Waveguide Division Multiple Access for Pinching-Antenna Systems (PASS)

I. INTRODUCTION

• Fixed-position MIMO antennas cannot fully exploit spatial degrees of freedom, while existing flexible antennas face high-frequency path-loss and blockage challenges and limited structural reconfiguration.PASS uses a dielectric waveguide with activatable pinching antennas to establish communication zones; flexible PA locations can bypass obstacles or enable last-meter transmission, and PAs can be added or removed for scalable deployment.Source: §I · pp. 1–2
• Prior PASS research analyzed single- and multiple-waveguide cases and studied PA activation, NOMA, hybrid beamforming, and machine-learning-based joint beamforming.The cited studies addressed rate maximization, weighted sum rate, or performance relative to a heuristic baseline.Source: §I · p. 2

A. Motivations and Contributions

• WDMA assigns each user a dedicated waveguide and uses pinching beamforming to enhance intended signals and mitigate inter-user interference, simplifying baseband processing to power allocation.The approach treats waveguides as radio resources and aims for near orthogonality in free space, avoiding superimposed user signals on each waveguide.Source: A. Motivations and Contributions · p. 2
• The joint sum-rate problem includes user minimum-rate and PA-position constraints; an alternating-optimization framework uses SCA for power allocation, gradient ascent for continuous activation, and matching for discrete activation.The discrete-case matching algorithm targets near-optimal pinching beamforming at low computational complexity.Source: A. Motivations and Contributions · pp. 2–3
• Numerical results show PASS outperforming conventional antenna systems in both activation cases, WDMA supporting near-orthogonal multi-user communication, and practical candidate-position counts alleviating the discrete–continuous performance gap.These are the paper’s reported findings for ideal continuous and practical discrete activation.Source: A. Motivations and Contributions · p. 3

B. Organization and Notations

• The paper proceeds from the WDMA system model and sum-rate formulation to continuous and discrete optimization, simulation results, and conclusions; its notation defines matrix operators and assumes zero-mean random variables.Scalars, vectors, and matrices use italic, bold lower-case, and bold upper-case letters, respectively.Source: B. Organization and Notations · p. 3

II. SYSTEM MODEL AND PROBLEM FORMULATION

• The system model has a base station with K dielectric waveguides serving K users, with each waveguide conveying one data stream to its intended user through N pinching antennas.The baseband allocates power to each user’s data stream.Source: A. System Description · p. 3, §II · p. 3

A. System Description

• The system considers ideal continuous PA activation subject to minimum spacing and practical discrete activation at preconfigured, uniformly spaced candidate locations.The practical candidate set has at least N locations, where N is the number of PAs on each waveguide.Source: A. System Description · p. 3

B. Channel and Signal Model

• The model defines the channel from PAs on waveguide k to user k′ and the propagation channel g_k on that waveguide.Source: B. Channel and Signal Model · pp. 3–4
• Waveguide loss is omitted as negligible relative to free-space path loss, and n_eff denotes the guided wavelength’s effective refractive index [16], as described in [13].Source: B. Channel and Signal Model · p. 4
• The received signal model uses a normalized transmitted signal, allocated user power p_k, and additive white Gaussian noise at user k.The transmitted signal has unit average power; the noise is modeled as complex Gaussian with noise power σ_k^2.Source: B. Channel and Signal Model · p. 4
• Power on each waveguide is assumed equally distributed among its N PAs, and the model then expresses user k’s achievable rate.Source: B. Channel and Signal Model · p. 4

C. Problem Formulation

• The objective is to maximize all users’ sum rate while satisfying each user’s minimum rate, feasible activation locations, and the base station’s maximum transmit power.Source: C. Problem Formulation · p. 4
• The problem is nonconvex because its objective and rate constraints couple power and beamforming variables; discrete activation also makes it a mixed-integer search over many locations.Source: C. Problem Formulation · p. 4

III. PROPOSED JOINT POWER ALLOCATION AND PINCHING BEAMFORMING SOLUTION

• The proposed AO-based framework decouples optimization into power allocation and pinching beamforming subproblems, iteratively optimizing each with others fixed; it uses SCA for power and GAA or matching-theory, for continuous and discrete activation, respectively.Source: §III · p. 4

A. Power Allocation

• With beamforming fixed, the power-allocation subproblem remains nonconvex, so SCA approximates each user’s rate with a concave lower bound at the current power iterate.Source: A. Power Allocation · p. 5
• Replacing the rates with their lower bounds yields a convex problem solvable by standard tools such as CVX; Algorithm 1 updates power until the sum-rate increment falls below ε.Source: A. Power Allocation · p. 5
• Because each SCA iteration does not decrease the objective and sum rate is power-bounded, the algorithm converges to a stationary point of the original power-allocation problem.Source: A. Power Allocation · p. 5

B. Pinching Beamforming with Continuous Activation

• With power allocation fixed, the rate is highly complex and nonconvex in activation positions X, motivating a gradient-ascent algorithm (GAA) for pinching beamforming.The positions affect both the SNR numerators and denominators and appear in the exponent of complex-valued terms.Source: B. Pinching Beamforming with… · p. 5
• The method converts the constrained optimization into an unconstrained problem with penalty terms, then uses log-sum-exp smoothing to obtain a differentiable formulation.The penalties account for constraint violations; the original formulation includes constraints transformed into equalities.Source: B. Pinching Beamforming with… · pp. 5–6
• The GAA updates the activation positions along a gradient-ascent direction, while equivalence to the original problem depends on penalty factor β and smoothing parameter ρ.The section first solves the smoothed problem for given β and ρ, then investigates a suboptimal solution to the original problem.Source: 1) Solve Problem (P1-2-3) With GAA · p. 6, B. Pinching Beamforming with… · p. 6

1) Solve Problem (P1-2-3) With GAA:

• Backtracking line search repeatedly shrinks the step size from initial τ̄ by factor ωτ until the Armijo condition is satisfied.The parameter δ ∈ (0, 1) controls the objective-function increment range.Source: 1) Solve Problem (P1-2-3) With GAA · p. 6
• The algorithm terminates when the increment of objective function (28) is below threshold ϵ or gradient-ascent steps reach upper bound l_max.Source: 1) Solve Problem (P1-2-3) With GAA · p. 6

• Equivalence between (P1-2-3) and (P1-2) depends on β and ρ, with β→+∞ and ρ→0 considered for constraints (10b).Source: 2) Find Sub-Optimal Solution for… · p. 6
• Algorithm 2 is presented for solving problem (P1-2-1) and calculates the next iterate ˜X(l+1) using (29).Source: 2) Find Sub-Optimal Solution for… · p. 7
• Equation (31) includes the expression 2(1+tanh(˜X(l))).Source: 2) Find Sub-Optimal Solution for… · p. 7

9 Output X = L/2

• Algorithm 3’s outer iteration proceeds until constraints (10b) and (17d) are satisfied.Source: §9 · p. 7, §8 · p. 7

8 Output X = L/2

• To avoid letting the penalty dominate sum-rate maximization, β starts small and increases gradually, while ρ decreases to improve approximation accuracy.Source: §8 · p. 7
• In the outer loop, β ← ωββ and ρ ← ωρρ, with ωβ > 1 and 0 < ωρ < 1, until constraints (10b) and (17d) are satisfied.Source: §8 · p. 7

3) Property Analysis of Algorithm 3:

• Algorithm 2’s objective q is guaranteed not to decrease after each gradient-ascent step when δ > 0 and τ(l) > 0.Source: 3) Property Analysis of Algorithm 3 · p. 7
• Backtracking line search for step-size selection has computational complexity O(IlsKN).Source: 3) Property Analysis of Algorithm 3 · p. 7
• Algorithm 3’s overall complexity estimate is parameterized by the maximum outer and inner iteration counts, Iout and Iin.The analysis identifies gradient calculation and backtracking line search as the main complexity components of Algorithm 2.Source: 3) Property Analysis of Algorithm 3 · p. 7

C. Pinching Beamforming with Discrete Activation

• Matching theory, is proposed to solve the intractable discrete-position integer program with a mathematically tractable, low-complexity method.The discrete PA positions are constrained to a limited set of available locations.Source: C. Pinching Beamforming with Discrete… · p. 7

1) One-Sided One-to-One Matching With Restriction:

• The model treats PAs and available locations on each waveguide as players seeking to maximize users’ sum rate, with PAs choosing their matched locations.Because locations view PAs identically, the per-waveguide game is one-sided.Source: 1) One-Sided One-to-One Matching With… · pp. 7–8
• Because a waveguide’s matching affects every user’s rate, the method jointly matches across all K waveguides but restricts PAs in Pk to locations in Lk.Fig. 5 illustrates the restricted PA-to-location matching.Source: 1) One-Sided One-to-One Matching With… · p. 8, Figure 5 · p. 8
• The matching assigns each PA exactly one location on its own waveguide, and each location can be assigned to at most one PA.These are the stated feasibility conditions for the matching.Source: 1) One-Sided One-to-One Matching With… · p. 8
• Each PA’s preference list ranks positions by the utility they achieve, where utility is the users’ sum rate under the matching state.The preference list is ordered in descending utility.Source: 1) One-Sided One-to-One Matching With… · p. 8
• The game has external effects: a PA’s utility depends on other PAs’ assignments, so its preference list changes with the matching state.Thus, preference is not fixed by the identity of the PA’s matched location alone.Source: 1) One-Sided One-to-One Matching With… · p. 8
• Algorithm 4 initializes a feasible matching randomly and considers rematching a PA when the updated state yields higher utility and satisfies the assignment restriction.The described update compares the sum-rate utility of the updated and current matching states.Source: 1) One-Sided One-to-One Matching With… · p. 8
• A matching is stable when no alternative state exists in which every PA weakly prefers its assignment and at least one PA strictly prefers it.The paper attributes the stability proof to.Source: 1) One-Sided One-to-One Matching With… · p. 8

2) Proposed PAs positions Optimization Algorithm Based on One-to-One Matching:

• Algorithm 4 initializes random PA-location matches, then lets each PA search its waveguide for a preferred unoccupied location until no PA can find one.The search checks whether a location is preferred to the PA’s current match.Source: 2) Proposed PAs positions Optimization… · p. 8

3) Property Analysis of Algorithm 4:

• The matching algorithm’s estimated complexity is O(ImatKNA), linear in K, N, and A, where Imat is the maximum number of outer-cycle iterations.The complexity mainly depends on the iterations spent searching for PA-location pairs that can improve the sum rate.Source: 3) Property Analysis of Algorithm 4 · pp. 8–9
• The users’ sum rate increases with each matching-state update during Algorithm 4’s relocation process.The analysis describes the matching states as Ω(0) → Ω(1) → · · · → Ω∗.Source: 3) Property Analysis of Algorithm 4 · p. 9
• Because the sum rate is upper bounded under limited power, Algorithm 4 is guaranteed to converge within a limited number of iterations.Source: 3) Property Analysis of Algorithm 4 · p. 9
• The final matching Ω∗ is a local optimum for (P1-3), since no PA can relocate to another location to further improve utility.Source: 3) Property Analysis of Algorithm 4 · p. 9
• The authors conclude that the output matching is stable: at termination, no PA can find an available location that increases utility over its current match.The stated termination condition is that all PAs fail to find an available location with utility increment.Source: 3) Property Analysis of Algorithm 4 · p. 9

D. Overall AO Algorithm Design

• Alternating optimization updates PA positions and power allocation in turn, stopping when the sum-rate increment falls below a threshold; its non-decreasing, upper-bounded sum rate guarantees convergence.The position and power subproblems are solved while holding the other variable fixed.Source: D. Overall AO Algorithm Design · p. 9

IV. SIMULATION RESULTS

• Numerical results are used to validate the feasibility and effectiveness of PASS-enabled multi-user communications with WDMA.Source: §IV · p. 9

A. Simulation Setup

• Simulations use two waveguides serving two users in separate rectangular coverage areas, averaging Figs. 7–11 over 500 channel realizations and using one realization for Fig. 6.The waveguides have height d = 3 meters and are separated by W along the y-axis.Source: A. Simulation Setup · p. 9, Figure 6 · p. 10

B. Baseline Schemes

• The conventional-antenna baseline uses two fixed-position BS antennas above the area centroid, with fully digital beamforming and antenna count matched to the number of waveguides.The matched antenna count supports comparison under the same installation cost.Source: B. Baseline Schemes · pp. 9–10
• The “MRT, continuous” baseline optimizes antenna locations first for desired-signal path loss, then at wavelength scale for constructive signal combining, following [12].Source: B. Baseline Schemes · p. 10
• The “GAA, discrete” baseline discretizes continuous GAA locations to the nearest feasible positions and moves a colliding PA to its next unoccupied nearest location.Source: B. Baseline Schemes · p. 10
• The “ES, discrete” baseline searches all discrete activation locations to find the highest-sum-rate solution, providing an upper bound for discrete activation.Source: B. Baseline Schemes · p. 10

C. Convergence of The Proposed Algorithms

• The proposed algorithms converge: continuous GAA reaches a steady point after about 30 iterations, discrete matching takes about 3–6 outermost iterations, and AO converges within a limited number of iterations.With A = 20, matching achieves nearly 91% of the “ES, discrete” upper bound; matching slows as N increases because the process involves more players.Source: C. Convergence of The Proposed… · p. 10

D. Sum Rate Versus BS Maximum Transmit Power

• With N = 4, PASS significantly outperforms conventional antenna systems for continuous and discrete activation; GAA beats “MRT, continuous,” and matching beats “GAA, discrete”.Flexible PA placement strengthens intended-user LoS links, while multiple PAs provide additional degrees of freedom; GAA balances intended signal strength against inter-user interference.Source: D. Sum Rate Versus BS Maximum Transmit… · pp. 10–11, Figure 7 · p. 11

E. Effectiveness of the Pinching Beamforming

• With GAA-selected PA positions and N = 4, each waveguide provides high channel gain to its intended user and much lower gain to the interfering user, enabling nearly orthogonal transmission.The illustrated users are at (3, −5, 0) meters and (7, 5, 0) meters.Source: Figure 8 · p. 11, E. Effectiveness of the Pinching… · p. 11

F. Sum Rate Versus Number of PAs on Each Waveguide

• As N increases, continuous GAA and “MRT, continuous” sum rates rise, while discrete matching and “GAA, discrete” rise then fall as the limited candidate positions constrain PA placement.At N = 10, matching and “GAA, discrete” perform the same because both activate PAs at all available positions; continuous placement retains greater flexibility.Source: Figure 9 · p. 11, F. Sum Rate Versus Number of PAs on… · p. 11

G. Sum Rate Versus Region Size

• As region size L grows, GAA sum rate increases while conventional-system sum rate decreases; matching declines because fixed candidate positions provide lower discretization accuracy.“MRT, continuous” increases with L as multi-user interference decreases; PASS can place PAs near intended users, whereas conventional antennas experience greater path loss over larger regions.Source: Figure 10 · p. 12, G. Sum Rate Versus Region Size · p. 12

H. Impact of Discrete PA Activation

• As candidate-position count A increases, discrete sum rate rises and the gap to continuous activation narrows: matching reaches about 90% of GAA at A = 200 and 97% at A = 5000.Higher discretization accuracy increases computational complexity, so practical implementation involves a performance-complexity tradeoff.Source: Figure 11 · p. 12, H. Impact of Discrete PA Activation · p. 12

V. CONCLUSIONS

• The paper formulates joint power allocation and pinching beamforming for WDMA-based PASS and uses AO with SCA for power allocation, penalty-based GAA for continuous activation, and matching for discrete activation.The conclusion reports improved sum rate over conventional antenna systems and a smaller discrete-continuous performance gap with more candidate positions.Source: §V · p. 12

• For i = k, the appendix gives the derivative expression in (39).Source: Appendix A · pp. 12–13
• For i ≠ k, the appendix gives the derivative expression in (40), using a derivation similar to the i = k case.Source: Appendix A · p. 13