State Transformations
In previous work, time-domain adaptors (digital filters) converting
between K variables and W variables have been devised
[223]. 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




The operator





The case of the finite string is identical to that of the infinite string when the matrix









Next Section:
Excitation Examples
Previous Section:
Introduction