State Transformations

In previous work, time-domain adaptors (digital filters) converting between K variables and W variables have been devised [224]. In this section, an alternative approach is proposed. Mapping Eq. (E.7) gives us an immediate conversion from W to K state variables, so all we need now is the inverse map for any time . This is complicated by the fact that non-local spatial dependencies can go indefinitely in one direction along the string, as we will see below. We will proceed by first writing down the conversion from W to K variables in matrix form, which is easy to do, and then invert that matrix. For simplicity, we will consider the case of an infinitely long string.

To initialize a K variable simulation for starting at time , we need initial spatial samples at all positions for two successive times and . From this state specification, the FDTD scheme Eq. (E.3) can compute for all , and so on for increasing . In the DW model, all state variables are defined as belonging to the same time , as shown in Fig.E.2.

From Eq.
(E.6), and referring to the notation defined in
Fig.E.2, we may write the conversion from W to K variables
as

where the last equality follows from the traveling-wave behavior (see Fig.E.2).

Figure E.3 shows the so-called ``stencil'' of the FDTD scheme.
The larger circles indicate the state at time
which can be used to
compute the state at time
. The filled and unfilled circles
indicate membership in one of two interleaved grids [55]. To
see why there are two interleaved grids, note that when
is even,
the update for
depends only on odd
from time
and even
from time
. Since the two W components of
are converted to
two W components at time
in Eq.
(E.8), we have that the update for
depends only on W components from time
and positions
.
Moving to the next position update, for
, the state used is
independent of that used for
, and the W components used are
from positions
and
. As a result of these observations, we
see that we may write the state-variable transformation separately for
even and odd
, *e.g.*,

Denote the linear transformation operator by and the K and W state vectors by and , respectively. Then Eq. (E.9) can be restated as

The operator can be recognized as the Toeplitz operator associated with the linear, shift-invariant filter . While the present context is not a simple convolution since is not a simple time series, the inverse of corresponds to the Toeplitz operator associated with

Therefore, we may easily write down the inverted transformation:

The case of the finite string is identical to that of the infinite string when the matrix is simply ``cropped'' to a finite square size (leaving an isolated 1 in the lower right corner); in such cases, as given above is simply cropped to the same size, retaining its upper triangular structure. Another interesting set of cases is obtained by inserting a 1 in the lower-left corner of the cropped matrix to make it

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