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Lossy Finite Difference Recursion

We will now derive a finite-difference model in terms of string displacement samples which correspond to the lossy digital waveguide model of Fig.C.5. This derivation generalizes the lossless case considered in §C.4.3.

Figure C.7 depicts a digital waveguide section once again in ``physical canonical form,'' as shown earlier in Fig.C.5, and introduces a doubly indexed notation for greater clarity in the derivation below [445,223,124,123].

Figure C.7: Lossy digital waveguide--frequency-independent loss-factors $ g$ .
\includegraphics{eps/wglossy}

Referring to Fig.C.7, we have the following time-update relations:

\begin{eqnarray*}
y^{+}_{n+1,m}&=& gy^{+}_{n,m-1}\;=\; g\cdot(y_{n,m-1}- y^{-}_{n,m-1})\\
y^{-}_{n+1,m}&=& gy^{-}_{n,m+1}\;=\; g\cdot(y_{n,m+1}- y^{+}_{n,m+1})
\end{eqnarray*}

Adding these equations gives

$\displaystyle y_{n+1,m}$ $\displaystyle =$ $\displaystyle g\cdot(y_{n,m-1}+y_{n,m+1})
- g\cdot(\underbrace{y^{-}_{n,m-1}}_{gy^{-}_{n-1,m}} +
\underbrace{y^{+}_{n,m+1}}_{gy^{+}_{n-1,m}})$  
  $\displaystyle =$ $\displaystyle g\cdot(y_{n,m-1}+y_{n,m+1}) - g^2 y_{n-1,m}
\protect$ (C.31)

This is now in the form of the finite-difference time-domain (FDTD) scheme analyzed in [223]:

$\displaystyle y_{n+1,m}=
g^{+}_my_{n,m-1}+
g^{-}_my_{n,m+1}+ a_my_{n-1,m},
$

with $ g^{+}_m= g^{-}_m= g$ , and $ a_m= -g^2$ . In [124], it was shown by von Neumann analysisD.4) that these parameter choices give rise to a stable finite-difference schemeD.2.3), provided $ \vert g\vert\leq 1$ . In the present context, we expect stability to follow naturally from starting with a passive digital waveguide model.



Subsections
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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA