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Two-Timescale Antenna Positioning for Pinching-Antenna System Based on Statistical Channel Information

Jingze Ding, Zijian Zhou, Bingli Jiao, Rui Zhang

arXiv:2610.30670v1cs.ITmath.IT

TL;DR

The paper addresses the difficulty of optimizing PA positions across an entire waveguide using instantaneous CSI. It proposes statistical-CSI-based wide-range placement followed by instantaneous-CSI local refinement, and reports robust performance gains across diverse user distributions.

Abstract

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This paper introduces a two-timescale antenna position optimization framework for the pinching-antenna system (PASS). The proposed framework exploits statistical channel state information (CSI) to determine large-timescale pinching-antenna (PA) positions over the entire waveguide, and utilizes instantaneous CSI to refine PA positions at the small timescale within a local region. This approach effectively captures the benefits of large-timescale PA positioning for long-term system performance and small-timescale PA movement for real-time adaptation to fast channel fluctuations. For the single-user case, we derive closed-form solutions for the optimal large-timescale PA positions, and further analyze a performance upper bound together with the conditions for achieving this bound, which guide the small-timescale position refinement. For the multi-user case, an upper bound on the minimum rate among all users is derived. To maximize their minimum rate, we develop an element-wise search algorithm for large-timescale PA positioning and a projected gradient ascent method for small-timescale position refinement. Simulation results show that the proposed scheme closely approaches the derived bounds and consistently outperforms conventional schemes by reducing path loss via large-timescale PA positioning and suppressing inter-user interference through small-timescale local movement.

I. Introduction

The paper motivates two-timescale PA positioning as a way to address the limits of centralized antenna layouts and impractical instantaneous-CSI optimization across an entire waveguide. It presents PASS and its positioning options, then proposes a statistical-CSI-based large-timescale design with local refinement.

A. Pinching-Antenna System

  • PASS uses dielectric waveguides with movable pinching antennas, enabling decentralized deployment near users to reduce propagation distance and large-scale path loss.The architecture also supports adding or removing dielectric particles.Source: A. Pinching-Antenna System · p. 1

1) Discrete PA Positioning:

  • Discrete positioning activates a subset of pre-installed, uniformly spaced PAs, offering fast reconfiguration and low hardware complexity but potentially limiting fine-grained spatial adaptation.The limitation arises from quantized feasible PA positions.Source: 1) Discrete PA Positioning · pp. 1–2

2) Continuous PA Positioning:

  • Continuous positioning provides high spatial resolution through mechanically actuated slide tracks, but adds hardware complexity, control overhead, latency, and energy costs.The paper states that this approach is best suited to slowly varying environments.Source: 2) Continuous PA Positioning · p. 2

3) Semi-continuous PA Positioning:

  • Semi-continuous positioning combines segment-based activation with local micro-adjustments, balancing positioning precision against refinement latency and hardware cost.It is described as a compromise between continuous positioning and discrete activation.Source: 3) Semi-continuous PA Positioning · p. 2

B. Motivations and Contributions

  • Existing single-timescale designs optimize over a large waveguide-wide space using instantaneous CSI, which is difficult to acquire at arbitrary PA positions and can yield many local optima.The paper identifies these issues as motivations for two-timescale PA-position optimization.Source: B. Motivations and Contributions · p. 2
  • The paper proposes a statistical channel model and joint two-timescale max-min-rate optimization, with closed-form large-timescale positions and parallel search for the single-user case.For multiple users, it derives a zero-interference max-min-rate upper bound and proposes element-wise large-timescale search and projected-gradient small-timescale refinement.Source: B. Motivations and Contributions · pp. 2–3
  • Simulations show that joint large- and small-timescale positioning improves rates through reduced path loss and user-channel correlation, with robust performance across diverse user distributions.Large-timescale positioning reduces large-scale path loss, while small-timescale movement exploits small-scale fading to reduce correlation among users’ channel vectors.Source: B. Motivations and Contributions · p. 3

II. System Model and Problem Formulation

  • The modeled downlink has one PA on each of N waveguides serving K single-antenna users, with large-timescale locations and local movement regions along the waveguides.The paper focuses on one PA per waveguide and notes that its model and algorithms can be extended to multiple PAs per waveguide.Source: §II · p. 3

A. Statistical Channel Model

The channel model separates slowly varying, position-dependent LoS behavior from more rapidly varying NLoS paths. The positioning scheme uses statistical CSI for wide-range PA placement and instantaneous CSI for local refinement, subject to a coherence-time condition.

A. Statistical Channel Model

  • Within the small-scale movement region, the far-field assumption keeps path angles and amplitudes unchanged, while PA movement changes path phases.The stated reason is that this movement region is much smaller than the signal propagation distance.Source: A. Statistical Channel Model · pp. 3–4
  • The Rician channel model treats the LoS path parameters as deterministic at large-timescale PA positions, while the NLoS paths vary dynamically.The model combines in-waveguide and free-space propagation; the free-space component uses path responses and position-dependent phase differences.Source: A. Statistical Channel Model · pp. 3–4

B. Two-Timescale Antenna Positioning

  • Because LoS varies slowly when PAs and users are fixed or move only within small regions, it can be characterized statistically, unlike the more rapidly varying random NLoS component.The paper uses the LoS-path CSI as statistical CSI for large-timescale positioning.Source: B. Two-Timescale Antenna Positioning · p. 4
  • Large-timescale PA positions are optimized over the waveguide using statistical CSI, then instantaneous CSI supports local refinement around those fixed positions.The paper describes sparse local measurements and channel reconstruction, and assumes measurement, reconstruction, and movement fit within channel coherence time; otherwise the scheme reduces to large-timescale positioning only.Source: B. Two-Timescale Antenna Positioning · pp. 4–5
  • Channel-estimation overheads are neglected, and practical estimation methods with reduced latency and energy consumption are left for future work.The omission is made to simplify the analysis and focus on two-timescale positioning performance.Source: B. Two-Timescale Antenna Positioning · p. 4

C. Signal Model

The model formulates user rates from the PASS channel and jointly optimizes PA positions across two timescales and transmit beamforming to maximize ergodic achievable rates. In the single-user case, MRT is optimal, and large-timescale PA positions place the antennas close to the user before local refinement.

C. Signal Model

  • The signal model represents the PASS-to-user channels with H(x), beamforming with W, and normalized information-bearing signals with s; user SINR determines instantaneous rate.The supplied passages define noise as additive Gaussian noise with average power σ^2.Source: C. Signal Model · p. 5, D. Problem Formulation · p. 5

D. Problem Formulation

  • The optimization maximizes the users’ ergodic achievable rates by jointly selecting two-timescale PA positions and the transmit beamforming matrix, subject to power and position-range constraints.The problem is non-convex with coupled variables.Source: D. Problem Formulation · p. 5
  • The formulation assumes perfect statistical CSI for large-timescale positioning and perfect instantaneous CSI within the small-scale movement region; imperfect instantaneous CSI is evaluated in simulations.Source: D. Problem Formulation · p. 5

III. Single-User System

  • The single-user case has no inter-user interference, making maximum ratio transmission optimal; large-timescale positioning places PAs as close to the user as possible, directly above and parallel to the y-axis, before local refinement.Each PA is then refined within its small-timescale movement range.Source: §III · p. 5

A. Performance Analysis

The analysis separates the channel gain into per-PA contributions and examines when its upper bound can be achieved. The one-path and two-path cases yield distinct conclusions for small-timescale position refinement.

A. Performance Analysis

  • Because each PA’s movement affects only its own channel gain, the channel-gain upper bound is achieved by maximizing each PA’s contribution independently.The analysis rewrites the total channel gain as a sum of per-PA gains.Source: A. Performance Analysis · p. 5

1) One-Path Case:

  • In the one-path case, small-timescale position refinement does not improve channel gain, so the optimal refined positions equal the stated x⋆ positions.Source: 1) One-Path Case · p. 5

2) Two-Path Case:

  • For the two-path case, the analysis derives optimal small-timescale positions under stated conditions, including a bound on the refinement-range length D_n.The supplied passage states that D_n is no greater than a period, but its expression is truncated.Source: 2) Two-Path Case · p. 6

3) Multiple-Path Case:

For multiple paths, the analysis approximates the channel-gain period using quantized virtual AoDs because random AoDs make explicit representation difficult. It also describes a parallel procedure for selecting small-timescale positions and beamforming.

3) Multiple-Path Case:

  • Random AoDs make the multiple-path channel-gain period difficult to represent explicitly; the analysis derives an approximation by assuming quantized virtual AoDs.Source: 3) Multiple-Path Case · p. 6
  • The algorithm obtains large-timescale positions, optimizes small-timescale positions of all PAs in parallel, and then computes the transmit beamforming vector.Source: 3) Multiple-Path Case · p. 7
  • Under quantized virtual AoDs, the optimal small-timescale position lies in a specified interval, and the maximum channel gain can be achieved when the movement region spans at least one period τ_n.The period is defined by the minimum real number satisfying the stated integer condition.Source: 3) Multiple-Path Case · p. 7

B. Optimization Algorithm

The single-user optimization discretizes each PA’s local movement region and searches candidate positions independently, enabling parallel refinement rather than alternating optimization.

B. Optimization Algorithm

  • Each PA’s small-timescale movement region is discretized into Q_n candidate points, and its position index identifies a candidate.Source: B. Optimization Algorithm · p. 7
  • Because each PA’s movement independently affects only its own channel gain, all small-timescale positions can be optimized in parallel instead of by alternating optimization.Source: B. Optimization Algorithm · p. 7

C. Overall Algorithm

For the multi-user case, the paper derives a max-min rate upper bound and analyzes when it can be attained. It also outlines the algorithm’s position-search stages.

C. Overall Algorithm

  • The overall algorithm obtains large-timescale positions in O(N) work, then searches small-timescale positions in parallel.It subsequently obtains the transmit beamforming vector.Source: C. Overall Algorithm · p. 7

IV. Multi-User System

  • The multi-user analysis derives an upper bound on the minimum achievable rate and identifies conditions for attaining the receive-SINR bound.Attainment requires nulling inter-user interference while maximizing each user’s received signal power.Source: §IV · p. 7, A. Performance Analysis · p. 7

A. Performance Analysis

  • Maximizing the minimum instantaneous rate calls for equal achievable rates across users, with power reallocated from higher-rate users to lower-rate users.The upper bound is denoted R_ub.Source: A. Performance Analysis · pp. 7–8
  • For a single path from each PA to each user, the optimal small-timescale PA positions achieve the minimum-rate upper bound.The general analysis also identifies orthogonal user channel vectors and MRT as equality conditions in its bounds.Source: A. Performance Analysis · pp. 7–8
  • The upper bound may be unattainable when the channel-condition set is empty, and explicit PA positions are difficult to derive for multipath channels.The paper therefore develops algorithms for both positioning timescales.Source: A. Performance Analysis · p. 8

B. Optimization Algorithm

The multi-user design uses zero-forcing beamforming and separates PA optimization across timescales. Large-timescale positions are optimized using statistical CSI and an element-wise search.

B. Optimization Algorithm

  • Zero-forcing beamforming provides a closed-form design that reduces PA-optimization complexity, trading optimality for a rate performance that approaches the upper bound.The method is used in light of the ability of PA positioning to mitigate channel correlation between users.Source: B. Optimization Algorithm · p. 8

1) Large-Timescale PA Position Optimization:

  • Rather than relying on a computationally expensive Monte Carlo estimate of expected rate, the method directly approximates the expectation.The Monte Carlo estimate uses a sufficiently large number of independent channel realizations.Source: 1) Large-Timescale PA Position… · p. 8
  • Large-timescale positions are optimized using statistical CSI, with an element-wise search over discretized candidate positions for each PA.The Sherman–Morrison lemma avoids recalculating a full matrix inverse for each candidate position.Source: 1) Large-Timescale PA Position… · pp. 8–9

2) Small-Timescale PA Position Optimization:

Small-timescale PA positions are refined using instantaneous CSI, starting from the large-timescale solution. Projected gradient ascent updates feasible positions with a backtracking line search.

2) Small-Timescale PA Position Optimization:

  • Small-timescale positions are refined from the large-timescale solution using instantaneous CSI to adapt to NLoS channel variations.The optimization maximizes the rate subject to the feasible-position constraint.Source: 2) Small-Timescale PA Position… · p. 9
  • Projected gradient ascent updates positions using the rate gradient, projects infeasible positions onto the nearest boundary, and shrinks the step size until sufficient ascent is ensured.Backtracking uses the Armijo-Goldstein condition.Source: 2) Small-Timescale PA Position… · p. 9

C. Overall Algorithm

  • The overall algorithm alternates statistical-CSI large-timescale updates with per-realization small-timescale refinement and stops when the objective increase falls below a threshold.Backtracking line search selects the step size during small-timescale optimization.Source: C. Overall Algorithm · p. 9, 2) Small-Timescale PA Position… · p. 9

V. Simulation Results

The simulations compare the proposed two-timescale scheme with positioning-scale ablations, fixed-antenna baselines, and upper bounds. In the single-user case, combining large- and small-timescale movement yields rates close to the upper bound, with performance depending on movement range and channel conditions.

A. Simulation Setup and Benchmark Schemes

  • The setup averages ergodic max-min rates over 100 instantaneous rates and 100 user-location realizations, with benchmark schemes including upper-bound and single-timescale variants.The benchmark set also includes sparse and dense fixed-position antenna arrays.Source: A. Simulation Setup and Benchmark… · p. 10

B. Single-User System

  • As movement-region length D increases, schemes with small-timescale PA movement gain rate, while schemes without it remain unchanged.The proposed scheme jointly benefits from reduced large-scale path loss and exploitation of small-scale fading.Source: B. Single-User System · pp. 10–11
  • The proposed two-timescale scheme achieves an ergodic rate close to the upper bound by combining large-timescale positioning with small-timescale movement.Reducing search resolution Q from 100 to 10 causes only slight degradation at large D while reducing Algorithm 1's complexity by about one order of magnitude.Source: B. Single-User System · p. 11
  • With L = 2 paths, rate first rises and then falls with Rician factor κ, peaking around κ = 0 dB; increasing L gradually removes this non-monotonic trend.At large κ, the proposed scheme approaches large-timescale-only positioning, while small-timescale-only movement approaches sparse FPA performance.Source: B. Single-User System · p. 11

C. Multi-User System

Multi-user simulations examine user count, spatial distribution, area size, and CSI error. Results show that large-timescale placement helps reduce path loss, while small-timescale movement can mitigate interference; the conclusion reports performance close to bounds and above benchmarks.

C. Multi-User System

  • For random and clustered users, optimization objectives rise monotonically and stabilize within 25 iterations; clustered users have lower objectives alongside stronger channel correlation and interference.Position optimization increases average channel gain and decreases normalized channel cross-correlation.Source: C. Multi-User System · pp. 11–12
  • As user count K grows, all schemes' max-min rates decline, and the gap to the no-interference upper bound widens; in rich multipath, small-timescale-only eventually beats large-timescale-only.In LoS-dominated channels, large-timescale positioning alone can achieve satisfactory performance without small-timescale movement overhead.Source: C. Multi-User System · p. 12
  • As the square-area side length S increases, rates degrade as users become more dispersed and PA-user distances and path loss increase.The proposed and large-timescale-only schemes are compared with the other schemes across random and clustered user distributions.Source: C. Multi-User System · p. 12
  • Increasing the random-user ratio ζ lowers the upper bound but raises the other schemes' rates, as reduced clustering alleviates interference while greater dispersion increases path loss.Small-timescale movement is particularly effective for clustered users, whereas large-timescale placement is more dominant for dispersed users.Source: C. Multi-User System · p. 13
  • With imperfect CSI, the proposed scheme's rate decreases slightly as maximum virtual AoD error ε rises and approaches large-timescale-only performance, yet remains above benchmarks.The test applies estimated CSI to PA positioning but actual channel vectors to beamforming; degradation is more pronounced at lower κ.Source: C. Multi-User System · pp. 13–14

VI. Conclusion

  • The proposed scheme closely approaches derived bounds and consistently outperforms benchmarks across the evaluated settings.The conclusion attributes large-timescale positioning's main role to reducing path loss and small-timescale refinement's role to exploiting fading and reducing inter-user interference.Source: §VI · p. 14

Appendix A Proof of (41)

The appendix establishes the stated condition for attaining the multi-user upper bound in a single-path case and derives gradient expressions used for position optimization.

Appendix A Proof of (41)

  • When each PA-user link has one path, equality in condition (37) always holds; satisfying channel orthogonality is the remaining condition for achieving the upper bound.The appendix states that the necessary and sufficient condition for achieving the bound is that equalities (36) and (37) hold.Source: Appendix A · p. 14

Appendix B Derivation of Gradient ∇xR (x)

  • The gradient derivation defines Z(x) = H(x)H^H(x), then obtains partial derivatives of R(x), Z(x), and f_k(x) with respect to position x_n.The supplied derivation also gives the derivative of g_n(x_n)f_{k,n}(x_n).Source: Appendix B · p. 14
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