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:

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

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:

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

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

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

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

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

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].

Figure: State-space realization in Fig.3 digitized via forward and backward Euler.