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Physical Scattering Junctions

All scattering matrices arising from the connection of $N$ scalar physical waveguides have the same basic structure which is an efficient subset of the set of all lossless scattering matrices in the sense of (26). To fix the setting, consider the parallel junction of $N$ lossless acoustic tubes, each with real, positive, scalar wave admittance $\Gamma_i$. The pressure at the junction is the same for all $N$ waveguides at the junction point, and the velocities from each branch sum to zero. These physical constraints yield scattering matrices of the form

$\displaystyle {{\mbox{\boldmath$A$}}}$ $\textstyle =$ $\displaystyle \left[ \begin{array}{cccc}
\frac{2 \Gamma_{1}}{\Gamma_J} - 1 & \f...
& \dots & \frac{2 \Gamma_{N}}{\Gamma_J} -1\\
  $\textstyle \stackrel{\triangle}{=}$ $\displaystyle {\mbox{\boldmath$1$}}{\mbox{\boldmath$\alpha$}}^T - {\mbox{\boldmath$I$}}$ (38)

where $\Gamma_J$ and ${\mbox{\boldmath$\alpha$}}$ are defined as follows:
\Gamma_J = \sum_{i=1}^N\Gamma_{i}
\end{displaymath} (39)

\begin{displaymath}{\mbox{\boldmath$\alpha$}}\stackrel{\triangle}{=}\frac{2}{\Gamma_J}[\Gamma_1,\dots,\Gamma_N]^T \end{displaymath} (40)

It is easy to verify that the matrix ${\mbox{\boldmath$A$}}$ satisfies the condition of losslessness (26). In the particular case of equal-impedance waveguides, we have

$\displaystyle {{\mbox{\boldmath$A$}}_e}$ $\textstyle =$ $\displaystyle \left[
\frac{2}{N} - 1 & \frac{2}{N} & \dots ...
...s \\
\frac{2}{N} & \frac{2}{N} & \dots & \frac{2}{N} -1\\
\end{array} \right]$  
  $\textstyle \stackrel{\triangle}{=}$ $\displaystyle {2\over N} {\mbox{\boldmath$1$}}{\mbox{\boldmath$1$}}^T - {\mbox{\boldmath$I$}}$ (41)

We notice that the elements of each row of matrix ${\mbox{\boldmath$A$}}$ in (38) and (41) are allpass complementary, i.e., $ {\mid \sum_{k=1}^{N}{ a_{i,k}} \mid} = 1$, for all $k$. Since the scattering matrix ${{\mbox{\boldmath$A$}}_e}$ for equal-impedance waveguides is symmetric, the elements of any row or column of matrix ${\mbox{\boldmath$A$}}_e$ are allpass complementary. Since ${{\mbox{\boldmath$A$}}_e}$ remains unchanged in the normalized case, the elements of each row and column are also power complementary.

The equal-impedance case is particularly important when $N$ is a power of two. In this case, all the multiplications can be replaced by right-shifts in fixed-point arithmetic, and considerable savings in circuit area can be achieved in a VLSI implementation. This case has been explored in the design of efficient digital waveguide meshes [91,122,125].

The scattering matrix for a connection of $N$ normalized physical waveguides is given by (32). It can be more explicitly written as

$\displaystyle \tilde {\mbox{\boldmath$A$}}$ $\textstyle =$ $\displaystyle \left[
\frac{2 \Gamma_{1}}{\Gamma_J} - 1
& \f...
& \dots
& \frac{2 \Gamma_{N}}{\Gamma_J} -1
\end{array} \right]$  
  $\textstyle \stackrel{\triangle}{=}$ $\displaystyle 2{\mbox{\boldmath$\gamma$}}{\mbox{\boldmath$\gamma$}}^T - {\mbox{\boldmath$I$}}$ (42)

where \( {\mbox{\boldmath$\gamma$}}{\tiny\stackrel{\triangle}{=}}\frac{2}{\sqrt {\Gamma_J}}[{\sqrt
\Gamma_1},\dots,{\sqrt \Gamma_N}]^T \), and the power-complementary property (37) can be verified for both rows and columns.

In the expressions (38)-(42), the elements of the matrices are real, since the one-variable waveguides intersecting at the junction have real, positive impedances.

Normalized $N$-way scattering junctions represented by real matrices can be implemented by $N(N-1)/2$ CORDIC processors [38]. Since the scattering matrix is orthogonal (real and unitary), it corresponds to a rotation in $N$ dimensions. Any such rotation can be obtained by $N(N-1)/2$ planar rotations [34], with each planar rotation being realizable using a CORDIC processor. This formulation can be useful for VLSI implementations.

The junction represented by the expressions (38) to (42) is formally a parallel junction. The corresponding series junction can be obtained by taking the dual of the parallel junction, i.e., by replacing admittance with impedance and interchanging pressure and velocity. In the series junction, the velocity is the same in all waveguides at the junction, and the pressures (or forces) sum to zero.

Parallel junctions occur naturally in the case of intersecting acoustic tubes, while series junctions arise at the intersection of $N$ transversely vibrating strings, where the transverse velocity must be the same for all $N$ strings. For waves in strings, force waves play the role of pressure waves in acoustic tubes. Note that the wave equation (1) for strings is usually written in terms of transverse displacement. However, choosing velocity and force as Kirchhoff variables unifies the treatment of vibrating strings with that of acoustic tubes.

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``Aspects of Digital Waveguide Networks for Acoustic Modeling Applications'', by Julius O. Smith III and Davide Rocchesso , December 19, 1997, Web published at
Copyright © 2007-02-07 by Julius O. Smith III and Davide Rocchesso
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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