Robust Beamforming for Pinching-Antenna Systems

Pinching-antenna system (PASS) mitigates large-scale path loss by enabling flexible placement of pinching antennas (PAs) along the dielectric waveguide. However, most existing studies assume perfect channel state information (CSI), overlooking the impact of channel uncertainty. This paper addresses this gap by proposing a robust beamforming framework for both lossy and lossless waveguides. For baseband beamforming, the lossy case yields an second-order cone programming-based solution, while the lossless case admits a closed-form solution via maximum ratio transmission. The PAs’ positions in both cases are optimized through the Gauss-Seidel-based method. Numerical results validate the effectiveness of the proposed algorithm and demonstrate that PASS exhibits superior robustness against channel uncertainty compared with conventional fixed-antenna systems. Notably, its worst-case achievable rate can even exceed the fixed-antenna baseline under perfect CSI.

💡 Research Summary

This paper tackles the practical problem of channel uncertainty in pinching‑antenna systems (PASS), which were originally proposed to mitigate large‑scale path loss by placing pinching antennas (PAs) at arbitrary locations along a dielectric waveguide. While prior works on PASS assumed perfect channel state information (CSI), the authors recognize that real‑world deployments inevitably suffer from estimation errors due to noise, limited pilot resources, and model mismatches. To address this gap, they develop a unified robust beamforming framework that covers both worst‑case (norm‑bounded) and probabilistic (Gaussian) error models.

The system model considers a downlink scenario where a base station activates N PAs on each of M waveguides. The signal radiated from each PA experiences deterministic attenuation inside the waveguide (parameterized by κ, the dB/m loss) and a stochastic wireless channel outside the guide, denoted by h(P). The overall received signal is y = h⁺(P) G(P) w s + z, where w is the baseband beamforming vector, G(P) captures waveguide propagation, and s is the data symbol.

Two uncertainty formulations are examined: (i) a norm‑bounded set ‖e(P)‖₂ ≤ δ, leading to a semi‑infinite worst‑case optimization (Problem P1); and (ii) a Gaussian error e(P) ∼ 𝒞𝒩(0, ε²I), leading to a chance‑constrained formulation (Problem P2) that requires the received SNR to exceed a threshold Γ with probability at least ρ. By applying the triangle inequality and Cauchy‑Schwarz bound, the authors lower‑bound the worst‑case SNR as |ĥ⁺GPw| − δ‖GPw‖₂.

A key theoretical contribution is Lemma 1, which shows that the two problems become equivalent if the norm bound δ and the Gaussian standard deviation ε satisfy δ = ε √{−ln(1−ρ)}. This equivalence allows the same solution machinery to be used for both formulations, turning the probabilistic design into a conservative robust design.

The joint optimization over the baseband vector w and the PA positions P is tackled via an alternating optimization (AO) scheme. For a given PA placement, the baseband problem reduces to minimizing δ‖Gw‖₂ − |ĥ⁺Gw| subject to a power constraint. In the lossy waveguide case (κ > 0) this is cast as a second‑order cone program (SOCP) by introducing an auxiliary variable and a phase‑alignment constraint, enabling efficient solution with standard convex solvers (e.g., CVX). In the lossless case (κ = 0) the waveguide matrix G becomes identity, and the problem simplifies to a classic maximum‑ratio transmission (MRT) solution: w_MRT = G⁺ĥ / ‖G⁺ĥ‖ · √Pₜ.

Optimizing the PA positions is more challenging because the objective is highly non‑convex and multimodal. The authors propose a Gauss‑Seidel‑based one‑dimensional (GS1D) search that updates each PA coordinate sequentially while keeping the others fixed. For lossy guides, the objective includes both the complex exponential term ηₘ,ₙ e^{−jk_g|pₘ,ₙ−oₘ|} (capturing phase and attenuation) and a penalty term proportional to δ. In the lossless case the penalty term becomes constant, so only the first term needs maximization, and ηₘ,ₙ = 1/N. Continuous placement is approximated by discretizing each guide into Nₛ sampling points; as Nₛ → ∞ the solution approaches the true continuous optimum. For discrete placement, the positions are restricted to a predefined set Sₘ.

Constraints include a total transmit power budget, a minimum spacing Δ_min between any two PAs on the same guide (to avoid mutual coupling), and feasibility constraints on the positions (either within the guide length or belonging to the discrete set).

Simulation results evaluate both lossy (κ = 2 dB/m) and lossless waveguides. In the lossy scenario, the SOCP‑based robust design achieves worst‑case SNR gains of roughly 15–20 % over a conventional fixed‑antenna system with the same power budget. The performance degrades gracefully as the error bound δ grows, and even for relatively large δ the PASS still outperforms the baseline. In the lossless case, the MRT solution combined with GS1D positioning yields near‑optimal performance with negligible computational overhead, making it attractive for real‑time implementation. The chance‑constrained formulation, when transformed via Lemma 1, matches the worst‑case design in both achieved SNR and outage probability, confirming the theoretical equivalence.

Overall, the paper delivers a comprehensive robust beamforming methodology for PASS, bridging the gap between idealized CSI assumptions and realistic uncertain channels. It demonstrates that, by jointly optimizing baseband weights and PA locations, PASS can maintain superior robustness and even surpass the performance of fixed‑antenna systems under perfect CSI. The work opens several avenues for future research, including multi‑user extensions, dynamic PA reconfiguration, and integration of learning‑based search strategies to further reduce computational complexity while preserving robustness.