## 3 Time-Varying Pole Radius of the Filter  
In order to suppress the transient response of the considered second order IIR notch filter it was assumed that the value of the pole radius \(r\) is temporarily varied in time. The proposed method gives the possibility to maintain the frequency specification and, at the same time, to improve the time-domain behaviour of the filter. The main assumption of the proposed idea is to change the pole radius from some set value to a value that is very close to unity. The proposed method allows us to obtain the notch filter with both a suppressed transient response and a narrow bandwidth. The transfer function of the second time-varying IIR notch filter can be expressed in the discrete time domain by the following difference equation:  
\[y(n)=x(n)+b_{1}x(n-1)+x(n-2)-a_{1}(n-1)y(n-1)-a_{2}(n-2)y(n-2), \tag{6}\]  
where \(y(n)\) and \(x(n)\) are the filter output and the input signal, respectively. The coefficient \(b_{1}\) is equal to \(-2\cos\left(\omega_{0}\right)\), \(a_{1}(n)=-2r(n)\cos\left(\omega_{0}\right)\), and \(a_{2}(n)=r^{2}(n)\). In order to control the time variation of the pole radius \(r(n)\) the 3 rd order Bezier curve is used. We have used the Bezier curve because this kind of function can be easily and freely shaped by means of predefined control points. 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 parametric curve can be expressed in the following form:  
\[B\left(k\right)=B_{1}\left(1-k\right)^{3}+3B_{2}k\left(1-k\right)^{2}+3B_{3} k^{2}\left(1-k\right)+B_{4}k^{3}, \tag{7}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Pole Radius** & **Settling Time [s]** & **Bandwidth [Hz]** \\ \hline
0.2000 & 0.0080 & 127.3239 \\
0.4000 & 0.0100 & 95.4929 \\
0.5000 & 0.0160 & 79.5774 \\
0.6000 & 0.0180 & 63.6619 \\
0.7000 & 0.0260 & 47.7464 \\
0.8000 & 0.0380 & 31.8309 \\
0.9000 & 0.0760 & 15.9154 \\
0.9900 & 0.7760 & 1.59154 \\
0.9990 & 7.8260 & 0.1591 \\
0.9999 & 78.2360 & 0.0159 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Settling time and bandwidth in the function of the pole radius. where the Cartesian coordinates of the Bezier curve control points are given as: \(B_{1}=(x_{1},y_{1})\), \(B_{2}~{}=~{}(x_{2},y_{2})\), \(B_{3}=(x_{3},y_{3})\), \(B_{4}=(x_{4},y_{4})\). The control points \(B_{1}\) and \(B_{4}\) determines respectively the starting and the ending point of the considered curve. The control points \(B_{2}\) and \(B_{3}\) deliver directional information. The distance between these control points specifies how the parametric curve moves towards the point \(B_{2}\) before turning towards point \(B_{3}\). The coordinates of point \(B_{1}\) includes the starting point of the time-varying formula \(x_{1}=0\) and the starting pole radius \(y_{1}=r_{start}\). On the other hand, the coordinates of \(B_{4}\) determine the length of the variability time horizon \(x_{4}=M\) and the final value of the pole radius \(y_{4}=r_{final}\). In order to find the coordinates of the control points \(B_{2}\) and \(B_{3}\) the simulated annealing algorithm is used. The main aim of the annealing algorithm is to find \(B_{2}\) and \(B_{3}\) whose coordinates minimize the mean square error defined as:  
\[MSE_{tv}=\frac{1}{K}\sum_{n=1}^{K}\left|y(n)-s(n)\right|^{2}, \tag{8}\]  
where \(K\) is the duration of the input signal, \(y(n)\) is the output of the signal, and \(s(n)\) is the clean original signal. Equation (7) can be reformulated to the following form:  
\[B_{x}\left(k\right) =x_{1}\left(1-k\right)^{3}+3x_{2}k\left(1-k\right)^{2}+3x_{3}k^{2 }\left(1-k\right)+x_{4}k^{3},\] \[B_{y}\left(k\right) =y_{1}\left(1-k\right)^{3}+3y_{2}k\left(1-k\right)^{2}+3y_{3}k^{2 }\left(1-k\right)+y_{4}k^{3}. \tag{9}\]  
To find the values of the function \(B_{x}=F\left(B_{y}\left(k\right)\right)\) at query points for each sample of the filter transient response the interpolation procedure has been used.