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Pinching-Antenna Systems (PASS): Power Radiation Model and Optimal Beamforming Design

Xiaoxia Xu, Xidong Mu, Zhaolin Wang, Yuanwei Liu, Arumugam Nallanathan

arXiv:2505.00218v2eess.SPeess.SYmath.OC

TL;DR

PASS addresses unresolved adjustable power radiation and globally optimal beamforming under discrete antenna activation. It introduces spacing-controlled radiation, jointly optimizes activation and beamforming with branch-and-bound, and adds a polynomial-time matching alternative. PASS outperforms conventional MIMO systems, while the matching method achieves near-optimal performance with marginal loss.

  • Problem

    PASS lacks adjustable power-radiation control and globally optimal joint beamforming for discrete activation, while existing designs often assume equal radiation or fixed antenna counts.

  • Method

    The paper tunes antenna-waveguide spacing for flexible or equal radiation, then jointly optimizes discrete activation and transmit/pinching beamforming using globally convergent branch-and-bound and a KKT-based many-to-many matching method.

  • Results

    PASS outperforms conventional MIMO systems in single-user and multi-user cases, while the matching algorithm achieves near-optimal performance with only marginal loss relative to the optimal algorithm.

  • Takeaways & Limitations

    Discrete PASS can combine adjustable radiation and joint beamforming optimization, with a lower-complexity matching design retaining near-optimal performance.

Abstract

from arXiv · show

Pinching-antenna systems (PASS) improve wireless links by configuring the locations of activated pinching antennas along dielectric waveguides, namely pinching beamforming. In this paper, a novel adjustable power radiation model is proposed for PASS, where power radiation ratios of pinching antennas can be flexibly controlled by tuning the spacing between pinching antennas and waveguides. A closed-form pinching antenna spacing arrangement strategy is derived to achieve the commonly assumed equal-power radiation. Based on this, a practical PASS framework relying on discrete activation is considered, where pinching antennas can only be activated among a set of predefined locations. A transmit power minimization problem is formulated, which jointly optimizes the transmit beamforming, pinching beamforming, and the numbers of activated pinching antennas, subject to each user's minimum rate requirement. (1) To solve the resulting highly coupled mixed-integer nonlinear programming (MINLP) problem, branch-and-bound (BnB)-based algorithms are proposed for both single-user and multi-user scenarios, which is guaranteed to converge to globally optimal solutions. (2) A low-complexity many-to-many matching algorithm is further developed. Combined with the Karush-Kuhn-Tucker (KKT) theory, locally optimal and pairwise-stable solutions are obtained within polynomial-time complexity. Simulation results demonstrate that: (i) PASS significantly outperforms conventional multi-antenna architectures, particularly when the number of users and the spatial range increase; and (ii) The proposed matching-based algorithm achieves near-optimal performance, resulting in only a slight performance loss while significantly reducing computational overheads. Code is available at https://github.com/xiaoxiaxusummer/PASS_Discrete

I. INTRODUCTION

PASS uses activated pinching antennas along dielectric waveguides to reshape wireless links, while this paper addresses adjustable radiation and globally optimal discrete-activation beamforming. It also develops a lower-complexity matching method that retains near-optimal performance.

  • Research challenges: Existing PASS designs leave globally optimal joint beamforming unexplored because digital and pinching beamforming form a highly coupled nonconvex problem.Prior studies commonly use suboptimal designs with a fixed number of activated antennas.
  • Research challenges: Existing PASS studies commonly assume equal-power radiation, leaving adjustable power radiation control unresolved.The proposed model instead controls radiation ratios by tuning coupling spacing between waveguides and pinching antennas.
  • Contributions: The paper derives closed-form spacing solutions for flexible or equal radiation and formulates transmit-power minimization over transmit beamforming, pinching beamforming, and activated-antenna counts.The optimization enforces users’ minimum rate requirements under practical discrete activation.
  • Algorithms: Branch-and-bound algorithms are proposed for globally optimal joint beamforming in both single-user and multi-user scenarios.The single-user MINLP is reduced to nonconvex QCQP, while the multi-user case uses McCormick-envelope relaxations for bound estimation.
  • Algorithms: A welfare-driven many-to-many matching algorithm reaches local optima in polynomial time and ensures pairwise equilibrium.Its preferences depend on beamforming solutions obtained through KKT theory.
  • Results: PASS outperforms conventional MIMO systems in single-user and multi-user cases, with gains increasing with multiplexed users and spatial range.The matching algorithm achieves near-optimal performance with only marginal loss relative to the optimal algorithm.
  • System model: PASS places dielectric waveguides across a service area and selectively activates pinching antennas at discrete locations for downlink MISO transmission.Activated antennas can adjust signal phases and large-scale path loss, enabling pinching beamforming.

B. PASS Signal Model

The PASS signal model combines digital transmit beamforming with waveguide propagation, pinching-antenna activation, and antenna-specific radiation responses. It incorporates power radiation ratios, propagation phases, and discrete activation through structured matrices.

  • Digital transmit beamforming multiplexes K users through W, with each wk representing user k’s transmit beamforming vector.
  • The propagated signal on each waveguide reflects feed-point transmission effects before reaching its pinching antennas.
  • Each pinching antenna contributes a power radiation ratio βl,n and an in-waveguide phase response determined by its feed-point distance.
  • The model assumes ideal waveguide propagation, while practical dielectric-waveguide attenuation is identified as a future consideration.
  • The activation matrix A is block diagonal, encoding which discrete pinching antennas are activated on each waveguide.

2) Signal Radiation in Free Space:

The free-space model uses LoS-dominant spherical-wave channels from activated pinching antennas to users, then formulates discrete activation and beamforming as a transmit-power minimization problem. The resulting problem is nonconvex and strongly coupled.

  • LoS-dominant channels connect the pinching antennas on each waveguide to each user through spherical-wave propagation.
  • The channel coefficient depends on wavelength, a reference gain φ, and the distance between each pinching antenna and user.
  • The received signal combines digital transmit beamforming with pinching beamforming through the activation matrix A.
  • The design jointly selects W, discrete activation A, and the numbers of activated antennas to minimize transmit power while satisfying every user’s SINR requirement.
  • The resulting MINLP is nonconvex because activation, activated-antenna counts, and transmit beamforming are strongly coupled.
  • For the single-user case, the transmit beamforming optimum follows maximum ratio transmission, after which antenna activation is optimized using BnB.

1) BnB Principles:

BnB solves the nonconvex design by partitioning the feasible domain into boxes and bounding each box through convex relaxation. As boxes shrink, the relaxation bounds converge toward the global optimum.

  • BnB partitions the solution space into smaller boxes and evaluates each box using convex-relaxation lower and upper bounds.
  • The global upper bound is the best feasible objective found, while the global lower bound is the minimum lower bound among candidate boxes.
  • As box sizes shrink, the bound gap vanishes, allowing the globally optimal objective to be approximated.
  • McCormick envelopes convexify bilinear terms in the relaxation and become tight as each box contracts to a point.
  • The relaxation provides feasibility tests, lower bounds, and upper bounds for the original nonconvex problem.

3) BnB Algorithm:

The PASS BnB algorithm branches on discrete activation variables, bounds child boxes using convex relaxations and feasible projections, and prunes boxes that cannot contain an optimum. It terminates with globally optimal beamforming and activation decisions.

  • The algorithm branches each selected box into two child boxes along a discrete activation-variable edge.
  • Best-bound-first selects the box with the strongest lower bound, while maximum-length-first selects its longest edge for branching.
  • Each child box receives a lower bound from relaxation and an upper bound from a projected feasible binary activation solution.
  • Algorithm 1 iterates branching, bounding, and pruning until the global bound gap meets the tolerance threshold.
  • The procedure outputs globally optimal activated-antenna counts, activation locations, transmit beamforming, and transmit power.
  • Boxes are pruned when relaxation is infeasible, fully explored, or provably nonoptimal relative to the global upper bound.
  • Pruning accelerates convergence without impacting global optimality, although the search complexity has best-case and worst-case bounds.

B. Optimal Solution for Multi-User Scenario

The multi-user formulation introduces auxiliary variables and convex relaxations to handle coupling among activation, beamforming, and bilinear terms. These transformations enable a tractable reformulation for optimization.

  • B. Optimal Solution for Multi-User Scenario: The multi-user transmit beamforming lacks explicit closed-form expressions, so the method constructs a tractable convex relaxation.The relaxation addresses coupling among W, A, and L_s^n.
  • B. Optimal Solution for Multi-User Scenario: The reformulated constraints include equivalent SOC constraints derived from the effective-channel representation.The phase-rotation invariance of modulus operations is used to reformulate minimum-SINR constraints.
  • B. Optimal Solution for Multi-User Scenario: An auxiliary matrix D represents effective transmit beamforming normalized by equal-power radiation ratios.Its n-th row describes the effective beamforming for signals radiated by pinching antennas on waveguide n.
  • B. Optimal Solution for Multi-User Scenario: The bilinear relation z_k = A d_k is introduced to represent products between activation variables and beamforming coefficients.The construction stacks the products a_n d_n,k into z_k.
  • B. Optimal Solution for Multi-User Scenario: McCormick envelopes replace bilinear constraints after binary constraints are relaxed, producing a convex relaxation of the multi-user problem.The real and imaginary parts of D are represented with auxiliary variables U and V.

2) BnB Algorithm:

The BnB procedure searches over binary activation and continuous beamforming variables using convex-relaxation bounds. Projection supplies feasible solutions, while bound comparisons drive pruning.

  • 2) BnB Algorithm:: The initial feasible region includes binary activation variables and bounded transmit-beamforming coefficients.The beamforming bounds use a sufficiently large power budget P0 to preserve feasibility.
  • 2) BnB Algorithm:: Each iteration selects a branching box and divides its longest edge, separating binary variables into 0 and 1 or bisecting continuous variables.The branching rule combines box selection with longest-edge division.
  • 2) BnB Algorithm:: Convex relaxation over child boxes provides local lower bounds for the original objective.Infeasible relaxed boxes are discarded during bounding.
  • 2) BnB Algorithm:: Projecting relaxed activation variables to binary values and solving the original problem yields feasible solutions and local upper bounds.Global upper and lower bounds are refined before unnecessary boxes are pruned.

3) Convergence and Optimality Analysis:

The paper analyzes convergence and optimality of the BnB method and then develops a KKT-based matching framework for low-complexity activation and beamforming. Matching preferences depend on the current activation state, creating externalities and non-substitutability.

  • 3) Convergence and Optimality Analysis:: Theorem 1 states that the proposed BnB converges in finite iterations to an ε-optimal solution.The achieved objective can be made arbitrarily close to the true optimum.
  • 3) Convergence and Optimality Analysis:: The multi-user BnB algorithm is organized around branching, convex-relaxation bounding, feasible-solution projection, bound updates, and pruning.The procedure terminates when the global upper–lower bound gap is within ε.
  • 3) Convergence and Optimality Analysis:: Lemma 3 establishes that the global upper–lower bound gap vanishes as the maximum box edge length decreases.This supports the tightening of bounds during the search.
  • 3) Convergence and Optimality Analysis:: The matching formulation pairs waveguides with pinching-antenna indices, with binary activation indicating whether each pair is matched.The associated beamforming is optimized for each matching state using KKT-based analysis and convex SOC programming.
  • 3) Convergence and Optimality Analysis:: Because beamforming changes with matching decisions and group composition, the matching game has externalities and non-substitutable preferences.Stable matching is therefore not guaranteed.

B. Proposed Low-Complexity Solution

The proposed low-complexity solution uses welfare-driven swaps to jointly improve matching decisions and total transmit power. It is designed to address externalities while retaining local optimality and pairwise stability.

  • B. Proposed Low-Complexity Solution: The algorithm uses welfare-improving swap operations so waveguides and pinching antennas can cooperatively reduce total transmit power while meeting users’ rate requirements.Unlike selfish swaps, an individual agent may accept higher power consumption when system-wide power decreases.
  • B. Proposed Low-Complexity Solution: Compared with vanilla swap matching, the proposed method guarantees both pairwise stability and local optimality.These guarantees are stated for the resulting matching solution.
  • B. Proposed Low-Complexity Solution: The matching procedure defines swaps and welfare-blocking pairs to identify changes that improve social welfare.Welfare-blocking pairs require feasible swaps that improve the social-welfare utility of all agents.
  • B. Proposed Low-Complexity Solution: Candidate operations add, replace, or exchange waveguide–pinching-antenna matches.These operations provide the search moves used by the welfare-driven matching game.

2) Performance and Complexity Analysis:

The proposed matching algorithm converges to a welfare-based pairwise-stable, locally optimal solution, while simulations evaluate coupling, radiation spacing, and conventional MIMO baselines under stated system settings.

  • Matching convergence: The welfare-driven matching algorithm terminates in finite steps at a welfare-based pairwise-stable matching and locally optimal solution.Each accepted swap strictly increases system utility, preventing cycles among the finite set of feasible matchings.
  • Matching convergence: The algorithm’s final matching contains no welfare-improving blocking pairs, so no feasible swap can further improve total utility.Under the fixed matching, the associated transmit beamforming is optimal, yielding a locally optimal joint solution.
  • Simulation setup: The simulation setup uses f = 15 GHz, noise power σ2 = −80 dBm, and refraction index neff = 1.4.The resource-limited experiments use N = K = {2, 4}, γmin = 20 dB, and spatial ranges Sx = {5, 10, 15, 20, 25, 30}.
  • Coupling and radiation: The fitted coupling model accurately approximates simulation data for a rectangular waveguide, supporting the effectiveness of model (3).The reported fitted parameters are α = 0.24615 mm−1 and Ω0 = 0.3300 mm−1.
  • Coupling and radiation: For Ls = 6, equal-power radiation uses spacings {5.554, 5.157, 4.633, 4.006, 3.016, 0.200} mm, decreasing along the waveguide.The spacing arrangement is reported for the equal-power radiation case.
  • Baselines: The simulations compare PASS with conventional MIMO and massive MIMO baselines, with massive MIMO using hybrid beamforming and phase shifters.The conventional MIMO baseline uses one antenna per RF chain, while massive MIMO connects each RF chain to L antennas.

A. Single-User Scenario

Single-user and multi-user experiments show that BnB converges to optimal solutions, while the matching approach achieves near-optimal transmit-power performance with substantially lower computational cost.

  • A. Single-User Scenario: The single-user BnB gap between GUB and GLB converges to 0 after only 10 branching operations.BnB-Equal achieves similar performance to BnB-Optimal while enumerating only equal activation counts across waveguides.
  • A. Single-User Scenario: PASS reduces over 30% power consumption compared to massive MIMO in single-user scenarios.Conventional MIMO requires the highest power, while BnB-Equal and the matching algorithm achieve near-optimal performance relative to BnB-Optimal.
  • B. Multi-User Scenario: The multi-user BnB gap between GUB and GLB narrows to zero, confirming convergence to an ε-optimal solution within finite iterations.The global lower bound increases monotonically and the global upper bound decreases with each branching step.
  • B. Multi-User Scenario: The matching algorithm’s transmit power decreases monotonically and converges within approximately Imatch = 10 outer loops.It reaches a pairwise-stable matching and near-optimal performance with marginal loss relative to globally optimal BnB.
  • B. Multi-User Scenario: PASS with globally optimal BnB reduces transmit power by over 22 dBm versus MIMO and 7.5 dBm versus massive MIMO at varying γmin.The reported reductions correspond to over 99% and 80% power savings, respectively.
  • B. Multi-User Scenario: When K = 4 and L = 14, PASS reduces transmit power by over 10 dBm compared with massive MIMO.PASS requires lower transmit power across all tested L, and its performance gain increases with the number of users.
  • B. Multi-User Scenario: As spatial range increases, discrete and continuous PASS activations require only slight transmit-power increases, unlike conventional massive MIMO.Continuous activation performs best, while discrete activation offers reduced implementation complexity.

APPENDIX A PROOF OF PROPOSITION 1

The appendix establishes analytical coupling expressions for rectangular and circular dielectric guides, then proves finite convergence and complexity bounds for the proposed branch-and-bound procedure.

  • Coupling-coefficient analysis: Proposition 1 extends prior analytical coupling results to rectangular and circular dielectric waveguides, with both cases reducing to the same expression.The rectangular derivation uses evanescent cladding fields, while the circular derivation uses exterior guided fields and modified Bessel functions.
  • Spacing construction: The closed-form spacing construction recursively solves the radiation-related equation and produces δ_l = 1/√(L_s−ρ_l) for l = 1, 2, ..., L.The derivation gathers activated antennas across candidate locations into the final closed-form solution.
  • BnB convergence: The algorithm terminates with an ε-accurate objective once ϕ_max ≤ ε/√(2MP_0B), because the global upper–lower bound gap is then at most ε.At termination, the returned objective satisfies f* ≤ f_true* + ε.
  • BnB convergence: Algorithm 2 satisfies BnB convergence conditions because its bounds tighten with shrinking boxes, branching is exhaustive, and GUB − f_true* vanishes as edge lengths approach zero.The proof uses McCormick envelopes for bilinear constraints and establishes GLB ≤ f_true* ≤ GUB.
  • Complexity bound: At depth N_tr, at most 2^(N_tr+1) − 1 candidate nodes exist, bounding the worst-case number of branching iterations.This count follows by summing the 2^n candidate nodes across all tree levels through N_tr.
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