PSO-ZF for NOMA-PASS MIMO

• The paper demonstrates a novel PSO-ZF framework that minimizes transmit power by jointly optimizing PA placement and beamforming in NOMA-assisted MIMO communications.
• The methodology employs a two-layer algorithm where PSO handles spatial antenna configuration and a closed-form ZF solution ensures rate and SINR constraints are met.
• Simulation results reveal significant power reductions over conventional methods, highlighting rapid convergence and robust performance in high rate scenarios.

Particle-Swarm Optimization with Zero-Forcing Beamforming (PSO-ZF) is a joint optimization approach for transmit power minimization in non-orthogonal multiple access (NOMA) assisted pinching antenna systems (PASS), designed for downlink multi-user multiple-input multiple-output (MIMO) communications. The methodology leverages outer particle-swarm optimization (PSO) for spatial configuration of pinching antennas (PAs) and inner zero-forcing (ZF) closed-form solutions for beamforming and power allocation. This combination allows rapid exploration of the nonconvex parameter space and circumvents poor local minima commonly encountered in gradient-based algorithms (Gan et al., 3 Jun 2025).

1. Problem Formulation and System Architecture

The PSO-ZF framework addresses the joint design of beamforming, pinching antenna placement, and power splitting in NOMA-assisted PASS architectures. The principal optimization problem is:

$$\begin{aligned} & P_0: \min_{W, X, \alpha} \sum_{q=1}^Q \| w_q \|_2^2 \ & \text{subject to:} \ & \quad R_{q,k} \geq R_{q,k}^{\min}, \; \forall q, k \quad \text{(rate/SINR constraints)} \ & \quad \sum_{k=1}^K \alpha_{q,k} = 1, \; 0 < \alpha_{q,k} < 1, \; \forall q,k \quad \text{(power-allocation simplex)} \ & \quad 0 \leq x_{n,1} < x_{n,2} < \cdots < x_{n,L} \leq x^{\max}, \; x_{n,l} - x_{n,l-1} \geq \Delta \; \forall n,l \end{aligned}$$

PA positioning and beamforming are jointly optimized, where $X$ is the matrix of PA positions, $W$ the beamforming matrix, and $\alpha$ the power allocation vector. The architecture comprises $N$ waveguides, each carrying $L$ PAs, for a total of $M = N \times L$ antenna elements. Each cluster serves $K$ users, with $Q$ clusters total.

2. Nested Algorithmic Structure and ZF Incorporation

PSO-ZF implements a two-layer optimization. The outer layer uses PSO to search the $L \times N$-dimensional space of PA positions $X = (x_{n, l})$, ensuring inter-PA spacing and box constraints. For each candidate $X$, a ZF-based closed-form solution solves the inner subproblems for $W$ and $\alpha$.

Given $X$, the equivalent pinching channel matrix $U(X) \in \mathbb{C}^{K \times M}$ is constructed as $u_{q,k}^H = h_{q,k}^H G(X)$, with $G(X)$ capturing the in-waveguide response. The ZF beamformer,

$W(X, P) = U(X) \big(U(X)^H U(X)\big)^{-1} \sqrt{P}$

nulls inter-cluster interference and conforms to required power allocation. The objective function is then evaluated as

$$f_0(X; \alpha) = \mathrm{Tr} \big[ \big(U^H U \big)^{-1} P \big]$$

and user SINRs are explicitly checked (cf. eq. (37)-(38)). Infeasible solutions are heavily penalized, steering the PSO away from regions violating minimum SINR constraints.

3. Particle Representation and Update Mechanisms

Each waveguide $n$ contains a swarm of $L$ particles, each representing a PA slot. At iteration $t$, the state of a particle is a triplet $$p_{n,l}^{(t)} = \{ x_{n,l}^{(t)}, v_{n,l}^{(t)}, \tilde{x}_{n,l}^{(t)} \}$$ encompassing position, velocity, and personal best. Updates employ standard PSO formulas:

$$\begin{aligned} v_{n,l}^{(t+1)} &= a_0 v_{n,l}^{(t)} + a_1 r_1 (\tilde{x}_{n,l}^{(t)} - x_{n,l}^{(t)}) + a_2 r_2 (o_{n,l}^{(t)} - x_{n,l}^{(t)}) \ x_{n,l}^{(t+1)} &= x_{n,l}^{(t)} + v_{n,l}^{(t+1)} \end{aligned}$$

where $a_0$ is inertia, $a_1$ and $a_2$ are cognitive and social coefficients, $r_1$ and $r_2$ are random scalars, and $o_{n,l}^{(t)}$ is the global best among the swarm. Post-update, strict box ($0 \leq x \leq x^{max}$) and minimum-spacing constraints ($x_{n,l} - x_{n,l-1} \geq \Delta$) are enforced.

4. Fitness Evaluation and Penalty Assignment

For each candidate position assembly $X$, the following evaluation loop is performed:

1. Build $U(X)$.
2. Compute ZF beamforming matrix $W$ via the closed-form solution.
3. Check SINR constraints for all users. If feasible, set fitness $$f_0(X) = \mathrm{Tr}[W W^H] = \sum_{q=1}^Q \|w_q\|^2$$; otherwise, assign a large penalty constant ($C_{penalty} \gg 1$).
4. Update personal and global bests based on fitness values.

This process ensures that the optimization landscape discourages nonviable regions, improving the robustness of the search against infeasible configurations.

5. Pseudocode Summary and Convergence Characteristics

The procedural execution of PSO-ZF comprises initialization, iterative PSO updates with ZF-evaluated fitness, and constraint enforcement. At termination, the best particle assembly yields the optimal PA layout $X^*$, corresponding ZF beams $W^*$, and $\alpha^*$ via SINR back-substitution or dynamic programming.

Convergence is empirically rapid; reductions in $f_0$ per iteration fall below $0.005$ dB by $t \approx 21$, and further improvement is negligible beyond $T=30$ iterations. Larger swarms ($P=30$) hasten convergence and reduce variance (Gan et al., 3 Jun 2025).

6. Key Parameters and Simulation Configuration

Experimental setups utilize typical system parameters:

Parameter	Value	Description
Number of waveguides ($N$)	4	Dielectric waveguides carrying PAs
PAs per waveguide ($L$)	4	Total PAs ($M = 16$)
Cluster users ($K$), clusters ($Q$)	2, 4	Multi-user, multi-cluster scenarios
Service area ($x^{max}$)	30 m	Maximum waveguide length
Inter-PA spacing ($\Delta$)	3 m	Minimum distance between PAs
Carrier frequency ($f_c$)	15 GHz
Noise, effective index	–80 dBm, $n_{eff}=1.4$
PSO Swarm size ($P$)	30	Number of particles per swarm
PSO parameters	$a_0=0.7$, $a_1=a_2=1.5$	Inertia, cognitive/social coefficients
Iterations ($T$)	30	Maximum number of PSO iterations
Penalty for infeasibility	Large constant

These parameters reflect operational constraints and deployment scenarios studied in simulations.

7. Performance Analysis and Comparative Results

PSO-ZF demonstrates marked improvements over conventional approaches:

• Convergence is achieved in approximately 20 iterations, significantly faster than the MM-PDD gradient-based method (≈50 iterations).
• Transmit power reductions at $S=30$ m, $L=4$ reach 39.1% over MM-PDD and up to 87.5% over massive MIMO-NOMA.
• Increasing PAs per waveguide $(L)$ consistently reduces transmit power, with PSO-ZF achieving –32.6% vs. MM-PDD and –96.4% vs. MIMO-NOMA at $20$ bps/Hz.
• Higher rate requirements enhance gains: at $40$ bps/Hz, $S=30$ m, PSO-ZF yields 95.2% power reduction versus MIMO-NOMA.
• With SINR targets at $20$ dB, reductions of 35.2% and 95.3% are observed versus MM-PDD and MIMO-NOMA, respectively.

This pattern suggests that PSO-ZF is highly effective in large-scale, high-rate, and strict-SINR scenarios. The use of closed-form ZF for fast fitness computation, nested within PSO-driven spatial search, enables robust escape from local optima and significant improvements in system power efficiency (Gan et al., 3 Jun 2025).