ISSN 1 746-7233, England, UK World Journal of Modelling and Simulation Vol. 4 (2008) No. 3, pp. 182-187 Design of digital differ entiators and integrators of order 1 2 B. T. Krishna ∗ , K. V . V . S. Reddy Department of ECE, GITAM University , Vis akhapatnam, India ( Received March 4 2008, Accepted May 1 2008) Abstract. In thi s pap er ,de sig n of digita l dif fer ent iat ors and int egrat ors of order 1 2 is prese nted. First, the ratio nal approximation for the fractional operator s ± 1 2 is calculated. Next, using s to z transforms it is digitized. The results obtained were closer to ideal characteristics. Keywords: continued fraction expansion, Al-Alaoui transform, fractional order, conve rgence, discretization 1 Introducti on Fractional order integrators and differentiators are used to calculate the fractional order integral and derivative of an input signal [8, 9] . These devices find applications in instrumentation,control systems,radar, digital Image processing, bio-medical engineering and other allied fields [3] . An ideal fractional order digital differentiator is defined by the following transfer function [4] , Hd ( jω ) = ( jω ) α (1) where α is fractional order and j = √ −1. Similarly an ideal fractional order integrator is defined as, HI( jω ) = 1 ( jω ) α (2) The key ste p in the digita l implementation of the fracti ona l order dif fer ent iat or/ inte gra tor is its discretization [4, 5, 11] . Direct and indirect discretization are the commonly used methods for discretization. Di- rect discretization method involves the application of the direct power series or continued fraction expansion ofs to z transform. In [4, 11], the diff erent methods of direct discret izati on of the fractional order controller are discussed. In [5], Dorcak et al. have compared all these direct discretization methods. In indirect discretization method,two steps are required, i.e., fitting the transfer function first and then discretizing the fit s-domain transfer function. In this paper, first, rational approximations for √ s and 1 √ s are obtained using continued fraction expansion. Then, using s to z transformations it is discretized. The paper is organised as follows. First order s to z transforms are discussed in Section 2. Design method is presented in section 3. Secti on 4 deals with Simulati on Results and conclus ions. 2 Fi rst order s to z transforms: Sto z transforms play major role in the discretization [6, 10] . The s to z trans form should be such that, • The imagina ry axis in the s-plane be mapped onto the unit circle in z plane. ∗ Corresponding author. E-mail address: [email protected]. Published by World Academic Press, World Academic Union
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7/28/2019 design of digital differentiators of order 1/2
Design of digital differentiators and integrators of order 1
2
B. T. Krishna∗ , K. V. V. S. Reddy
Department of ECE, GITAM University, Visakhapatnam, India
( Received March 4 2008, Accepted May 1 2008 )
Abstract. In this paper,design of digital differentiators and integrators of order 1
2is presented.First,the rational
approximation for the fractional operator s±1
2 is calculated. Next, using s to z transforms it is digitized. The
results obtained were closer to ideal characteristics.
Keywords: continued fraction expansion, Al-Alaoui transform, fractional order, convergence, discretization
1 Introduction
Fractional order integrators and differentiators are used to calculate the fractional order integral and
derivative of an input signal[8, 9]. These devices find applications in instrumentation,control systems,radar,
digital Image processing, bio-medical engineering and other allied fields[3].
An ideal fractional order digital differentiator is defined by the following transfer function[4],
H d( jω ) = ( jω)α (1)
where α is fractional order and j =√ −1. Similarly an ideal fractional order integrator is defined as,
H I ( jω) =1
( jω)α(2)
The key step in the digital implementation of the fractional order differentiator/integrator is its
discretization[4, 5, 11]. Direct and indirect discretization are the commonly used methods for discretization. Di-
rect discretization method involves the application of the direct power series or continued fraction expansion
of s to z transform. In [4, 11], the different methods of direct discretization of the fractional order controller
are discussed. In [5], Dorcak et al. have compared all these direct discretization methods.In indirect discretization method,two steps are required, i.e., fitting the transfer function first and then
discretizing the fit s-domain transfer function. In this paper, first, rational approximations for√
s and 1√ s
are
obtained using continued fraction expansion. Then, using s to z transformations it is discretized.
The paper is organised as follows. First order s to z transforms are discussed in Section 2. Design method
is presented in section 3. Section 4 deals with Simulation Results and conclusions.
2 First order s to z transforms:
S to z transforms play major role in the discretization[6, 10]. The s to z transform should be such that,
• The imaginary axis in the s-plane be mapped onto the unit circle in z plane.∗ Corresponding author. E-mail address: [email protected].
Published by World Academic Press, World Academic Union
7/28/2019 design of digital differentiators of order 1/2
World Journal of Modelling and Simulation, Vol. 4 (2008) No. 3, pp. 182-187 187
4 Results and conclusions
This section presents pole-zero diagrams, magnitude and phase responses of the designed digital differ-
entiators and integrators evaluated at, sampling time, T = 1 sec.
From the magnitude plots it is to be noted that Al-Alaoui transform has shown superior performance
compared to bilinear transform. The phase response is more nearer to ideal using bilinear transform. The
percent relative error is very less. Poles and zeros were lying inside of the unit circle and alternate on negative
real axis. So, the indirect discretization produced stable,minimum phase differentiators and integrators.
References
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