FDTD State Space Model

Let denote the FDTD state for one of the two subgrids at time , as defined by Eq.(P.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 [55,149]. 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 denotes a point in a space of countably infinite dimensionality. A proper treatment of this case would be in terms of operator theory [331]. However, matrix notation is also clear and will be used below. Boundary conditions are taken up in §P.4.3.

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