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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Equivalence of Digital Waveguide and Finite Difference Schemes
       State Space Formulation
          Boundary Conditions
             Boundary Conditions as Perturbations

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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 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... ... 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$.

Previous: Resistive Terminations
Next: Reactive Terminations

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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