# Transient Time Reduction in Time-Varying Digital Filters via Second-Order Section Optimization
## 4 Results
### Robustness Analysis  
A natural way to highlight the benefits of introducing time-varying coefficients into a single second-order section is to compare how both the classical, fully LTI filter and the partially time-varying design perform under changing signal conditions. Specifically, analyzing their time-domain behavior when subjected to different noise levels or varying signal frequencies provides clear evidence of how transient response and overall performance are impacted. By overlaying their outputs on a noisy input signal, one can observe which approach settles faster and better suppresses undesired oscillations. This direct, side-by-side comparison helps quantify whether allowing even a single SOS can significantly reduce the transient period without degrading the filter's long-term characteristics. Figure 7 illustrates an exemplary performance of both the classical and the proposed structures within a noised environment. In the first experiment, we added a high-frequency (300 Hz) noise component on top of an otherwise simple step input signal. The dashed black trace in Figure 7 shows how the noise significantly distorts the step's clean rise. A conventional time-invariant filter (blue line) reduces much of the noise but still exhibits a notable transient, including an overshoot and oscillations as it settles. By contrast, modifying a single second-order section to have time-varying coefficients (red line) shortens the settling period while maintaining effective noise suppression. This illustrates how the proposed approach can outperform a purely LTI filter when faced with strong high-frequency interference. To further evaluate the robustness of the proposed method, we examined a more challenging case where the interfering signal lies just above the passband of the elliptic filter. Specifically, we added a 110 Hz sinusoidal component--which is only 10 Hz above the cutoff frequency of the sixth-order low-pass elliptic filter (Fc = 100 Hz)--to a step \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Transient Time Threshold** & **Sample Index (LTI)** & **Time (LTI)** & **Sample Index (LTV)** & **Time (LTV)** \\ \hline
1\% & 198 & 0.197 s & 118 & 0.117 \\ \hline
2\% & 176 & 0.175 s & 68 & 0.067 \\ \hline
5\% & 86 & 0.085 s & 16 & 0.015 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Transient time comparison between classical (LTI—linear time-invariant) and the proposed (LTV—linear time-varying) method. Figure 7: Classical linear time-invariant filtering structure compared with proposed linear time-varying SoS on exemplary test signal. signal. This setup stresses the filter's ability to suppress high-frequency content close to the transition band, while still preserving a rapid transient response. Figure 8 illustrates an exemplary performance of both the classical and the proposed structures within a challenging noised environment. As shown in Figure 8, the classical time-invariant filter (blue) exhibits a relatively slow settling behavior with noticeable oscillations caused by the nearby frequency interference. In contrast, the proposed design with one time-varying SOS (red) converges more quickly and cleanly, effectively dampening the transient while maintaining similar steady-state accuracy. This result highlights the method's strength in handling tight spectral proximity scenarios, which are common in dense signal environments. Figure 9 presents the classical notch-type IIR filter compared to another implementation of the proposed method. In this experiment, we employ an IIR notch filter designed to attenuate a 100 Hz disturbance while preserving both the step component and any relevant lower frequencies. The dashed black curve again shows the input signal, now containing a step, a 5 Hz \"useful\" component, and strong 100 Hz noise. The notch filter's time-invariant version (blue) substantially reduces the 100 Hz amplitude but still suffers from a longer transient and overshoot.