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
In the time domain, we interpret
as the impulse response
. We may now derive the frequency-dependent
counterpart of Eq.
) 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
. For computational efficiency,
two spatial lines should be stored in memory at time
, for all
. These state variables
enable computation of
, after which each sample of
) is filtered
for the next iteration, and
is filtered by
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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