Generalized Pickup/Pick-Position Modeling

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Generalized Pickup/Pick-Position Modeling

Given a digital-waveguide model of a vibrating string, using velocity waves (§H.7), we have that a virtual pickup is given by the sum of two delay-line taps, while a virtual excitation is given by summing into two delay-line cells. Thus, an excitation point is formally the transpose [460] of a pickup point (see §1.5.2 ^D.17 for related discussion regarding transposed tapped delay lines).

Since the pick-position comb-filter may be implemented as a pickup comb-filter (i.e., moved from the input side of the filtered delay loop to the output side), it follows that both excitation points and pickup points correspond to delay-line taps that are summed. If $\underline{x}(n) = [x_0(n),x_1(n),\ldots,x_N(n)]$ denotes the state of the delay line in the filtered delay loop at time , then the output signal is given as a $1\times N$ mixing matrix^D.18 $\mathbf{C}$ times the state vector:

$\displaystyle y(n) = \mathbf{C}\underline{x}(n)$

The mixing matrix $\mathbf{C}$ typically contains a

and a

for each input or output point in a velocity-wave simulation. (The

implements the effect of the inverting reflections at the string terminations for velocity waves.)

This suggests a generalized string simulation in which the string is driven and observed at a

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Physical Audio Signal Processing
    Making Virtual Electric Guitars and Guitar Effects Using Faust and Octave
       Pickup Modeling
          Generalized Pickup/Pick-Position Modeling