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The preceding derivation generalizes immediately to
frequency-dependent losses. First imagine each in Fig. G.7
to be replaced by , where for passivity we require
In the time domain, we interpret as the impulse response
corresponding to . We may now derive the frequency-dependent
counterpart of Eq. (G.31) as follows:
where denotes convolution (in the time dimension only).
Define filtered node variables by
Then the frequency-dependent FDTD scheme is simply
We see that generalizing the FDTD scheme to frequency-dependent losses
requires a simple filtering of each node variable by the
per-sample propagation filter . For computational efficiency,
two spatial lines should be stored in memory at time : and
, for all . These state variables enable computation of
, after which each sample of () is filtered
by to produce
for the next iteration, and
is filtered by to produce for the next iteration.
The frequency-dependent generalization of the FDTD scheme described in
this section extends readily to the digital waveguide mesh. See
§G.12.5 for the structure of the derivation.
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