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Derivation of the Wave Digital Inductor

It is instructive to see what happens when this change of variables in applied to the inductor.

  1. We have the following difference equation for the inductor (using the trapezoidal rule of numerical integration, or bilinear transform, as you prefer):

    $\displaystyle i_n = i_{n-1} + \frac{T}{2L} (v_n + v_{n-1})
$

  2. Perform the wave-variable substitution

    $\displaystyle v_n = \frac{a_n+b_n}{2} \qquad
i_n = \frac{a_n-b_n}{2R_0}
$

    to get

    $\displaystyle \frac{a_n - b_n}{2R_0} = \frac{a_{n-1} - b_{n-1}}{2R_0} +
\frac{T}{2L}\left(\frac{a_n + b_n}{2} + \frac{a_{n-1} + b_{n-1}}{2}\right)
$

  3. Now choose $ R_0 = 2L/T$ to obtain

    \begin{eqnarray*}
a_n - b_n &=& a_{n-1} - b_{n-1} + (a_n + b_n + a_{n-1} + b_{n-1}) \\
&=& 2 a_{n-1} + a_n + b_n
\end{eqnarray*}

    which further simplifies to

    $\displaystyle \zbox{b_n = - a_{n-1}} \qquad \hbox{(Wave Digital Inductor)}
$

Now there is no direct path from input to output.

In terms of wave variables, with simplest choices of the port resistances, we obtain the following wave digital filter elements (the elementary one-ports):

\epsfig{file=eps/WD1ports.eps,width=6.5in}
Elementary wave-digital one-ports. The port impedances for the wave-digital inductor, capacitor, and resistor (on the right) are defined as $ 2L/T$ , $ T/(2C)$ , and $ R$ , respectively.


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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 © 2014-03-25 by Stefan Bilbao and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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