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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 $ n$. 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 $ n+1$, we need initial spatial samples at all positions $ m$ for two successive times $ n-1$ and $ n$. From this state specification, the FDTD scheme Eq.$ \,$(E.3) can compute $ y(n+1,m)$ for all $ m$, and so on for increasing $ n$. In the DW model, all state variables are defined as belonging to the same time $ n$, as shown in Fig.E.2.
Figure E.2: DW flow diagram.
\includegraphics{eps/wglossless}
From Eq.$ \,$(E.6), and referring to the notation defined in Fig.E.2, we may write the conversion from W to K variables as
$\displaystyle y_{n,m+1}$ $\displaystyle =$ $\displaystyle y^{+}_{n,m+1}+ y^{-}_{n,m+1}$  
$\displaystyle y_{n,m-1}$ $\displaystyle =$ $\displaystyle y^{+}_{n,m-1}+ y^{-}_{n,m-1}$  
$\displaystyle y_{n-1,m}$ $\displaystyle =$ $\displaystyle y^{+}_{n-1,m}+ y^{-}_{n-1,m}$  
  $\displaystyle =$ $\displaystyle y^{+}_{n,m+1}+ y^{-}_{n,m-1} \protect$ (E.8)

where the last equality follows from the traveling-wave behavior (see Fig.E.2).
Figure E.3: Stencil of the FDTD scheme.
\includegraphics{eps/stencil}
Figure E.3 shows the so-called ``stencil'' of the FDTD scheme. The larger circles indicate the state at time $ n$ which can be used to compute the state at time $ n+1$. 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 $ m$ is even, the update for $ y_{n+1,m}$ depends only on odd $ m$ from time $ n$ and even $ m$ from time $ n-1$. Since the two W components of $ y_{n-1,m}$ are converted to two W components at time $ n$ in Eq.$ \,$(E.8), we have that the update for $ y_{n+1,m}$ depends only on W components from time $ n$ and positions $ m\pm1$. Moving to the next position update, for $ y_{n+1,m+1}$, the state used is independent of that used for $ y_{n+1,m}$, and the W components used are from positions $ m$ and $ m+2$. As a result of these observations, we see that we may write the state-variable transformation separately for even and odd $ m$, e.g.,

$\displaystyle \left[\! \begin{array}{c} \vdots \\ y_{n,m-1}\\ y_{n-1,m}\\ y_{n,... ...n,m+3}\\ y^{+}_{n,m+5}\\ y^{-}_{n,m+5}\\ \vdots \end{array} \!\right]. \protect$ (E.9)

Denote the linear transformation operator by $ \mathbf{T}$ and the K and W state vectors by $ \underline{x}_K$ and $ \underline{x}_W$, respectively. Then Eq.$ \,$(E.9) can be restated as

$\displaystyle \underline{x}_K= \mathbf{T}\underline{x}_W. \protect$ (E.10)

The operator $ \mathbf{T}$ can be recognized as the Toeplitz operator associated with the linear, shift-invariant filter $ H(z)=1+z^{-1}$. While the present context is not a simple convolution since $ \underline{x}_W$ is not a simple time series, the inverse of $ \mathbf{T}$ corresponds to the Toeplitz operator associated with

$\displaystyle H(z) = \frac{1}{1+z^{-1}} = 1 - z^{-1}+ z^{-2} - z^{-3} + \cdots. $

Therefore, we may easily write down the inverted transformation:

$\displaystyle \left[\! \begin{array}{c} \vdots \\ y^{+}_{n,m-1}\\ y^{-}_{n,m-1}... ... y_{n,m+3}\\ y_{n-1,m+4}\\ y_{n,m+5}\\ \vdots \\ \end{array} \!\right] \protect$ (E.11)

The case of the finite string is identical to that of the infinite string when the matrix $ \mathbf{T}$ is simply ``cropped'' to a finite square size (leaving an isolated 1 in the lower right corner); in such cases, $ \mathbf{T}^{-1}$ as given above is simply cropped to the same size, retaining its upper triangular $ \pm 1$ structure. Another interesting set of cases is obtained by inserting a 1 in the lower-left corner of the cropped $ \mathbf{T}$ matrix to make it circulant; in these cases, the $ M\times M$ matrix $ \mathbf{T}^{-1}$ contains $ \pm1/2$ in every position for even $ M$, and is singular for odd $ M$ (when there is one zero eigenvalue).
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