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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^... ... &=& 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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