## 3 Filter Dynamics  
The dynamic properties of the second-order lowpass analog and discretized filters are described by the natural frequency \(\omega_{0}\), the damping ratio \(\zeta\), and the gain coefficient \(k\). All these parameters affect the duration of the transient response of the considered filter. Figure 3 presents the step response of the digital filter with three different values of the gain \(k\), \(\omega_{0}=8.41\cdot 10^{-1}\), \(\zeta=3.83\cdot 10^{-1}\), and \(t_{s}=1/10\) s.  
Table 1 presents the detailed characteristics of the presented step responses. As one can notice, the \(k\) parameter does not have a direct influence on the settling time values. On the other hand, let us assume that the \(k\) parameter is initially larger than the base value but, further on (with the right timing), is set back to the aforementioned pre-determined number. One could expect the final value of the response as expected with this design, but with a reduced settling time characteristic. As the change of the parameter \(k\) would happen only for a brief moment and only at the beginning of the processing, one can expect the frequency characteristic to be as designed. Figure 4 presents the step response of the filter with three different values of \(w_{0}\), \(k=1\), \(\zeta=3.83\cdot 10^{-1}\), \(t_{s}=1/10\) s.  
Figure 2: Bode diagram (**a**) of the analog model, bode diagram (**b**) of the digital model. Figure 1: Comparison of the analog and digital models of the oscillatory system for \(\omega_{0}=8.41\cdot 10^{-1}\), \(\zeta=3.83\cdot 10^{-1}\), normalized gain \(k=1\), and \(t_{s}=1/10\) s. \begin{table}
\begin{tabular}{l c c c} \hline \hline  & \(k\) = 0.5 & \(k\) = 1 & \(k\) = 2 \\ \hline RiseTime & 16.9859 & 16.9859 & 16.9859 \\ SettlingTime & 132.1759 & 132.1759 & 132.1759 \\ SettlingMin & 0.4630 & 0.9260 & 1.8520 \\ SettlingMax & 0.6360 & 1.2720 & 2.5440 \\ Overshoot & 27.9031 & 27.9031 & 27.9031 \\ PeakTime & 41 & 41 & 41 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Step response characteristics for \(k\) = 0.5, 1, and 2. RiseTime—time it takes for the response to rise from 10% to 90% of the steady-state response; SettlingTime—time it takes for the error to fall to within 2% of steady-state response; SettlingMin—minimum value once the response has risen; SettlingMax—maximum value once the response has risen; Overshoot—percentage overshoot; and PeakTime—time at which the peak value occurs. Figure 4: Comparison of the digital oscillatory system with \(\omega_{0}\) = 4.205\(\cdot\)10\({}^{-1}\) (orange), 8.41\(\cdot\)10\({}^{-1}\) (blue), and 1.682 (red); \(\zeta\) = 3.83\(\cdot\)10\({}^{-1}\); \(k\) = 1; and \(t_{s}\) = 1/10 s.  
Figure 3: Comparison of the digital oscillatory system with \(\omega_{0}\) = 8.41\(\cdot\)10\({}^{-1}\); \(\zeta\) = 3.83\(\cdot\)10\({}^{-1}\); \(k\) = 0.5 (orange), 1 (blue), and 2 (red); and \(t_{s}\) = 1/10 s.  
Table 2 presents the detailed characteristic of the step responses, similar to Table 1.  
This example shows that, by increasing \(\omega_{0}\), the settling time reduced. Similar to the gain coefficient, one can deduce that it is desirable for the filter to behave as a model with large \(\omega_{0}\) at the beginning of the response and, once reaching a close proximity to the steady state, \(\omega_{0}\) should be set back to the predefined value to preserve the frequency characteristic demands. Figure 5 presents the step response of the filter with three different values of the \(\zeta\) parameter, \(k=1\), \(\omega_{0}=8.41\cdot 10^{-1}\), and \(t_{s}=1/10\) s.  
Table 3 presents the detailed characteristic of the step responses, similar to Tables 1 and 2. Similar to the \(\omega_{0}\) parameter, one can draw the conclusion that increasing the value of \(\zeta\) improves the settling time of the system response.