Volume IV, Issue XI, November 2015 IJLTEMAS ISSN 2278 - 2540 www.ijltemas.in Page 22 Sensitivity Analysis of Fractional Order Filter Dr. Madhab Chandra Tripathy # # Department of Instrumentation and Electronics Engineering, College of engineering and technology, Bhubaneswar, Abstract— In this paper the sensitivity analysis of continuous- time fractional order filters has been done to investigate ‘how much the filter’s behavior changes as a component value changes’. In particular, a KHN type fractional biquad filter has been taken where two fractional order elements of order α and β (0 < α, β ≤ 1) are used for realization. It is seen that pole frequency (ωn) and pole quality factor of fractional order filter is more sensitive towards component variations as compared to classical integer order filter; however transfer function sensitivities are less affected due to component variations. Keywords— KHN filter, pole frequency, Quality factor, Constant Phase Element, Biquad I. INTRODUCTION enerally in signal processing, much of what is done in the analog domain is amplification and filtering [1]. Hence, there is a great importance for filter design in analog signal processing. However, because of component tolerance and opamp no idealities, the response of a practical filter is likely to deviate from the theoretical value. Even if some of the components are made adjustable, deviation will still arises because of component aging and thermal drift. It is therefore interest to know how sensitive a given filter is to component variation. Simply, sensitivity shows how much one thing/parameter changes as a function of another thing which is changing. Mathematically, Sensitivity [i] is defined as 0 / lim[ ] / y x x y y x dy S xx y dx      (i) Where x is the component that is varied and y is the filter characteristic (ωn or Q) in our case) that we wish to evaluate as x is varied. Sensitivities are the exponents in the circuit equations. For example, Sensitivity of a pole frequency (ωn) w. r. t. capacitance c in a filter circuit is 0.5 n c S   . This implies if capacitance increases by factor 4, then pole frequency (ωn) increases by factor 0.5 4 i.e.2. In this paper the sensitivity of the fractional order filter circuits have been investigated. Fractional calculus is an effective mathematical tool to describe the dynamics of fractional order filter circuit. Among the various applications of fractional calculus, modeling of real world phenomena, and physical systems, that is, voltage–current relationship in a nonideal capacitor [2], fractal behavior of metal-insulator- solution interface [3], biological systems [4-6] are few important ones. Recent research works show a trend in generalizing integer-order dynamics to fractional-order and study the performances through both simulation and experimentation [7]. Similar approach has been adopted to investigate the sensitivity of the fractional order filter. This paper is organized as follows: Sect.2 presents the background of fractional order calculus. Sect.3 presents the basics of fractional order filter and their parameters. The sensitivity analysis of fractional order filter has been presented in Section 4 and concluding remarks are summarized in Sect.5. II. BACKGROUND OF FRACTIONAL ORDER CALCULUS Different definitions of fractional derivatives and fractional Integrals (Diffrintegrals) are considered. By means of them explicit formula and some special functions are derived. Fractional order derivative and integrals provide a powerful instrument for the description of memory and hereditary properties of different substances. This is the most significant advantage of fractional order models in comparison with integer models. Because of absence of appropriate mathematical models, fractional order systems were studied. Fractional differential equations (also known as extraordinary differential equations) are a generalization of differential equations through the application of fractional calculus. The use of fractional calculus can improve and generalize well-established control methods and strategies. There are two main approaches for defining a fractional derivative. The first considers differentiation and integration as limits of finite differences. The Grunwald-Letnikov definition follows this approach. The other approach generalizes a convolution type representation of repeated integration [5]. The Riemann-Liouville and Caputo definitions take this approach. Riemann-Liouville and Caputo fractional derivatives are fundamentally related to fractional integration operators. Consequently, the initial conditions of fractional derivatives are the frequency distributed and infinite dimensional state vector of fractional integrators. G