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Why Is Cubic-Phase Airy Beamforming Sufficient for Blockage Recovery?

Yi Wang, Linglong Dai

arXiv:2609.01313v1eess.SP

TL;DR

Near-field blockage makes reliable transmission difficult, while trajectory-based Airy interpretations leave the sufficiency and quantitative role of the cubic phase unresolved. The paper extracts the blockage-induced mismatch relative to conventional focusing and evaluates successive phase-order compensation. Linear and quadratic terms recover most available gain, and the cubic term brings the Airy family to 99.77% of the phase-only upper bound.

  • Problem

    Trajectory-based interpretations do not quantitatively explain why cubic-phase Airy beamforming is sufficient for blockage recovery or how much gain the cubic term contributes.

  • Method

    The paper derives the blockage-induced phase mismatch beyond conventional focusing, evaluates successive linear, quadratic, and cubic compensation, and realizes the cubic basis through weighted phase projection.

  • Results

    3.223 dB of the 3.376-dB mean phase-only gain is recovered by linear and quadratic compensation; the cubic adds 0.1433 dB and reaches 99.77% of the phase-only upper bound.

  • Takeaways & Limitations

    Lower-order phase terms provide most blockage-recovery gain, while extending the phase to cubic order recovers nearly all remaining available gain.

Abstract

from arXiv · show

Blockage is a critical challenge for near-field communications, where reliable transmission depends heavily on the line-of-sight (LoS) path and can suffer severe power degradation when that path is obstructed. Near-field Airy beams offer a promising solution for blockage mitigation by forming curved trajectories that guide energy around obstacles, and can be practically generated with phased arrays by imposing a cubic source phase. However, trajectory-based interpretations explain how Airy beams propagate, but not why cubic-phase Airy beamforming is sufficient for blockage recovery or how much received-power gain the cubic term itself contributes. To answer these questions, we identify the blockage-induced phase mismatch relative to conventional near-field focusing and quantify how successive phase orders compensate it. The resulting analysis reveals that the linear and quadratic degrees of freedom, originally used to compensate the free-space geometric phase, can be reoptimized under blockage to provide resteering and refocusing, respectively. The quadratic term can compensate the dominant quadratic component of the additional mismatch, while the Airy cubic provides the first independent correction to the remaining non-quadratic mismatch. Simulations show that linear and quadratic compensation recover most of the available gain. The Airy cubic adds only $0.1433$ dB on average, yet enables the cubic-phase family to attain $99.77\%$ of the phase-only upper bound. Residual-phase analysis further determines when the remaining higher-order components are negligible within a prescribed received-power tolerance. These results explain why cubic-phase Airy beamforming is sufficient: lower-order phase terms provide most of the recovery, while the cubic term closes nearly all of the remaining gap.

I. INTRODUCTION

The paper reframes Airy blockage recovery as phase compensation: blockage creates an additional mismatch beyond conventional focusing, and successive phase orders approximate it. Results show lower-order terms recover most gain, while the cubic term closes nearly all of the remaining gap.

  • LoS blockage severely degrades near-field communication reliability because mmWave and THz links have weak NLoS propagation.
  • Existing trajectory-based interpretations do not quantify why cubic-phase Airy beamforming is sufficient or how much gain the cubic term contributes.
  • The paper identifies blockage-induced phase mismatch relative to conventional near-field focusing and uses it to quantify compensation by successive phase orders.
  • Linear and quadratic terms are reoptimized for resteering and refocusing, while the Airy cubic independently corrects the remaining non-quadratic mismatch.
  • Residual-phase analysis determines when higher-order components are negligible within a prescribed received-power tolerance, and weighted projection realizes the compensation in the standard Airy family.
  • 3.223 dB of the 3.376-dB mean phase-only gain is recovered by linear and quadratic compensation, while the cubic adds 0.1433 dB and reaches 99.77% of the upper bound.

B. Single-Edge Blockage Model

The model uses a single dominant obstacle edge in a scalar Fresnel near-field link and represents the transmit field with a hardware-compatible Gaussian-cubic phase basis. Fixed aperture amplitude isolates source-phase effects within a phase-only architecture.

  • The single-edge blockage model places an opaque obstacle between transmitter and receiver, with edge (zo, xe) defining the visible half-plane.
  • The transmitter knows the obstacle geometry through sensing or a higher-layer map, but complete post-blockage channel state information is not assumed.
  • Scalar Fresnel propagation models the blocked receiver field through two propagation segments coupled by integration over the obstacle edge.
  • C. Hardware-Compatible Airy Phase Basis: The Gaussian-cubic Airy realization uses a Gaussian aperture envelope and varies only source phase to form a hardware-compatible polynomial phase basis.
  • Amplitude control is excluded, and the phase-matched reference defines the upper bound attainable with the prescribed aperture amplitude.
  • C. Hardware-Compatible Airy Phase Basis: The Airy parameters B, F, and θ encode bending, focusing, and steering degrees of freedom, respectively, while conventional generation interprets them through geometry-defined trajectories.
  • C. Hardware-Compatible Airy Phase Basis: The triplet (B, F, θ) is retained as a phase-basis parameterization, with any generation plane or trajectory treated as a propagated-field consequence.

III. BLOCKAGE-INDUCED PHASE MISMATCH

The paper combines the blocked Fresnel segments into an effective channel and isolates the additional phase mismatch beyond conventional free-space focusing. This mismatch supplies the basis for phase-order compensation and phase-only performance evaluation.

  • The blocked propagation model is reduced to a blockage-aware effective channel that captures edge-induced amplitude attenuation and phase variation.
  • For fixed aperture amplitude, coherently aligning surviving aperture contributions gives the phase-only upper bound on received power.
  • The additional mismatch is defined as Δϕe(t) = ϕideal(at) −ϕF(t) = −arg Te(at; xr), representing the correction beyond conventional focusing.
  • The mismatch vanishes without obstruction but generally varies nonlinearly across the aperture under blockage.
  • The extracted mismatch is progressively approximated by linear, quadratic, and cubic components, with the Airy family providing a hardware-compatible realization.

IV. PHASE-COMPENSATION HIERARCHY AND AIRY PHASE INTERPRETATION

The blockage-induced phase mismatch can be approximated hierarchically by linear, quadratic, and cubic terms beyond conventional focusing. Linear and quadratic terms provide resteering and refocusing, while the Airy cubic supplies the first independent non-quadratic correction.

  • Phase-mismatch extraction: The blockage-induced mismatch ∆ϕe is approximated globally by successive constant, linear, quadratic, and cubic phase components.The constant phase is irrelevant because common phase offsets do not affect received power.
  • Hierarchical compensation: Linear compensation provides additional steering, while quadratic compensation adjusts focusing curvature to address the dominant blockage-induced curvature mismatch.These are additional compensations beyond the receiver-directed linear and quadratic terms already present in conventional near-field focusing.
  • Hierarchical compensation: The Airy cubic component is the first non-quadratic correction remaining after linear and quadratic compensation.The cubic term therefore adds a phase degree of freedom not reproduced by steering and refocusing.
  • Phase-mismatch extraction: The phase-matched reference isolates the additional blockage requirement by comparing it with conventional free-space focusing.Removing the free-space focusing phase yields ∆ϕe, with common offsets discarded as power-irrelevant.
  • Hierarchical compensation: The compensation spaces are nested, with S1 adding linear compensation, S2 adding refocusing, and S3 introducing the independent Airy cubic correction.The physically feasible subset of S3 is the Gaussian-cubic Airy phase family.

B. Weighted Phase Projection and Airy Realization

The Airy realization replaces iterative parameter search with a weighted phase projection that approximates the blockage-aware target in a fixed-dimensional polynomial basis. A small-mismatch theorem connects this projection to received-power optimization.

  • Weighted projection: The contribution-weighted projection penalizes phase errors more heavily at aperture elements that contribute more strongly to the received field.Weights combine the fixed Gaussian amplitude with the magnitude of the effective channel.
  • Polynomial fitting: Fitting through quadratic or cubic order combines conventional focusing with blockage-induced linear and quadratic compensation, while cubic order adds the first non-quadratic correction.The constant coefficient absorbs the irrelevant common phase.
  • Weighted projection: The nonlinear coherent-combining objective becomes a fixed-dimensional weighted phase projection under the small-mismatch approximation.The derivation expands the weighted coherent sum around zero phase mismatch and requires a nonsingular weighted Gram matrix.
  • Airy realization: The fitted coefficients map one-to-one to Airy controls (B, F, θ), producing a hardware-compatible realization of the identified compensation.The mapping converts the polynomial phase fit into the Gaussian-cubic Airy family.

C. Airy Cubic as the First Non-Quadratic Correction

The Airy cubic addresses only the residual non-quadratic mismatch left after lower-order compensation. Its gain is therefore a measure of residual mismatch capture, not the source of the dominant blockage-recovery gain.

  • Physical interpretation: The Airy cubic is not the source of the dominant blockage-recovery gain; it compensates the residual non-quadratic mismatch after quadratic compensation.The quadratic stage has already removed the dominant blockage-induced curvature mismatch.
  • Cubic isolation: The independent cubic component is obtained by removing weighted constant, linear, and quadratic components from t3.This projection isolates cubic action that steering and refocusing cannot reproduce.
  • Residual capture: η3 quantifies the fraction of residual phase-mismatch variance after refocusing that the independent Airy cubic can capture.The residual after cubic correction is characterized relative to the variance remaining after quadratic compensation.
  • Physical interpretation: A small cubic gain reflects that the dominant quadratic stage leaves only a small residual phase mismatch, rather than indicating a weak cubic correction.The cubic remains the first Airy-specific phase action beyond steering and refocusing.
  • Residual reduction: Linear compensation removes first-order variation, quadratic compensation removes dominant curvature variation, and the Airy cubic reduces the remaining non-quadratic residual from V2 to V3.Fig. 6 visualizes this successive residual reduction for a representative geometry.

V. ANALYTICAL AIRY REALIZATION FRAMEWORK

The analytical framework converts sensed blockage geometry into feasible Gaussian-cubic Airy controls through target construction, weighted polynomial projection, and coefficient conversion. This provides a direct geometry-to-Airy realization rather than iterative Airy-parameter search.

  • Framework: The framework extracts the blockage-aware phase compensation, projects it onto the polynomial Airy basis, and maps the coefficients to realizable controls.These three stages form the practical Gaussian-cubic Airy realization.
  • Target construction: The first stage constructs a discrete phase-matched target from the sensed geometry, wavelength, antenna positions, and blocked-channel response.The target combines the conventional free-space phase with the blockage-induced mismatch.
  • Airy control conversion: The final stage converts the fitted phase coefficients into feasible Airy controls and constructs the corresponding beamforming vector.If coefficients fall outside admissible intervals, a finite active-set procedure is applied before conversion.
  • Weighted projection: The second stage continuously unwraps the sampled phase and applies contribution-weighted projection to obtain constant, linear, quadratic, and cubic coefficients.Unwrapping prevents artificial 2π discontinuities from appearing as high-order aperture-phase variation.

B. Complexity Analysis

The framework evaluates successive phase compensation under blockage, showing that lower-order terms recover most available power while cubic compensation closes nearly all of the remaining gap. It also characterizes cubic relevance, sufficiency, and realization accuracy across blockage geometries.

  • Effectiveness of Phase Compensation: Linear and quadratic compensation recover 3.223 dB of the 3.376-dB mean phase-only gain, leaving a 0.1532-dB residual gap.The quadratic term provides the dominant additional gain after linear compensation.
  • Effectiveness of Phase Compensation: The Airy cubic adds 0.1433 dB on average, reduces the residual gap to 0.00991 dB, and reaches 99.77% of the phase-only upper bound.This result quantifies the cubic term as a final refinement rather than the main source of recovery.
  • Field-Level Mechanism: Linear compensation removes first-order phase variation, quadratic compensation removes dominant curvature variation, and the cubic component corrects the remaining non-quadratic residual.The propagated curved trajectory appears after source-phase compensation and is described as its spatial manifestation.
  • Cubic Relevance and Sufficiency: Cubic relevance depends jointly on edge-transition sharpness and clear-side aperture span, rather than blockage ratio alone.The 0.1-dB contour separates geometries where cubic correction is negligible from those where it is beneficial.
  • Cubic Relevance and Sufficiency: At a 0.01-dB tolerance, the post-cubic residual-variance certificate covers 528 of 720 evaluated scenes, or 73.33%.The residual-variance bound provides a prescribed-tolerance certificate for cubic sufficiency.
  • Analytical Realization: Weighted phase projection achieves a mean gap of 9.19×10^-5 dB to continuous phase optimization across all 720 geometries.On a 72-scene stratified subset, the mean and maximum gaps to independently initialized broad search are 8.62 × 10^-5 dB and 2.93 × 10^-4 dB.

APPENDIX A LOCAL EDGE APPROXIMATION AND GEOMETRY

The local edge-intersection approximation reduces cubic-phase relevance to edge geometry and provides a low-cost indication of when cubic correction matters, while the full weighted fit remains the accurate design.

  • Local approximation: The edge-intersection expansion applies when the normalized edge transition lies within the illuminated aperture and physical-branch conditions hold.Otherwise, the full weighted phase fit is used.
  • Approximation boundary: The local third-order phase-matching rule is not the exact finite-aperture optimum; the full weighted fit remains more accurate for analytical design.The local rule determines bending direction from the visible side and magnitude from a Fresnel geometry scale.
  • Two-parameter reduction: The reduced geometry uses edge-transition sharpness κ and normalized edge position d, eliminating separate dependence on visible-side sign for normalized fitted quantities.The mirrored coordinate combines both visible-side cases, while d locates the edge transition within the aperture.
  • Stage sufficiency: The strict stage-sufficiency certificate is evaluated from remaining phase-mismatch variance after the weighted phase fit, whereas the local third-derivative rule is only a preliminary cubic-relevance indicator.The certificate can guarantee a prescribed received-power tolerance; the local rule estimates when cubic correction is relevant more cheaply.

C. Physical Interpretation of Cubic Relevance

Cubic relevance is interpreted through edge Fresnel geometry and vanishes continuously when edge diffraction becomes aperture-constant, restoring focused Gaussian transmission.

  • Physical interpretation: The effective edge distance and edge Fresnel number provide the geometric quantities used to interpret the cubic-relevance indicator.Equation (62) connects cubic relevance to edge Fresnel number and aperture geometry, while strict sufficiency remains certified by (59).
  • Free-space degeneration: If edge diffraction approaches a constant complex gain across the illuminated aperture, no independent Airy cubic correction remains.This is the free-space degeneration limit of the phase-compensation interpretation.
  • Phase fitting: The formal phase analysis derives coefficient fitting by centering the weighted residual and minimizing its weighted second-order loss.The resulting normal equation yields the fitted coefficients when the weighted Gram matrix is nonsingular.
  • Cubic correction: The cubic-stage projection adds one orthogonal direction beyond the linear-quadratic phase space, with the resulting gain characterized through a Pythagorean decomposition.The cubic coefficient is then mapped to the physical bending parameter.

B. Discrete Airy Realization

Discrete Airy realization replaces continuous weighted moments with array sums while preserving the phase-compensation mechanism, and finite receiver windows introduce a fourth-order power-loss perturbation.

  • Discrete realization: For a physical ULA, continuous unwrapped phase samples are projected using element weights formed from Gaussian amplitude and effective-channel magnitude.The discrete basis uses normalized element coordinates and a diagonal contribution-weight matrix.
  • Discrete realization: Discretization changes the weighted moments, not the phase-compensation mechanism.The discrete-array corollary implements the same compensation structure through sampled sums.
  • Feasible controls: Physical control constraints convert the unconstrained coefficient fit into a fixed-dimensional convex projection over the feasible Airy-control image.For box-bounded controls, finite active-set enumeration solves the constrained problem.
  • Finite receiver window: A finite receiver window averages point-receiver power over its width, and using the point-receiver maximizer instead of the window maximizer incurs O(W_r^4) loss.The perturbation result connects the point-receiver phase analysis to finite-window simulations.
  • Finite receiver window: The finite-window perturbation result provides the analytical link between point-receiver optimization and the receiver-window evaluation used in simulations.Its numerical accuracy is assessed in Appendix C-C.

APPENDIX C NUMERICAL VALIDATION AND SCOPE CHECKS

Numerical checks validate the effective-channel reduction, one-shot phase fitting, discrete realization, and cubic near-saturation within the adopted Fresnel regime and scope.

  • Reference and order checks: Adding the quartic phase term beyond the Airy cubic provides only 0.00815 dB mean gain, supporting cubic near-saturation without claiming exact higher-order sufficiency.Deterministic scans and independently initialized searches found audit advantages below 10^-14 dB over reported references.
  • Reference and order checks: The broad search exceeds the one-shot phase fit by mean and maximum advantages of 8.62 × 10^-5 dB and 2.93 × 10^-4 dB, respectively.All analytical solutions remain inside the narrower reference domain, so bounded fallback is not activated in the reported experiments.
  • Propagation scope: The Fresnel criterion requires the maximum omitted phase remainder across modeled propagation segments to remain at most 0.5 rad, satisfied by 92.6% of scenes.The remaining scenes are not used to claim validity beyond the paraxial regime.
  • Independent propagation check: Under ASM, mean phase-fit gains over focus are 3.2320 dB for Fresnel and 3.2404 dB for ASM, with 0.0126 dB mean absolute difference.ASM evaluation used the same Gaussian amplitudes and phase coefficients without reoptimization over 72 Fresnel-valid scenes.
  • Finite-window check: Doubling the receiver-window width to 20 mm increases the gap by approximately 16×, while the mean gap remains 0.0196 dB at 40 mm.This agrees with the predicted O(W_r^4) scaling.
  • Discrete-array check: For a 256-element half-wavelength ULA, applying continuous analytical controls causes only 1.97 × 10^-5 dB mean loss, while the largest adjacent phase step is 0.571 rad.Sampling changes numerical moments without changing the underlying compensation mechanism.
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