A simple one-dimensional digital waveguide string simulation is depicted in Fig. 4 below.
Figure 4:
Discrete-time simulation of one-dimensional string vibrations.
In previous sections, we modeled displacement or velocity excitations by appropriate delay-line initializations.
Piano strings are excited by hammer strikes, which produce velocity input profiles. When the hammer hits (and remains in contact with) the string, the string is effectively divided into two parts.
Based on this simple analysis, a more physical implementation for a piano synthesis system is diagrammed in Fig. 5, where the delay-line length .
Figure 5:
Hammer input in a discrete-time simulation of a piano string.
It is possible to rearrange the block diagram components of Fig. 5 to produce the equivalent system shown in Fig. 6.
Figure 6:
Equivalent system depicting hammer input in a discrete-time simulation of a piano string.
These block-diagram components can be further re-distributed to produce the equivalent system shown in Fig. 7.
Figure 7:
Another equivalent piano string simulation with comb-filtered input factored out of the string feedback loop.
Figure 7 makes explicit the fact that a feedforward comb filter can be placed in series with a digital waveguide string simulation to account for the effects of pluck position.