Digitizing Bridge Reflectance

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Physical Audio Signal Processing

Bridge Modeling

Bridge Transmittance

A Two-Resonance Guitar Bridge

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The wave impedance in a propagation medium is the force variable (such as pressure for acoustic waves) divided by the velocity variable. In ideal plane waves and traveling waves, the wave impedance is a real, positive number. For spherical waves in air, the wave impedance is different at each frequency. — Click for https://ccrma.stanford.edu/~jos/OnePorts/Impedance.html

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A digital filter can be visualized as a 'black box' that accepts a sequence of numbers and emits a new sequence of numbers. In digital audio signal processing applications, such number sequences usually represent sounds. For example, digital filters are used to implement graphic equalizers and other digital audio effects. — Click for https://ccrma.stanford.edu/~jos/filters/

A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

The transfer function is defined for LTI filters as the z transform of the filter output signal, divided by the z transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Transfer_Function_Analysis.html

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Physical Audio Signal Processing

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Bridge Transmittance

A Two-Resonance Guitar Bridge

Digitizing Bridge Reflectance

Converting continuous-time transfer functions such as $\hat{\rho}_b(s)$ and $\hat{\tau}_b(s)$ to the digital domain is analogous to converting an analog electrical filter to a corresponding digital filter--a problem which has been well studied [346]. For this task, the bilinear transform (§7.3.2) is a good choice. In addition to preserving order and being free of aliasing, the bilinear transform preserves the positive-real property of passive impedances (§C.11.2).

Digitizing $\hat{\rho}_b(s)$ via the bilinear transform (§7.3.2) transform gives

$\displaystyle \hat{\rho}_b^d(z) \isdefs \hat{\rho}_b\left(c\frac{1-z^{-1}}{1+z^{-1}}\right)$

which is a second-order digital filter having gain less than one at all frequencies--i.e., it is a Schur filter that becomes an allpass as the damping $\mu$ approaches zero. The choice of bilinear-transform constant $c=1/\tan(\omega_0T/2)$ maps the peak-frequency $\omega_0$ without error (see Problem 4).

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Digitizing Bridge Reflectance

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University