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## State Transformations

In previous work, time-domain adaptors (digital filters) converting between K variables and W variables have been devised . In this section, an alternative approach is proposed. Mapping Eq. (8) 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. (4) 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. 3. From Eq. (7), and referring to the notation defined in Fig. 3, we may write the conversion from W to K variables as           (9)

where the last equality follows from the traveling-wave behavior (see Fig. 3). Figure 4 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 . 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. (9), 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., (10)

Denote the linear transformation operator by and the K and W state vectors by and , respectively. Then Eq. (10) can be restated as (11)

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: (12)

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 circulant; in these cases, the matrix contains in every position for even , and is singular for odd (when there is one zero eigenvalue).

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