A Bi-Stage Framework for Automatic Development of Pixel-Based Planar Antenna Structures

Development of modern antennas is a cognitive process that intertwines experience-driven determination of topology and tuning of its parameters to fulfill the performance specifications. Alternatively, the task can be formulated as an optimization problem so as to reduce reliance of geometry selection on engineering insight. In this work, a bi-stage framework for automatic generation of antennas is considered. The method determines free-form topology through optimization of interconnections between components (so-called pixels) that constitute the radiator. Here, the process involves global optimization of connections between pixels followed by fine-tuning of the resulting topology using a surrogate-assisted local-search algorithm to fulfill the design re-quirements. The approach has been demonstrated based on two case studies concerning development of broadband and dual-band monopole antennas.

💡 Research Summary

The paper introduces a two‑stage optimization framework for the fully automated design of pixel‑based planar antennas. The authors address two major challenges that have limited the practical use of pixel‑based representations: (1) the combinatorial explosion of possible inter‑pixel connections, and (2) the high computational cost of fine‑tuning continuous geometric parameters. In the first stage, the antenna is modeled using the Internal Multi‑Port Method (IMPM). A single multi‑port electromagnetic (EM) simulation of the entire pixel lattice yields an impedance matrix Z that captures the electromagnetic behavior of the structure. Once Z is obtained, the binary connection vector y (which indicates whether each internal port is open or shorted) can be altered without any further EM simulations, allowing rapid evaluation of millions of topologies. A global search algorithm (exhaustive search in the paper, but any meta‑heuristic could be employed) minimizes a scalar objective U (e.g., the maximum reflection coefficient above a threshold) to identify the optimal connection pattern y*.

The second stage fixes y* and optimizes the continuous design variables x (e.g., pixel lengths, spacings, feed line dimensions). This is performed with a trust‑region (TR) based local search. At each iteration i, a first‑order Taylor model R_sx(i) = R_f(x(i),y*) + J_f(x(i),y*)(x – x(i)) is constructed, where R_f represents the full‑wave response of the concrete geometry obtained by replacing the open/short ports with metal or etched connections, and J_f is a Jacobian estimated via large‑step finite differences. The TR radius δ is adapted according to the gain ratio ρ, expanding when the model predicts improvement and shrinking when it does not. Convergence is declared when δ or the change in x falls below a small tolerance.

To further reduce non‑linearity when the design goal involves specific resonant frequencies (e.g., dual‑band antennas), the authors replace the raw S‑parameter response with a feature‑based surrogate F(x). F(x) extracts a small set of points (frequency‑level pairs) that capture the most relevant aspects of the antenna’s performance. Optimizing over these features is considerably smoother than optimizing the full response, enabling efficient frequency scaling (e.g., shifting the second band by a factor K = f₂/f₁).

The framework is validated on two case studies using a 3 × 3 pixel lattice (12 internal ports) printed on an FR‑4 substrate. In the broadband example, the goal was to achieve a reflection coefficient below –10 dB from 3.8 GHz to 10 GHz. The global stage produced a binary connection pattern y* that already yielded a reasonable response; the TR‑based local stage refined the geometry in only six iterations (31 full‑wave simulations total), delivering the required bandwidth with a total computational effort of roughly 0.56 CPU‑hours on a dual‑AMD EPYC 7282 system. In the dual‑band example, the target resonances were 3 GHz and 6 GHz. By enforcing a scaling factor K = 2, the global stage identified a suitable topology, and the local stage shifted the second resonance from an initial 9.1 GHz down to the desired 6 GHz while maintaining a reflection below –15 dB at both bands. The entire design required fewer than 36 full‑wave simulations.

Key contributions of the work are: (i) decoupling of the binary topology search from continuous geometry tuning via the IMPM model, (ii) a low‑cost surrogate for rapid evaluation of inter‑pixel connections, (iii) a trust‑region gradient‑based local optimizer that converges in very few EM evaluations, and (iv) the use of feature‑based surrogates to handle frequency‑specific objectives. The authors demonstrate that the proposed bi‑stage approach can dramatically reduce the computational burden of free‑form antenna synthesis while still delivering high‑performance designs. Future directions include scaling to larger pixel arrays, extending the method to multi‑port (MIMO) antennas, and integrating more sophisticated global optimizers to further explore the vast design space.