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*
corresponding to

. We may now derive the frequency-dependent
counterpart of Eq.

(

C.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
§

C.14.5 for the outline of the derivation.

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