Dynamic Pinching Beamforming

• Dynamic pinching beamforming is a reconfigurable radiation mechanism that adaptively tunes antenna positions and phase parameters to optimize the radiated signal.
• It employs advanced algorithmic frameworks including alternating optimization, metaheuristics, and learning-based control to jointly adjust PA positions and beamforming weights.
• The approach offers significant energy efficiency and robust interference management while addressing trade-offs in computational complexity and hardware precision.

Dynamic pinching beamforming is the reconfigurable radiation mechanism of pinching-antenna systems (PASS), in which dielectric “pinching antennas” (PAs) coupled to low-loss waveguides are adaptively positioned, activated, or tuned so that the radiated field jointly reshapes large-scale path loss and signal phase. In its baseline form, the beamforming variable is the PA location along a waveguide; in later extensions, the controllable variables also include radiation coefficients, power-splitting ratios, activation states, and phase-mismatch parameters that realize complex radiation weights without mechanical motion. Across the literature, dynamic pinching beamforming appears both as a standalone analog beamforming layer and as part of hybrid or tri-hybrid architectures that combine digital precoding, analog phase shifting, and pinching-domain reconfiguration (Wang et al., 9 Feb 2025, Zhao et al., 18 Nov 2025, Altinoklu et al., 26 May 2026).

1. Fundamental operating principle

At the hardware level, PASS consists of dielectric waveguides carrying guided waves and a set of small dielectric radiators that extract part of the guided energy and re-radiate it into free space. In a representative fully-connected tri-hybrid formulation, PASS contains $M$ parallel dielectric waveguides of length $L$, each carrying $N$ movable PAs, while the transmitted vector is

$z=W_{PB}(X)\,W_{RF}\,W_{BB}\,s,$

with

$$W_{PB}(X)=\operatorname{diag}[g_1(X_1),g_2(X_2),\ldots,g_M(X_M)], \qquad g_{m,n}=\frac{1}{\sqrt N}\exp(-j\,2\pi x_{m,n}/\lambda_g).$$

This makes the pinching stage an explicit beamforming block whose coefficients are governed by the PA positions $X$ (Zhao et al., 18 Nov 2025).

The physical basis of this mechanism is commonly modeled through an open-ended directional coupler and coupled-mode theory. In one foundational formulation, the modal amplitudes in the guide and pinching element satisfy

$$\frac{dA}{dx}=-j\,\kappa\,B\,e^{-j\Delta\beta x},\qquad \frac{dB}{dx}=-j\,\kappa\,A\,e^{+j\Delta\beta x},$$

with $\kappa$ the mode-coupling coefficient and $\Delta\beta=\beta_p-\beta_g$. The resulting radiated baseband signal from a single PA is

$$s_{\mathrm{rad}}=\sin(\kappa L)\,e^{-j\beta_g x_p}\,c_0,$$

which already shows the dual role of pinching: a PA location changes both the guided-path phase and the free-space propagation geometry (Wang et al., 9 Feb 2025).

A more recent extension replaces or complements motion with phase-mismatch control. Under single-mode excitation in amplitude-tunable PASS, the $L$0-th element has radiation weight $L$1, with

$L$2

By tuning $L$3, both $L$4 and $L$5 are controllable, so PASS becomes a weight-adaptive analog beamforming architecture rather than a purely equal-power radiation system (Altinoklu et al., 26 May 2026).

2. Signal models and optimization criteria

Most PASS formulations write the received signal as the cascade of a free-space channel and an in-waveguide response. In a multiuser downlink LoS model, user $L$6 receives

$L$7

and, with composite beamformer $L$8, the SINR is

$L$9

A standard objective is weighted sum-rate,

$N$0

subject to total power, unit-modulus analog phases, PA-position bounds, and minimum inter-PA spacing (Zhao et al., 18 Nov 2025).

The same structural idea appears in earlier PASS models without an analog phase-shifter network. With $N$1 waveguides, $N$2 pinching antennas per waveguide, and digital beamformer $N$3, the user-$N$4 SINR becomes

$N$5

and one prominent design problem is transmit-power minimization under SINR targets and PA-spacing constraints (Wang et al., 9 Feb 2025).

Beyond sum-rate and power minimization, dynamic pinching beamforming has been embedded into several other objective classes. Multicast formulations maximize the worst-user rate $N$6 (Shan et al., 31 May 2025). Secure formulations maximize secrecy rate by enforcing constructive combination at the legitimate user and destructive combination at the eavesdropper, optionally with artificial noise (Zhu et al., 18 Apr 2025). Cognitive-radio formulations maximize the sum of primary and secondary average spectral efficiencies under power budgets, minimum antenna separation, feasible PA deployment regions, and an interference temperature constraint (Sun et al., 17 Nov 2025). Mobility-aware formulations maximize average sum rate or discounted long-term spectral efficiency while including movement costs, QoS constraints, and blockage-dependent channels (Zhao et al., 28 Feb 2026, Amhaz et al., 8 May 2026).

These formulations differ in objective and channel assumptions, but they share the same core design variable: a spatially reconfigurable radiating structure whose geometry enters both path-loss terms and phase terms. This suggests that dynamic pinching beamforming is best understood not as a single algorithm, but as a class of coupled geometry-and-precoding optimization problems.

3. Algorithmic frameworks

A major strand of work solves the joint design problem by alternating optimization. In fully-connected tri-hybrid PASS, an FP-based algorithm removes the power constraint by reformulation, applies a Lagrangian-dual transform and a quadratic transform, and then alternates updates of auxiliary variables $N$7 and $N$8, the digital beamformer $N$9, the analog beamformer $z=W_{PB}(X)\,W_{RF}\,W_{BB}\,s,$0, and the PA positions $z=W_{PB}(X)\,W_{RF}\,W_{BB}\,s,$1. The analog subproblem is handled on the complex torus via Riemannian conjugate gradient, while the PA positions are optimized by a success-history based adaptive differential evolution (SHADE) method using the current-to-pbest mutation strategy and crossover-selection updates. A lower-complexity alternative replaces the transmit design with zero forcing and reuses SHADE for the PA positions (Zhao et al., 18 Nov 2025).

Earlier PASS beamforming studies used penalty-based alternating optimization and ZF-based simplifications. One representative method reformulates the power-minimization problem through normalized beamformers, introduces auxiliary variables to separate the effective channel decomposition, and alternately updates the transmit beamformer by convex SOC optimization, auxiliary variables by SCA or closed form, and PA positions by element-wise one-dimensional search. The companion ZF-based design chooses

$z=W_{PB}(X)\,W_{RF}\,W_{BB}\,s,$2

which decouples SINR constraints and reduces the optimization to the geometry variable $z=W_{PB}(X)\,W_{RF}\,W_{BB}\,s,$3 (Wang et al., 9 Feb 2025).

A second class of methods combines block-coordinate optimization with surrogate construction. The MM-PDD framework introduced for PASS-enabled downlink MU-MISO handles nonconvex complex exponential terms by a Lipschitz-gradient surrogate and invokes penalty dual decomposition to update beamformers, PA positions, and auxiliary variables toward a stationary point (Xu et al., 12 Feb 2025). Closely related PDD constructions were later used for max-min fairness in multi-user PASS with waveguide multiplexing, waveguide division, and waveguide switching, where augmented Lagrangian relaxation and block decomposition were applied to the resulting coupled constraints (Zhao et al., 20 Aug 2025). In NOMA-assisted PASS, an MM-PDD algorithm was paired with a swarm-based PSO-ZF alternative that evaluates candidate PA layouts through a ZF fitness function (Gan et al., 3 Jun 2025).

A third class exploits problem-specific geometry. The PA-wise successive tuning (PAST) algorithm for secure PASS first computes a coarse, symmetric PA layout around the legitimate user and then applies outward PA-wise fine tuning so that the legitimate user sees constructive superposition while the eavesdropper sees destructive combination (Zhu et al., 18 Apr 2025). The cognitive-radio extension follows a similar three-stage structure: coarse waveguide-level placement, wavelength-level refinement, and closed-form secondary-transmit-power control (Sun et al., 17 Nov 2025). In multi-antenna reception, a two-layer placement strategy first optimizes a central radiation point using large-scale channel characteristics and then applies a heuristic compressed placement algorithm based on a sliding-window scan over candidate phase-alignment points (Zhou et al., 2 Sep 2025).

Metaheuristics are also pervasive. Genetic algorithms optimize $z=W_{PB}(X)\,W_{RF}\,W_{BB}\,s,$4-vectors, activations, and movable setups in amplitude-tunable PASS (Altinoklu et al., 26 May 2026). PSO is used for secrecy-oriented waveguide multiplexing (Zhu et al., 18 Apr 2025) and for continuous-position 2D-PASS, while a discrete MILP formulation handles grid-constrained 2D layouts (Zhong et al., 12 Nov 2025). The diversity of these solvers reflects the multimodal, nonconvex, and mixed discrete-continuous character of dynamic pinching beamforming.

4. Adaptation, beam training, and mobility-aware control

When instantaneous optimization is impractical, PASS can be configured through beam training. In single-waveguide single-user PASS, a scalable codebook and a three-stage beam training (3SBT) scheme are used: a coarse one-dimensional search with one activated PA, a finer search with an increased number of activated antennas, and a final exhaustive refinement. In the reported setup, two-dimensional exhaustive search requires $z=W_{PB}(X)\,W_{RF}\,W_{BB}\,s,$5 pilot slots, whereas 3SBT requires $z=W_{PB}(X)\,W_{RF}\,W_{BB}\,s,$6 slots, and for SWSU at $z=W_{PB}(X)\,W_{RF}\,W_{BB}\,s,$7 GHz/$z=W_{PB}(X)\,W_{RF}\,W_{BB}\,s,$8 GHz the method converges to within $z=W_{PB}(X)\,W_{RF}\,W_{BB}\,s,$9 bit/s/Hz of ideal phase-aligned gain in $$W_{PB}(X)=\operatorname{diag}[g_1(X_1),g_2(X_2),\ldots,g_M(X_M)], \qquad g_{m,n}=\frac{1}{\sqrt N}\exp(-j\,2\pi x_{m,n}/\lambda_g).$$0 layers (Lv et al., 9 Feb 2025).

Dynamic optimization studies also report explicit adaptation times. In fully-connected tri-hybrid PASS, the outer AO loop adapts $$W_{PB}(X)=\operatorname{diag}[g_1(X_1),g_2(X_2),\ldots,g_M(X_M)], \qquad g_{m,n}=\frac{1}{\sqrt N}\exp(-j\,2\pi x_{m,n}/\lambda_g).$$1 within a few iterations per channel realization, FP-AO converges in $$W_{PB}(X)=\operatorname{diag}[g_1(X_1),g_2(X_2),\ldots,g_M(X_M)], \qquad g_{m,n}=\frac{1}{\sqrt N}\exp(-j\,2\pi x_{m,n}/\lambda_g).$$2 outer iterations with $$W_{PB}(X)=\operatorname{diag}[g_1(X_1),g_2(X_2),\ldots,g_M(X_M)], \qquad g_{m,n}=\frac{1}{\sqrt N}\exp(-j\,2\pi x_{m,n}/\lambda_g).$$3 of final WSR in $$W_{PB}(X)=\operatorname{diag}[g_1(X_1),g_2(X_2),\ldots,g_M(X_M)], \qquad g_{m,n}=\frac{1}{\sqrt N}\exp(-j\,2\pi x_{m,n}/\lambda_g).$$4 iteration, ZF-AO converges in $$W_{PB}(X)=\operatorname{diag}[g_1(X_1),g_2(X_2),\ldots,g_M(X_M)], \qquad g_{m,n}=\frac{1}{\sqrt N}\exp(-j\,2\pi x_{m,n}/\lambda_g).$$5 inner iterations, and SHADE converges in $$W_{PB}(X)=\operatorname{diag}[g_1(X_1),g_2(X_2),\ldots,g_M(X_M)], \qquad g_{m,n}=\frac{1}{\sqrt N}\exp(-j\,2\pi x_{m,n}/\lambda_g).$$6 generations for near-global $$W_{PB}(X)=\operatorname{diag}[g_1(X_1),g_2(X_2),\ldots,g_M(X_M)], \qquad g_{m,n}=\frac{1}{\sqrt N}\exp(-j\,2\pi x_{m,n}/\lambda_g).$$7. The same study notes that real-time deployment may require warm-start, reduced population, or learning-based surrogates (Zhao et al., 18 Nov 2025).

A formal two-timescale decomposition was later proposed for downlink MU-MISO PASS. There, the long-term variable is the pinching layout $$W_{PB}(X)=\operatorname{diag}[g_1(X_1),g_2(X_2),\ldots,g_M(X_M)], \qquad g_{m,n}=\frac{1}{\sqrt N}\exp(-j\,2\pi x_{m,n}/\lambda_g).$$8, updated once per channel-statistics block by stochastic successive convex approximation, while the short-term variable is the transmit beamformer $$W_{PB}(X)=\operatorname{diag}[g_1(X_1),g_2(X_2),\ldots,g_M(X_M)], \qquad g_{m,n}=\frac{1}{\sqrt N}\exp(-j\,2\pi x_{m,n}/\lambda_g).$$9, updated every fast-fading slot using a Karush-Kuhn-Tucker-guided dual learning approach. The short-term stage reconstructs the beamformer from learned dual variables rather than predicting $X$0 directly, and the long-term stage optimizes $X$1 through sampled gradients of the ergodic sum-rate surrogate (Zhang et al., 13 Apr 2025).

Under explicit mobility and blockage, reinforcement learning becomes the control mechanism. In an urban-micro setting with $X$2 PAs at $X$3 m and $X$4 users, a bilevel framework uses a soft actor-critic policy to place the PAs and ZF precoding to compute the instantaneous digital beamformer. The MDP state is $X$5, the action is a normalized displacement vector in $X$6, and the reward is the sum spectral efficiency minus a movement penalty (Zhao et al., 28 Feb 2026). A single-user mobility formulation uses DDPG to jointly output the beamforming vector and all PA $X$7-coordinates under Random Waypoint mobility and probabilistic blockage, with reward penalties enforcing QoS, feasible domains, and minimum PA spacing (Amhaz et al., 8 May 2026).

These results indicate two distinct operating modes. One mode relies on direct optimization or beam training per realization; the other relies on learned control policies or two-timescale decompositions that amortize computation over many realizations. The literature treats both as valid realizations of dynamic pinching beamforming.

5. Architectural variants and communication settings

Dynamic pinching beamforming has expanded beyond the original line-shaped, end-fed, equal-power PASS. Fully-connected tri-hybrid beamforming introduces a tunable phase-shifter network between RF chains and waveguides, so the pinching stage becomes the third layer after digital and analog beamforming (Zhao et al., 18 Nov 2025). Capacity analysis of tri-hybrid PASS further shows that, under idealized assumptions without in-waveguide attenuation, the optimal precoder aligns with the effective channel $X$8, and the resulting SNR exhibits small-$X$9 linear scaling and large-$$\frac{dA}{dx}=-j\,\kappa\,B\,e^{-j\Delta\beta x},\qquad \frac{dB}{dx}=-j\,\kappa\,A\,e^{+j\Delta\beta x},$$0 $$\frac{dA}{dx}=-j\,\kappa\,B\,e^{-j\Delta\beta x},\qquad \frac{dB}{dx}=-j\,\kappa\,A\,e^{+j\Delta\beta x},$$1 decay, implying an optimal PA count $$\frac{dA}{dx}=-j\,\kappa\,B\,e^{-j\Delta\beta x},\qquad \frac{dB}{dx}=-j\,\kappa\,A\,e^{+j\Delta\beta x},$$2 rather than indefinite monotone gain (Cheng et al., 2 Nov 2025).

The geometry itself has also diversified. Two-dimensional PASS extends the line-shaped structure into a continuous dielectric waveguide plane $$\frac{dA}{dx}=-j\,\kappa\,B\,e^{-j\Delta\beta x},\qquad \frac{dB}{dx}=-j\,\kappa\,A\,e^{+j\Delta\beta x},$$3, so the design variable is the 2D PA position $$\frac{dA}{dx}=-j\,\kappa\,B\,e^{-j\Delta\beta x},\qquad \frac{dB}{dx}=-j\,\kappa\,A\,e^{+j\Delta\beta x},$$4 subject to a minimum separation $$\frac{dA}{dx}=-j\,\kappa\,B\,e^{-j\Delta\beta x},\qquad \frac{dB}{dx}=-j\,\kappa\,A\,e^{+j\Delta\beta x},$$5; the objective is max-min SNR across users (Zhong et al., 12 Nov 2025). Center-fed PASS distributes $$\frac{dA}{dx}=-j\,\kappa\,B\,e^{-j\Delta\beta x},\qquad \frac{dB}{dx}=-j\,\kappa\,A\,e^{+j\Delta\beta x},$$6 input ports along a single waveguide and adds power-splitting ratios $$\frac{dA}{dx}=-j\,\kappa\,B\,e^{-j\Delta\beta x},\qquad \frac{dB}{dx}=-j\,\kappa\,A\,e^{+j\Delta\beta x},$$7, radiation coefficients $$\frac{dA}{dx}=-j\,\kappa\,B\,e^{-j\Delta\beta x},\qquad \frac{dB}{dx}=-j\,\kappa\,A\,e^{+j\Delta\beta x},$$8, and small PA micro-adjustments to the optimization variables. Its effective channel is $$\frac{dA}{dx}=-j\,\kappa\,B\,e^{-j\Delta\beta x},\qquad \frac{dB}{dx}=-j\,\kappa\,A\,e^{+j\Delta\beta x},$$9, and the architecture achieves $\kappa$0 under symmetric choices of $\kappa$1 and $\kappa$2 (Gan et al., 16 Feb 2026).

Application settings are equally broad. PASS has been developed for multicast rate maximization (Shan et al., 31 May 2025), secure communications with artificial noise and waveguide division or multiplexing (Zhu et al., 18 Apr 2025), multi-group multicast and unicast under WM/WD/WS transmission structures (Zhao et al., 20 Aug 2025), NOMA-assisted downlink MIMO (Gan et al., 3 Jun 2025), multiple-access uplink capacity analysis with multiple pinching beamforming vectors across time (Chen et al., 7 Aug 2025), cognitive radio with simultaneous primary and secondary transmission (Sun et al., 17 Nov 2025), and downlink systems with multiple receive antennas (Zhou et al., 2 Sep 2025). Amplitude-tunable PASS further unifies weight tuning, antenna movability, and discrete activation within a common hardware model based on phase-mismatch control (Altinoklu et al., 26 May 2026).

A notable theoretical distinction appears in multiple access. For a PASS-assisted MAC with an asymptotically large number of pinching beamforming vectors, the optimal transmission scheme is alternating transmission among each user with its channel power gain maximized by dynamic pinching beamforming, which implies that the NOMA-based transmission scheme is not needed. The optimal time-sharing factors are proportional to the maximized user channel gains, and at most $\kappa$3 beamforming vectors are needed (Chen et al., 7 Aug 2025). This result sharply contrasts with settings where NOMA is explicitly optimized as part of the beamforming design.

6. Reported performance, trade-offs, and recurrent design issues

Across the literature, dynamic pinching beamforming is repeatedly reported to outperform fixed-position arrays and several hybrid-MIMO baselines, but the gains depend strongly on architecture, objective, and operating regime. In the early transmit-power minimization study, PASS reduces transmit power by over $\kappa$4 compared to conventional and massive MIMO; at $\kappa$5 dB and $\kappa$6, continuous PASS requires $\kappa$7 dBm, versus $\kappa$8 dBm for conventional MIMO and $\kappa$9 dBm for massive MIMO, while discrete activation incurs minimal loss but needs $\Delta\beta=\beta_p-\beta_g$0/m to reach within $\Delta\beta=\beta_p-\beta_g$1 dB of continuous activation (Wang et al., 9 Feb 2025).

For tri-hybrid PASS, the fully-connected design achieves WSR comparable to the sub-connected architecture while delivering superior energy efficiency with fewer RF chains. In the reported simulations, FC-PASS with FP achieves $\Delta\beta=\beta_p-\beta_g$2 of SC-PASS WSR with half the RF chains, ZF-PASS lags by $\Delta\beta=\beta_p-\beta_g$3 dB, partially-connected massive MIMO is $\Delta\beta=\beta_p-\beta_g$4 below SC-PASS, and both PASS variants exceed hybrid MIMO in energy efficiency by a factor $\Delta\beta=\beta_p-\beta_g$5. Under CSI error $\Delta\beta=\beta_p-\beta_g$6, PASS remains robust and still above MIMO until $\Delta\beta=\beta_p-\beta_g$7 becomes large (Zhao et al., 18 Nov 2025).

Amplitude-tunable PASS changes the performance ranking across regimes. At $\Delta\beta=\beta_p-\beta_g$8 dBm, AT-PASS achieves $\Delta\beta=\beta_p-\beta_g$9 bps/Hz versus $$s_{\mathrm{rad}}=\sin(\kappa L)\,e^{-j\beta_g x_p}\,c_0,$$0 bps/Hz for MOV-PASS, $$s_{\mathrm{rad}}=\sin(\kappa L)\,e^{-j\beta_g x_p}\,c_0,$$1 bps/Hz for DAC-PASS, and $$s_{\mathrm{rad}}=\sin(\kappa L)\,e^{-j\beta_g x_p}\,c_0,$$2 bps/Hz for fixed $$s_{\mathrm{rad}}=\sin(\kappa L)\,e^{-j\beta_g x_p}\,c_0,$$3 MISO; for $$s_{\mathrm{rad}}=\sin(\kappa L)\,e^{-j\beta_g x_p}\,c_0,$$4, AT-PASS reaches $$s_{\mathrm{rad}}=\sin(\kappa L)\,e^{-j\beta_g x_p}\,c_0,$$5 bps/Hz versus $$s_{\mathrm{rad}}=\sin(\kappa L)\,e^{-j\beta_g x_p}\,c_0,$$6 bps/Hz for MOV-PASS. The same study reports that in the noise-limited regime spatial reconfiguration offers similar gains, whereas in the interference-limited regime amplitude tunability provides $$s_{\mathrm{rad}}=\sin(\kappa L)\,e^{-j\beta_g x_p}\,c_0,$$7 additional sum-rate by fine-tuning weights for interference nulling. Quantization is also mild: at $$s_{\mathrm{rad}}=\sin(\kappa L)\,e^{-j\beta_g x_p}\,c_0,$$8 dBm, $$s_{\mathrm{rad}}=\sin(\kappa L)\,e^{-j\beta_g x_p}\,c_0,$$9 gives $L$00 bps/Hz and the continuous case gives $L$01 bps/Hz (Altinoklu et al., 26 May 2026).

Two-dimensional and mobility-aware variants emphasize robustness rather than only peak rate. In 2D-PASS, the continuous design outperforms line-PASS by $L$02 dB and fixed-position antennas by $L$03 dB at $L$04 dBm, while a discrete design with $L$05 m loses only $L$06 dB relative to continuous placement (Zhong et al., 12 Nov 2025). In the SAC-based urban-micro study, the mean sum-SE is $L$07 for the proposed method, versus $L$08 for fixed PA placement and $L$09 for a random policy (Zhao et al., 28 Feb 2026). In the DDPG mobility-and-blockage study, dynamic PASS provides up to $L$10 rate gain over static PASS over the $L$11 dBm range and maintains $L$12 QoS satisfaction against $L$13 for the static baseline (Amhaz et al., 8 May 2026).

Several recurrent design issues appear consistently. One is computational burden: SHADE, PSO, GA, and PDD-based methods are effective on multimodal landscapes but are explicitly described as computationally heavy or dominated by repeated QCQP, CVX, or fitness evaluations (Zhao et al., 18 Nov 2025, Zhao et al., 20 Aug 2025). Another is hardware precision: secure and cognitive-radio studies emphasize sub-wavelength positioning, wavelength-level phase refinement, and minimum spacing such as $L$14 to avoid coupling (Zhu et al., 18 Apr 2025, Sun et al., 17 Nov 2025). A third is that more reconfigurability does not imply a single universally optimal architecture. The MAC capacity result shows that NOMA is not needed in the asymptotic dynamic-beamforming limit (Chen et al., 7 Aug 2025), while tri-hybrid capacity analysis shows that the SNR does not increase indefinitely with $L$15, but instead admits an optimal $L$16 in the idealized single-user setting (Cheng et al., 2 Nov 2025).

Taken together, these studies portray dynamic pinching beamforming as a general reconfigurable-waveguide paradigm rather than a single technique. Its essential feature is the direct optimization of radiating geometry—or, in amplitude-tunable implementations, equivalent complex radiation weights—as part of the communication design. The resulting beamforming layer can be integrated with digital precoding, analog phase-shifter networks, time-sharing policies, or learning-based control, and the achievable gains depend on how that extra spatial degree of freedom is exploited under the specific channel, hardware, and latency constraints of the target system.