## 4 Results
### ECG Filtering  
In the first step, the evaluation and quantitative comparison of the performance of ECG denoising processes for three various ECG signals were carried out. The results are presented in Table 3. For each signal four simulations have been done. The ECGs start at a different signal feature point in each simulation. The main aim of the proposed filtering algorithm is to work out the denoised ECG signal as fast as possible. The portion of the ECG signal lost on the transient response may impede appropriate measures.  
As one can notice, the proposed IIR notch filter with time-varying pole radius outperformed the traditional time-invariant filter structure in all considered cases. The value of index SNR\({}_{imp}\) for the proposed time-varying filter was greater for each simulation than in the time-invariant filter case. Moreover, the values of the cross-correlation coefficient \(\rho\), percentage root mean square difference index, and mean square error index show that the proposed parameter-varying notch filter is definitely better if compared to the traditional filter with time-invariant pole radius. In Figures 4-6 the results of simulations are presented.  
\begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline \multirow{2}{*}{**ECG Signal**} & \multicolumn{1}{c}{\begin{tabular}{c} **SNR\({}_{imp}\)** \\ **Time** \\ **Invariant** \\ \end{tabular} } & \multicolumn{1}{c}{\begin{tabular}{c} **SNR\({}_{imp}\)** \\ **Time** \\ **Varying** \\ \end{tabular} } & \multicolumn{1}{c}{\begin{tabular}{c} **\(\rho\) Time** \\ **Invariant** \\ \end{tabular} } & \multicolumn{1}{c}{\begin{tabular}{c} **PRD** \\ **Time** \\ **Invariant** \\ \end{tabular} } & \multicolumn{1}{c}{\begin{tabular}{c} **MSE** \\ **Time** \\ **Invariant** \\ \end{tabular} } & \multicolumn{1}{c}{\
\begin{tabular}{c} **MSE** \\ **Time** \\ **Varying** \\ \end{tabular} } & \multicolumn{1}{c}{
\begin{tabular}{c} **MSE** \\ **Time** \\ **Invariant** \\ \end{tabular} } \\ \hline ECG 1, Q point start & 3.1419 & 4.1937 & 0.9888 & 0.9912 & 0.1520 & 0.1346 & 0.0231 & 0.0181 \\ ECG 1, R point start & 3.3275 & 4.5226 & 0.9893 & 0.9919 & 0.1479  
Figure 4: Comparison between time-invariant infinite impulse response (IIR) notch filter and the proposed time-varying filter. Testing signal ECG 1 starts at various featured points.  
Figure 5: Comparison between time-invariant IIR notch filter and the proposed time-varying filter. Testing signal ECG 2 starts at various featured points.  
Figure 6: Comparison between time-invariant IIR notch filter and the proposed time-varying filter. Testing signal ECG 3 starts at various featured points.  
In the second phase of the comparative analysis all the considered filters are tested using the ECG signal distorted by the 50 Hz powerline interference. In Figures 9 and 10 the results of filtering are presented. As one can notice, the proposed notch filter based on the Bezier parametric curve is characterized by a much shorter transient response if compared to the filter based on the exponential pole radius variability. The advantages of the filter based on the Bezier curve are also reflected in all quality indices listed in Table 5.