We define general DW inputs as follows:
The
th
block of the input
matrix
driving state
components
and multiplying
is then given by

(E.35) 
Typically, input
signals are injected equally to the left and right
along the string, in which case
Physically, this corresponds to applied
forces at a single,
nonmoving, string position over time. The state update with this
simplification appears as
Note that if there are no inputs driving the adjacent subgrid
(
), such as in a halfrate staggered grid
scheme, the input reduces to
To show that the directly obtained FDTD and DW
statespace models
correspond to the same dynamic system, it remains to verify that
. It is somewhat easier to show that
A straightforward calculation verifies that the above identity holds,
as expected. One can similarly verify
, as expected.
The relation
provides a recipe for translating any
choice of input signals for the FDTD model to equivalent inputs for
the DW model, or vice versa.
For example, in the
scalar input case (
), the DW inputweights
become FDTD inputweights
according to
The left and rightgoing inputweight superscripts indicate the
origin of each coefficient. Setting
results in

(E.36) 
Finally, when
and
for all
, we obtain the
result familiar from Eq.
(
E.23):
Similarly, setting
for all
, the weighting
pattern
appears in the second column, shifted down one row.
Thus,
in general (for physically stationary
displacement inputs)
can be seen as the superposition of weight patterns
in the
left column for even
, and the right column for odd
(the other
subgrid), where the
is aligned with the driven sample.
This is the general collection of displacement inputs.
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ThirdOrder Time Derivative is Ill Posed