Mathematical Problems in Engineering

Volume 2015, Article ID 832468, 7 pages

http://dx.doi.org/10.1155/2015/832468

## Approximated Fractional Order Chebyshev Lowpass Filters

^{1}Tangent Design Engineering Ltd., 2719 7 Avenue Northeast, Calgary, AB, Canada T2A 2L9^{2}Department of Electrical and Computer Engineering, University of Calgary, 2500 University Dr. N. W., Calgary, AB, Canada T2N 1N4^{3}Department of Electrical and Computer Engineering, University of Sharjah, P.O. Box 27272, UAE

Received 13 June 2014; Accepted 21 August 2014

Academic Editor: Guido Maione

Copyright © 2015 Todd Freeborn et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose the use of nonlinear least squares optimization to approximate the passband ripple characteristics of traditional Chebyshev lowpass filters with fractional order steps in the stopband. MATLAB simulations of , , and order lowpass filters with fractional steps from = 0.1 to = 0.9 are given as examples. SPICE simulations of 1.2, 1.5, and 1.8 order lowpass filters using approximated fractional order capacitors in a Tow-Thomas biquad circuit validate the implementation of these filter circuits.

#### 1. Introduction

Fractional calculus, the branch of mathematics concerning differentiations and integrations to noninteger order, has been steadily migrating from the theoretical realms of mathematicians into many applied and interdisciplinary branches of engineering [1]. From the import of these concepts into electronics for analog signal processing emerged the field of fractional order filter design. This import into filter design has yielded much recent progress in theory [2–6], noise analysis [7], stability analysis [8], implementation [9–13], and applications [14, 15]. These filter circuits have all been designed using the fractional Laplacian operator, , because the algebraic design of transfer functions is much simpler than solving the difficult time domain representations of fractional derivatives. One definition of a fractional derivative of order is given by the Caputo derivative [16] aswhere is the gamma function and . The Caputo definition of a fractional derivative is often used over other approaches because the initial conditions for this definition take the same form as the more familiar integer order differential equations. Applying the Laplace transform to the fractional derivative of (1) with lower terminal yieldswhere is also referred to as the fractional Laplacian operator. With zero initial conditions (2) can be simplified to

Therefore it becomes possible to define a general fractance device with impedance proportional to [17], where the traditional circuit elements are special cases of the general device when the order is , , and for a capacitor, resistor, and inductor, respectively. The expressions of the voltage across a traditional capacitor are defined by integer order integration of the current through it. This element can be expanded to the fractional domain using noninteger order integration which results in the time domain expression for the voltage across the fractional order capacitor given bywhere is the fractional orders of the capacitor, is the current through the element, is the capacitance with units , and is a unit of time not to be mistaken with the Laplacian operator. Note that we will refer to the units of these devices as for simplicity.

By applying the Laplace transform to (4) with zero initial conditions the impedance of this fractional order element is given as . Using this element in electrical circuits increases the range of responses that can be realized, expanding them from the narrow integer subset to the more general fractional domain. While these devices are not yet commercially available, recent research regarding their manufacture and production shows very promising results [18–20]. Therefore, it is becoming increasingly important to develop the theory behind using these fractional elements so that when they are available their unique characteristics can be fully taken advantage of.

In traditional filter design, ideal filters are approximated using methods that include Butterworth, Chebyshev, Elliptic, and Bessel filters. These filters attempt to approximate the ideal frequency response given by for a lowpass filter that passes all frequencies below the cutoff frequency () with no attenuation and removes all frequencies above. A necessary condition for physically realizable filters though is to satisfy the Paley-Wiener criterion [21] which requires a nonzero magnitude response. Hence, ideal filters are not physically realizable because they have a magnitude of zero in a certain frequency range. However, in [21] it was suggested that ideal filters when viewed from the fractional order perspective might not require satisfying the Paley-Wiener criterion to be physically realizable. If fractional order filters do not require satisfying the Paley-Wiener criterion it marks another significant different over their integer order counterparts; which requires further investigation to determine conclusively.

In this work we use a nonlinear least squares optimization routine to determine the coefficients of a fractional order transfer function required to approximate the passband ripple characteristics of traditional Chebyshev lowpass filters. MATLAB simulations of , , and order lowpass filters with fractional steps from to designed using this process are given as examples. SPICE simulations of , , and order lowpass filters using approximated fractional order capacitors in a Tow-Thomas biquad circuit validate the implementation of these filter circuits.

##### 1.1. Approximated Chebyshev Response

Fractional order lowpass filters with order have previously been designed in [9, 22] using the transfer function given byand realized using various topologies including a Tow-Thomas biquad [9], fractional circuits, and field programmable analog arrays (FPAAs) [22]. In [22] the coefficients of (6) were selected to approximate the flat passband response of the Butterworth filters. These coefficients were selected using a numerical search that compared the passband of the fractional filter to the Butterworth approximation over the frequency range rad/s to rad/s and returned the coefficients that yielded the lowest error over this region.

A similar method can be applied to determine the coefficients of (6) required to approximate the ripple characteristics in the passband of the Chebyshev approximation. Here a nonlinear least squares fitting is used that attempts to solve the problemwhere is the vector of filter coefficients, is the magnitude response using (6) calculated using , is the normalized order Chebyshev magnitude response, and are the magnitude responses of (6) and order Chebyshev approximation at frequency , and is the total number of data points in the collected magnitude response. This routine aims to find the coefficients that minimize the error between the magnitude response of (6) and the Chebyshev approximation. The constraint () is added for this problem because negative coefficients are not physically possible and to return values will be easily realized in hardware. This is not the first application of optimization routines in the field of fractional filters. Previously, optimization routines have been employed in [23] to generate approximations of for simulation and further realization for audio applications.

Applying the nonlinear least squares fitting over the frequency range rad/s to rad/s using (6) and the second order Chebyshev filter designed with a ripple of 3 dB with transfer functionyields the coefficients given in Table 1 for orders , , and . The 3 dB ripple was selected over smaller ripple magnitudes to highlight the difference in ripple size using the fractional order response over the integer order response. The coefficients were determined in MATLAB using the* lsqcurvefit* function to implement the NLSF described by (7). This function uses the trust-region-reflective algorithm [24] with termination tolerances of the function value and the solution set to .