Input Locality

The DW state-space model is given in terms of the FDTD state-space model by Eq.$\,$(O.31). 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.$\,$(O.11) 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 §O.3. We can say that local K-variable excitations may correspond to non-local W-variable excitations. From Eq.$\,$(O.35) and Eq.$\,$(O.36), 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.$\,$(O.37) is used, all inputs are ultimately displacement inputs, and the distinction disappears.