### DW State Space Model

As discussed in §E.2, the traveling-wave decomposition Eq.(E.4) defines a linear transformation Eq.(E.10) from the DW state to the FDTD state:

Since is invertible, it qualifies as a linear transformation for performing a

*change of coordinates*for the state space. Substituting Eq.(E.27) into the FDTD state space model Eq.(E.24) gives

Multiplying through Eq.(E.28) by gives a new state-space representation of the same dynamic system which we will show is in fact the DW model of Fig.E.2:

(E.30) |

where

To verify that the DW model derived in this manner is the computation diagrammed in Fig.E.2, we may write down the state transition matrix for one subgrid from the figure to obtain the permutation matrix ,

and displacement output matrix :

#### DW Displacement Inputs

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

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

Finally, when and for all , we obtain the result familiar from Eq.(E.23):

####

DW Non-Displacement Inputs

Since a displacement input at position corresponds to
symmetrically exciting the right- and left-going traveling-wave
components and , it is of interest to understand what
it means to excite these components *antisymmetrically*. As
discussed in §E.3.3, an antisymmetric excitation of
traveling-wave components can be interpreted as a *velocity*
excitation. It was noted that localized velocity excitations in the
FDTD generally correspond to non-localized velocity excitations in the
DW, and that velocity in the DW is proportional to the *spatial
derivative* of the difference between the left-going and right-going
traveling displacement-wave components (see Eq.(E.13)). More
generally, the antisymmetric component of displacement-wave excitation
can be expressed in terms of any wave variable which is linearly
independent relative to displacement, such as acceleration, slope,
force, momentum, and so on. Since the state space of a vibrating
string (and other mechanical systems) is traditionally taken to be
position and velocity, it is perhaps most natural to relate the
antisymmetric excitation component to velocity.

In practice, the simplest way to handle a velocity input in a DW simulation is to first pass it through a first-order integrator of the form

to convert it to a displacement input. By the equivalence of the DW and FDTD models, this works equally well for the FDTD model. However, in view of §E.3.3, this approach does not take full advantage of the ability of the FDTD scheme to provide localized velocity inputs for applications such as simulating a piano hammer strike. The FDTD provides such velocity inputs for ``free'' while the DW requires the external integrator Eq.(E.37).

Note, by the way, that these ``integrals'' (both that done internally by the FDTD and that done by Eq.(E.37)) are merely sums over discrete time--not true integrals. As a result, they are exact only at dc (and also trivially at , where the output amplitude is zero). Discrete sums can also be considered exact integrators for impulse-train inputs--a point of view sometimes useful when interpreting simulation results. For normal bandlimited signals, discrete sums most accurately approximate integrals in a neighborhood of dc. The KW-converter filter has analogous properties.

#### 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.

**Next Section:**

Boundary Conditions

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FDTD State Space Model