To save computation in the filtering, we can make use of the observation that, under the assumption of a string initially at rest, each interaction pulse is smoother than the one before it. That suggests applying the force-pulse filtering progressively, as was done with Leslie cabinet reflections in §5.7.6. In other words, the second force-pulse is generated as a filtering of the first force-pulse. This arrangement is shown in Fig.9.36.
With progressive filtering, each filter need only supply the mild smoothing (and perhaps dispersion) associated with traveling from the hammer to the agraffe and back, plus the mild attenuation associated with reflection from the felt-covered hammer (a nonlinear mass-spring system as described in §9.3.2).
Referring to Fig.9.36, The first filter LPF1 can shape a velocity-independent excitation signal to obtain the appropriate ``shock spectrum'' for that hammer velocity. Alternatively, the Excitation Table itself can be varied with velocity to produce the needed signal. In this case, filter LPF1 can be eliminated entirely by applying it in advance to the excitation signal. It is possible to interpolate between tables for two different striking velocities; in this case, the tables should be pre-processed to eliminate phase cancellations during cross-fade.
Assuming the first filter in Fig.9.36 is ``weakest'' at the highest hammer velocity (MIDI velocity ), that filtering can be applied to the excitation table in advance, and the first filter then becomes no computation for MIDI velocity , and as velocity is lowered, the filter only needs to make up the difference between what was done in advance to the table and what is desired at that velocity.
Since, for most keys, only a few interaction impulses are observed, per hammer strike, in real pianos, this computational model of the piano achieves a high degree of realism for a price comparable to the cost of the strings only. The soundboard and enclosure filtering have been eliminated and replaced by a look-up table using a few read-pointers per note, and the hammer costs only one or a few low-order filters which in principle convert the interaction impulse into an accurate force pulse.