Input Locality

The DW state-space model is given in terms of the FDTD state-space model by Eq.$ \,$(E.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.$ \,$(E.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 §E.3. We can say that local K-variable excitations may correspond to non-local W-variable excitations. From Eq.$ \,$(E.35) and Eq.$ \,$(E.36), we see that displacement inputs are always local in both systems. Therefore, local FDTD and non-local DW excitations can only occur when a variable dual to displacement is being excited, such as velocity. If the external integrator Eq.$ \,$(E.37) is used, all inputs are ultimately displacement inputs, and the distinction disappears.

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DW Non-Displacement Inputs