Constrained Pinching Antenna Array Design for Sum-Rate Maximization in Multi-User PASS

Technical Summary: Constrained Pinching Antenna Array Design for Sum-Rate Maximization in Multi-User PASS

Problem Statement
Pinching Antenna Systems (PASS) have emerged as a flexible architecture for indoor wireless communications, utilizing dielectric pinches along a waveguide to radiate signals. While existing research has explored PASS for multi-user scenarios, practical deployment faces significant challenges. Current designs generally fall into two categories: fixed-spacing arrays that lack precise user-specific beam adaptation, or unconstrained user-specific repositioning schemes that are either too costly (requiring dedicated hardware per user) or physically impractical due to mechanical delays and potential overlaps in dense environments. Furthermore, existing constrained array concepts (e.g., movable modules with fixed spacing) lack a comprehensive framework for optimizing performance in multi-user downlink scenarios.

This paper addresses the gap between theoretical optimality and practical implementation by investigating a Constrained Pinching Antenna Array (C-PAA). The core problem is to maximize the system sum-rate in a multi-user downlink scenario by jointly optimizing the position of the array's center along the waveguide and the fine-grained, wavelength-scale distribution of individual antennas within the array, subject to physical aperture constraints.

Methodology
The authors propose a system model where $M$ pinching antennas (PAs) are grouped into a movable array deployed on a ceiling-mounted dielectric waveguide. The array center, $Q_c$, can move along the waveguide, while individual PAs within the array can be adjusted at the wavelength scale to ensure phase alignment at the user's location.

1. Structural Characterization: The paper first characterizes the C-PAA structure under an adjacent-PA phase-alignment condition. It derives an explicit upper bound on the array aperture, demonstrating that the array remains physically compact even when PAs are adaptively reconfigured for different users.
2. Channel and Rate Approximation: To make the optimization tractable, the authors develop approximations for the effective channel gain and achievable user rate.
   • They derive an integral-based approximation for the effective channel gain by treating the discrete antenna summation as a continuous integral.
   • A further simplified closed-form approximation is derived under the assumption that the user-array distance is significantly larger than the array aperture.
3. Optimization Analysis: The sum-rate maximization problem is reformulated as a one-dimensional optimization over the array center position ($x_c$).
   • The authors analyze the structural properties of the sum-rate function, proving that while individual user rates are not globally concave, the expected sum-rate function is strictly concave under uniform user distributions.
   • This unimodal property allows for the use of efficient one-dimensional search algorithms (e.g., bisection) to find the optimal array center.
4. Low-Complexity Solution: To further reduce computational complexity, a closed-form approximate solution for the optimal array-center position is derived using expectation-based analysis, effectively modeling the problem as a logarithmic variant of the Weber–Fermat problem.

Key Contributions

• Architecture Proposal: Introduction of a C-PAA architecture for downlink PASS that balances physical aperture constraints with the ability to perform fine-grained, wavelength-scale antenna adjustments for phase alignment.
• Theoretical Bounds: Derivation of an explicit upper bound on the C-PAA aperture and tractable approximations for effective channel gain and achievable rates that facilitate analytical optimization.
• Optimization Framework: Demonstration that the multi-user sum-rate function exhibits a favorable unimodal structure (strictly concave in expectation), enabling efficient optimization.
• Algorithmic Solutions: Development of both an iterative one-dimensional search algorithm and a closed-form approximate solution for determining the optimal array center, offering a trade-off between optimality and computational complexity.
• Geometric Insight: Provision of a geometric interpretation of the optimality condition, showing that the optimal position corresponds to a state where the weighted sum of user-to-center vectors has zero projection onto the waveguide direction.

Results
Numerical results, validated via Monte Carlo simulations, confirm the following:

• Accuracy of Approximations: The derived approximations for effective channel gain and sum-rate closely match exact calculations across various carrier frequencies (15 GHz and 28 GHz) and transmit power levels. The error remains low even as the number of antennas increases.
• Optimization Efficacy: The proposed iterative algorithm (Algorithm 1) yields results identical to exhaustive search, confirming the unimodal nature of the sum-rate function.
• Closed-Form Performance: The closed-form approximate solution performs nearly identically to the iterative optimal solution, particularly as the number of users increases, validating the expectation-based derivation.
• Performance Gains: The proposed C-PAA scheme significantly outperforms conventional fixed-spacing arrays and priority-oriented (PO) benchmark schemes (where the array is steered only toward the nearest user).
  • The C-PAA closely approaches the theoretical ideal upper bound (where mutual coupling and physical constraints are ignored).
  • Unlike the PO scheme, which degrades in dense user scenarios due to the suboptimality of prioritizing a single user, the C-PAA maintains high performance by balancing the spatial distribution of gains across all users.
  • The system benefits from increased numbers of PAs, whereas the PO scheme suffers performance degradation as more antennas create narrower beams that favor only the prioritized user.

Significance
The paper claims to provide a practical, low-complexity design framework for future indoor multi-user PASS deployments. By addressing the limitations of both rigid fixed-spacing arrays and impractical unconstrained repositioning, the C-PAA architecture offers a viable path to realizing the spatial flexibility and high spectral efficiency promised by PASS. The work bridges the gap between theoretical performance limits and implementable hardware constraints, offering analytical tools (bounds, approximations, and closed-form solutions) that enable efficient system optimization without the need for computationally expensive exhaustive searches.