## 2 Approximated Lowpass Elliptic Response
### Symmetrical Fittings  
It was noted previously in [20] that the frequency band used in the fitting procedure could impact the coefficients. However, this impact has not yet been formally investigated, providing the motivation for this exploration. This work quantifies the differences (in terms of coefficients and overall magnitude characteristics) that result from using different frequency bands in the optimization procedure. The least squares fitting procedure described by Equation (9) was applied to three symmetrical test cases. The three frequency bands were \(\omega_{B1}\,\epsilon\,\left[10^{-5},10^{5}\right]\) rad/s, \(\omega_{B2}\,\epsilon\,\left[10^{-3},10^{3}\right]\) rad/s, and \(\omega_{B3}\,\epsilon\,\left[10^{-1},10^{1}\right]\) rad/s. These frequency bands, labeled in Figure 1a, were selected because they each capture the passband ripple and transition to the stopband (which occurs between approximately \(10^{-1}\) rad/s and \(10^{1}\) rad/s). The wider ranges (\(\omega_{B1}\), \(\omega_{B2}\)) capture more data in the flat regions of the stopband and passband. The coefficients determined for the fitting with each frequency band for \(\alpha=0.1\) to \(\alpha=0.9\) in 0.01 steps are given in Figure 2. The solid lines in Figure 2 represent the coefficients using \(\omega_{B1}\), dashed lines those using \(\omega_{B2}\), and dash-dotted those using \(\omega_{B3}\). The coefficients (\(a_{4}\), \(a_{3}\), \(a_{2}\), \(a_{1}\)) are very similar for \(\alpha>0.7\) for all fitting cases. The greatest differences between coefficients determined using the different frequency bands are observed at lower \(\alpha\) values. Therefore, the difference in fitted frequency bands do not have a significant effect on the determined coefficients for \(\alpha>0.7\) in these specific cases. Coefficients (\(a_{2}\), \(a_{3}\), \(a_{4}\)) all display a general trend of decreasing values for the cases using smaller frequency fitting bands (i.e., the fitting for \(\omega_{B3}\) shows the lowest coefficient values for \(a_{2}\), \(a_{3}\), and \(a_{4}\)). MATLAB simulations of Equation (8) using the \(\omega_{B1}\) coefficients for \(\alpha=0.1\) to \(\alpha=0.9\) in steps of 0.1 are given in Figure 1a as dashed lines. From these simulations, it is clear that this approximation method does realize the fractional-step attenuations (with the slope in the transition band strictly dependent on \(\alpha\)) but that there are further differences compared to the elliptic response. The most significant difference is that the HFG of each fractional-order case is higher than the \(-50\) dB magnitude of the elliptic response (detailed in Figure 1a). Further, the passband ripple is smaller for each fractional-order case (detailed in Figure 1b) with the smallest passband ripple (i.e., the lowest passband gain) occurring for the \(\alpha=0.1\) case. This case, given in Figure 1b, reaches approximately \(-3.6\) dB compared to 0 dB of the second-order elliptic response. This is understandable, as \(\alpha\) approaches zero the filter order is approaching 1, with first-order filters not able to provide any ripple or higher \(Q\). The differences in DC gain and HFG for each set of coefficients are further detailed in Figure 3 for \(\alpha=0.1\) to \(\alpha=0.9\) in steps of 0.01. The solid, dashed, and dash-dotted lines correspond to those cases using the coefficients from the \(\omega_{B1}\), \(\omega_{B2}\), and \(\omega_{B3}\) frequency bands, respectively. Observing the DC gains, the magnitude for each fitting is very close to the theoretical value of \(-5\) dB for \(\alpha>0.7\), but it decreases at lower values of \(\alpha\). These deviations are most significant for the \(\omega_{B3}\) fitting, which has a DC gain of approximately \(-19\) dB for the \(\alpha=0.1\) case. A similar trend is observed in the HFG, with the greatest deviations observed for the \(\omega_{B3}\) fitting; though the most significant differences occur near \(\alpha=0.5\) for the HFG.