## 3 Fitted Frequency Range Comparison  
To quantify the differences using coefficients from the different fitted frequency ranges, MATLAB simulations of the magnitude responses of Equation (7) for \(\alpha=0.75\) and \(\alpha=0.25\) are given in Figure 8. The solid, dashed, and dash-dotted lines represent the simulated responses using the coefficients from the optimization fittings applied to the \(\omega_{B1}\), \(\omega_{B2}\), and \(\omega_{B3}\) frequency ranges, respectively. Each of these simulations used the coefficient values detailed in Figure 2. For the \(\alpha=0.75\) case, the most significant differences are observed above 5 rad/s. Above this frequency, the stopband ripple occurs at the lowest frequency with the least attenuation when the coefficients from the smallest frequency band, \(\omega_{B3}\) are used. These simulations support the results in Figure 3, with the \(\omega_{B1}\) case yielding the closest high frequency gain to the elliptic case. Figure 8: _Cont._  
Figure 7: Minimum phase angle of roots of Equation (10) using coefficients from asymmetrical frequency fittings with: (**a**) lower limits of \(\omega=10^{-5}\) rad/s; and (**b**) upper limits of \(\omega=10^{5}\). The magnitude characteristics of Equation (7) using the extracted coefficients from the asymmetrical fittings in Figure 4 for \(\alpha=0.25\) and \(\alpha=0.75\) are given in Figures 9 and 10, respectively. These magnitude responses highlight the previous DC and HFG differences in Figure 5. For the coefficients derived using frequency bands with greater passband data (\(\omega_{A2,A3}\)), the magnitude responses show very good agreement for both \(\alpha=0.25\) and \(\alpha=0.75\) up to 1 rad/s. At higher frequencies, the responses with fewer stopband data show poorer agreement with the elliptic response. This behavior is reversed in the magnitude responses that use the coefficients derived from the \(\omega_{A4,A5}\). That is, the magnitude characteristics above 1 rad/s show very good agreement for all sets of coefficients for both \(\alpha=0.25\) and \(\alpha=0.75\) cases, resulting from the stopband containing the greatest number of data in the fittings, which results in the best fit of these region. Figure 8: Simulated magnitude responses of (**a**) \((1+0.75)\) and (**b**) \((1+0.25)\) lowpass fractional order transfer function using coefficients from different symmetrical frequency fitting ranges. Figure 9: Simulated magnitude responses of \((1+0.25)\) lowpass fractional order transfer function using coefficients from different asymmetrical frequency fitting ranges with: (**a**) lower limits of \(\omega=10^{-5}\) rad/s; and (**b**) upper limits of \(\omega=10^{5}\). Comparing the \(\alpha=0.25\) cases of the symmetrical fittings in Figure 8, the coefficients have a significant impact on the HFG, DC gain and passband ripples of the magnitude characteristics. While the \(\omega_{B3}\) case yields the most significant low- and high-frequency differences compared to the elliptic response, it has the closest approximation of the passband ripple. The elliptic response reaches a maximum of 0 dB, which is 5 dB above the DC gain, while the \(\omega_{B3}\) case reaches \(-0.40\) dB. For comparison, the \(\omega_{B2}\) and \(\omega_{B1}\) cases reach \(-1.79\) dB and \(-2.26\) dB, respectively. This is expected to be an impact of the frequency range, with the \(\omega_{B2}\) and \(\omega_{B1}\) fittings having a larger weighting (based on the distribution of data) to fit the high and low frequency bands which reduces the weighting of the interior band. As a result, the \(\omega_{B3}\) case provides the coefficients that best approximate the magnitude characteristics in this band. This is also observed in the asymmetrical fittings of Figure 9, with the \(\omega_{AA,A5}\) cases showing better agreement with the passband ripple than the \(\omega_{A2,A3}\) cases. Again, this is a result of the \(\omega_{AA,A5}\) cases having fewer data in the flat passband region to influence the fitting process.