International Journal of Electronics and Computer Science Engineering 2095 Available Online at www.ijecse.org ISSN- 2277-1956 ISSN 2277-1956/V1N4-2095-2099 Design of Stabilized operator based 1/3 rd Order Digital Differentiator Mausmi Verma Shri Ramswaroop Group of Professional Colleges, Lucknow (U.P.)-India mausmi.7nov@gmail.com Abstract- In this paper, the design of Simpson and Alaoui operator based 1/3rd order digital differentiators are presented. To improve the design accuracy of the Simpson operator based 1/3 rd digital differentiator, firstly, the stability criterion is considered, and then the fractional order digital differentiator is designed by using Continued Fraction Expansion (CFE) discretization scheme. Next Alaoui operator is used to design 1/3 rd order digital differentiator. Lastly, the MATLAB simulation results of both the designed fractional order digital differentiators are compared with the ideal fractional order digital differentiator. Keywords- Continued Fraction Expansion (CFE), Simpson operator, Alaoui operator, Fractional order differentiator I. INTRODUCTION The fractional-order (FO) calculus is near about 300-years old topic. The theory of Fractional Order derivative was developed in the 19th century. If we talk about fractional order filters then it can be both i.e. fractional order differentiator and fractional order integrator. Differentiators are very useful in the processing of signals in different fields, such as digital image processing [1], digital control [2], biomedical applications [3] and communications [4]. In this paper, Simpson and Al-Alaoui operators are used to design fractional order digital differentiators. Initially a stable differentiator is obtained by inverting the transfer function of the Simpson integrator using the approach in [5]. Then Continued Fractional Expansion (CFE) discretization scheme is applied on the obtained stable differentiator to design 1/3 rd order digital differentiator. Similarly CFE discretization scheme is also applied on Al-Alaoui operator to design another 1/3 rd order digital differentiator. Finally, both the above designed fractional order digital differentiators simulation results are compared with the ideal fractional order digital differentiator. By using above procedure 1/3 rd order digital differentiators are designed, as improved design of fractional order differential filter is important topic for research [6-10]. II- DESIGN OF 1/3 rd ORDER DIGITAL DIFFERENTIATOR USING STABILIZED OPERATOR- A differentiator is obtained by inverting the transfer function of the Simpson's integrator [6],[12]. Now if stability criterion is considered i.e., the operator has all the poles and zeros inside the unit circle, then it is clear that the Simpson differentiator derived by reversing the Simpson Integrator has one of its poles (3.7321) outside the unit circle. So this pole must be reflected inside the unit circle by using method suggested by Steigglitz. K. in [11], [5] for stability and minimum phase. The transfer function of Simpson Integrator [6], [12] is given as, 1 2 2 (1 4 ) ( ) 3(1 ) T z z H z z - - - + + = - (1) Reversing above equation to derive the differentiator. The transfer function for the differentiator is given as,