A Single-Tuning-Element Loaded Pixelated Tunable Band-Pass Filters with Low Loss via Inverse Synthesis on Massive-Scale Dataset

Technical Summary: Single-Tuning-Element Loaded Pixelated Tunable Band-Pass Filters via Inverse Synthesis

Problem Statement
The proliferation of wireless standards (3G/4G/5G, Wi-Fi 5/6/7) necessitates tunable band-pass filters (BPFs) to reduce the number of required filters in RF front-ends. Conventional tunable filter designs typically rely on transmission-line or coupled-resonator topologies loaded with multiple active elements (e.g., 4–10 varactors or p-i-n diodes). While designs with fewer elements (e.g., two) exist, they often offer only moderate tuning ranges. Furthermore, the inverse design paradigm—using pixelated traces on ground planes to synthesize complex electromagnetic responses—has largely been applied to static passive components or power amplifier bandwidth optimization, but has not been fully extended to reconfigurable devices. The challenge lies in exploring the vast design space formed by active components and complex pixelated structures to achieve high-performance tunable filters with minimal tuning elements and low insertion loss.

Methodology
The authors propose a novel three-phase inverse-design workflow to synthesize tunable pixelated BPFs using a single varactor. The process avoids neural network surrogate models in favor of direct search within a massive pre-computed dataset.

1. Massive Dataset Acquisition and Augmentation:

   • A dataset of approximately 300,000 unloaded 5-port S-parameter samples is generated using custom electromagnetic solvers accelerated by method-of-moments pre-computation on eight NVIDIA L40S GPUs.
   • The base geometry consists of 50 × 50 pixel maps (0.2 mm pixel size) with fill densities between 0.4 and 0.8, simulated from DC to 10 GHz.
   • Augmentation: To expand the search space by a factor of 120, the authors dynamically reassign input, output, and tuning ports. The remaining two auxiliary ports are randomly terminated with either open or short circuits.
   • Mathematical Transformation: A procedure converts the unloaded 5-port scattering matrix ($[S]_{unloaded,5p}$) into a loaded 3-port matrix by removing rows/columns corresponding to open/short terminations in the impedance or admittance domain. A final loaded 2-port network is derived by terminating the varactor port with a reflection coefficient $\Gamma_L(f)$ based on the varactor impedance.

2. Initial Seed Selection (Brute-Force Search):

   • The augmented dataset is searched to identify "seeds" that exhibit a wide-tuning-range band-pass response.
   • A mono-objective Figure of Merit (FoM) evaluates candidates based on the spread of peak frequencies ($\Delta f$) and average peak transmission ($S_{21,peak}$) across seven sampled capacitance states.
   • Hard Constraints: A candidate is valid only if at least 40% of its loading states show a single passband above -3 dB, a bandwidth < 1 GHz, and spurious peaks below -8 dB.

3. Fine-Tuning (Direct Binary Search):

   • Selected seeds undergo extensive optimization using a Direct Binary Search (DBS) algorithm.
   • The FoM is refined to include penalty terms for rejection levels ($P_{rej}$), insertion loss ($P_{IL}$), and return loss ($P_{RL}$), weighted to prioritize frequency spread and low loss.
   • Hard constraints are enforced during optimization to ensure a single passband above -6 dB with bandwidth < 1 GHz for every capacitance state.

Key Contributions

• Workflow Innovation: Introduction of a three-step inverse-design flow that directly searches a massive augmented dataset rather than relying on trained neural networks.
• Single-Element Tuning: Demonstration that a single varactor, combined with complex pixelated traces, can achieve reconfigurable passband and stopband responses.
• Augmentation Strategy: A technique to generate a 120× larger search space from a single 5-port dataset by randomizing port assignments and terminations.
• First-of-its-Kind Application: The first reported demonstration of continuously reconfigurable inverse design in the microwave regime.

Results
The methodology was applied to two target bands:

1. 4G Mid-/High-Band (1.7–2.69 GHz):

   • Design 1 (Shorting Pins): Achieved a 33% tuning range (2.4–2.8 GHz) with insertion loss (IL) maintained below -10 dB return loss. The design utilized an additional shorting port to create a notch left of the passband.
   • Design 2 (Radial Stub): Replaced shorting pins with a radial stub, achieving a tuning range of 2.45–2.95 GHz with IL between -1.2 dB and -1.3 dB.
   • Both designs demonstrated wide stopbands with rejection levels of -20 dB extending up to 10 GHz.

2. 5G n79/UNII-Band (4.64–6.16 GHz):

   • A filter was synthesized targeting the 5G n79 and UNII bands using open/open terminations for the auxiliary ports.
   • Performance: Achieved a 28% tuning range (4.64 to 6.16 GHz).
   • Loss: Insertion loss ranged from 1.3 dB to 1.8 dB in simulation.
   • Reconfiguration: The single varactor successfully tuned both the passband position and the notches on either side of the passband.

Significance and Claims
The paper claims that these inversely designed filters achieve lower insertion loss than state-of-the-art counterparts that utilize a significantly larger number of tuning elements. By relying on only a single varactor, the work demonstrates that complex pixelated traces can effectively manage the trade-off between tuning range and loss. The authors state that this is the first report of continuously reconfigurable inverse design in the microwave regime, marking a significant step in applying inverse synthesis to active, reconfigurable passive devices. The work suggests that the vast design space of pixelated structures, when coupled with active components, can realize high-performance tunable BPFs that were previously unexplored using conventional topologies.