--- title: PSO-ZF for NOMA-PASS MIMO url: https://www.emergentmind.com/topics/particle-swarm-optimization-with-zero-forcing-beamforming-pso-zf type: topic --- # PSO-ZF for NOMA-PASS MIMO 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 [2506.03063]. ## 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 [2506.03063]. ## 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 [2506.03063]. Source: https://www.emergentmind.com/topics/particle-swarm-optimization-with-zero-forcing-beamforming-pso-zf