#### 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 is bidiagonal, so that and are both approximately diagonal when the output is string displacement for all . However, since given in Eq.(E.11) is upper triangular, the input matrix can replace sparse input matrices with only half-sparse , unless successive columns of 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