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Frequency-Dependent Losses

The preceding derivation generalizes immediately to frequency-dependent losses. First imagine each $ g$ in Fig.C.7 to be replaced by $ G(z)$ , where for passivity we require

$\displaystyle \left\vert G(e^{j\omega T})\right\vert\leq 1.
$

In the time domain, we interpret $ g(n)$ as the impulse response corresponding to $ G(z)$ . We may now derive the frequency-dependent counterpart of Eq.$ \,$ (C.31) as follows:

\begin{eqnarray*}
y^{+}_{n+1,m}&=& g\ast y^{+}_{n,m-1}\;=\; g\ast (y_{n,m-1}- y^{-}_{n,m-1})\\
y^{-}_{n+1,m}&=& g\ast y^{-}_{n,m+1}\;=\; g\ast (y_{n,m+1}- y^{+}_{n,m+1})\\ [10pt]
\Rightarrow\quad
y_{n+1,m}&=& g\ast (y_{n,m-1}+y_{n,m+1})
- g\ast (\underbrace{y^{-}_{n,m-1}}_{g\ast y^{-}_{n-1,m}} +
\underbrace{y^{+}_{n,m+1}}_{g\ast y^{+}_{n-1,m}})\nonumber \\
&=& g\ast (y_{n,m-1}+y_{n,m+1}) - g\ast g\ast y_{n-1,m}\\
&=& g\ast \left[(y_{n,m-1}+y_{n,m+1}) - g\ast y_{n-1,m}\right]
\end{eqnarray*}

where $ \ast $ denotes convolution (in the time dimension only). Define filtered node variables by

\begin{eqnarray*}
y^f_{n,m}&=& g\ast y_{n,m}\\
y^{ff}_{n,m}&=& g\ast y^f_{n,m}.
\end{eqnarray*}

Then the frequency-dependent FDTD scheme is simply

$\displaystyle y_{n+1,m}= y^f_{n,m-1}+ y^f_{n,m+1}- y^{ff}_{n-1,m}.
$

We see that generalizing the FDTD scheme to frequency-dependent losses requires a simple filtering of each node variable $ y_{n,m}$ by the per-sample propagation filter $ G(z)$ . For computational efficiency, two spatial lines should be stored in memory at time $ n$ : $ y^f_{n,m}$ and $ y^{ff}_{n-1,m}$ , for all $ m$ . These state variables enable computation of $ y_{n+1,m}$ , after which each sample of $ y^f_{n,m}$ ($ \forall m$ ) is filtered by $ G(z)$ to produce $ y^{ff}_{n-1,m}$ for the next iteration, and $ y_{n+1,m}$ is filtered by $ G(z)$ to produce $ y^f_{n,m}$ for the next iteration.

The frequency-dependent generalization of the FDTD scheme described in this section extends readily to the digital waveguide mesh. See §C.14.5 for the outline of the derivation.


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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