Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Input Locality

The DW state-space model is given in terms of the FDTD state-space model by Eq. (32). The similarity transformation matrix $ \mathbf{T}$ is bidiagonal, so that $ \mathbf{C}_K$ and $ \mathbf{C}_W=\mathbf{C}_K\,\mathbf{T}$ are both approximately diagonal when the output is string displacement for all $ m$. However, since $ \mathbf{T}^{-1}$ given in Eq. (12) is upper triangular, the input matrix $ {\mathbf{B}_W}=\mathbf{T}^{-1}\mathbf{B}_K$ can replace sparse input matrices $ \mathbf{B}_K$ with only half-sparse $ {\mathbf{B}_W}$, unless successive columns of $ \mathbf{T}^{-1}$ are equally weighted, as discussed in §4. We can say that local K-variable excitations may correspond to non-local W-variable excitations. From Eq. (36) and Eq. (37), we see that displacement inputs are always local in both systems. Therefore, local FDTD and non-local DW excitations can only occur when a variable dual to displacement is being excited, such as velocity. If the external integrator Eq. (38) is used, all inputs are ultimately displacement inputs, and the distinction disappears.


Next  |  Prev  |  Up  |  Top  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Download wgfdtd.pdf

``On the Equivalence of the Digital Waveguide and Finite Difference Time Domain Schemes'', by Julius O. Smith III, version published at http://arXiv.org/abs/physics/0407032 (in PDF and PostScript formats only).
Copyright © 2005-12-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]