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
![$\displaystyle \left\vert G(e^{j\omega T})\right\vert\leq 1.
$](http://www.dsprelated.com/josimages_new/pasp/img3372.png)
![$ g(n)$](http://www.dsprelated.com/josimages_new/pasp/img451.png)
![$ G(z)$](http://www.dsprelated.com/josimages_new/pasp/img13.png)
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
![\begin{eqnarray*}
y^{+}_{n+1,m}&=& g\ast y^{+}_{n,m-1}\;=\; g\ast (y_{n,m-1}- y^...
...
&=& g\ast \left[(y_{n,m-1}+y_{n,m+1}) - g\ast y_{n-1,m}\right]
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img3373.png)
where denotes convolution (in the time dimension only).
Define filtered node variables by
![\begin{eqnarray*}
y^f_{n,m}&=& g\ast y_{n,m}\\
y^{ff}_{n,m}&=& g\ast y^f_{n,m}.
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img3374.png)
Then the frequency-dependent FDTD scheme is simply
![$\displaystyle y_{n+1,m}= y^f_{n,m-1}+ y^f_{n,m+1}- y^{ff}_{n-1,m}.
$](http://www.dsprelated.com/josimages_new/pasp/img3375.png)
![$ y_{n,m}$](http://www.dsprelated.com/josimages_new/pasp/img3376.png)
![$ G(z)$](http://www.dsprelated.com/josimages_new/pasp/img13.png)
![$ n$](http://www.dsprelated.com/josimages_new/pasp/img146.png)
![$ y^f_{n,m}$](http://www.dsprelated.com/josimages_new/pasp/img3377.png)
![$ y^{ff}_{n-1,m}$](http://www.dsprelated.com/josimages_new/pasp/img3378.png)
![$ m$](http://www.dsprelated.com/josimages_new/pasp/img6.png)
![$ y_{n+1,m}$](http://www.dsprelated.com/josimages_new/pasp/img3379.png)
![$ y^f_{n,m}$](http://www.dsprelated.com/josimages_new/pasp/img3377.png)
![$ \forall m$](http://www.dsprelated.com/josimages_new/pasp/img3380.png)
![$ G(z)$](http://www.dsprelated.com/josimages_new/pasp/img13.png)
![$ y^{ff}_{n-1,m}$](http://www.dsprelated.com/josimages_new/pasp/img3378.png)
![$ y_{n+1,m}$](http://www.dsprelated.com/josimages_new/pasp/img3379.png)
![$ G(z)$](http://www.dsprelated.com/josimages_new/pasp/img13.png)
![$ y^f_{n,m}$](http://www.dsprelated.com/josimages_new/pasp/img3377.png)
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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