FDTD State Space Model
Let
denote the FDTD state for one of the two subgrids at time
, as defined by Eq.
(E.10). The other subgrid is handled
identically and will not be considered explicitly. In fact, the other
subgrid can be dropped altogether to obtain a half-rate,
staggered grid scheme [55,147]. However, boundary
conditions and input signals will couple the subgrids, in general. To
land on the same subgrid after a state update, it is necessary to
advance time by two samples instead of one. The state-space model for
one subgrid of the FDTD model of the ideal string may then be written
as
To avoid the issue of boundary conditions for now, we will continue working with the infinitely long string. As a result, the state vector

When there is a general input signal vector
, it is necessary to
augment the input matrix
to accomodate contributions over both
time steps. This is because inputs to positions
at time
affect position
at time
. Henceforth, we assume
and
have been augmented in this way. Thus, if there are
input
signals
,
, driving the full
string state through weights
,
, the vector
is of dimension
:
![\begin{displaymath}
\underline{u}(n+2) =
\left[\!
\begin{array}{c}
\underline{\upsilon}(n+2)\\
\underline{\upsilon}(n+1)
\end{array}\!\right]
\end{displaymath}](http://www.dsprelated.com/josimages_new/pasp/img4608.png)






forms the output signal as an arbitrary linear combination of
states. To obtain the usual displacement output for the subgrid,
is the matrix formed from the identity matrix by deleting every
other row, thereby retaining all displacement samples at time
and
discarding all displacement samples at time
in the state vector
:
![\begin{displaymath}
\underbrace{\left[\!
\begin{array}{c}
\vdots \\
y_{n,m-2} \...
..._{n,m+4}\\
\vdots
\end{array}\!\right]}_{\underline{x}_K(n)}
\end{displaymath}](http://www.dsprelated.com/josimages_new/pasp/img4612.png)




The intra-grid state update for even

For odd




![$ [1,-1,1,-1,1]$](http://www.dsprelated.com/josimages_new/pasp/img4624.png)


![$ [1,-1,1]$](http://www.dsprelated.com/josimages_new/pasp/img4625.png)
![\begin{displaymath}
\underbrace{\left[\!
\begin{array}{l}
\qquad\vdots\\
y_{n+1...
...+3}\\
\qquad\vdots
\end{array}\!\right]}_{\underline{x}_K(n)}
\end{displaymath}](http://www.dsprelated.com/josimages_new/pasp/img4626.png)


![\begin{displaymath}
\underbrace{\left[\!
\begin{array}{l}
\qquad\vdots\\
y_{n+1...
...ine{\upsilon}(n+1)
\end{array}\!\right]}_{\underline{u}(n+2)}.
\end{displaymath}](http://www.dsprelated.com/josimages_new/pasp/img4627.png)
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DW State Space Model
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Interpretation of the Time-Domain KW Converter