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Boundary Conditions as Perturbations

To study the effect of boundary conditions on the state transition matrices $ \mathbf{A}_W$ and $ \mathbf{A}_K$ , it is convenient to write the terminated transition matrix as the sum of the ``left-clamped'' case $ \mbox{$\stackrel{{\scriptscriptstyle \vdash}}{\mathbf{A}}$}$$ _W$ (for which $ g_l=-1$ ) plus a series of one or more rank-one perturbations. For example, introducing a right termination with reflectance $ g_r$ can be written

$\displaystyle \mbox{$\stackrel{{\scriptscriptstyle \vdash\!\!\dashv}}{\mathbf{A}}$}$$\displaystyle _W=$   $\displaystyle \mbox{$\stackrel{{\scriptscriptstyle \vdash}}{\mathbf{A}}$}$$\displaystyle _W+ g_r{\bm \delta}_{8,7} = \mathbf{A}_W- {\bm \delta}_{1,2} + g_r{\bm \delta}_{8,7}, \protect$ (E.39)

where $ {\bm \delta}_{ij}$ is the $ M\times
M$ matrix containing a 1 in its $ (i,j)$ th entry, and zero elsewhere. (Following established convention, rows and columns in matrices are numbered from 1.)

In general, when $ i+j$ is odd, adding $ {\bm \delta}_{ij}$ to $ \mbox{$\stackrel{{\scriptscriptstyle \vdash}}{\mathbf{A}}$}$$ _W$ corresponds to a connection from left-going waves to right-going waves, or vice versa (see Fig.E.2). When $ i$ is odd and $ j$ 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 $ i$ is even and $ j$ is odd. The spatial sample numbers involved in the connection are $ 2\lfloor (i-1)/2\rfloor$ and $ 2\lfloor (j-1)/2\rfloor$ , where $ \lfloor x\rfloor$ denotes the greatest integer less than or equal to $ x$ .

The rank-one perturbation of the DW transition matrix Eq.$ \,$ (E.39) corresponds to the following rank-one perturbation of the FDTD transition matrix $ \mbox{$\stackrel{{\scriptscriptstyle \vdash}}{\mathbf{A}}$}$$ _K$ :

   $\displaystyle \mbox{$\stackrel{{\scriptscriptstyle \vdash\!\!\dashv}}{\mathbf{A}}$}$$\displaystyle _K\;\isdef \;$   $\displaystyle \mbox{$\stackrel{{\scriptscriptstyle \vdash}}{\mathbf{A}}$}$$\displaystyle _K+ g{\bm \Delta}_{8,7}
$

where
$\displaystyle {\bm \Delta}_{8,7}$ $\displaystyle \isdef$ \begin{displaymath}\mathbf{T}{\bm \delta}_{8,7}\mathbf{T}^{-1}
=
\left[\!
\begin{array}{rrrrrrrrrrr}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots &
\vdots & \vdots & \vdots & \vdots\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & -1
\end{array}\!\right].
\protect\end{displaymath} (E.40)

In general, we have

$\displaystyle {\bm \Delta}_{ij} = \sum_{\kappa=j}^M (-1)^{\kappa-j} \left({\bm \delta}_{i\kappa}+{\bm \delta}_{i-1,\kappa}\right). \protect$ (E.41)

Thus, the general rule is that $ {\bm \delta}_{ij}$ transforms to a matrix $ {\bm \Delta}_{ij}$ which is zero in all but two rows (or all but one row when $ i=1$ ). The nonzero rows are numbered $ i$ and $ i-1$ (or just $ i$ when $ i=1$ ), and they are identical, being zero in columns $ 1:j-1$ , and containing $ [1,-1,1,-1,\ldots]$ starting in column $ j$ .


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2015-05-22 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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