Optimal Approximation of Fractional-Order Butterworth Filter Based on Weighted Sum of Classical Butterworth Filters

Abstract

In this paper, a new two-steps design strategy is introduced for the optimal rational approximation of the fractional-order Butterworth filter. At first, the weighting factors of the summation between the n th -order and the (n + 1) th -order Butterworth filters are optimally determined. Subsequently, this model is employed as an initial point for another optimization routine, which minimizes the magnitudefrequency error relative to the (n + α) th -order, where α ∈ (0, 1), Butterworth filter. The proposed approximant demonstrates improved performance about the magnitude mean squared error compared to the state-of-the-art design for six decades of bandwidth, but the introduced approach does not require a fractional-order transfer function model and the approximant of the s α operator. The proposed strategy also avoids the use of the cascading technique to yield higher-order fractional-order Butterworth filter models. The performance of the proposed 1.5 th -order Butterworth filter in follow-the-leader feedback topology is verified through SPICE simulations and its hardware implementation based on Analog Devices AD844ANtype current feedback operational amplifier.

A Fractional-Order Transitional Butterworth-Butterworth Filter and Its Experimental Validation

“…Therefore, the roll-off rate for the FO-TBBF is -32.02 decibel/decade (dB/dec), which is close to the theoretical value of -32.0 dB/dec obtained for the 1.6 thorder FOBF. However, the maximum group delay achieved for the proposed FO-TBBF (1.217 s) is substantially smaller as compared to the reported FOBF model [22] (1.392 s). This is due to the fact that the dominating response in the passband for the FO-TBBF depends on the (n 2 + β) th -order Butterworth filter, which is of order 0.8 in the present case.…”

“…The effectiveness of the proposed models in attaining a smaller group delay in the passband as compared to the FOBF is also demonstrated. For this purpose, the magnitude (top) and group delay (bottom) responses of the 1.6 th -order FOBF reported in [22] are compared with the proposed [22]. Note that the numbers within the parenthesis represent (n1, n2, α, β).…”

“…Optimization techniques were employed to approximate the characteristics of the FO Butterworth Filter (FOBF) [21], [22]. However, to the best of the authors' knowledge, no literature exists on the FO modeling of TBBFs.…”

“…Thus, inequality constraints are avoided to meet the s-domain stability criteria. Table 1 compares the advantages and limitations of the proposed method with those of the FOBF [9], [21], [22], and TBBF [20] design techniques. Several design cases are considered to evaluate the performance of the proposed technique.…”

“…FIGURE 4. Comparisons of MATLAB-simulated magnitude and group delay plots of the proposed FO-TBBF with the FOBF reported in[22]. Note that the numbers within the parenthesis represent (n1, n2, α, β).…”