Nulling Control Beamforming (NCBF)

• Nulling Control Beamforming (NCBF) is a method to maximize useful signal power toward desired directions while suppressing signals in undesired spatial regions.
• It employs multiple formulations—including point nulling, low exposure regions, and learning-based approaches—to optimize beam performance in near-field and multi-user scenarios.
• NCBF balances signal gain with null constraints under strict hardware calibration and computational limits, impacting SNR and operational efficiency.

Searching arXiv for recent and foundational papers on nulling control beamforming and closely related beam-nulling methods. arxiv_search("nulling control beamforming near-field beamnulling low exposure region dominant subspace projection") Nulling Control Beamforming (NCBF) denotes beamforming methods that maximize useful signal power toward a desired user, direction, or focal point while enforcing nulls, low-power constraints, or leakage suppression toward undesired users, directions, or spatial regions. In conventional array processing this objective is often posed as point nulling in the far field, whereas recent work extends it to near-field beampointing and beamnulling, continuous low exposure regions, multi-user interference mitigation in XL-MIMO, and standards-constrained beam selection in 5G systems (Defraigne et al., 13 Feb 2026, Harp, 2012, Tusha et al., 24 Apr 2026). A related but distinct usage of “beam-nulling” in adaptive MIMO excludes the weakest eigenmode and transmits in the orthogonal subspace, showing that nulling can be formulated either as spatial suppression in physical space or as exclusion of unfavorable transmit subspaces (0807.4881).

1. Foundations and problem regimes

Traditional NCBF aims to steer beams toward desired users while nulling or suppressing power at specific locations—historically as point constraints, often in the far-field. In this regime, the nulling target is usually specified by direction, and the array response is primarily angle-dependent. In the near field, by contrast, steering vectors depend not only on angle, but also on range, which creates additional beamforming flexibility and simultaneously increases constraint dimensionality and correlation (Defraigne et al., 13 Feb 2026). This near-field dependence underlies recent formulations of beam focusing, beam nulling, and low exposure region design for large arrays and XL-MIMO.

A second foundational distinction concerns the type of nulling constraint. Some formulations require exact zero power at specific points or directions; others impose relaxed thresholds, average leakage penalties, or region-wide suppression. The Allen Telescope Array formulation emphasizes custom beam shaping through complex antenna coefficients for RFI mitigation, wide-area nulling, and wideband nulls (Harp, 2012). In multi-user communication systems, the same design principle appears as interference suppression: LCMV-based nulling control beam focusing in XL-MIMO, analog-only beamforming that minimizes leakage to non-target users, passive zero forcing with beyond-diagonal RIS, and frequency-domain nullspace projection in time-reversal beamforming (Karimi et al., 23 Sep 2025, Wang et al., 2024, Yahya et al., 2024, Viteri-Mera et al., 2015).

A third distinction is architectural. NCBF may be implemented with fully digital arrays, analog or hybrid phase-shifter networks, passive RIS hardware, movable antennas, or mobile platforms such as UAVs. The common objective is unchanged—focus energy where it is useful and suppress it where it is harmful—but the feasible weight sets, calibration sensitivity, and computational mechanisms differ substantially (Wang et al., 2024, Yang et al., 25 Jan 2026, Fotiadis et al., 24 Aug 2025).

2. Canonical mathematical formulations

A classical narrowband beam pattern model writes the synthesized response in direction $\vec{k}$ as

$$S(\vec{k}-\vec{k}_0)=\sum_i \omega_i e^{-i(\vec{k}-\vec{k}_0)\cdot \vec{r}_i},$$

where $\omega_i$ are the complex antenna weights, $\vec{k}_0$ is the beam pointing direction, and $\vec{r}_i$ is the position of antenna $i$ (Harp, 2012). For a single point null, the condition is $S(\vec{k}_1-\vec{k}_0)=0$. For multiple nulls, the ATA iterative update is

$$\omega_i^{(p)}=\omega_i^{(p-1)}-\sum_{m=1}^M S^{(p-1)}(\vec{k}_m-\vec{k}_0)e^{i(\vec{k}_m-\vec{k}_0)\cdot \vec{r}_i},$$

which progressively suppresses gain in the null region (Harp, 2012).

Modern multi-user NCBF often uses linearly constrained formulations. In the near-field XL-MIMO setting, the LCMV beamformer is written as

$$\boldsymbol{w}_\mathrm{LCMV}(\boldsymbol{p}_1,\ldots,\boldsymbol{p}_K) = \mathbf{R}^{-1}\boldsymbol{C}\left(\boldsymbol{C}^H\mathbf{R}^{-1}\boldsymbol{C}\right)^{-1}\boldsymbol{d},$$

where $\mathbf{R}$ is the signal covariance, $$S(\vec{k}-\vec{k}_0)=\sum_i \omega_i e^{-i(\vec{k}-\vec{k}_0)\cdot \vec{r}_i},$$0 collects steering vectors, and $$S(\vec{k}-\vec{k}_0)=\sum_i \omega_i e^{-i(\vec{k}-\vec{k}_0)\cdot \vec{r}_i},$$1 encodes the desired response, typically $$S(\vec{k}-\vec{k}_0)=\sum_i \omega_i e^{-i(\vec{k}-\vec{k}_0)\cdot \vec{r}_i},$$2 for the desired user and $$S(\vec{k}-\vec{k}_0)=\sum_i \omega_i e^{-i(\vec{k}-\vec{k}_0)\cdot \vec{r}_i},$$3 for interferers (Karimi et al., 26 Sep 2025, Karimi et al., 23 Sep 2025). This formulation is central in recent supervised learning approaches because it provides ground-truth nulling weights across collinear and non-collinear user configurations.

For near-field low exposure region design, the discretized optimization problem is

$$S(\vec{k}-\vec{k}_0)=\sum_i \omega_i e^{-i(\vec{k}-\vec{k}_0)\cdot \vec{r}_i},$$4

subject to

$$S(\vec{k}-\vec{k}_0)=\sum_i \omega_i e^{-i(\vec{k}-\vec{k}_0)\cdot \vec{r}_i},$$5

where $$S(\vec{k}-\vec{k}_0)=\sum_i \omega_i e^{-i(\vec{k}-\vec{k}_0)\cdot \vec{r}_i},$$6 is the user steering vector, $$S(\vec{k}-\vec{k}_0)=\sum_i \omega_i e^{-i(\vec{k}-\vec{k}_0)\cdot \vec{r}_i},$$7 is the steering vector for the $$S(\vec{k}-\vec{k}_0)=\sum_i \omega_i e^{-i(\vec{k}-\vec{k}_0)\cdot \vec{r}_i},$$8-th sampled point in the low exposure region, and $$S(\vec{k}-\vec{k}_0)=\sum_i \omega_i e^{-i(\vec{k}-\vec{k}_0)\cdot \vec{r}_i},$$9 is typically much greater than $\omega_i$0 (Defraigne et al., 13 Feb 2026). By leveraging phase invariance, this is reformulated as an SOCP with constraints on $\omega_i$1, $\omega_i$2, and $\omega_i$3.

Other architectures impose additional feasibility constraints. In analog or hybrid mmWave systems, the combining vector satisfies constant-modulus and phase-quantization constraints,

$\omega_i$4

with $\omega_i$5 drawn from a quantized phase set (Zhang et al., 2022). In analog-only near-field multiuser MIMO, the design can be reframed through the Signal-to-Leakage-plus-Noise Ratio and then through the surrogate objective

$\omega_i$6

under constant-modulus constraints on each RF weight (Wang et al., 2024). In wideband multi-user mmWave massive MIMO, interference-nulling time-reversal solves, at each frequency, a projection problem that finds the closest time-reversal pre-filter lying in the nullspace of the other users’ channel vectors (Viteri-Mera et al., 2015). Across these formulations, NCBF is best viewed as a constrained power-shaping problem whose precise geometry depends on propagation regime and hardware.

3. Near-field, region-constrained, and geometry-aware NCBF

Near-field NCBF introduces two coupled complications: the wavefront is spherical, and practical protection requirements may concern a continuous two-dimensional region rather than isolated sample points. The dominant-subspace projection method addresses this by noting that steering vectors for a compact, densely sampled low exposure region are highly correlated, so the constraint matrix $\omega_i$7 has low effective rank (Defraigne et al., 13 Feb 2026). After computing the SVD,

$\omega_i$8

the method projects the user steering vector onto the orthogonal complement of the dominant singular-vector subspace, normalizes the result, and searches over the retained subspace dimension $\omega_i$9 to balance user power and low exposure region suppression. Empirically, a small fraction ($\vec{k}_0$0) of singular values captures nearly all the energy. For $\vec{k}_0$1, $\vec{k}_0$2, and $\vec{k}_0$3 dB relative to MRT, the optimal SOCP/interior-point solution achieves $\vec{k}_0$4 received power at the user with $\vec{k}_0$5 hr real-time compute, while DoSP achieves $\vec{k}_0$6 with $\vec{k}_0$7 s precompute and $\vec{k}_0$8 s real-time compute; the paper also reports more homogeneous power mitigation over the continuous low exposure region (Defraigne et al., 13 Feb 2026).

Near-field beam nulling also benefits from jointly optimizing physical array geometry. With movable antennas, the objective is to maximize the beam gain toward a desired user while nulling multiple undesired users by jointly optimizing antenna positions $\vec{k}_0$9 and beamforming vector $\vec{r}_i$0 (Yang et al., 25 Jan 2026). For fixed positions, the zero-forcing beamformer is

$\vec{r}_i$1

where

$\vec{r}_i$2

The paper shows that, with proper positioning of the movable antennas, directing the beam toward a specific direction can lead to nulls in other directions in the beam nulling scenario. For finite movement regions it proposes a discrete sampling method and a sequential update algorithm with complexity $\vec{r}_i$3, and for position uncertainty it derives a Taylor-series-based QCQP handled via SDR (Yang et al., 25 Jan 2026). This suggests that in the near field, NCBF is increasingly a joint optimization over weights and geometry rather than a weight-design problem alone.

Analog-only near-field multiuser MIMO adopts a related philosophy but keeps the reconfiguration entirely in the RF domain. Two AoBF schemes based on the majorization-minimization algorithm alternate beam focusing for the target user and leakage suppression for non-target users. With perfect CSI, the algorithm nulls at exact user locations; with imperfect CSI, it samples multiple auxiliary points over a codeword area and broadens the focusing and nulling region accordingly (Wang et al., 2024). Simulation results demonstrate that the two AoBF schemes can approach the sum rate of the HBF schemes but outperform HBF schemes in terms of energy efficiency, and at SNR $\vec{r}_i$4 dB the AoBF energy efficiency exceeds that of HBF by over $\vec{r}_i$5 and basic analog-only beam steering by over $\vec{r}_i$6 (Wang et al., 2024).

4. Learning-based and adaptive NCBF

Recent learning-based NCBF replaces repeated online optimization with direct prediction of beamforming weights from user geometry. A dual-estimator framework for near-field nulling control beam focusing in XL-MIMO uses two fully connected deep neural networks to separately predict the phase and magnitude components of the NCBF weights from the locations of desired and interfering users (Karimi et al., 23 Sep 2025). The training labels are produced by the LCMV beamformer, the phase loss is Circular Mean Absolute Error, and the magnitude loss is Root Mean Squared Error in dB. In a three-user $\vec{r}_i$7 ULA scenario, the reported test errors are $\vec{r}_i$8 radians for phase estimation and $\vec{r}_i$9 dB for magnitude estimation. Full-wave simulations with a metamaterial-based $i$0 patch array at $i$1 GHz show close agreement between DNN-predicted and LCMV-based NCBF patterns, with average multi-user interference suppression of $i$2 dB, minimum suppression of $i$3 dB, angular null deviation of less than $i$4, and prediction time below $i$5 ms on standard CPUs for thousands of samples (Karimi et al., 23 Sep 2025).

A complementary approach partitions the Fresnel region using correlation-based sampling and assigns a lightweight fully connected DNN model to each subsection (Karimi et al., 26 Sep 2025). Each sector has one phase-estimator network and one magnitude-estimator network, trained independently on $i$6 LCMV-generated samples. For a codebook with $i$7 sectors, the reported mean phase error is approximately $i$8 radians with standard deviation approximately $i$9 radians, and the mean magnitude error is approximately $S(\vec{k}_1-\vec{k}_0)=0$0 dB with standard deviation approximately $S(\vec{k}_1-\vec{k}_0)=0$1 dB. Full-wave simulations in Ansys HFSS with a $S(\vec{k}_1-\vec{k}_0)=0$2 real patch ULA at $S(\vec{k}_1-\vec{k}_0)=0$3 GHz report interference suppression between $S(\vec{k}_1-\vec{k}_0)=0$4 and $S(\vec{k}_1-\vec{k}_0)=0$5 dB, with a performance gap within $S(\vec{k}_1-\vec{k}_0)=0$6 dB of the LCMV method, for both collinear and non-collinear user scenarios (Karimi et al., 26 Sep 2025). The architectural significance is that partitioning by physical channel correlation replaces a single large model with many smaller models, improving scalability and memory efficiency.

Not all learning-based NCBF is supervised. In mmWave MIMO with analog or hybrid constraints, an online deep reinforcement learning algorithm learns beam patterns that null interfering directions using only power measurements and without explicit channel knowledge or coordination with interferers (Zhang et al., 2022). The state is the current vector of phase settings, the action is a new phase assignment, and the reward is binary according to whether SINR improves. A surrogate model approximates the environment response and reduces hardware interactions needed for convergence by more than $S(\vec{k}_1-\vec{k}_0)=0$7. In simulation, SIR increases by over $S(\vec{k}_1-\vec{k}_0)=0$8 dB compared to interference-unaware beams, and by around $S(\vec{k}_1-\vec{k}_0)=0$9 dB after around $$\omega_i^{(p)}=\omega_i^{(p-1)}-\sum_{m=1}^M S^{(p-1)}(\vec{k}_m-\vec{k}_0)e^{i(\vec{k}_m-\vec{k}_0)\cdot \vec{r}_i},$$0 learning iterations; hardware experiments with a $$\omega_i^{(p)}=\omega_i^{(p-1)}-\sum_{m=1}^M S^{(p-1)}(\vec{k}_m-\vec{k}_0)e^{i(\vec{k}_m-\vec{k}_0)\cdot \vec{r}_i},$$1-antenna linear phased array at $$\omega_i^{(p)}=\omega_i^{(p-1)}-\sum_{m=1}^M S^{(p-1)}(\vec{k}_m-\vec{k}_0)e^{i(\vec{k}_m-\vec{k}_0)\cdot \vec{r}_i},$$2 GHz and $$\omega_i^{(p)}=\omega_i^{(p-1)}-\sum_{m=1}^M S^{(p-1)}(\vec{k}_m-\vec{k}_0)e^{i(\vec{k}_m-\vec{k}_0)\cdot \vec{r}_i},$$3-bit phase shifters show at least $$\omega_i^{(p)}=\omega_i^{(p-1)}-\sum_{m=1}^M S^{(p-1)}(\vec{k}_m-\vec{k}_0)e^{i(\vec{k}_m-\vec{k}_0)\cdot \vec{r}_i},$$4 dB SIR and INR improvement versus the best fixed beam in every scenario (Zhang et al., 2022). The contrast with supervised DNN methods is direct: the latter amortize LCMV-like optimization offline, whereas DRL learns interference-aware codebooks online under measurement-only feedback.

5. Hardware realization, calibration, and standards-constrained nulling

Deep nulls are intrinsically narrow, and even small phase, timing, or gain mismatches across RF chains can significantly degrade suppression (Kim et al., 2 Apr 2026). The ATA study already identified hardware control as central: high-speed amplitude control of each array element over the full range $$\omega_i^{(p)}=\omega_i^{(p-1)}-\sum_{m=1}^M S^{(p-1)}(\vec{k}_m-\vec{k}_0)e^{i(\vec{k}_m-\vec{k}_0)\cdot \vec{r}_i},$$5–$$\omega_i^{(p)}=\omega_i^{(p-1)}-\sum_{m=1}^M S^{(p-1)}(\vec{k}_m-\vec{k}_0)e^{i(\vec{k}_m-\vec{k}_0)\cdot \vec{r}_i},$$6 is critically important to allow testing of wide area and wide bandwidth nulling, with an update rate of approximately $$\omega_i^{(p)}=\omega_i^{(p-1)}-\sum_{m=1}^M S^{(p-1)}(\vec{k}_m-\vec{k}_0)e^{i(\vec{k}_m-\vec{k}_0)\cdot \vec{r}_i},$$7 ms (Harp, 2012). It also reported that deep nulls require highly precise gain and phase calibration; even modest phase greater than $$\omega_i^{(p)}=\omega_i^{(p-1)}-\sum_{m=1}^M S^{(p-1)}(\vec{k}_m-\vec{k}_0)e^{i(\vec{k}_m-\vec{k}_0)\cdot \vec{r}_i},$$8 radians or gain greater than $$\omega_i^{(p)}=\omega_i^{(p-1)}-\sum_{m=1}^M S^{(p-1)}(\vec{k}_m-\vec{k}_0)e^{i(\vec{k}_m-\vec{k}_0)\cdot \vec{r}_i},$$9 errors directly degrade null depth (Harp, 2012). These constraints distinguish NCBF from ordinary beam steering: a main beam can remain acceptable while nulls collapse.

A recent SDR implementation addresses this calibration bottleneck through a self-calibrating architecture tailored for high-fidelity null forming (Kim et al., 2 Apr 2026). A compact reference transmitter is directionally coupled to all antenna feeds through a length-matched Wilkinson divider, and per-channel offsets are compensated digitally with lightweight FIR filters. The channel frequency response is modeled as

$$\boldsymbol{w}_\mathrm{LCMV}(\boldsymbol{p}_1,\ldots,\boldsymbol{p}_K) = \mathbf{R}^{-1}\boldsymbol{C}\left(\boldsymbol{C}^H\mathbf{R}^{-1}\boldsymbol{C}\right)^{-1}\boldsymbol{d},$$0

and the null-forming vector is

$$\boldsymbol{w}_\mathrm{LCMV}(\boldsymbol{p}_1,\ldots,\boldsymbol{p}_K) = \mathbf{R}^{-1}\boldsymbol{C}\left(\boldsymbol{C}^H\mathbf{R}^{-1}\boldsymbol{C}\right)^{-1}\boldsymbol{d},$$1

In simulation, the average nulling ratio improves from $$\boldsymbol{w}_\mathrm{LCMV}(\boldsymbol{p}_1,\ldots,\boldsymbol{p}_K) = \mathbf{R}^{-1}\boldsymbol{C}\left(\boldsymbol{C}^H\mathbf{R}^{-1}\boldsymbol{C}\right)^{-1}\boldsymbol{d},$$2 before calibration to $$\boldsymbol{w}_\mathrm{LCMV}(\boldsymbol{p}_1,\ldots,\boldsymbol{p}_K) = \mathbf{R}^{-1}\boldsymbol{C}\left(\boldsymbol{C}^H\mathbf{R}^{-1}\boldsymbol{C}\right)^{-1}\boldsymbol{d},$$3 after the proposed two-stage FIR compensation, whereas one-stage direct equalization reaches only $$\boldsymbol{w}_\mathrm{LCMV}(\boldsymbol{p}_1,\ldots,\boldsymbol{p}_K) = \mathbf{R}^{-1}\boldsymbol{C}\left(\boldsymbol{C}^H\mathbf{R}^{-1}\boldsymbol{C}\right)^{-1}\boldsymbol{d},$$4. On a $$\boldsymbol{w}_\mathrm{LCMV}(\boldsymbol{p}_1,\ldots,\boldsymbol{p}_K) = \mathbf{R}^{-1}\boldsymbol{C}\left(\boldsymbol{C}^H\mathbf{R}^{-1}\boldsymbol{C}\right)^{-1}\boldsymbol{d},$$5–$$\boldsymbol{w}_\mathrm{LCMV}(\boldsymbol{p}_1,\ldots,\boldsymbol{p}_K) = \mathbf{R}^{-1}\boldsymbol{C}\left(\boldsymbol{C}^H\mathbf{R}^{-1}\boldsymbol{C}\right)^{-1}\boldsymbol{d},$$6 GHz SDR platform, the measured average nulling ratio improves from $$\boldsymbol{w}_\mathrm{LCMV}(\boldsymbol{p}_1,\ldots,\boldsymbol{p}_K) = \mathbf{R}^{-1}\boldsymbol{C}\left(\boldsymbol{C}^H\mathbf{R}^{-1}\boldsymbol{C}\right)^{-1}\boldsymbol{d},$$7 before calibration to $$\boldsymbol{w}_\mathrm{LCMV}(\boldsymbol{p}_1,\ldots,\boldsymbol{p}_K) = \mathbf{R}^{-1}\boldsymbol{C}\left(\boldsymbol{C}^H\mathbf{R}^{-1}\boldsymbol{C}\right)^{-1}\boldsymbol{d},$$8 after calibration (Kim et al., 2 Apr 2026).

At the system level, standards-compliant nulling may incur losses larger than those predicted by idealized theory. In a 3GPP Release-18 FR-1 analysis of the 3D EIRP profile of a 5G gNB, two beam nulling methods are introduced: threshold-based precoding-matrix selection and HPBW-based precoding-matrix selection (Tusha et al., 24 Apr 2026). Both methods achieve approximately $$\boldsymbol{w}_\mathrm{LCMV}(\boldsymbol{p}_1,\ldots,\boldsymbol{p}_K) = \mathbf{R}^{-1}\boldsymbol{C}\left(\boldsymbol{C}^H\mathbf{R}^{-1}\boldsymbol{C}\right)^{-1}\boldsymbol{d},$$9 dB reduction of radiation toward a target direction, with up to $\mathbf{R}$0 dB reduction in some configurations, but the restriction of the available precoding-matrix set causes a $\mathbf{R}$1–$\mathbf{R}$2 dB degradation in SNR at BER $\mathbf{R}$3 depending on modulation order. The paper explicitly notes that these practical, standards-based schemes exhibit notably greater SNR and BER penalties than most theoretical or analytical treatments predict (Tusha et al., 24 Apr 2026). This is a recurring theme in NCBF: feasibility in ideal array models does not automatically imply benign deployment loss under codebook, port-mapping, and channel-estimation constraints.

6. Trade-offs, misconceptions, and application domains

A persistent misconception is that nulling is nearly free whenever an array has many elements. The literature consistently shows otherwise. In the ATA analysis, each independent narrowband null reduces overall SNR and main beam gain according to

$\mathbf{R}$4

so each narrowband null costs one antenna in SNR terms (Harp, 2012). For area and wideband nulls, the relevant cost is the occupied $\mathbf{R}$5-space volume, written as

$\mathbf{R}$6

In the near-field low exposure region problem, exact zero forcing at $\mathbf{R}$7 requires $\mathbf{R}$8, and with $\mathbf{R}$9 the only solution is typically $$S(\vec{k}-\vec{k}_0)=\sum_i \omega_i e^{-i(\vec{k}-\vec{k}_0)\cdot \vec{r}_i},$$00 unless the threshold is relaxed (Defraigne et al., 13 Feb 2026). These results clarify why relaxed zero forcing, leakage penalties, dominant-subspace projection, or region sampling are often necessary.

A second misconception is that beam steering accuracy and nulling accuracy are comparable engineering tasks. The hardware literature states the opposite: null forming is far more demanding than conventional beam steering because nulls are intrinsically narrow, and small mismatches in phase, timing, or gain can significantly degrade suppression (Kim et al., 2 Apr 2026). The 5G EIRP study reaches a similar conclusion at the network level: 3GPP-compliant beam nulling achieves coexistence benefits, but the associated SNR loss is higher than theoretical analyses suggest (Tusha et al., 24 Apr 2026). A plausible implication is that NCBF evaluation must include architecture-specific constraints, channel estimation, and calibration fidelity rather than only ideal array factors.

The application space is correspondingly broad. NCBF is used for RFI mitigation and arbitrary beam shaping in radio astronomy arrays (Harp, 2012), for low exposure regions in near-field beampointing (Defraigne et al., 13 Feb 2026), for multi-user interference suppression in near-field XL-MIMO and analog-only multiuser MIMO (Karimi et al., 23 Sep 2025, Wang et al., 2024), for passive interference nulling in MU-MISO through beyond-diagonal RIS (Yahya et al., 2024), for frequency-selective indoor mmWave systems through interference-nulling time-reversal (Viteri-Mera et al., 2015), and for contested-spectrum operation, spectrum sharing, anti-jamming, and covert communications in SDR arrays (Kim et al., 2 Apr 2026). The UAV jamming formulation extends the idea to mobile platforms: with two omnidirectional antennas, a closed-form phase relation

$$S(\vec{k}-\vec{k}_0)=\sum_i \omega_i e^{-i(\vec{k}-\vec{k}_0)\cdot \vec{r}_i},$$01

guarantees zero jamming impact on the client while orientation and trajectory are optimized to maximize jamming on the eavesdropper (Fotiadis et al., 24 Aug 2025). Across these settings, NCBF is less a single algorithm than a family of constrained spatial power-control methods whose central design question is always the same: how much useful gain can be preserved while nulls are imposed where the system cannot tolerate radiation or interference.