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ISSN 1 746-7233, England, UK World Journal of Modelling and Simulation Vol. 4 (2008) No. 3, pp. 182 187 Design of digital differentiators and integrators of order 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 is presented.First,the rational approximation for the fractional operator is calculated. Next, using to 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 . These devices ﬁnd applications in instrumentation,control systems,radar, digital Image processing, bio-medical engineering and other allied ﬁelds An ideal fractional order digital differentiator is deﬁned by the following transfer function j ) = ( j (1) where is fractional order and . Similarly an ideal fractional order integrator is deﬁned as, j ) = j (2) The key step in the digital implementation of the fractional order differentiator/integrator is its discretization 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 to transform. In [ 11 ], the different methods of direct discretization of the fractional order controller are discussed. In [ ], 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 -domain transfer function. In this paper, ﬁrst, rational approximations for and are obtained using continued fraction expansion. Then, using to transformations it is discretized. The paper is organised as follows. First order to transforms are discussed in Section 2. Design method is presented in section 3. Section 4 deals with Simulation Results and conclusions. 2 First order to transforms: to transforms play major role in the discretization 10 . The to transform should be such that, The imaginary axis in the -plane be mapped onto the unit circle in plane. Corresponding author. E-mail address : tkbattula@gmail.com. Published by World Academic Press, World Academic Union

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World Journal of Modelling and Simulation, Vol. 4 (2008) No. 3, pp. 182 187 183 A stable analog transfer function be transformed into a stable digital transfer function. Property 1 preserves the frequency selective properties of the continuous system, whereas property 2 ensures that stable continuous systems are mapped into stable discrete systems 10 . Bilinear transform and backward difference transform were the most widely used to transforms. In case of backward difference transform, the imaginary axis of the -plane maps as a circle of radius centered at . Bilinear transform satisﬁes the above two conditions but it produces warping effect.The following is the expression for bilinear transform, 2( 1) + 1) (3) A new type of to transform called as Al-Alaoui transform is obtained from the interpolation of backward and bilinear transforms with a tuning factor, of value varying from 0 to 1 ) = aTz 1) + (1 + 1) 2( 1) (4) The resulting to transform is, 2( 1) [(1 ) + (1 + (5) Substituting , in Eq. ( ) it simpliﬁes to, 8( 1) (6) The above transform has been proved to be less warped than bilinear transformation and less linear than backward difference transform. The Al-Alaoui Transform has shown superior performance in digital ﬁlter design compared to previously existing to transforms 3 Design method Continued fraction expansion is used to obtain the rational approximations of the irrational functions It helps in terminating an inﬁnite order transfer function to ﬁnite order. Since the fractional order systems were characterized by inﬁnite memory , they will have an inﬁnite order rational approximation. So, in order to have practical realization of the system its transfer function has to be terminated to a ﬁnite order. We have the continued fraction expansion for as [ (7) Consequently, ...... Rogers ][4] (8) One can re-write the continued fraction as, 1+ 1+ 1 + ...... (9) The above continued fraction expansion converges in the ﬁnite complex -plane along the negative real axis satisfying the inequality < s . Considering number of terms of Eq. ( ), the rational approx- imations obtained for were summarised in Tab. . In order to get the rational approximation of the WJMS email for subscription : info@wjms.org.uk

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184 B. Krishna & K. Reddy Design of digital differentiators Table 1. Rational approximations for S. NO No. of Terms Rational Approximation +1 +3 +10 +1 +10 +5 +35 +21 +1 +21 +35 +7 +84 +126 +36 +1 +36 +126 +84 +9 10 11 +165 +462 +330 +55 +1 +55 +330 +462 +165 +11 Fig. 1. Comparison of Magnitude and Phase Responses of Rational approximation functions with ideal expressions has to be simply reversed. Fig. compares the magnitude and phase responses of rational approx- imations with the ideal one. From the magnitude and phase response Plots it is evident that ﬁfth order rational approximation is best ﬁt in -domain. Higher order rational approximations can be obtained by increasing the number of terms in Eq. ( ). So, 11 + 165 + 462 + 330 + 55 + 1 + 55 + 330 + 462 + 165 + 11 (10) + 55 + 330 + 462 + 165 + 11 11 + 165 + 462 + 330 + 55 + 1 (11) In order to check for the stability of the rational approximations as deﬁned in Eq. ( ) and ( ), pole-zero plot are drawn and are shown in Fig. and Fig. From pole-zero plots,it is evident that pole and zeros interlace on negative real axis making the system as stable. By digitizing the Eq. ( ) and ( ) using transforms as deﬁned in Eq. ( ) and ( ), the following transfer functions were obtained. WJMS email for contribution : submit@wjms.org.uk

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World Journal of Modelling and Simulation, Vol. 4 (2008) No. 3, pp. 182 187 185 Fig. 2. Pole-Zero Plot of Fig. 3. Pole-Zero Plot of Fig. 4. Magnitude Response, Phase Response and error plots of digital differentiators dT ) = 1 414 443 123 + 0 776 181 024 443 067 + 0 43 + 0 187 045 (12) dA ) = 1 069 533 + 2 214 748 + 0 067 004 962 + 1 175 179 021 + 0 002 (13) IT ) = 0 707 443 067 + 0 43 + 0 187 045 443 123 + 0 776 181 024 (14) IA ) = 0 935 962 + 1 175 179 021 + 0 002 533 + 2 214 748 + 0 067 004 (15) WJMS email for subscription : info@wjms.org.uk

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186 B. Krishna & K. Reddy Design of digital differentiators Fig. 5. Magnitude Response, Phase Response and error plots of digital integrators where dT IT are the transfer functions of digital differentiator and integrator obtained using bilinear transform, and dA IA are the transfer functions of digital differentiator and integrator when Al- Alaoui transform is used. It is to be noted that sampling time, = 1 sec is used in all calculations. Fig. 6. Pole-zero diagrams WJMS email for contribution : submit@wjms.org.uk

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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, = 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 [1] M. Al-Alaoui. Novel digital integrator and differentiator. Electronics Letters , 1993, 29 (4): 376378. [2] M. Al-Alaoui. Filling the gap between the bilinear and the backward difference transforms: An interactive design approach. Int. J. of Electrical Engineering Education , 1997, 34 (4): 331337. [3] J. Bruce, B. Mauro, G. Paolo. Physics of Fractal operators . Springer Verilog, 2003. [4] Y. Chen, B. Vinagre. A new IIR-type digital fractional order differentiator. Signal Processing , 2003, 83 : 2359 2365. [5] L. Dorcak, I. Petras, M. Zborovjan. Comparison of the methods for discrete approximation of the fractional-order operator. Acta Montanistica Slovaca , 2003, 236239. [6] E. Ifeachor, B. Jervis. Digital signal processing-a practical approach, pearson education. 2004. [7] A. Khovanskii. The application of continued fractions and their generalizations to problems in approximation theory. 1993. [8] K. Miller, B. Ross. An introduction to the fractional calculus and fractional differential equations . John Wiley sons, 1993. [9] K. Oldham, J. Spanier. The Fractional Calculus . Academic Press, 1974. [10] J. Proakis, D. Manolakis. Digital signal processing, principles, algorithms, and applications . PHI Publications, 1999. [11] B. Vinagre, Y. Chen, I. Petras. Two direct tustin discretization methods for fractional-order differentiator and integrator. J. franklinInst , 2003, 340 (5): 349362. WJMS email for subscription : info@wjms.org.uk

4 2008 No 3 pp 182 187 Design of digital differentiators and integrators of order 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 paperdesign of digital dif ID: 23732

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ISSN 1 746-7233, England, UK World Journal of Modelling and Simulation Vol. 4 (2008) No. 3, pp. 182 187 Design of digital differentiators and integrators of order 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 is presented.First,the rational approximation for the fractional operator is calculated. Next, using to 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 . These devices ﬁnd applications in instrumentation,control systems,radar, digital Image processing, bio-medical engineering and other allied ﬁelds An ideal fractional order digital differentiator is deﬁned by the following transfer function j ) = ( j (1) where is fractional order and . Similarly an ideal fractional order integrator is deﬁned as, j ) = j (2) The key step in the digital implementation of the fractional order differentiator/integrator is its discretization 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 to transform. In [ 11 ], the different methods of direct discretization of the fractional order controller are discussed. In [ ], 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 -domain transfer function. In this paper, ﬁrst, rational approximations for and are obtained using continued fraction expansion. Then, using to transformations it is discretized. The paper is organised as follows. First order to transforms are discussed in Section 2. Design method is presented in section 3. Section 4 deals with Simulation Results and conclusions. 2 First order to transforms: to transforms play major role in the discretization 10 . The to transform should be such that, The imaginary axis in the -plane be mapped onto the unit circle in plane. Corresponding author. E-mail address : tkbattula@gmail.com. Published by World Academic Press, World Academic Union

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World Journal of Modelling and Simulation, Vol. 4 (2008) No. 3, pp. 182 187 183 A stable analog transfer function be transformed into a stable digital transfer function. Property 1 preserves the frequency selective properties of the continuous system, whereas property 2 ensures that stable continuous systems are mapped into stable discrete systems 10 . Bilinear transform and backward difference transform were the most widely used to transforms. In case of backward difference transform, the imaginary axis of the -plane maps as a circle of radius centered at . Bilinear transform satisﬁes the above two conditions but it produces warping effect.The following is the expression for bilinear transform, 2( 1) + 1) (3) A new type of to transform called as Al-Alaoui transform is obtained from the interpolation of backward and bilinear transforms with a tuning factor, of value varying from 0 to 1 ) = aTz 1) + (1 + 1) 2( 1) (4) The resulting to transform is, 2( 1) [(1 ) + (1 + (5) Substituting , in Eq. ( ) it simpliﬁes to, 8( 1) (6) The above transform has been proved to be less warped than bilinear transformation and less linear than backward difference transform. The Al-Alaoui Transform has shown superior performance in digital ﬁlter design compared to previously existing to transforms 3 Design method Continued fraction expansion is used to obtain the rational approximations of the irrational functions It helps in terminating an inﬁnite order transfer function to ﬁnite order. Since the fractional order systems were characterized by inﬁnite memory , they will have an inﬁnite order rational approximation. So, in order to have practical realization of the system its transfer function has to be terminated to a ﬁnite order. We have the continued fraction expansion for as [ (7) Consequently, ...... Rogers ][4] (8) One can re-write the continued fraction as, 1+ 1+ 1 + ...... (9) The above continued fraction expansion converges in the ﬁnite complex -plane along the negative real axis satisfying the inequality < s . Considering number of terms of Eq. ( ), the rational approx- imations obtained for were summarised in Tab. . In order to get the rational approximation of the WJMS email for subscription : info@wjms.org.uk

Page 3

184 B. Krishna & K. Reddy Design of digital differentiators Table 1. Rational approximations for S. NO No. of Terms Rational Approximation +1 +3 +10 +1 +10 +5 +35 +21 +1 +21 +35 +7 +84 +126 +36 +1 +36 +126 +84 +9 10 11 +165 +462 +330 +55 +1 +55 +330 +462 +165 +11 Fig. 1. Comparison of Magnitude and Phase Responses of Rational approximation functions with ideal expressions has to be simply reversed. Fig. compares the magnitude and phase responses of rational approx- imations with the ideal one. From the magnitude and phase response Plots it is evident that ﬁfth order rational approximation is best ﬁt in -domain. Higher order rational approximations can be obtained by increasing the number of terms in Eq. ( ). So, 11 + 165 + 462 + 330 + 55 + 1 + 55 + 330 + 462 + 165 + 11 (10) + 55 + 330 + 462 + 165 + 11 11 + 165 + 462 + 330 + 55 + 1 (11) In order to check for the stability of the rational approximations as deﬁned in Eq. ( ) and ( ), pole-zero plot are drawn and are shown in Fig. and Fig. From pole-zero plots,it is evident that pole and zeros interlace on negative real axis making the system as stable. By digitizing the Eq. ( ) and ( ) using transforms as deﬁned in Eq. ( ) and ( ), the following transfer functions were obtained. WJMS email for contribution : submit@wjms.org.uk

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World Journal of Modelling and Simulation, Vol. 4 (2008) No. 3, pp. 182 187 185 Fig. 2. Pole-Zero Plot of Fig. 3. Pole-Zero Plot of Fig. 4. Magnitude Response, Phase Response and error plots of digital differentiators dT ) = 1 414 443 123 + 0 776 181 024 443 067 + 0 43 + 0 187 045 (12) dA ) = 1 069 533 + 2 214 748 + 0 067 004 962 + 1 175 179 021 + 0 002 (13) IT ) = 0 707 443 067 + 0 43 + 0 187 045 443 123 + 0 776 181 024 (14) IA ) = 0 935 962 + 1 175 179 021 + 0 002 533 + 2 214 748 + 0 067 004 (15) WJMS email for subscription : info@wjms.org.uk

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186 B. Krishna & K. Reddy Design of digital differentiators Fig. 5. Magnitude Response, Phase Response and error plots of digital integrators where dT IT are the transfer functions of digital differentiator and integrator obtained using bilinear transform, and dA IA are the transfer functions of digital differentiator and integrator when Al- Alaoui transform is used. It is to be noted that sampling time, = 1 sec is used in all calculations. Fig. 6. Pole-zero diagrams WJMS email for contribution : submit@wjms.org.uk

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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, = 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 [1] M. Al-Alaoui. Novel digital integrator and differentiator. Electronics Letters , 1993, 29 (4): 376378. [2] M. Al-Alaoui. Filling the gap between the bilinear and the backward difference transforms: An interactive design approach. Int. J. of Electrical Engineering Education , 1997, 34 (4): 331337. [3] J. Bruce, B. Mauro, G. Paolo. Physics of Fractal operators . Springer Verilog, 2003. [4] Y. Chen, B. Vinagre. A new IIR-type digital fractional order differentiator. Signal Processing , 2003, 83 : 2359 2365. [5] L. Dorcak, I. Petras, M. Zborovjan. Comparison of the methods for discrete approximation of the fractional-order operator. Acta Montanistica Slovaca , 2003, 236239. [6] E. Ifeachor, B. Jervis. Digital signal processing-a practical approach, pearson education. 2004. [7] A. Khovanskii. The application of continued fractions and their generalizations to problems in approximation theory. 1993. [8] K. Miller, B. Ross. An introduction to the fractional calculus and fractional differential equations . John Wiley sons, 1993. [9] K. Oldham, J. Spanier. The Fractional Calculus . Academic Press, 1974. [10] J. Proakis, D. Manolakis. Digital signal processing, principles, algorithms, and applications . PHI Publications, 1999. [11] B. Vinagre, Y. Chen, I. Petras. Two direct tustin discretization methods for fractional-order differentiator and integrator. J. franklinInst , 2003, 340 (5): 349362. WJMS email for subscription : info@wjms.org.uk

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