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Equivalent Forms

In a physical piano string, as a specific example, the hammer strikes the string between its two inputs, some distance from the agraffe and far from the bridge. This corresponds to the diagram in Fig.6.15, where the delay lines are again arranged for clarity of physical interpretation. Figure 6.15 is almost identical to Fig.6.14, except that the delay lines now contain samples of traveling force waves, and the bridge is allowed to vibrate, resulting in a filtered reflection at the bridge (see §9.2.1 for a derivation of the bridge filter). The hammer-string interaction force-pulse is summed into both the left- and right-going delay lines, corresponding to sending the same pulse toward both ends of the string from the hammer. Force waves are discussed further in §C.7.2.

Figure 6.15: Model of a piano string struck in its interior by a hammer.
\includegraphics[width=\twidth]{eps/pianoInteriorStringExcitation}

Figure: Diagram equivalent to Fig.6.15, obtained by combining upper- and lower-rail delay lines.
\includegraphics[width=\twidth]{eps/pianoSimplifiedInteriorStringExcitation}

By commutativity of linear, time-invariant elements, Figure 6.15 can be immediately simplified to the form shown in Fig.6.16, in which each delay line corresponds to the travel time in both directions on each string segment. From a structural point of view, we have a conventional filtered delay loop plus a second input which sums into the loop somewhere inside the delay line. The output is shown coming from the middle of the larger delay line, which gives physically correct timing, but in practice, the output can be taken from anywhere in the feedback loop. It is probably preferable in practice to take the output from the loop-delay-line input. That way, other response latencies in the overall system can be compensated.

Figure: Diagram equivalent to Fig.6.15, obtained by replacing the second string input by a separate comb-filter applied to a single input.
\includegraphics[width=\twidth]{eps/pianoSimplifiedISEExtracted}

An alternate structure equivalent to Fig.6.16 is shown in Fig.6.17, in which the second input injection is factored out into a separate comb-filtering of the input. The comb-filter delay equals the delay between the two inputs in Fig.6.16, and the delay in the feedback loop equals the sum of both delays in Fig.6.16. In this case, the string is modeled using a simple filtered delay loop, and the striking-force signal is separately filtered by a comb filter corresponding to the striking-point along the string.



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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA