State Transformations

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State Transformations

In previous work, time-domain adaptors (digital filters) converting between K variables and W variables have been devised [227]. In this section, an alternative approach is proposed. Mapping Eq. $\,$ (P.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. $\,$ (P.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.P.2.

**Figure P.2:** DW flow diagram.
$\begin{figure}\input fig/wglossless.pstex_t \end{figure}$

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

$\displaystyle y_{n,m+1}$	$\displaystyle =$	$\displaystyle y^{+}_{n,m+1}+ y^{-}_{n,m+1}$
$\displaystyle y_{n,m-1}$	$\displaystyle =$	$\displaystyle y^{+}_{n,m-1}+ y^{-}_{n,m-1}$
$\displaystyle y_{n-1,m}$	$\displaystyle =$	$\displaystyle y^{+}_{n-1,m}+ y^{-}_{n-1,m}$
	$\displaystyle =$	$\displaystyle y^{+}_{n,m+1}+ y^{-}_{n,m-1} \protect$	(P.8)