npick — Browser-based Multiplexer Synthesis & Impedance Matching (Nevanlinna-Pick)

npick is a free, open-source, browser-based synthesis tool for two long-standing problems in RF and microwave engineering:

1. Manifold-coupled multiplexer synthesis — duplexers, triplexers, and N-channel multiplexers with ideal or user-supplied T-junctions, following the Martínez et al. (EuMC 2019) continuation method.
2. Broadband impedance matching — Chebyshev-type matching networks for a complex-valued load specified by Touchstone S-parameter files, following Baratchart–Olivi–Seyfert's generalized Nevanlinna-Pick interpolation.

Both run entirely client-side in the browser via WebAssembly at jedrzejmichalczyk.github.io/npick. No installation, no license server, no data leaves your machine.

As far as we know, this is the only openly-available browser tool that synthesizes manifold-coupled multiplexers from a per-channel specification — commercial RF-design packages (CST, AWR, HFSS, Microwave Office, …) can do it but require paid licenses and desktop installs; published research code is mostly single-channel MATLAB scripts.

Live demo

https://jedrzejmichalczyk.github.io/npick/

• Impedance Matching tab: drop a .s1p / .s2p / CSV file, pick a passband, get the coupling matrix and matched response.
• Multiplexer Synthesis tab: choose a duplexer or triplexer preset (or define channels manually), optionally upload .s3p T-junction S-parameters, click Synthesize. Output: per-channel coupling matrices plus the full frequency response of the assembled network.

Key features

Multiplexer synthesis (Martínez 2019):

• Manifold-coupled N-channel synthesis via homotopy continuation λ = 0 → 1
• Correct Wirtinger-based complex Newton for the non-holomorphic residual F(P) = f(p) − conj(L(P))
• Wilson's iterative spectral factorization (root-free, numerically robust; see Wilson 1969, Sayed & Kailath 2001)
• Joint N·n equiripple optimization across all channels' interpolation points
• Custom T-junction S-parameters accepted via .s3p upload
• Duplexer example hits −20 dB worst-case RL in ~1 s, triplexer in ~2 min including equiripple tuning

Impedance matching (Baratchart-Olivi-Seyfert):

• Equiripple Newton optimizer with backtracking line search
• Homotopy continuation with adaptive Dormand-Prince 8(5,3) predictor-corrector
• All-pole and cross-coupled filters with prescribed transmission zeros
• Symmetric load detection and reduced parameterization
• Direct coupling-matrix output (transversal form) ready for physical realization

Portable:

• Single dependency: Eigen (header-only)
• Builds with GCC, Clang, MSVC, and Emscripten
• Same numerical behavior on native and WASM (carefully handles portability traps like std::pow(complex(0,0), 0) which differs between libstdc++ and libc++)

Building

Requires a C++17 compiler and CMake 3.16+.

# Clone with Eigen (or place Eigen headers in external/eigen/)
git clone <this-repo>
cd nevanlinna-pick-cpp

# Build
cmake -B build -DCMAKE_BUILD_TYPE=Release
cmake --build build -j8

# Run the benchmark suite
./build/benchmark_suite

If Eigen is not found in external/eigen/, CMake will look for a system installation via find_package(Eigen3).

Quick Example

#include "impedance_matching.hpp"
using namespace np;

// Define a load: Gamma_L(omega)
auto load = [](double omega) -> Complex {
    Complex j(0, 1);
    return 0.45 * std::exp(j * omega - 0.25 * omega * omega);
};

// Solve: order 8, two transmission zeros, 16 dB return loss, passband [-1, 1]
std::vector<Complex> tzs = {Complex(2, 0), Complex(3, 0)};
ImpedanceMatching matcher(load, 8, tzs, 16.0, -1.0, 1.0);

MatrixXcd cm = matcher.run();  // Returns (N+2) x (N+2) coupling matrix

std::cout << "Achieved: " << matcher.achieved_return_loss_db() << " dB\n";

Benchmarks

Selected results from benchmark_suite (Release build, GCC 14, Ryzen 7):

Test case	Time	Achieved RL
Gaussian order 2, all-pole	40 ms	-21.8 dB
Gaussian order 4, TZ={2,3}	184 ms	-17.0 dB
Gaussian order 6, TZ={2,3}	573 ms	-16.8 dB
Gaussian order 8, TZ={2,3}	1233 ms	-16.7 dB
Resistive order 6	14 ms	-23.0 dB
Reactive RL order 6	15 ms	-14.6 dB
Asymmetric order 8, TZ={2,3}	2176 ms	-22.4 dB
Gaussian order 6, narrow band	149 ms	-52.1 dB

The full suite runs 34 test cases covering orders 2-8, six load types, varied transmission zero configurations, and target return losses from 10 to 20 dB.

Theory and references

Multiplexer synthesis. The continuation algorithm for manifold-coupled multiplexers follows:

D. Martínez Martínez, S. Bila, F. Seyfert, M. Olivi, O. Tantot, L. Carpentier, "Synthesis Method for Manifold-Coupled Multiplexers," 49th European Microwave Conference (EuMC), Paris, 2019. HAL hal-02377002

The residual for the coupled matching problem is F(P) = f(p_i)(ξ_{i,m}) − conj(L_i(P)(ξ_{i,m})) where L_i is the load seen at filter i through the manifold. It is not holomorphic in p, so Newton's method in complex variables fails silently; npick uses proper Wirtinger derivatives to assemble a real 2N × 2N Jacobian and converges quadratically.

Impedance matching. The single-channel matcher implements:

L. Baratchart, M. Olivi, F. Seyfert, "Generalized Nevanlinna-Pick interpolation on the boundary. Application to impedance matching," Proceedings of the 22nd International Symposium on Mathematical Theory of Networks and Systems (MTNS), 2016. HAL hal-01249330

The INRIA PUMA project solves the same problem using convex relaxation and SDP. npick takes the complementary homotopy continuation approach, which is faster but provides a local (rather than globally certified) optimum. In practice both methods converge to the same equiripple solution for well-behaved loads.

Spectral factorization. Implemented via Wilson's Newton iteration on the factor coefficients (root-free, quadratically convergent):

G. T. Wilson, "Factorization of the covariance generating function of a pure moving-average process," SIAM J. Numer. Anal. 6(1), 1–7, 1969. doi:10.1137/0706001

A. H. Sayed, T. Kailath, "A survey of spectral factorization methods," Numer. Linear Algebra Appl. 8(6–7), 467–496, 2001. doi:10.1002/nla.250

Web demo source

The browser UI lives in docs/ (served by GitHub Pages) and web/ (build source). The two directories are kept in sync; docs/ ships the prebuilt npick.js/npick.wasm artifacts. See web/README.md for rebuild instructions via Emscripten.

Keywords

multiplexer synthesis, manifold-coupled multiplexer, duplexer synthesis, triplexer synthesis, RF multiplexer design, diplexer, impedance matching, Nevanlinna-Pick interpolation, Martinez 2019, Baratchart Olivi Seyfert, coupling matrix synthesis, microwave filter design, broadband matching, S-parameter solver, Touchstone, Feldtkeller equation, Wilson spectral factorization, equiripple matching, homotopy continuation, Wirtinger derivatives, C++ scientific computing, WebAssembly, browser-based EDA, free filter synthesis, open-source RF design tool

License

MIT