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There is one final weakness of the digital waveguide model that we
have developed so far. Energy
at higher frequencies in vibrating strings tends to decay more quickly than at
lower frequencies. This is true of physical systems in general because any
mass vibrating at an infinitely-large frequency would have infinite energy.
However, the current model does not implement this decay
phenomenon. The simplest way to ensure that high-frequency energy decays
quickly is to insert what is called a lowpass filter into the loop. Lowpass
filters pass more energy at lower frequencies than they do at higher
frequencies. In a loose sense, this is equivalent to assigning smaller 's
to higher frequencies than to lower frequencies. Figure 7 depicts
the finalized basic digital waveguide model that incorporates a lowpass
filter.
Figure 7:
Digital waveguide model with lowpass filter
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Sharp edges (or transients) in signals are characterized by energy at high
frequencies. Since lowpass filters remove high-frequency energy, they
tend to smooth out sharp edges in signals. This is precisely what the lowpass
filter in the digital waveguide model does. For instance, if the waveguide is
initialized with a rectangular pluck, the rectangular signal not only becomes slightly smaller due to , but the sharp
edges become smoothed out more and more after each trip around the waveguide.
The result is visualized in Figure 8 with one animation frame
taken every period of samples.
Figure 8:
Visualization of a damped vibrating string
simulation with lowpass filter (one animation frame per period of samples in
time) where the string is initialized according to a rectangular pluck
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This improved digital waveguide physical model more accurately captures
the behavior of a vibrating string, and so, as you will experience in the
following section, it also produces much more realistic simulated string
sounds. Much more information is available on digital waveguide modeling [1].
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Download waveguideintro.pdf