## 2 The Existing Algorithms Based on Pole and Zero Recursive (Geometric) Distribution
Figure 1 highlights this distribution and also compares the asymptotic Bode plots of \(I^{v}(s)\), \(I^{v}_{a}(s)\) and \(I^{v}_{N}(s)\). It also shows that the high and low asymptotic frequency behaviors are constant. In [22], to obtain an integer integral like asymptotic behavior at low and high frequency, relation (2) is replaced by:

\[I^{v}(s)\approx I^{v}_{a}(s)=C_{0}\frac{1}{s}\left(\frac{1+\frac{s}{\omega_{h }}}{1+\frac{s}{\omega_{b}}}\right)^{v-1}\qquad\text{with}\qquad C_{0}=\left(For a fractional differentiation operator defined in the Laplace domain by

\[D^{\
u}(s)=s^{\
u}\quad\text{with}\quad 0<\
u<1 \tag{11}\]

the frequency truncation leads to

\[D^{\
u}(s)\approx D^{\
u}_{a}(s)=\mathcal{C}_{0}\Bigg{(}\frac{1+\frac{s}{\omega _{b}}}{1+\frac{s}{\omega_{b}}}\Bigg{)}\qquad\text{with}\qquad C_{0}=\bigg{(} \frac{\omega_{b}}{\omega_{h}}\bigg{)}^{\
u}\bigg{(}\frac{{\omega_{h}}^{2}+1}{{ \omega_{b}}^{2}+1}\bigg{)}^{\frac{\
u}{2}} \tag{12}\]

and permits the following algorithm (Algorithm 3). ```
1. Initialize \[r=\alpha\eta=\bigg{(}\frac{\omega_{h}}{\omega_{b}}\bigg{)}^{\frac{1}{h}}\quad \alpha=r^{\
u}\quad\eta=r^{1-\
u}\quad\omega^{\prime}{}_{1}=\eta^{1/2}\omega_{ b}\quad\omega_{1}=\alpha\omega^{\prime}{}_{1}\] (13)
2. For\(i\in[1\ldots N]\)do \[\omega^{\prime}{}_{i+1}=\kappa\omega^{\prime}{}_{i}\qquad\omega_{i+1}=r \omega_{i}\] (14) End for 3. Compute \(\mathcal{C}^{\prime}{}_{0}\) with relation (5)
4. Define the fractional differentiator (12) approximation in the frequency band [\(\omega_{b}\), \(\omega_{h}\)], by the transfer function \[I^{\
u}(s)\approx I^{\
u}_{a}(s)=C_{0}\left(\frac{1+\frac{s}{\omega_{h}}}{1+ \frac{s}{\omega_{h}}}\right)^{\
u}\qquad\text{with}\qquad C_{0}=\left(\frac{ \omega_{h}}{\omega_{b}}\right)^{\
u}\left(\frac{{\omega_{b}}^{2}+1}{{\omega_{ h}}^{2}+1}\right)^{\frac{\
u}{2}}. \tag{2}\]0 (15)  
**Algorithm 3** Fractional differentiator approximation--first method  
By analogy with Algorithm 2, the following algorithm (Algorithm 4) permits to get an integer differentiation like asymptotic behavior in low and high frequency. The set of algorithms previously established for a real fractional differentiation order can be extended to a complex fractional differentiation order \(\
u=a+\mathrm{i}b\)[21]. Approximation of a Fractional Integrator by a Recursive Distribution of Poles and Zeros: An Analytical Approach  
For the approximation given in relation (1) using Algorithm 1, the analytical demonstration of the graphical approach presented in the previous section was done in [21] by considering the limit case where \(\omega_{b}\) tends towards 0 and \(\omega_{\mathrm{h}}\) tends towards infinity. However, a simpler demonstration based on the impulse response of a fractional operator is now used. This impulse response is defined by:  
\[I^{\
u}(s)\approx I^{\
u}_{a}(s)=C_{0}\left(\frac{1+\frac{s}{\omega_{h}}}{1+ \frac{s}{\omega_{h}}}\right)^{\
u}\qquad\text{with}\qquad C_{0}=\left(\frac{ \omega_{h}}{\omega_{b}}\right)^{\
u}\left(\frac{{\omega_{b}}^{2}+1}{{\omega_{ h}}^{2}+1}\right)^{\frac{\
u}{2}}. \tag{2}\]1  
where \(\mathcal{E}^{-1}\) denotes the inverse Laplace-Transform.