Validation of Fractional-Order Lowpass Elliptic Responses of (\(1+\alpha\))-Order Analog Filters +
Footnote †: journal: Applied Physics Letters B  
[PERSON]  
1 Department of Telecommunications, Brno University of Technology, Technicka 12, 616 00 Brno, Czech Republic; [EMAIL_ADDRESS] (D.K. ); [PERSON]L_ADDRESS] (J.K.); [EMAIL_ADDRESS] (J.D.) [PERSON]  
2 Department of Electrical and Computer Engineering, The University of Alabama,  
Tuscaloosa, AL 35487, USA  
derivative of order \(0<\alpha<1\). Therefore, the current-voltage relationship for this component is defined as:  
\[i(t)=C\frac{d^{\alpha}v(t)}{dt^{\alpha}} \tag{1}\]  
where \(i(t)\) and \(v(t)\) are the time-dependent current and voltage, respectively. One definition of a fractional derivative of order \(\alpha\) is given by the Grunwald-Letnikov definition  
\[{}_{a}D_{t}^{\alpha}f(t)=\lim_{h\to 0}\frac{1}{h^{\alpha}}\sum_{m=0}^{ \left[\frac{t-a}{h}\right]}\left(-1\right)^{m}\frac{\Gamma\left(\alpha+1 \right)}{m!\Gamma\left(\alpha-m+1\right)}f\left(t-mh\right) \tag{2}\]  
where \(\Gamma(\cdot)\) is the gamma function, \(n-1\leq\alpha\leq n\), and \(a\) and \(t\) are the terminals of fractional differentiations [2]. The Grunwald-Letnikov definition is presented here because this definition leads to a correct generalization of the current linear systems theory [3], but it is important to note that other definitions such as the Riemann-Liouville and Caputo definitions are also available for describing fractional-derivatives. Applying the Laplace transform to the fractional derivative of Equation (2) with zero initial conditions with lower terminal \(a=0\) yields  
\[\mathcal{L}\left\{{}_{0}D_{t}^{\alpha}f(t)\right\}=s^{\alpha}F(s) \tag{3}\]  
The transfer-function representations of fractional-order differential equations are widely used during the design of fractional-order analog filters. Using these representations does not require the computation of their time-domain fractional-order differential equation representations, which reduces their design complexity. The numerical complexity for simulations of fractional-order differential equations stems from the number of computations required to capture the significant number of addends [2]. Consider Equation (2), which is a series that requires a greater number of operations for greater values of \(t\) to capture the entire history of the function \(f(t)\), especially for \(t\gg a\)[2]. [PERSON] noted, however, that for large \(t\) the history of the function at the lower terminal (\(t=a\)) can be neglected under certain assumptions. This reduces the numerical complexity required to simulate a fractional-order differential equation by applying the \"short memory\" principle. The \"short memory\" principle approximates the lower limit \(a\) with a moving lower limit \((t-L)\) in cases where the behavior of the function is driven by the memory of the \"recent past\" [2], that is:  
\[{}_{a}D_{t}^{\alpha}f(t)\approx_{(t-L)}D_{t}^{\alpha}f(t) \tag{4}\]  
where \(t>a+L\). This approximation does reduce the accuracy of the simulations, although the \"memory length\" (\(L\)) can be determined to meet a required accuracy (\(\epsilon\)) with details given in [2]. Further, [PERSON] noted that using the short-memory principle leads to reductions in accumulated rounding errors during simulations as a result of the fewer addends [2]. The \"short memory\" principle has recently been applied in electronics for FPGA implementations of fractional-order systems [16; 17]. These works highlight the impact of different window sizes on the accuracy and necessary hardware to realize Grunwald-Letnikov implementations. While the complexities of simulating fractional-order differential equations are discussed here, this work employs transfer-function representations of fractional-order systems and does not implement the time-domain simulations of the underlying fractional-order differential equations.