Lossy Finite Difference Recursion

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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.H.5. This derivation generalizes the lossless case considered in §H.4.3.

Figure H.7 depicts a digital waveguide section once again in ``physical canonical form,'' as shown earlier in Fig.H.5, and introduces a doubly indexed notation for greater clarity in the derivation below [453,226,124,123].

**Figure H.7:** Lossy digital waveguide--frequency-independent loss-factors .
$\begin{figure}\input fig/wglossy.pstex_t \end{figure}$

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

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

Adding these equations gives

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

This is now in the form of the finite-difference time-domain (FDTD) scheme analyzed in [226]:

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

with $g^{+}_m= g^{-}_m= g$ , and

. In [124], it was shown by von Neumann analysis (§N.4) that these parameter choices give rise to a stable finite-difference scheme (§N.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

Frequency-Dependent Losses

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written by 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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Physical Audio Signal Processing
    Digital Waveguide Theory
       The Lossy 1D Wave Equation
          Lossy Finite Difference Recursion