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Chapters

Physical Audio Signal Processing
    Making Virtual Electric Guitars and Guitar Effects Using Faust and Octave
       Speaker and Cabinet Modeling
          Modeling Low-Frequency Loudspeaker Roll-Off

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Modeling Low-Frequency Loudspeaker Roll-Off

Working up from dc, the first feature we need to model is a slope of $ +12$ dB/octave in the Bode plot, ``breaking'' to flat at around 130 Hz, according to Fig.D.15. (We ignore response below 70 Hz since the low E string of a guitar is tuned to $ 82.4$ Hz.) An overview of relevant acoustics is given by Dan A. Russel at http://www.kettering.edu/~drussell/Demos/BaffledPiston/BaffledPiston.html, and in basic textbooks on acoustics [325,326,358].

A $ +12$ dB/octave slope in the Bode plot can be simply obtained using a series combination of two dc blockers [460], where each dc blocker has the $ s$-plane transfer function

$\displaystyle H_{0a}(s) = \frac{s}{s+\omega_c}, $

where in our problem we desire $ \omega_c \approx 2\pi 130$. As in §D.2.5, we may digitize via the standard low-frequency-matching bilinear transform

$\displaystyle s = \frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}} $ (D.10)

to obtain

$\displaystyle H_{0d}(z) = \frac{1}{1+\tilde{\omega_1}}\frac{1-z^{-1}}{1-\left(\frac{1-\tilde{\omega_1}}{1+\tilde{\omega_1}}\right)z^{-1}} $

where $ \tilde{\omega_1}\isdef \omega_c T/2$ and $ T$ is the sampling interval in seconds as usual. For our problem, with $ \omega_c = 2\pi 130$, we obtain