DW State Space Model

As discussed in §E.2, the traveling-wave decomposition Eq.(E.4) defines a linear transformation Eq.(E.10) from the DW state to the FDTD state:

$$\underline{x}_K= \mathbf{T}\, \underline{x}_W \protect$$ (E.27)

Since $\mathbf{T}$ is invertible, it qualifies as a linear transformation for performing a change of coordinates for the state space. Substituting Eq.(E.27) into the FDTD state space model Eq.(E.24) gives

$$\mathbf{T}\,\underline{x}_W(n+2) = \mathbf{A}_K\, \mathbf{T}\,\underline{x}_W(n) + \mathbf{B}_K\, \underline{u}(n+2)\protect$$ (E.28)

$$\underline{y}(n) = \mathbf{C}_K\, \mathbf{T}\,\underline{x}_W(n). \protect$$ (E.29)

Multiplying through Eq.(E.28) by $\mathbf{T}^{-1}$ gives a new state-space representation of the same dynamic system which we will show is in fact the DW model of Fig.E.2:

$$\underline{x}_W(n+2) = \mathbf{A}_W\, \underline{x}_W(n) + {\mathbf{B}_W}\, \underline{u}(n+2)$$

$$\underline{y}(n) = \mathbf{C}_W\, \underline{x}_W(n)$$ (E.30)

where

$$\mathbf{A}_W \isdef \mathbf{T}^{-1}\mathbf{A}_K\,\mathbf{T}$$

$${\mathbf{B}_W} \isdef \mathbf{T}^{-1}\mathbf{B}_K$$

$$\mathbf{C}_W \isdef \mathbf{C}_K\,\mathbf{T} \protect$$ (E.31)

To verify that the DW model derived in this manner is the computation diagrammed in Fig.E.2, we may write down the state transition matrix for one subgrid from the figure to obtain the permutation matrix $\mathbf{A}_W$ ,

$$\underbrace{\left[\! \begin{array}{l} \qquad\vdots \\ y^{+}_{n+2,m-2} \\ y^{-}_{n+2,m-2} \\ y^{+}_{n+2,m} \\ y^{-}_{n+2,m} \\ y^{+}_{n+2,m+2} \\ y^{-}_{n+2,m+2} \\ \qquad\vdots \end{array} \!\right]}_{\underline{x}_W(n+2)} \!\leftarrow\! \underbrace{\left[\! \begin{array}{cccccccccccc} \cdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \cdots & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots\\ \cdots & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \cdots\\ \cdots & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots\\ \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & \cdots\\ \cdots & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \cdots\\ \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & \cdots\\ \cdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \end{array} \!\right]}_{\mathbf{A}_W} \! \underbrace{\left[\! \begin{array}{l} \quad\vdots \\ y^{+}_{n,m-4} \\ y^{-}_{n,m-4} \\ y^{+}_{n,m-2} \\ y^{-}_{n,m-2} \\ y^{+}_{n,m} \\ y^{-}_{n,m} \\ y^{+}_{n,m+2} \\ y^{-}_{n,m+2} \\ y^{+}_{n,m+4} \\ y^{-}_{n,m+4} \\ \quad\vdots \end{array} \!\right]}_{\underline{x}_W(n)} \protect$$ (E.32)

and displacement output matrix $\mathbf{C}_W$ :

$$\underbrace{\left[\! \begin{array}{c} \vdots \\ y_{n,m-2} \\ y_{n,m} \\ y_{n,m+2} \\ y_{n,m+4}\\ \vdots \end{array}\!\right]}_{\underline{y}(n)} = \underbrace{\left[\! \begin{array}{cccccccccc} & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \\ \cdots & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots\\ \cdots & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & \cdots\\ \cdots & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & \cdots\\ \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & \cdots\\ & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \end{array}\!\right]}_{\mathbf{C}_W} \underbrace{\left[\! \begin{array}{l} \quad\vdots \\ y^{+}_{n,m-2} \\ y^{-}_{n,m-2} \\ y^{+}_{n,m} \\ y^{-}_{n,m} \\ y^{+}_{n,m+2} \\ y^{-}_{n,m+2} \\ y^{+}_{n,m+4} \\ y^{-}_{n,m+4} \\ \quad\vdots \end{array}\!\right]}_{\underline{x}_W(n)}$$