Null Steering Beamforming: Methods & Applications

• NSB is a spatial filtering algorithm that imposes zero response constraints to create deep nulls for suppressing interference while enhancing coherent gain in a desired direction.
• It employs a closed-form weight solution under linear constraints and is extended using movable and rotatable array designs to overcome fixed-position limitations.
• NSB is applied in diverse systems—from wireless communications to speech enhancement—requiring precise array geometry, robust synchronization, and efficient optimization methods.

Null Steering Beamforming (NSB) is a class of spatial filtering algorithms that constrain the array synthesis such that the beampattern exhibits deep nulls, typically at known or estimated interferer directions, while maximizing coherent gain toward a desired look direction. NSB is a cornerstone methodology in multi-antenna and microphone array processing—enabling interference suppression, protected-user communication, and robust sensing—by directly exploiting the structure of array manifold vectors. Unlike classic adaptive beamformers which minimize output power under distortionless-response constraints, NSB explicitly imposes “zero response” constraints, resulting in deterministic spatial nulls whose precise depth and angular placement are guaranteed by the beamformer geometry.

1. Fundamental Formulation and Closed-Form Analysis

The essential NSB setup considers an M-element array, a desired look angle $\theta_0$, and $K$ interferer or null-steer angles $\{\phi_1, \dots, \phi_K\}$. The NSB optimization seeks to maximize directivity at $\theta_0$ while exactly nulling synthesis in all $\phi_k$ directions. The general joint problem can be formalized as:

$$\begin{aligned} &\max_{x\in\mathbb{R}^M,\, w\in\mathbb{C}^M} && G_{x, w}(\theta_0) = |w^H a(x, \theta_0)|^2 \ &\,\,\text{s.t.} && |w^H a(x, \phi_k)|^2 = 0,\,\,\,\, k=1, \ldots, K \ & && |x_m - x_n| \geq d_\text{min},\,\,\,\forall m\ne n \ & && \|w\|_2^2 = 1 \end{aligned}$$

where $a(x, \theta)$ is the steering vector for positions $x$ at angle $\theta$, and $d_\text{min}$ is a practical element separation lower bound. For fixed-position arrays, NSB is typically realized in the weight space, yielding the well-known closed-form for unit-norm weights $w^*$ under linear constraints:

$w^* = C\, (C^H C)^{-1} g$

with $C = [a(\theta_0), a(\phi_1), ..., a(\phi_K)]$ and $g = [1, 0, ..., 0]^T$. This result gives optimal minimum-power weights subject to unity gain at $\theta_0$ and nulls at the interference angles (Ting et al., 2023, Bhattacharyya et al., 2024). In dual-microphone and other low-channel-count arrays, the nulling subspace projection method is frequently used, formalized as:

$$w^* = \frac{\Phi\, a(\theta_0)}{a(\theta_0)^H \Phi\, a(\theta_0)}$$

where $\Phi = I - \frac{a(\phi) a(\phi)^H}{\|a(\phi)\|^2}$ annihilates the interferer (Ting et al., 2023).

2. Geometry-Enhanced Null Steering: Movable and Rotatable Arrays

NSB trade-offs in fixed-position arrays—viz., the loss of array gain under aggressive nulling—can be fundamentally overcome by introducing geometric degrees of freedom. In "Movable-Antenna Array Enhanced Beamforming: Achieving Full Array Gain with Null Steering", closed-form construction is given for a movable array where both the positions (APV) and weights (AWV) are optimized (Zhu et al., 2023). If $M$ admits a factorization $M = \prod_{i=1}^{I_M} f_i$, one can allocate independent placement parameters per null so that, under the steering-vector orthogonality (SVO) condition $a(x, \phi_k)^H a(x, \theta_0) = 0$, each null is implemented through geometry, and $w^* = a(x, \theta_0)/\|a(x, \theta_0)\|_2$—phase-only weighting. The necessary and sufficient condition for perfect nulls with full array gain is $K \leq I_M$.

Similarly, Rotatable Antenna Arrays (RAAs) introduce three rotational DoF (roll, pitch, yaw) to a conventional array, yielding substantially relaxed angular separation requirements for zero-forcing (ZF) null steering (Wen et al., 13 Dec 2025). The joint optimization over rotation and weights is non-convex, but iterative (sequential update + Gibbs sampling) algorithms enable global exploration of the rotational search space, restoring full array gain under much broader conditions than fixed-orientation arrays.

3. NSB in Advanced System Scenarios: Distributed, Hybrid, and Robust Configurations

In multi-node and hybrid systems, NSB principles are extended to accommodate distributed spatial geometries, analog-digital partitioning, and environmental uncertainty. Multi-objective distributed beamforming with high-accuracy synchronization and localization enables application of NSB to open-loop distributed arrays. Precise time transfer and node localization (e.g., dual-LFM ranging yielding $\sim$4 mm accuracy, $<$5 ps jitter) facilitate formation of steering vectors, after which transmission weights are solved via Linear-Constrained Minimum Power (LCMP) or LCMV beamforming (Bhattacharyya et al., 2024):

$w^*_{LCMP} = C (C^H C)^{-1} g$

This construct enables a non-cooperative set of radios to form a beam at a desired receiver and a null at another, even in the near-field.

Hybrid analog-digital NSB architectures, such as those designed for high dynamic range ambient backscatter receivers, deploy an RF-domain null-steering stage to reject strong direct-path signals before digitization, followed by digital covariance-based beamforming (Duan et al., 2019). This front-end NSB reduces the analog dynamic range requirement, suppressing direct path by $\sim$30 dB while enabling detection of weak, backscattered signals via subsequent digital processing.

Frequency-invariant, robust differential beamforming is realized via convex quadratic optimization subject to null constraints and white-noise-gain (WNG) lower bounds, e.g.:

$$\min_h\, J(h) \qquad \text{s.t. } D h = \gamma,\,\, h^H h \leq 10^{-\zeta_{WNG}/10}$$

where $D$ embodies both distortionless and null constraints, $J(h)$ is MSE to a specified beampattern, and $\zeta_{WNG}$ tunes robustness (Zhang et al., 25 Aug 2025).

4. Applications: Communications, Security, and Speech Enhancement

NSB is operationally central in both communication and acoustic domains:

• In covert wireless communication, NSB is used by multi-antenna jammers to maximize confusion at an adversary (warden) while placing intended receivers in a spatial null, thus preserving achievable rates (Forouzesh et al., 2019). Optimization proceeds under strict null-space (e.g., $h_{jb}^H w_j = 0$), power, and covertness constraints, solved using alternating search and semidefinite relaxation.
• In multi-hop relay security, the eavesdropper’s channel is placed in the transmit null-space of a multi-antenna relay subject to relay power constraints, maximizing the minimum achievable secrecy rate. The resulting design is a non-convex optimization, addressed by lifting to a positive semi-definite (PSD) matrix variable and imposing linear matrix inequalities, solved via bisection and semidefinite programming (Khordad et al., 2016).
• For speech and signal enhancement, NSB is the foundation for intelligibility-aware spatial filters (e.g., IANS), where a grid of null-steering beamformer outputs is evaluated with non-intrusive intelligibility predictors (e.g., STOI-Net) to select the maximal-intelligibility output with no prior knowledge of DOAs or RTFs (Ting et al., 2023).

5. Performance Criteria and Theoretical Guarantees

Under sufficient geometric degrees of freedom (movable/rotatable arrays), perfect nulls and full array gain are attainable even for several null directions; e.g., with $M=8$ elements, $K=3$ nulls, the MA design yields $G(\theta_0) = 8$ ($9$ dB gain) and $G(\phi_k)=0$ (exact nulls), whereas fixed geometry analog beamforming suffers up to $7$ dB loss (Zhu et al., 2023).

Distributed NSB experiments realize 4 mm localization, $\sim$5 ps sync, $>$90% main-lobe coherence, and $\geq$15 dB null depth, preserved under node or target movement (Bhattacharyya et al., 2024). In hybrid analog-digital receivers, NSB architectures enable bit error rates (BER) near $10^{-3}$ for ambient backscatter at practical ranges without instantaneous CSI (Duan et al., 2019).

Frequency-invariant NSB with direct linear null constraints achieves deep nulls ($<-40$ dB) and maintains robust WNG ($>0$ dB) over broad frequency bands, outperforming harmonic expansion methods (Zhang et al., 25 Aug 2025). In speech enhancement, NSB-based IANS approaches match or exceed performance of oracle DOA-informed procedures (Ting et al., 2023).

6. Implementation and Practical Considerations

The realization of NSB is subject to degrees-of-freedom sufficiency, array geometry constraints, and robustness to hardware and environmental uncertainties. For movable arrays, realization of the optimal positions requires mechanical or other actuators, precise spatial resolution ($\ll \lambda$), and mitigation of mutual coupling (Zhu et al., 2023). Rotatable array architectures necessitate global mounting and fine-grained 3D orientation control (Wen et al., 13 Dec 2025).

In hybrid architectures, efficient analog nulling enables use of standard ADCs and obviates the need for instantaneous channel estimates (Duan et al., 2019). For frequency-invariant and robust designs, convex programming with explicit WNG bounds is essential. Distributed systems require picosecond time synchronization and centimeter-scale localization, achievable via specialized ranging waveforms and protocols (Bhattacharyya et al., 2024).

Algorithmic frameworks for NSB range from direct closed-form weight computation under linear constraints, sequential/Gibbs hybrid optimization for non-convex geometric parameters, to convex optimization (e.g., QP or SDP) when additional robustness or performance metrics (e.g., MSE, WNG, secrecy rate) are imposed.

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References:

(Zhu et al., 2023, Forouzesh et al., 2019, Duan et al., 2019, Wen et al., 13 Dec 2025, Zhang et al., 25 Aug 2025, Bhattacharyya et al., 2024, Ting et al., 2023, Khordad et al., 2016)