#### Boundary Conditions as Perturbations

To study the effect of boundary conditions on the state transition matrices and , it is convenient to write the terminated transition matrix as the sum of of the left-clamped'' case (for which ) plus a series of one or more rank-one perturbations. For example, introducing a right termination with reflectance can be written

 (E.39)

where is the matrix containing a 1 in its th entry, and zero elsewhere. (Following established convention, rows and columns in matrices are numbered from 1.) In general, when is odd, adding to corresponds to a connection from left-going waves to right-going waves, or vice versa (see Fig.E.2). When is odd and is even, the connection flows from the right-going to the left-going signal path, thus providing a termination (or partial termination) on the right. Left terminations flow from the bottom to the top rail in Fig.E.2, and in such connections is even and is odd. The spatial sample numbers involved in the connection are and , where denotes the greatest integer less than or equal to . The rank-one perturbation of the DW transition matrix Eq.(E.39) corresponds to the following rank-one perturbation of the FDTD transition matrix :

where
 (E.40)

In general, we have

 (E.41)

Thus, the general rule is that transforms to a matrix which is zero in all but two rows (or all but one row when ). The nonzero rows are numbered and (or just when ), and they are identical, being zero in columns , and containing starting in column .
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