In typical string models for virtual musical instruments, the ``nut
end'' of the string is rigidly clamped while the ``bridge end'' is
terminated in a
passive reflectance . The condition
for passivity of the reflectance is simply that its gain be bounded
by 1 at all frequencies [
447]:

(E.42) 
A very simple case, used, for example, in the KarplusStrong
pluckedstring algorithm, is the twopointaverage
filter:
To impose this
lowpassfiltered reflectance on the right in the chosen
subgrid, we may form
which results in the FDTD transition
matrix
This gives the desired filter in a halfrate, staggered grid case.
In the fullrate case, the termination filter is really
which is still passive, since it obeys Eq.
(
E.42), but it does not have
the desired
amplitude response: Instead, it has a notch (gain of 0)
at onefourth the
sampling rate, and the gain comes back up to 1 at
half the
sampling rate. In a fullrate scheme, the twopointaverage
filter must straddle both subgrids.
Another oftenused string termination filter in
digital waveguide
models is specified by [
447]
where
is an overall gain factor that affects the decay
rate of all frequencies equally, while
controls the
relative decay rate of lowfrequencies and high frequencies. An
advantage of this termination filter is that the delay is
always one sample, for all frequencies and for all parameter settings;
as a result, the tuning of the string is invariant with respect to
termination filtering. In this case, the perturbation is
and, using Eq.
(
E.41),
the order
FDTD state transition matrix is given by
where
The filtered termination examples of this section generalize
immediately to arbitrary finite
impulse response (FIR) termination
filters
. Denote the
impulse response of the termination filter
by
where the length
of the filter does not exceed
. Due to
the DWFDTD equivalence, the general
stability condition is stated
very simply as
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