Online Correction of Dispersion Error in 2D Waveguide Meshes

TL;DR

This paper proposes to reduce numerical dispersion by embedding warping elements, i.e., properly tuned allpass filters, in the structure, and exhibits a significant reduction in dispersion, and requires less computational resources than a regular mesh structure having comparable accuracy.

Abstract

An elastic ideal 2D propagation medium, i.e., a membrane, can be simulated by models discretizing the wave equation on the time-space grid (finite difference methods), or locally discretizing the solution of the wave equation (waveguide meshes). The two approaches provide equivalent computational structures, and introduce numerical dispersion that induces a misalignment of the modes from their theoretical positions. Prior literature shows that dispersion can be arbitrarily reduced by oversizing and oversampling the mesh, or by adpting offline warping techniques. In this paper we propose to reduce numerical dispersion by embedding warping elements, i.e., properly tuned allpass filters, in the structure. The resulting model exhibits a significant reduction in dispersion, and requires less computational resources than a regular mesh structure having comparable accuracy.

Figures (5)

• Figure 1: Plot of the dispersion error versus temporal frequency magnitude
• Figure 2: Frequency response taken at the center of a TWM (size $24\times 24$) excited by an impulse at the same point. Theoretical positions of the odd modes resonating in a mebrane below $\sqrt{2}\pi /\sqrt{3}$ [rad$/$sample], weighted by the nominal propagation speed factor ($\circ$). Positions of the same modes affected by dispersion ($\times$).
• Figure 3: Mapping functions $\tilde{z}^{-1}=z^{-1}A(z)$ for equally-spaced values of the parameter $\alpha$ of the allpass filter $A(z)$. Top line: $\alpha = 0$. Bottom line: $\alpha = -0.9$.
• Figure 4: Impulse response taken at the center of a warped TWM (size $24\times 24$) excited by an impulse at the same point. Theoretical positions of the odd modes resonating in a mebrane, rescaled to align the fundamentals ($\circ$). Positions of the same modes affected by residual dispersion ($\times$).
• Figure 5: Plot of the dispersion error versus temporal frequency magnitude in the warped TWM.