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.