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Digitization of Second-Order Continuous-Time Lowpass Filters in State Space Form

The only elements in Fig.3 needing modification for digitization are the two integrators, each having transfer function $ 1/\tilde{s}$. Normally the preferred digitization method is the bilinear transform:

$\displaystyle \tilde{s}\leftarrow \frac{1-z^{-1}}{1+z^{-1}}
$

However, the bilinear transform cannot be used here due to the presence of feedback which would give a delay-free loop.

The Forward Euler (FE) finite-difference scheme introduces a sample of delay in the digitization of $ 1/s$ that avoids a delay-free loop:

$\displaystyle \tilde{s}\leftarrow \frac{z-1}{T} \;=\;\frac{1-z^{-1}}{T\,z^{-1}}
$

where $ T$ denotes the sampling interval in seconds. There is also a Backward Euler (BE) finite-difference scheme:

$\displaystyle \tilde{s}\leftarrow \frac{1-z^{-1}}{T}
$

It can be effective to use FE and BE together in alternation to avoid delay build-up in either direction:

$\displaystyle \tilde{s}^2 = \tilde{s}\cdot \tilde{s}\leftarrow \frac{z-1}{T} \cdot \frac{1-z^{-1}}{T} = \frac{z - 2 + z^{-1}}{T^2}
$

Digitizing the two integrators in Fig.3 via FE and BE respectively and removing the frequency normalization yields

\begin{eqnarray*}
\frac{1}{s} &\leftarrow& \omega_c T \frac{z^{-1}}{1-z^{-1}}\\
&\mbox{and}& \omega_c T \frac{1}{1-z^{-1}}.
\end{eqnarray*}

The resulting digital filter is drawn in Fig.4. This structure is normally called the Chamberlin form digital filter section [1].

\includegraphics{eps/dssm}
State-space realization in Fig.3 digitized via forward and backward Euler.


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``Digital State-Variable Filters'', by Julius O. Smith III.
Copyright © 2013-05-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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