## 2 Approximated Lowpass Elliptic Response
### Asymmetrical Fittings  
In the previous fittings, each of the selected frequency bands were symmetrical in terms of the number of frequency decades included above and below 1 rad/s. However, it is possible to choose an asymmetrical frequency band to manipulate the distribution of data in the passband and stopband within the fitting routine. To evaluate the differences this causes in the returned coefficients, the fitting routine was applied to five different frequency bands: (i) \(\omega_{B1}\,\epsilon\,\left[10^{-5},10^{5}\right]\) rad/s; (ii) \(\omega_{A2}\,\epsilon\,\left[10^{-5},10^{3}\right]\) rad/s; (iii) \(\omega_{A3}\,\epsilon\,\left[10^{-5},10^{1}\right]\) rad/s; (iv) \(\omega_{A4}\,\epsilon\,\left[10^{-3},10^{5}\right]\) rad/s; and (v) \(\omega_{A5}\,\epsilon\,\left[10^{-1},10^{5}\right]\) rad/s. The frequency bands \(\omega_{A2,A3}\) were selected to introduce a greater passband frequency range into the fitting procedure and \(\omega_{A4,A5}\) were selected to introduce a greater stopband frequency range. These ranges are all within the frequency bands explored for the symmetrical fittings. Applying the previously described optimization fitting routine, the coefficients extracted from each frequency band for \(\alpha=0.1\) to \(\alpha=0.9\) in 0.01 steps are given in Figure 4a,b. In Figure 4a, the solid lines represent the coefficients using \(\omega_{B1}\), dashed lines those using \(\omega_{A2}\), and dash-dotted those using \(\omega_{A3}\). Note that in these cases the set of returned coefficients shows very few differences based on the different frequency bands. This suggests that the frequency range of the stopband has little effect on the coefficients when a large sampling of the passband is used. In Figure 4b, the solid lines represent the coefficients using \(\omega_{B1}\), dashed lines those using \(\omega_{A4}\), and dash-dotted those using \(\omega_{A5}\). Similar to the symmetric results, the coefficients (\(a_{4}\), \(a_{3}\), \(a_{2}\), \(a_{1}\)) are very similar for \(\alpha>0.7\) for all fitting cases. That is, the greatest differences between the coefficients extracted using the different frequency bands are observed at lower \(\alpha\) values, with a decreasing trend for coefficients (\(a_{2}\), \(a_{3}\), \(a_{4}\)) observed at lower \(\alpha\) values.  
To further evaluate the differences of the asymmetric fitting frequency bands, the DC gain and HFG for \(\alpha=0.1\) to \(\alpha=0.9\) in steps of 0.01 are detailed in Figure 5. Figure 5a presents the gains for \(\omega_{B1}\) (solid), \(\omega_{A2}\) (dashed), and \(\omega_{A3}\) (dash-dotted). Observing the DC gains, the magnitude for each fitting is very close to the theoretical value of \(-5\) dB for \(\alpha>0.2\). This is not unexpected, since the DC gain is related to coefficient \(a_{4}\), which is noted in Figure 4 to show very little variation for this set of asymmetrical fittings. Similar to the symmetric fittings, the HFG shows variation based on the frequency band, with the frequency band that includes the greatest number of data in the stopband (\(\omega_{A3}\)) showing the closest agreement with the elliptic case. Figure 5b presents the gains for \(\omega_{B1}\) (solid), \(\omega_{A4}\) (dashed), and \(\omega_{A5}\) (dash-dotted). In these cases, the HFG show similar values for all frequency fittings and the DC gains showing the greatest deviations. This confirms the previous expectations that the distribution of data in the passband and stopbands impacts the fitting process. With those frequency bands that have a larger representation of data yielding filter characteristics closer to the elliptic characteristics.  
Figure 4: Coefficients of fractional-order transfer function in Equation (7) to approximate second-order elliptic characteristics applying least squares fitting to asymmetrical frequency ranges with: (**a**) lower limits of \(\omega=10^{-5}\) rad/s; and (**b**) upper limits of \(\omega=10^{5}\).