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Digitized Reflectances Without Delay-Free Paths

Plan:

  1. Fix the bilinear-transform frequency-scaling parameter $ c$ once for the whole system (so there is only one frequency-warping)
  2. Set the ``connector'' wave impedance $ R_0$ separately for each circuit element to eliminate the delay-free path in its reflectance
  3. We will then get scattering when we connect different elements together

This yields the following elementary reflectances:

\begin{eqnarray*}
\hbox{\textbf{\underline{Element}}} && \hbox{\textbf{\underline{Reflectance}}}\\
\hbox{\emph{ideal spring (capacitor)}} &\leftrightarrow& \hbox{\emph{unit delay}}\\
\hbox{\emph{ideal mass (inductor)}} &\leftrightarrow& \hbox{\emph{unit delay and sign inversion}}\\
\hbox{\emph{ideal dashpot (resistor)}} &\leftrightarrow& 0
\end{eqnarray*}

The original element values remain only in the waveguide-interface impedances $ R_0=k/c,mc,\mu$


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Download WaveDigitalFilters.pdf
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``Wave Digital Filters'', by Stefan Bilbao and Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2017-06-05 by Stefan Bilbao and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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