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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