## 5 Conclusions  
In the paper, a concept of the second order time-varying IIR notch filter with reduced transient response based on the Bezier curve pole radius variability is presented. Thanks to the Bezier curve we are able to generate the function that changes the pole radius in a more flexible manner if compared to the exponential function. The proposed filtering algorithm is able to suppress the transient response much better than the traditional time-invariant filter and the filter based on the exponential pole radius variability. The proposed filter with suppressed transient response is able to process input signals faster than traditional solutions. As an example, the proposed notch filter has been successfully used to suppress the powerline interference from ECG signals. The performance of the considered filter structure has been evaluated using four different indices. Future works will be devoted to the implementation of the proposed filter structure. For example we are going to use the concept of coupled form second order sections [16; 17; 18] in our analysis and implementation works, in particular in case of using short word-length fixed-point binary arithmetic.  
Conceptualization, J.P. and S.K.; methodology, J.P. and S.K.; software, S.K.; validation, J.P., and S.K.; formal analysis, J.P.; investigation, S.K.; resources, S.K.; data curation, S.K.; writing--original draft preparation, S.K.; writing--review and editing, J.P.; visualization, S.K.; supervision, J.P.  
This research received no external funding.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**SNR Ratio** & \begin{tabular}{c} **Time-Invariant** \\ Filter** \\ \end{tabular} & \
\begin{tabular}{c} **Time-Varying Filter** \\ **Based on Exponential Function** \\ \end{tabular} &
\begin{tabular}{c} **Proposed Time-Varying Filter** \\ **Based on Bézier Curve** \\ \end{tabular} \\ \hline \(-40\) & 0.1274 & 0.1612 & 0.2622 \\ \(-30\) & 0.3708 & 0.4523 & 0.6438 \\ \(-20\) & 0.7793 & 0.8434 & 0.9303 \\ \(-10\) & 0.9633 & 0.9742 & 0.9861 \\ \(-5\) & 0.9833 & 0.9869 & 0.9908 \\ \(0\) & 0.9899 & 0.9910 & 0.9923 \\ \(5\) & 0.9920 & 0.9923 & 0.9927 \\ \(10\) & 0.9926 & 0.9928 & 0.9929 \\ \(20\) & 0.9929 & 0.9929 & 0.9929 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Comparison of the \(\rho\) index for the time-invariant notch filter and time-varying filters based on the Bézier parametric curve and exponential function for various values of the SNR.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**SNR Ratio** & \begin{tabular}{c} **Time-Invariant** \\ Filter** \\ \end{tabular} & \
\begin{tabular}{c} **Time-Varying Filter** \\ **Based on Exponential Function** \\ \end{tabular} &
\begin{tabular}{c} **Proposed Time-Varying Filter** \\ **Based on Bézier Curve** \\ \end{tabular} \\ \hline \(-40\) & 16.9566 & 19.0389 & 23.4573 \\ \(-30\) & 16.9484 & 19.0244 & 23.4205 \\ \(-20\) & 16.8650 & 18.8861 & 23.0565 \\ \(-10\) & 16.1040 & 17.7133 & 20.4802 \\ \(-5\) & 14.6987 & 15.7821 & 17.3341 \\ \(0\) & 12.0470 & 12.5789 & 13.2365 \\ \(5\) & 8.4580 & 8.6596 & 8.8949 \\ \(10\) & 4.8698 & 4.9360 & 5.0142 \\ \(20\) & 0.8136 & 0.8190 & 0.8265 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Comparison of the SNR improvement index for the time-invariant notch filter and time-varying filters based on the Bézier parametric curve and exponential function for various values of the SNR.  
**Acknowledgments:** The paper is dedicated to the memory of Professor [PERSON], who was one of the pioneers of the theory of time-varying filters.  
**Conflicts of Interest:** The authors declare no conflict of interest.