Fast Numerical Methods for Rigid Adaptors in Wave Digital Filters

Soule G. Fast Numerical Methods for Rigid Adaptors in Wave Digital Filters. In: AES Show 2026 Nashville. Audio Engineering Society; 2026.pp. 44:1-10. Available from: https://aes.org/publications/elibrary-page/?id=23475

@inproceedings{Soule2026_23475,
  author = {Soule, Gabriel},
  title = {{Fast Numerical Methods for Rigid Adaptors in Wave Digital Filters}},
  booktitle = {AES Show 2026 Nashville},
  pages = {44:1-10},
  url = {https://aes.org/publications/elibrary-page/?id=23475},
  year = {2026},
  month = oct,
  publisher = {Audio Engineering Society},
  address = {Gaylord Opryland Resort}
}

TY  - CPAPER
TI  - Fast Numerical Methods for Rigid Adaptors in Wave Digital Filters
AU  - Soule, Gabriel
T2  - AES Show 2026 Nashville
SP  - 1
EP  - 10
M1  - Paper 44
PY  - 2026
DA  - 2026/10/30
UR  - https://aes.org/publications/elibrary-page/?id=23475
PB  - Audio Engineering Society
CY  - Gaylord Opryland Resort
LA  - en
AB  - Wave digital filters (WDFs) are widely used for real-time emulation of analog audio circuits. Subcircuits with arbitrary topology are represented by R-type (rigid) adaptors, which relate incident and reflected waves via a scattering matrix. Modulating a circuit parameter, such as a potentiometer setting, often changes the port resistances of these adaptors, which necessitates reconstruction of the scattering matrix. In existing WDF libraries, this matrix is derived symbolically and its algebra is baked into compiled source code, so modulation reduces to evaluating a compiled arithmetic expression. A WDF assembled dynamically at runtime must instead recompute its R-type scattering matrices numerically, via a circuit analysis formulation such as modified nodal analysis (MNA), whose cost is cubic in the size of the resultant system. We show that changing m port resistances is a rank-m structured update of the MNA matrix and derive two exact methods whose reduced linear solve costs O(m3), independent of the size of the underlying adaptor circuit. One method rebuilds the scattering matrix for block-rate modulation, whereas the other corrects the reflected wave directly at sample rate. In synthetic numerical benchmarks, the faster of our two methods outperforms inverse MNA reconstruction in every tested configuration. Implemented in complete circuits, the numerical WDF adaptors match or outperform otherwise identical compiled-symbolic baselines in a majority of configurations. These results clear a path toward performant runtime WDF assembly and real-time modulation.
ER  -