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Physical Audio Signal Processing
    Digital Waveguide Theory
       The Lossy 1D Wave Equation
          Lossy Finite Difference Recursion
             Frequency-Dependent Losses

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Frequency-Dependent Losses

The preceding derivation generalizes immediately to frequency-dependent losses. First imagine each $ g$ in Fig.H.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)$. We may now derive the frequency-dependent counterpart of Eq.$ \,$(H.31) as follows:

\begin{eqnarray*} y^{+}_{n+1,m}&=& g\ast y^{+}_{n,m-1}\;=\; g\ast (y_{n,m-1}- y^... ... &=& 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 §H.12.5 for the structure of the derivation.

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Previous: Lossy Finite Difference Recursion
Next: The Dispersive 1D Wave Equation
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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