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Delay Lines And Gain Blocks

From the animations, we see that a digital waveguide model must essentially consist of at least two elements: 1) a delay element that models the time delay seen by traveling waves between the string ends and 2) an element that inverts the sign of the wave as it reflects back from an end. The delay element is known as a delay line, which delays an input signal by a specified number of samples, and is depicted as a wide rectangle in Figure 3. The sign-inverting element may be implemented more generally as a gain block with a gain of -1. In general, a gain block takes a signal as input and outputs a scaled version of the signal. The gain blocks are depicted as triangles in Figure 3, which is a block diagram of the most basic digital waveguide model of a vibrating string.

Figure 3: Most basic digital waveguide model of a vibrating string
\includegraphics{figures/waveguidenoloopfilt.eps}

Note that the delay due to each of the delay lines is $N/2$ samples. This means that the total delay around the feedback loop consisting of the delay lines and gain blocks is $N$ samples, or $NT$ seconds, which is the period of the vibrating string model. The fundamental frequency $f_0$ of the vibrating string model is the reciprocal of the period, so $f_0 = \frac{1}{NT}$.

This model was used to generate the animation in Figure 2, which never comes to rest. This is okay for the animation since the string's behavior is slowed down so much for visualization purposes. From physical reality, you are aware that vibrating strings eventually come to rest (often after hundreds of periods) until they are excited again via plucking, striking, picking, bowing, etc. This is because traditional physical musical instruments have damping, meaning that various frictional forces eventually suck all of the energy out of the vibrating string.



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Download waveguideintro.pdf

``Plucked String Digital Waveguide Model'', by Edgar J. Berdahl, and Julius O. Smith III,
REALSIMPLE Project — work supported by the Wallenberg Global Learning Network .
Released 2008-06-05 under the Creative Commons License (Attribution 2.5), by Edgar J. Berdahl, and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA