Fractional Order Butterworth Filter

Fractional order Butterworth filter derived from the conventional 3 order Butterworth filter by the transition of the ordinary derivative to the fractional one as in [1] is investigated in this paper. The change of the filter characteristics is studied depending on the order of the fractional derivative. Effects of the transformation on the components’ behaviors of the filter is formulated. Design of the filter in the sense of choosing the filter parameters to satisfying the given specifications is described. Keywords—Butterworth filter, control system, fractional order.


I. INTRODUCTION
Fractional calculus (FC) [2,3]provides more accurate models than classical calculus for many of the physical systems showing intrinsic fractional order (FO) behavior [4][5][6]. Fractional derivative takes into account the past history when memory like behaviors exhibit in components [7][8][9][10]. The same advantages appear in some electrical circuits as well [11][12][13][14][15][16]. In [1], a method is given for obtaining fractional order differential equation (FODE) from ordinary differential equation (ODE) keeping the consistency of dimensionality in physical systems. This method is used to find the natural response of a FO RLC circuit by F. Gomez at al. [17]. In this paper, FO Butterworth filter (FOBF) is derived by using a method similar to Gomez's method. The ordin ary derivative is replaced by FO derivative) by the transformation where is an arbitrary parameter defining the order of derivative and the auxiliary parameter has the dimension of second. Time and frequency response characteristics of the obtained FOBF is investigated according to the transformation parameters and . The paper is organized as follows: In Section 2, a brief summary of the FC relevant to the content of this paper is presented. Section 3 covers the short introduction of the conventional Butterworth filter. Section 4 describes the evaluation of the FOBF and the related formulas. In Section 5, the time and frequency domain characteristics of the FOBF is investigated. Finally, conclusions are included in Section 6.

II. FRACTIONAL CALCULUS
It is well known that Caputo fractional time derivative [16,17] of a function ( ) is defined by where ∈ , − 1 < ≤ ∈ = { 1, 2, … } , represents the ordinary derivative of order , and Γ is the Gamma function. For = 1, Eq. (2) gives the usual derivative. It is assumed in the scope of this paper that represents the time in seconds (s), and = 1 so that (2) reduces to We note that Eq. (2b) yields Considering the inverse Laplace transform, a few useful formulas are listed as follows:  One of the circuit realizations of this transfer function is done as the voltage ratio transfer function as it is shown in Fig.1. The analysis of the circuit results the following transfer function: The frequency domain characteristics, namely gain and phase responses of the filter are obtained replacing in Eq. (5a); the result is 1 − 2 2 , (7c ) are the phase and phase characteristic, respectively. From Eq. (7a), it is obvious that the gain decreases from its value 1 = 0, and it becomes equal to its half power value = 1/√2 = 1, it decays to zero as → ∞. And the phase start from 0 = 0, it decreases monotonically to −270 → ∞. The time and frequency characteristics are not plotted at this stage since they will appear as the special case of the fractional Butterworth filter considered in the next section.

IV. FRACTIONAL ORDER LOW PASS
BUTTERWORTH FILTER Applying the fractional transformation to the components of the LPBF in Fig. 1, we have the following time and Laplace domain element behavior equations in the circuit: The frequency and time domain analysis of the circuit with these component behavior equations, or direct respectively. The characteristic equation is obtained by equating the denominator polynomial of the transfer function in Eq. (9a) to zero: The characteristic polynomial appearing in the left side of the equality in Eq. (9c) is s a commensurate polynomial in power . So the time domain responses (such as step and impulse responses) of the filter can be obtained analytically by using Mittag-Leffler function [5]. But, these solutions are not included within the content of the paper and it is satisfied by their plots only, instead we are confined to the frequency domain responses .

V. FILTER CHARACTERIS TICS
To find the gain and phase characteristics of the FO LPBF derived in the previous section, we let = in Eq. (9a) and use the identity https: //dx.doi.org/10.22161/ijaers.5.12.25  ISSN: 2349-6495(P) | 2456-1908(O) www.ijaers.com The result is where A = 3 cos ( 1.5 ) + 2 2 cos ( ) , (11e , f) The gain and phase characteristics on the logarithmic scale (Bode plots) of the filter for = 1 and different values of is shown in Fig. 2 where the conventional 3 rd order Butterworth filter characteristics is shown by the thick dashed red line ( = 1); The details of the numerical data is given in Table 1 where (in dB) is the peak gain at the peak frequency (in rad/s) , 1 = 2 = − 3are the gains at the cut off frequencies 1 and 2 , is the bandwidth which is defined as 2 − 1 if a peak exist (the case = 1); otherwise = 2 (the cases = 1, 0.75, 0.50) The quality factor = 2 /( 2 − 1 )is defined only for = 1.25 for which the gain characteristics has a peak exceeding the 0 dB level and the filter can be considered as of bandpass type as well. It is observed from Fig. 2 and the data given in Table 1 that the first cut off frequency 1 and the associated gain 1 , and are defined only when the characteristic is handled as a BP type for which BW=0.221. In general, the bandwidth ( ) and hence the cut off frequency ( 2 = ) decreases with decreasing values for the LP filter. The phase characteristics decrease from 0 − 180 as : 0 → ∞ but with a faster rate at the intermediate frequencies with increasing .   The step response of the filter is shown in Fig. 3 for values of = 1.25, 1.00, 0.75, 0.50 . It is seen that all the responses start from 0 = 0 and approaches to ∞ → ∞, which is a typical property for a LP filter.
The response is highly oscillatory for = 1.25, the case in which the filter can be interpreted as a BP filter as well.
There is a smaller overshoot for the ordinary 3 rd order Butterworth filter (case = 1.00, thick red dashed line). The rise time and the settling time decreases as takes smaller values. See Table 2 for the detailed numerical data, where : rise time, : rise time, 1 , 2 , 3 : peak times, 1 , , 3 : peak values, : settling time all in seconds. Some of the data could not be detected since the simulations are done up to 40 s. Rise time increases with decreasing and the minimum settling time occurs for the conventional filter ( ).

Fig.3:
Step responses of the transfer function when =1.00.   Fig. 4 and Table 3. The general arguments discussed for the case = 1 hold. For = 1.25, all the critical frequencies are increased whilst the critical gains and the quality factor remain the same. For = 1.00 the responses are hardly affected. For ≤ 1, all the critical frequencies decrease with decreasing . Considering the phase characteristic, it is slightly increased (decreased) for = 1.25 ( < 1.00) and almost unaffected for = 1.00.  /dx.doi.org/10.22161/ijaers.5.12.25  ISSN: 2349-6495(P) | 2456-1908(O) www.ijaers.com Page | 182 Table.3: Numerical data for frequency response curves when = 0.1 for the filter.
Step response of the filter for = 0.1 is shown in Fig. 5. It is seen that the conventional Butterworth filter response is not affected by when = 1.00 (see the legends = 1.25 = 1.25 * ). Further, the peak values of oscillations are not affected by changing from 1 0.1.
But there is an apparent time lead (squeeze) for = 1.25 >, and time lag (spread) for = 0.75 < 1. The lag (spread) is higher for = 0.50. See Table 4 for numerical details. The time lead (lag) is used in the sense of faster (slower) motion.
Similar frequency characteristics for = 10 and the same different values of considered before are presented in Fig. 2     Step response of the filter for = 10 is shown in Fig. 5. It is seen that the conventional Butterworth filter response is not affected by when = 1.00 (see the legends = 1.25 = 1.25 * ). Further, the peak values of oscillations are not affected by changing from 1 to 10 .

VI. CONCLUSION
A FO Butterworth filter derived from the conventional 3 rd order Butterworth filter by the IO derivative to FO derivative transformation with two parameters as in [17] is investigated in this paper. It is arrived the following conclusions. 1. A variety of low pass filters can be obtained by changing the parameters and . 2. For > 1, the classical frequency response curve of the Butterworth filter disappears , and a peak occur in the frequency response. 3. When = 1, does not effect on the characteristics. 4.
dominantly affect the time characteristics, critical magnitudes of the time responses are not changed, but increase of cuuses a slower (faster) response for > 1 ( < 1). That is effects like a time scaling operator. 5. Effect of on the phase characteristics is that increase of decreases (increases the phase if > 1 ( < 1). 6. For > 1 and as → 1.38 the filter can be used a high-Q narrow band pass filter as well. But in the limit case the filter becomes unstable. Having in mind these conclusions, fractional order low pass filters and/or band pass filers can be designed.