We define general DW inputs as follows:

(E.33) | |||

(E.34) |

The th block of the input matrix driving state components and multiplying is then given by

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, non-moving, 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 half-rate staggered grid scheme, the input reduces to

To show that the directly obtained FDTD and DW state-space 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 input-weights become FDTD input-weights according to

The left- and right-going input-weight superscripts indicate the origin of each coefficient. Setting results in

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University