flat-Top Beamfocusing with Concentric Circular Vertical Polarized Dipole Array For Large Intelligent Surface Applications" (Li, 9 Jun 2025) shows that small-radius rings produce a monotonic increase in field strength near the focus, whereas large-radius rings produce a monotonic decrease. By superimposing rings with different radii and then fine-tuning the ring excitations, the method achieves a flat-top beam with ripple below $\mathbf{R}$7 dB in theory, while full-wave FEKO simulations validate the overall trend and show practical degradation to about $\mathbf{R}$8 dB flatness (Li, 9 Jun 2025).

6. Array design choices, robustness, and unresolved issues

NCBF performance depends strongly on array geometry and implementation constraints. "Sparse Arrays Enable Near-Field Constant-Distance Focusing with Reduced Focal Shift" (Li et al., 12 May 2025) revisits near-field focusing from a degrees-of-freedom perspective and argues that optimized sparse spacing can suppress focal shift while maintaining nearly uniform energy distribution during beam scanning. In the tested case of a $\mathbf{R}$9-element array at $\mathbf{d}$0 GHz with focus distance $\mathbf{d}$1, the optimal spacing is reported as about $\mathbf{d}$2; dense arrays with $\mathbf{d}$3 exhibit severe focal shift, whereas the optimized sparse array keeps the field maxima at the target spots and the energy nearly uniform (Li et al., 12 May 2025).

A different route is to introduce positional degrees of freedom directly. In "Movable Antenna-Enhanced Near-Field Flexible Beamforming: Performance Analysis and Optimization" (Yang et al., 25 Jan 2026), beam nulling is posed as joint optimization of antenna positions and beamforming weights, with the gain expression

$\mathbf{d}$4

For fixed positions, the optimal beamforming vector is the zero-forcing solution projected onto the nullspace of interference steering vectors; for general cases, the paper proposes discrete sampling, sequential updates, alternating optimization, and a Taylor-series-based sensitivity analysis for position errors. The reported findings state that errors of $\mathbf{d}$5 already start to degrade nulls and degrade focusing, underscoring the precision demands of near-field spatial control (Yang et al., 25 Jan 2026).

Several broader challenges remain structural rather than algorithm-specific. Near-field beam focusing is highly sensitive to CSI errors; Fourier or angular dictionaries are insufficient and polar-domain models are required. Wideband systems suffer beam misfocus or beam split because practical phased shifters are often frequency-flat. Very large arrays can exhibit non-stationarity and hybrid near-field/far-field behavior across the aperture. These issues indicate that NCBF is not only an optimization problem over weights, but also a modeling, calibration, and hardware co-design problem for 6G-scale systems (Zhang et al., 2022).

In the current literature, NCBF is therefore best understood as a family of location-selective beamforming methods enabled by spherical-wave propagation. Its classical core is LCMV-style null-constrained focusing; its modern realizations include dual-estimator DNNs, region-specialized codebooks, dominant-subspace nulling for continuous low-exposure regions, RIS codebook compilation under mutual coupling and reflections, sparse-array designs that reduce focal shift, and movable-antenna formulations that treat array geometry as an optimization variable. This suggests that the central research trajectory is toward physically consistent, low-latency, and robust near-field beam control rather than toward a single universal architecture (Karimi et al., 23 Sep 2025, Karimi et al., 26 Sep 2025).