⚡ electrical engineering

Max-Min Rate Fairness Optimization for Multi-User Pinching-Antenna NOMA Systems

This paper proposes a two-stage structured optimization framework that jointly designs pinching-antenna positions and transmit precoding to maximize the minimum user rate in multi-user NOMA systems, effectively addressing the non-convex challenges of inter-antenna interference and significantly outperforming heuristic benchmarks and traditional MIMO-NOMA systems.

Original authors: Mahmoud AlaaEldin, Amy Inwood, Xidong Mu, Michail Matthaiou

Published 2026-06-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mahmoud AlaaEldin, Amy Inwood, Xidong Mu, Michail Matthaiou

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Fixing the "Blocked Signal" Problem

Imagine you are trying to talk to a group of friends in a large, crowded room. Some friends are standing right next to you, while others are far away. Worse yet, there are giant pillars (obstacles) blocking the direct path to some of them. In the world of wireless internet, these "pillars" are walls or furniture that block signals, especially the high-speed signals needed for future 6G networks.

This paper introduces a new way to fix this problem using a technology called Pinching-Antenna Systems (PAS).

The Hardware: The "Flexible Light String"

Think of a traditional cell tower antenna as a rigid, heavy spotlight fixed in one spot. If you can't see the light, you can't get the signal.

The Pinching-Antenna System (PAS) is different. Imagine a long, flexible string (a waveguide) running along the ceiling of a room. Attached to this string are many small, movable "beads" (the Pinching Antennas).

The Magic: You can slide these beads anywhere along the string.
The Goal: You can slide a bead right next to a friend who is hiding behind a pillar, creating a clear, direct line of sight (LoS) to them.

The paper looks at a system with multiple strings (waveguides) and many beads (antennas) on each string, all working together to serve multiple users at once.

The Challenge: The "Choir" Problem

The system uses a technique called NOMA (Non-Orthogonal Multiple Access). Imagine the base station is a choir director trying to sing different songs to different people at the same time.

The Problem: To do this, the choir (the antennas) must sing in perfect harmony. If the singers are slightly out of sync (phase mismatch), their voices cancel each other out, and the listener hears silence or noise.
The Difficulty: Because the beads can move, the "distance" and "timing" of the signal change constantly. If you move one bead just a tiny bit, it might ruin the harmony for everyone else. This makes finding the perfect spot for every bead incredibly hard mathematically—it's like trying to tune a piano while the keys are moving.

The Solution: A Two-Step "Rough Draft" and "Fine-Tuning" Strategy

The authors realized that trying to solve the perfect harmony problem all at once is too messy. So, they invented a two-stage algorithm to find the best solution.

Stage 1: The "Blind" Sketch (Coarse Optimization)

Imagine you are trying to arrange furniture in a room to make the most space, but you are wearing blindfolds. You can't see the details, but you can feel the general shape of the room.

What they do: The algorithm ignores the complex "timing" (phases) of the signals for a moment. It only looks at the "distance" (how far the signal has to travel).
The Result: It quickly finds a "good enough" arrangement of the beads and how much power to send. It's not perfect, but it gets the beads in the right general neighborhood. This is like drawing a rough sketch of the furniture layout.

Stage 2: The "Fine-Tuning" (Phase Alignment)

Now, you take off the blindfolds and look at the details.

Step A (Phase Zeroing): The algorithm nudges each bead slightly left or right until the signal waves from that specific bead line up perfectly (like tuning a guitar string) so they don't cancel each other out.
Step B (The Dance): The algorithm then plays a game of "hot and cold."
1. It moves the beads to find a better spot.
2. It adjusts the "volume" and "timing" of the signals (precoding) to match the new bead positions.
3. It repeats this back and forth until the system stops improving.

Why This Matters (The Results)

The paper tested this method against other ways of solving the problem (like random guessing or "heuristic" methods).

Better Performance: The new method found a solution that gave the "worst-off" user the best possible speed (Max-Min Fairness). It performed almost as well as the theoretical "perfect" limit, which is impossible to reach in real life.
Faster: Unlike other methods that try thousands of random guesses (which takes a long time), this method is a smart, step-by-step guide. It found the solution much faster.
Beating the Competition: When compared to traditional MIMO systems (which use fixed antennas), the Pinching-Antenna system was much better. Because the beads can move right next to the user, they avoid the "walls" that block fixed antennas.

Summary Analogy

Think of the old way of doing this as trying to light up a dark room with a single, fixed flashlight. If someone stands behind a chair, they stay in the dark.

This paper proposes a system with a long, flexible rope of lights hanging from the ceiling.

First, you roughly guess where to hang the lights to cover the room.
Then, you slide each light bulb until the light hits the people perfectly, adjusting the brightness so everyone sees clearly without blinding each other.

The result is a room where everyone gets the best possible light, and you figured it out quickly without wasting time guessing randomly.

Problem Statement

The paper addresses the design of a downlink (DL) Non-Orthogonal Multiple Access (NOMA) system assisted by Pinching-Antenna Systems (PASs). In this architecture, multiple dielectric waveguides (WGs), each containing multiple repositionable dielectric radiating elements (Pinching Antennas or PAs), serve multiple single-antenna users. Each waveguide is driven by a single Radio-Frequency (RF) chain, necessitating the use of power-domain NOMA to support more users than available RF chains.

The core optimization challenge is to jointly optimize the continuous positions of all PAs across all waveguides and the transmit precoding matrix at the base station. The objective is to maximize the minimum user rate (max-min fairness) subject to:

Total transmit power constraints.
Successive Interference Cancellation (SIC) decoding constraints required for NOMA.
Minimum spacing constraints between adjacent PAs to mitigate mutual coupling.

The problem is identified as highly non-smooth and non-convex. This difficulty arises because the received signal at each user is a coherent sum of signals from all PAs. Small changes in PA positions cause rapid phase oscillations in the objective function, leading to a landscape riddled with local optima that are difficult to navigate using standard gradient-based methods or heuristic searches.

Methodology

To overcome the non-smooth nature of the problem, the authors propose a structured two-stage optimization algorithm:

Stage 1: Coarse Optimization (Phase-Relaxed)

Concept: The authors observe that while coherent combining creates rapid small-scale fluctuations, the channel gains' magnitudes exhibit a smoother large-scale structure governed by path loss and geometry.
Approach: The complex channel phases are temporarily neglected (assumed to be perfectly aligned). The problem is transformed into optimizing real-valued amplitude coefficients and PA positions.
Algorithm: An Interior-Point Algorithm (IPA) is used to solve this smooth, convexified problem.
- The min-rate operator is approximated using a log-sum-exp surrogate to ensure differentiability.
- The solution provides a "phase-relaxed" upper bound and a reliable initialization for PA locations and power allocations.
- A post-processing step refines the real-valued precoding vectors using a dual Semi-Definite Programming (SDP) approach and backward recursion to ensure the target SINR is met with minimum power.

Stage 2: Fine-Tuning (Phase-Aware)

Concept: This stage reintroduces the complex channel phases to find a feasible, near-optimal solution that respects the actual coherent combining effects.
Sub-stage 1: Phase-Zeroing: Starting from the coarse solution, each PA is sequentially repositioned within a local neighborhood to align its channel phase toward zero (modulo $2\pi$ ). This mitigates destructive interference before the main optimization begins.
Sub-stage 2: Alternating Optimization: The algorithm alternates between:
1. PA Position Refinement: A forward-backward coordinate search sweeps through all PAs. For each PA, the position is updated to maximize the worst-user rate within a feasible local interval, given fixed precoding.
2. Transmit Precoding Optimization: With fixed PA positions, the precoding matrix is optimized to minimize transmit power for a target common SINR. This is solved using Successive Convex Approximation (SCA) to handle non-convex SIC constraints, embedded within an outer bisection search to maximize the target SINR.

Key Contributions

First Max-Min Framework for Multi-WG PAS-NOMA: The paper formulates the first max-min rate fairness optimization problem for a system with multiple waveguides, multiple PAs per waveguide, and multiple users, where all PAs jointly contribute to all users' signals.
Structured Two-Stage Algorithm: The authors develop a deterministic optimization framework that exploits the physical structure of PAS channels. It separates the problem into a smooth, phase-relaxed coarse stage and a phase-aware fine-tuning stage, avoiding the "black-box" nature of heuristic methods.
Novel Optimization Techniques:
- Application of an Interior-Point Algorithm for the coarse joint optimization of geometry and power.
- A phase-zeroing initialization procedure to align channel phases.
- An alternating optimization scheme combining forward-backward coordinate search for PA placement and SCA-based convex optimization for complex precoding.
Complexity Analysis: The paper provides a detailed computational complexity analysis, demonstrating that the proposed dual-SDP/eigenvector approach for precoding refinement is more efficient than generic black-box SOCP implementations.

Simulation Results

The proposed framework was evaluated against three benchmarks:

Perfect Combining Upper Bound (UB): The theoretical limit assuming perfect phase alignment (Stage 1 result).
Heuristic Optimization (HO): A pattern-search algorithm (representing state-of-the-art metaheuristics like PSO).
Hybrid Beamforming MIMO-NOMA: A conventional system with a Uniform Linear Array (ULA).

Key Findings:

Performance: The proposed scheme achieves max-min rates nearly indistinguishable from the Perfect Combining Upper Bound, significantly outperforming the Heuristic Optimization benchmark.
Efficiency: The proposed method requires significantly less computational time than the HO benchmark, especially as the number of PAs increases. The HO method suffers from frequent convergence to local optima and irregular performance growth.
PAS vs. MIMO: PAS-NOMA systems provide substantial gains over comparable MIMO-NOMA systems. This is attributed to the ability of PAs to be repositioned close to users, drastically reducing path loss, which fixed MIMO arrays cannot achieve.
Parameter Impact:
- Number of PAs ( $N$ ): Increasing $N$ improves performance logarithmically for both PAS and MIMO, but PAS gains are more pronounced due to proximity.
- Number of Users ( $K$ ): As $K$ increases, the max-min rate decreases due to power sharing and increased inter-user interference, which limits the effectiveness of SIC.
- Transmit Power ( $P_T$ ): Rate increases monotonically with power. However, for higher $K$ , the system becomes interference-limited, causing the rate to scale less favorably with power.
- Spatial Range: As users are distributed over a larger area, the max-min rate decreases, though the drop is mitigated in PAS because PAs can move closer to individual users.

Significance and Claims

The paper claims that its primary significance lies in providing a deterministic, efficient, and physically interpretable solution to a highly non-convex optimization problem in emerging 6G technologies. Unlike population-based metaheuristics (e.g., PSO) that rely on random initialization and extensive function evaluations, the proposed algorithm leverages the specific structure of PAS channels (separating large-scale path loss from small-scale phase fluctuations).

The authors assert that this approach not only yields superior performance (near-optimal fairness rates) but also offers a scalable solution for practical deployment where computational resources and convergence reliability are critical. The results demonstrate that the flexibility of repositioning PAs offers a fundamental performance advantage over fixed-antenna MIMO systems in high-frequency, blockage-prone environments.