We will now derive a finite-difference 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
Lossy digital waveguide--frequency-independent loss-factors .
Referring to Fig.C.7, we have the following time-update
Adding these equations gives
This is now in the form of the finite-difference time-domain
scheme analyzed in [222
], it was shown by von Neumann analysis
) that these parameter choices give rise to a stable
. In the
present context, we expect stability
to follow naturally from starting
with a passive digital waveguide model.
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.
Next Section: Higher Order TermsPrevious Section: Digital Filter Models of Damped Strings