We will now derive a finitedifference model in terms of string
displacement samples which correspond to the lossy
digital waveguide
model of Fig.
C.5. This derivation generalizes the lossless case
considered in §
C.4.3.
Figure
C.7 depicts a
digital waveguide section once again in
``physical canonical form,'' as shown earlier in Fig.
C.5, and
introduces a doubly indexed notation for greater clarity in the
derivation below
[
442,
222,
124,
123].
Figure C.7:
Lossy digital waveguidefrequencyindependent lossfactors .

Referring to Fig.
C.7, we have the following timeupdate
relations:
Adding these equations gives
This is now in the form of the
finitedifference timedomain (FDTD)
scheme analyzed in [
222]:
with
, and
. In
[
124], it was shown by
von Neumann analysis
(§
D.4) that these parameter choices give rise to a stable
finitedifference scheme (§
D.2.3), provided
. In the
present context, we expect
stability to follow naturally from starting
with a passive digital waveguide model.
The preceding derivation generalizes immediately to
frequencydependent 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 frequencydependent
counterpart of Eq.
(
C.31) as follows:
where
denotes
convolution (in the time dimension only).
Define
filtered node variables by
Then the frequencydependent FDTD scheme is simply
We see that generalizing the FDTD scheme to frequencydependent losses
requires a simple filtering of each node variable
by the
persample
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 frequencydependent 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.
Next Section: Higher Order TermsPrevious Section: Digital Filter Models of Damped Strings