Pinching Beamforming in PASS Systems

• Pinching beamforming is a propagation-shaping technique in PASS systems that configures PA positions, activation states, and radiation coefficients to reconfigure both guided-wave and free-space channels.
• It optimizes system performance by jointly designing antenna geometries and beamforming weights to maximize sum-rate, minimize transmit power, or enhance security and ISAC capabilities.
• Advanced algorithms such as branch-and-bound, alternating optimization, and learning-based methods address the nonconvex, MINLP challenges inherent in pinching beamforming design.

Pinching beamforming is the PASS-specific beamforming mechanism in which the locations, activation states, and, in newer formulations, the radiation coefficients of pinching antennas (PAs) on dielectric waveguides are configured to reshape the end-to-end propagation channel. In the PASS literature, the design variable is therefore not limited to digital or RF weights; it also includes the spatial distribution of radiating points, which changes guided-wave phase, free-space phase, large-scale path loss, and, in amplitude-tunable architectures, the complex radiation weight of each element (Xu et al., 30 Apr 2025, Wang et al., 9 Feb 2025, Altinoklu et al., 26 May 2026).

1. Conceptual definition and architectural scope

In its most common form, pinching beamforming means beamforming by configuring which PAs are activated and where they are placed or selected on dielectric waveguides. PASS is typically modeled as a downlink MISO architecture in which each waveguide is connected to one RF chain and supports multiple PAs, either movable along the waveguide or selectable from predefined locations. The resulting control acts directly on the physical aperture rather than only on a fixed-array excitation vector. This distinguishes pinching beamforming from conventional transmit beamforming, which assumes a fixed antenna geometry and optimizes only baseband or RF coefficients (Xu et al., 30 Apr 2025, Guo et al., 3 Feb 2025, Zhang et al., 13 Apr 2025).

The literature uses the term in several closely related senses. In continuous-position formulations, pinching beamforming is the optimization of PA coordinates along one or more waveguides. In discrete-activation formulations, it is the binary selection of activated PAs from a predefined set of mounting points. In secure, cognitive, multicast, and ISAC variants, the same principle is specialized to different system objectives, but the underlying operation remains geometry-driven channel shaping through reconfigurable radiation points (Shan et al., 31 May 2025, Zhu et al., 18 Apr 2025, Li et al., 3 May 2026, Guo et al., 3 Dec 2025).

A broader interpretation has emerged in later hardware models. Amplitude-tunable PASS treats pinching beamforming not only as antenna placement or activation, but as controllable complex weighting of radiating elements via phase-mismatch manipulation under single-mode excitation. Under that view, PASS becomes a weight-adaptive analog beamforming architecture, while conventional equal-power, movable, and discrete-activation models appear as special cases of a unified physics-based framework (Altinoklu et al., 26 May 2026).

2. Electromagnetic and signal-theoretic foundations

A recurring system representation writes the received signal through an effective channel of the form

$$y_k=\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\sum_{i=1}^K \mathbf{w}_i c_i+n_k,$$

with the corresponding multiuser SINR

$$\mathrm{SINR}_k=\frac{|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_k|^2}{\sum_{i\neq k}|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_i|^2+\sigma^2}.$$

Here, $\mathbf{G}(\mathbf{X})$ models guided-wave propagation from feed points to PAs, while $\mathbf{h}_k(\mathbf{X})$ models PA-to-user propagation. The dependence on $\mathbf{X}$ is fundamental: moving or selecting PAs modifies both the in-waveguide phase accumulation and the free-space spherical-wave channel, so the effective channel itself becomes a design variable (Zhang et al., 13 Apr 2025, Xu et al., 12 Feb 2025).

Physics-based PASS modeling often begins with coupled-mode theory. One widely used radiation model expresses the power radiation ratio of antenna $l$ on waveguide $n$ as

$$\beta_{l,n}=a_{l,n}\sin(\kappa_{l,n}L^{\mathrm{PA}})\prod_{i=1}^{l-1}\sqrt{1-a_{i,n}\sin^2(\kappa_{i,n}L^{\mathrm{PA}})},$$

showing that the radiated power of one PA depends on the coupling state of all previously activated PAs on the same waveguide. The same work makes the coupling coefficient physically adjustable through the spacing $S_{l,n}$ between waveguide and PA,

$\kappa_{l,n}=\Omega_0 e^{-\alpha S_{l,n}},$

thereby turning spacing into a direct control knob for radiation strength. On that basis, the equal-power model is no longer merely assumed: if $$\mathrm{SINR}_k=\frac{|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_k|^2}{\sum_{i\neq k}|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_i|^2+\sigma^2}.$$0 PAs are activated on waveguide $$\mathrm{SINR}_k=\frac{|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_k|^2}{\sum_{i\neq k}|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_i|^2+\sigma^2}.$$1, equal-power radiation with $$\mathrm{SINR}_k=\frac{|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_k|^2}{\sum_{i\neq k}|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_i|^2+\sigma^2}.$$2 can be realized in closed form by a sequential spacing arrangement (Xu et al., 30 Apr 2025).

Another branch of the literature starts from a directional-coupler interpretation of the pinching antenna. In that model, the coupled-mode equations

$$\mathrm{SINR}_k=\frac{|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_k|^2}{\sum_{i\neq k}|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_i|^2+\sigma^2}.$$3

show that power transfer depends on the phase mismatch $$\mathrm{SINR}_k=\frac{|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_k|^2}{\sum_{i\neq k}|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_i|^2+\sigma^2}.$$4 between the main waveguide and the pinching antenna. By electrically tuning $$\mathrm{SINR}_k=\frac{|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_k|^2}{\sum_{i\neq k}|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_i|^2+\sigma^2}.$$5, the transfer coefficient $$\mathrm{SINR}_k=\frac{|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_k|^2}{\sum_{i\neq k}|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_i|^2+\sigma^2}.$$6 acquires both tunable magnitude and phase. The resulting radiated coefficient is inherently complex, so the beamforming variable is no longer only location or activation but also a directly tunable analog weight (Altinoklu et al., 26 May 2026).

These models imply two complementary physical mechanisms. The first is geometric channel reconfiguration through PA placement, which changes path loss and phase. The second is radiation-control reconfiguration through coupling or phase mismatch, which changes the distribution of power extracted from the waveguide. This suggests that pinching beamforming is best understood as a propagation-shaping layer spanning aperture geometry, radiation coefficients, and guided-wave transport.

3. Canonical optimization formulations

The dominant formulation in early PASS work is joint transmit-and-pinching beamforming for downlink multiuser MISO. Typical objectives include sum-rate maximization and transmit-power minimization under QoS constraints. In sum-rate form, the variables are the PA positions $$\mathrm{SINR}_k=\frac{|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_k|^2}{\sum_{i\neq k}|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_i|^2+\sigma^2}.$$7 and the digital precoder $$\mathrm{SINR}_k=\frac{|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_k|^2}{\sum_{i\neq k}|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_i|^2+\sigma^2}.$$8; in power-minimization form, the variables may additionally include binary activation matrices and the number of activated PAs per waveguide (Xu et al., 12 Feb 2025, Xu et al., 30 Apr 2025).

Several specialized formulations extend this template. In multicast PASS, the objective is max–min user rate, so pinching beamforming is optimized to improve the worst-user channel rather than the sum throughput. In the single-waveguide, single-PA regime, the optimal activated location can even be derived in closed form, and for linearly distributed users the multicast-optimal position is the midpoint of the leftmost and rightmost users, i.e., a one-dimensional Chebyshev-center solution (Shan et al., 31 May 2025).

Security-oriented formulations optimize secrecy rate by choosing PA positions so that signals combine constructively at the legitimate receiver and destructively at the eavesdropper. Cognitive-radio formulations add power-budget and interference-temperature constraints with respect to primary users, so pinching beamforming must jointly strengthen intended channels and suppress leakage into protected links. In NOMA-assisted PASS, the optimization further couples PA placement with cluster beamformers and power-allocation coefficients under SIC feasibility constraints (Zhu et al., 18 Apr 2025, Li et al., 3 May 2026, Gan et al., 3 Jun 2025).

More recent formulations move beyond communication-only objectives. PASS-assisted symbiotic radio maximizes the primary sum rate under a backscatter-device detection constraint at an IoT receiver. PASS-enabled ISAC maximizes sensing rate while enforcing communication QoS for UAV users and placement constraints on segmented waveguides. In the uplink multiple-access setting, dynamic pinching beamforming treats the PA configuration as time-varying over transmission slots and optimizes beam patterns, slot durations, and powers jointly (Wang et al., 9 Aug 2025, Guo et al., 3 Dec 2025, Chen et al., 7 Aug 2025).

4. Algorithmic design methodologies

Because pinching beamforming couples geometry, channel, and beamforming variables nonlinearly, the resulting problems are generally MINLPs or highly nonconvex continuous programs. One exact approach is branch-and-bound. For discrete PASS with equal-power radiation, globally convergent BnB algorithms have been developed for both single-user and multiuser cases. The single-user problem reduces to a binary QCQP after MRT substitution, while the multiuser case is reformulated with SOC constraints and McCormick relaxations, yielding finite $$\mathrm{SINR}_k=\frac{|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_k|^2}{\sum_{i\neq k}|\mathbf{h}_k^H(\mathbf{X})\mathbf{G}(\mathbf{X})\mathbf{w}_i|^2+\sigma^2}.$$9-optimal convergence (Xu et al., 30 Apr 2025).

A second family uses alternating optimization with convex surrogates. MM-PDD and related methods introduce auxiliary variables to decouple the position-dependent exponential or distance terms, construct Lipschitz-gradient or first-order surrogates, and alternate among beamforming, auxiliary-variable, and position updates. WMMSE-based AO is especially common because it turns rate objectives into blockwise tractable subproblems; digital beamforming subproblems often become SOC programs or convex QPs, whereas PA updates remain one-dimensional searches or nonconvex coordinate updates (Xu et al., 12 Feb 2025, Li et al., 3 May 2026, Zhang et al., 13 Apr 2025).

Low-complexity position search is a distinct methodological theme. Element-wise sequential optimization updates one PA at a time while holding the rest fixed, using one-dimensional search over continuous or discretized candidate sets. Variants appear in downlink and uplink multiuser PASS, multicast PASS, robust PASS under CSI uncertainty, and cognitive-radio PASS. The appeal is that it avoids expensive inner–outer alternating loops between full beamformer updates and full geometry updates, while still exploiting the highly structured dependence of the objective on a single coordinate (Sun et al., 4 Jun 2025, Shan et al., 31 May 2025, Sun et al., 19 Dec 2025).

Heuristic global search is used when the position landscape is strongly multimodal. PSO, GA, and SHADE have been applied to pinching beamforming in symbiotic radio, secure PASS, amplitude-tunable PASS, and fully connected tri-hybrid PASS. These methods are typically embedded in outer AO loops: beamforming is updated by WMMSE, SCA, or ZF, and PA positions are then searched by population-based heuristics (Wang et al., 9 Aug 2025, Zhu et al., 18 Apr 2025, Altinoklu et al., 26 May 2026, Zhao et al., 18 Nov 2025).

Learning-based approaches exploit structure rather than replacing it. GPASS uses a staged pair of sub-GNNs: a PBF-sub-GNN predicts antenna positions and a TBF-sub-GNN predicts transmit beamforming conditioned on those positions. KDL-Transformer reconstructs KKT-structured beamforming solutions by learning dual variables instead of full primal beamforming matrices. SWISAC-GNN extends the same philosophy to ISAC by learning PA positions and the low-dimensional parameters underlying the structured beamformer. A common design choice is permutation-aware or graph-based modeling, reflecting the symmetry of users, waveguides, and PAs (Guo et al., 3 Feb 2025, Xu et al., 12 Feb 2025, Guo et al., 3 Dec 2025).

5. Major variants and generalizations

The earliest PASS models focused on line-shaped waveguides with position optimization and equal-power or proportional-power radiation. Subsequent work introduced discrete activation, in which PAs are not continuously moved but activated from a predefined set of mounting points. This discrete model is important because it converts pinching beamforming from continuous geometry optimization into combinatorial activation-and-beamforming co-design (Wang et al., 9 Feb 2025, Xu et al., 30 Apr 2025).

Amplitude-tunable PASS extends the concept further by allowing electrically tunable complex radiation weights through phase-mismatch manipulation under single-mode excitation. The feasible phase-mismatch set

$\mathbf{G}(\mathbf{X})$0

supports a unified hardware model covering amplitude-tunable PASS, discrete activation PASS, movable PASS, and the equal-power baseline. This generalization makes explicit that pinching beamforming is not intrinsically restricted to position control (Altinoklu et al., 26 May 2026).

Dynamic pinching beamforming introduces a time dimension. In a PASS-assisted multiple-access channel, multiple pinching beamforming vectors are used within one transmission period. For an asymptotically large number of beamforming patterns, the optimal strategy is alternating transmission among users with each user served under the PA configuration that maximizes its channel power gain, implying that the NOMA-based transmission scheme is not needed in that regime (Chen et al., 7 Aug 2025).

Architectural generalizations also alter the meaning of the beamforming layer. Two-dimensional PASS replaces the line-shaped waveguide with a continuous dielectric waveguide plane, so beamforming becomes analog spatial control over a 2D aperture. Fully connected tri-hybrid PASS adds a tunable RF phase-shifter network above the waveguides, making pinching beamforming the third layer after digital and analog beamforming. Center-fed C-PASS replaces the single end-fed structure by multiple center-fed input ports, introducing forward- and backward-propagation components and enabling a single-waveguide architecture with $\mathbf{G}(\mathbf{X})$1 and received-power scaling of order $\mathbf{G}(\mathbf{X})$2 under the symmetric configuration (Zhong et al., 12 Nov 2025, Zhao et al., 18 Nov 2025, Gan et al., 16 Feb 2026).

6. Performance characteristics, trade-offs, and interpretive issues

Across application domains, the reported performance trend is consistent: PASS gains increase when path-loss heterogeneity becomes more severe or when multiuser geometry becomes harder. Discrete PASS with jointly optimized transmit and pinching beamforming significantly outperforms conventional multi-antenna architectures, particularly when the number of users and the spatial range increase. In multicast systems, the rate advantage similarly becomes more pronounced as user count and coverage enlarge. Two-dimensional PASS is reported to improve the minimum SNR over both line-shaped PASS and fixed-position antenna benchmarks under dispersed user layouts (Xu et al., 30 Apr 2025, Shan et al., 31 May 2025, Zhong et al., 12 Nov 2025).

The magnitude of the reported gains is often large. A physics-based PASS model reports transmit-power reductions of over $\mathbf{G}(\mathbf{X})$3 compared to conventional and massive MIMO, with discrete activation causing minimal performance loss and the proportional power model performing comparably to the equal power model. In discrete PASS with exact and matching-based optimization, reported examples include over $\mathbf{G}(\mathbf{X})$4 power reduction versus massive MIMO in a single-user case and over $\mathbf{G}(\mathbf{X})$5 dBm versus conventional MIMO and $\mathbf{G}(\mathbf{X})$6 dBm versus massive MIMO in a multiuser example. In NOMA-assisted PASS, the reported reduction exceeds $\mathbf{G}(\mathbf{X})$7 relative to conventional massive MIMO-NOMA (Wang et al., 9 Feb 2025, Xu et al., 30 Apr 2025, Gan et al., 3 Jun 2025).

Complexity trade-offs are equally prominent. Globally optimal BnB delivers exactness or $\mathbf{G}(\mathbf{X})$8-optimality but scales poorly. Matching, ZF-based approximations, PSO/SHADE heuristics, and element-wise searches sacrifice optimality guarantees for tractability. Learning-based methods target inference latency: GPASS reports an average inference time of $\mathbf{G}(\mathbf{X})$9 ms for $\mathbf{h}_k(\mathbf{X})$0, $\mathbf{h}_k(\mathbf{X})$1, while KDL-Transformer is reported to improve system performance by over $\mathbf{h}_k(\mathbf{X})$2 relative to MM-PDD and achieve a millisecond-level response on modern GPUs (Guo et al., 3 Feb 2025, Xu et al., 12 Feb 2025).

Several misconceptions are corrected by the recent literature. First, pinching beamforming is not only “moving antennas around”; it may also mean discrete activation, spacing-controlled radiation allocation, or electrically controlled complex weighting via phase mismatch. Second, the equal-power assumption frequently used in early PASS analysis is physically realizable by explicit spacing design rather than being a purely abstract idealization. Third, more PAs are not universally beneficial: in PASS-enabled tri-hybrid beamforming, capacity scaling analyses show a small-$\mathbf{h}_k(\mathbf{X})$3 linear SNR gain but an asymptotic large-$\mathbf{h}_k(\mathbf{X})$4 decay of order $\mathbf{h}_k(\mathbf{X})$5, implying an optimal PA count rather than monotonic growth (Altinoklu et al., 26 May 2026, Xu et al., 30 Apr 2025, Cheng et al., 2 Nov 2025).

A plausible implication is that pinching beamforming should not be classified as a narrow variant of array steering. The research trajectory instead presents it as a hierarchy of propagation-control mechanisms operating across geometry, coupling physics, and hybrid precoding, with the physical channel itself elevated to a beamforming variable.