Airy Null-Steering Method in Beamforming

• Airy Null-Steering Method is a beamforming and imaging strategy that exploits the phase structure of Airy beams to create directional nulls for interference suppression.
• In wireless communications, it aligns phase-induced nulls to enhance signal clarity in shadowed and interference-prone scenarios while preserving main-lobe energy.
• In digital imaging, the method uses phase masks to suppress π-out-of-phase sidelobes, achieving higher resolution and improved contrast.

The Airy Null-Steering Method encompasses a family of beamforming and image processing strategies that exploit the structured phase properties of Airy wavefields to selectively suppress interference or artifacts by aligning field nulls with undesired directions or spatial regions. Initially developed to overcome radiative near-field (RNF) blockage in multi-user wireless channel scenarios and to remove Airy noise in high-resolution digital imaging, this approach harnesses the oscillatory phase structure of Airy patterns—arising from cubic or quadratic phase profiles—to engineer constructive and destructive interference in both analog (physical array) and digital (post-processing) contexts (Qin et al., 14 Jan 2026, Solanki, 2020). The method fundamentally differs from classical amplitude apodization by employing phase-domain discrimination, thereby preserving main-lobe energy while directly zeroing out unwanted side lobes.

1. Conceptual Foundation

The Airy Null-Steering Method is predicated on the unique field structure of Airy beams, which are generated by modulating the aperture phase with cubic terms. This modulation induces self-accelerating main lobes and oscillatory tails, the latter naturally containing localized spatial nulls. In RNF wireless communications, these properties enable "edge riding," where the main lobe bends into geometric shadows created by half-space obstacles, while the nulls in the oscillatory tail are deliberately aligned with bright-region users to suppress inter-user interference (Qin et al., 14 Jan 2026). In digital imaging, Airy null-steering targets phase singularity between lobes, using phase masks to remove the π-out-of-phase side lobes without sacrificing resolution (Solanki, 2020).

2. Mathematical Formulation

RNF Communications

The free-space, element-wise channel is described as:

$$h_{k,n} = \lambda/(4\pi r_{k,n}) \cdot \exp(-j k_0 r_{k,n})$$

where $k_0=2\pi/\lambda$ and $r_{k,n}=\|p_k-(x_n,0)\|_2$. In the presence of half-space blockage, the complex field evolution is governed by Fresnel diffraction:

$$E(x,z_k) = \mathcal{P}_{\Delta z_2}\{M(x)\mathcal{P}_{\Delta z_1}\{E_0(x)\}\}$$

where $\mathcal{P}_z$ denotes the Fresnel propagator and $M(x)$ is a binary mask.

The transmit aperture phase for Airy beams is:

$$\varphi_{Airy}(x_n) = (k_0/(2F)) x_n^2 - k_0 \sin\theta x_n + (2\pi/(3\lambda)) B (x_n/F)^3$$

with analog weights:

$$w_{Airy}^{(n)} = (1/\sqrt{N}) \exp[j\varphi_{Airy}(x_n)]$$

where $B$ is the transverse acceleration coefficient, $F$ is the virtual focal distance, and $\theta$ is the launch angle.

The null-steering optimization seeks parameters $\psi=(B, F, \theta_1)$ subject to the service constraint $|h_{11}(\psi)|^2 \geq \eta |h_{11}^{geo}|^2$, maximizing the sum-rate $R(\psi)=2\log_2(1+|\alpha|^2/\sigma^2)$.

Imaging Implementation

For a circular pupil (radius $a$), with wavelength $\lambda$, the Airy PSF amplitude and phase are:

$$h_{amp}(r) = A \left[ \frac{2J_1(\alpha r)}{\alpha r} \right]$$, $\alpha = \pi a/(\lambda z)$

$h_\phi(r) = \exp \Big\{ j \frac{k}{2z} r^2 \Big\}$, $k=2\pi/\lambda$

so $h(r) = h_{amp}(r)\cdot h_\phi(r)$.

Phase separation between lobes (at $r_n=\xi_n/\alpha$, $\xi_n\approx(n+1/2)\pi$) is:

$$\Phi_n(u,v) = \Phi_0 + n\pi + \frac{\pi}{\lambda z} \big[ (u-u_n)^2 + (v-v_n)^2 \big]$$

A digital phase mask is constructed:

$$H(\omega_x,\omega_y) = \Pi\left( \frac{\Phi(\omega_x,\omega_y) - \Phi_0}{\Delta\Phi} \right)$$

where $\Delta\Phi$ bounds lobes to the main signal by phase.

3. Algorithmic Implementation

RNF Wireless (Coarse-to-Fine Airy-Null Search)

1. Initialization: Compute geometrical steering angle $\theta_{geo}=\mathrm{atan2}(x_1,z_1)$, baseline gain $G_{geo}$, and set service threshold $\tau=\eta G_{geo}$.
2. Coarse Search: For $B \in \mathcal{B}_c$, $F \in \mathcal{F}_c$, $\Delta\theta \in \Delta\Theta_c$:
   • Set $\theta_1=\theta_{geo}+\Delta\theta$.
   • Propagate to obtain $h_1$; apply discard criterion $|h_{11}|^2<\tau$.
   • Compute $R$ and retain maximizer.
3. Fine Search: Refine parameter grids around the coarse optimum; repeat above.
4. Return: Optimized tuple $\psi^*=(B^*,F^*,\theta_1^*)$.

Computational complexity per propagation is $O(N_x\log N_x)$ via FFT-based angular spectrum; offline codebook reduces online complexity to $O(1)$.

Digital Imaging (Phase-based Null-Steering)

1. Acquire and digitally reconstruct complex field $U(x,y)$.
2. Fourier-transform to $\widehat{U}(\omega_x,\omega_y)$, extract phase map $\Phi(\omega_x,\omega_y)$.
3. Construct binary mask centered at $\Phi_0$ with width $\Delta\Phi$.
4. Apply mask to reject $\pi$-out-of-phase sidelobes.
5. Inverse transform to yield phase-filtered intensity, preserving full bandwidth.

4. Design Guidelines and Trade-offs

Both domains require tuning to maximize effectiveness while balancing main-lobe gain against interference or artifact suppression.

Parameter	Role	Trade-off
$B$	Transverse acceleration, main lobe curvature	Higher $|B|$ deepens bend, widens tails—greater nulling but risk of interference
$F$	Virtual focal distance	$F \gtrsim$ obstacle depth to form main lobes just beyond knife-edge
$\theta$	Launch angle	Small offsets $\Delta\theta$ align nulls with interfering users

Larger $B$ improves reach into shadow regions, but broadens beam and may demand stronger nulling. The optimization ensures channel gain above service ratio $\eta$ while minimizing leakage. For phase masks in imaging, $\Delta\Phi$ must be narrow enough to exclude side lobes, yet wide enough to encompass main lobe phase spread.

5. Performance Benchmarks

RNF Wireless

• In mixed shadow–bright user scenarios, traditional focusing yields outage for shadowed users and severe interference for bright users.
• Airy Null-Steering (Airy-Opt) achieves $\sim 4$ bps/Hz sum-rate gain, $+21$ dB restored shadowed link strength, $-3.7$ dB reduced bright user interference, and sharp drops in channel condition number $\kappa(H_{eff})$ at optimal null-alignment offsets.
• Robustness: Positioning error of $\pm 3\lambda$ at bright user preserves sum-rate advantage ($\geq 3.5$ bps/Hz) and low $\kappa(H_{eff})$ ($\sim 3$) versus ill-conditioned traditional schemes (Qin et al., 14 Jan 2026).

Digital Imaging

• Side-lobe suppression depth: $-40$ dB (phase null-steering) vs. $-20$ dB (Gaussian apodization).
• Central lobe FWHM: $0.50$ mm (phase null-steering) vs. $0.58$ mm (Gaussian); peak intensity is $1.3$ (phase) vs. $1.0$ (Gaussian).
• SNR at sharp edge: $\sim 26$ dB (phase null-steering) vs. $\sim 18$ dB (Gaussian).
• Removal of odd-order lobes restores contrast and enables cleaner, higher-resolution reconstructions; FWHM reduction of $14\%$ observed (Solanki, 2020).

6. Limitations and Further Directions

The Airy Null-Steering Method is inherently nonconvex, relying on highly oscillatory and phase-sensitive optimizations. Global closed-form solutions are unknown; the employed coarse-to-fine search is effective heuristically but not guaranteed optimal.

Direct geometric steering is sub-optimal in blockage-prone environments; virtual-source driven designs may further enhance performance. The 2D ULA strategy generalizes to 2D/3D arrays, though complex phase mask synthesis and real-time adaptation present open challenges.

Hardware realization in ELAA wireless systems requires high-resolution analog phase shifters; quantization effects and practical calibration demand further study. In imaging contexts, accurate phase estimation and sufficient digital sampling are necessary to prevent lobe misclassification and leakage. Real-time GPU implementations are feasible for live applications.

This suggests ongoing research will focus on robust real-time parameter tracking, learning-based adaptation for dynamic environments, and hardware-efficient phase mask synthesis across wireless and imaging domains.