Frequency-Dependent Losses
The preceding derivation generalizes immediately to
frequency-dependent losses. First imagine each in Fig.C.7
to be replaced by
, where for passivity we require




![\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*}](http://www.dsprelated.com/josimages_new/pasp/img3373.png)
where denotes convolution (in the time dimension only).
Define filtered node variables by

Then the frequency-dependent FDTD scheme is simply















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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