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 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

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

In general, we have

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University