Studies on fractional order differentiators and integrators: a survey.

*(English)*Zbl 1203.94035Summary: Studies on analysis, design and applications of analog and digital differentiators and integrators of fractional order is the main objective of this paper. Time and frequency domain analysis, different ways of realization of fractance device is presented. Active and passive realization of fractance device of order \(\frac 12\) using continued fraction expansion is carried out. Later, time and frequency domain analysis of fractance based circuits is considered. The variations of rise time, peak time, settling time, time constant, percent overshoot with respect to fractional order \(\alpha \) is presented.

Digital differentiators and integrators of fractional order can be obtained by using direct and indirect discretization techniques. The \(s\) to \(z\) transforms used for this purpose are revisited. In this paper by using indirect discretization technique fractional order differentiators and integrators of order \(\frac 12\) and \(\frac 14\) are designed. These digital differentiators and integrators are implemented in real time using TMS320C6713 DSP processor and tested using National instruments education laboratory virtual instrumentation system (NIELVIS). The designed fractional order differentiators have been used for the detection of QRS sequences as well as the occurrence of Sino Atrial Rhythms in an ECG signal and also for the detection of edges in an image. The obtained results are in comparison with the conventional techniques.

Digital differentiators and integrators of fractional order can be obtained by using direct and indirect discretization techniques. The \(s\) to \(z\) transforms used for this purpose are revisited. In this paper by using indirect discretization technique fractional order differentiators and integrators of order \(\frac 12\) and \(\frac 14\) are designed. These digital differentiators and integrators are implemented in real time using TMS320C6713 DSP processor and tested using National instruments education laboratory virtual instrumentation system (NIELVIS). The designed fractional order differentiators have been used for the detection of QRS sequences as well as the occurrence of Sino Atrial Rhythms in an ECG signal and also for the detection of edges in an image. The obtained results are in comparison with the conventional techniques.

##### MSC:

94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |

34A08 | Fractional ordinary differential equations |

26A33 | Fractional derivatives and integrals |

##### Keywords:

fractional order; Mittag-Leffler function; \(s\) to \(z\) transform; digital differentiator; digital integrator; discretization; QRS complex; edge detection
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\textit{B. T. Krishna}, Signal Process. 91, No. 3, 386--426 (2011; Zbl 1203.94035)

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##### References:

[1] | Oldham, K. B.; Spanier, J.: The fractional calculus, (1974) · Zbl 0292.26011 |

[2] | Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008 |

[3] | Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993) · Zbl 0789.26002 |

[4] | West, B. J.; Bologna, M.; Grigolini, P.: Physics of fractal operators, (2003) |

[5] | Khovanskii, A. N.: The application of continued fractions and their generalizations to problems in approximation theory, (1963) · Zbl 0106.27204 |

[6] | Moshrafitorbati, M.; Hammond, J. K.: Physical and geometrical interpretation of fractional operators, Journal of franklin institute 335B, No. 6, 1077-1086 (1998) · Zbl 0989.26004 |

[7] | Mainardi, F.; Gorenflo, R.: On Mittag–Leffler type functions in fractional evolution process, Journal of computational and applied mathematics 118, No. 1, 283-299 (2000) · Zbl 0970.45005 |

[8] | Srivastava, H. M.; Saxena, R. K.: Operators of fractional integartion and their applications, Journal of applied mathematics and computation 118, No. 1, 1-12 (2001) · Zbl 1022.26012 |

[9] | Podlubny, I.: Geometrical and physical interpretation of fractional integration and differentiation, Fractional calculus and applied analysis 5, No. 4, 367-386 (2002) · Zbl 1042.26003 |

[10] | Gorenflo, P.; Loutchko, J.; Luchko, Y.: Computation of the Mittag–Leffler function \(E{\alpha},{\beta}(z)\) and its derivative, Fractional calculus and applied analysis 5, No. 4, 491-519 (2002) · Zbl 1027.33016 |

[11] | Debnath, L.: Recent applications of fractional calculus to science and engineering, International journal of mathematics and mathematical sciences 2003, No. 54, 3413-3442 (2003) · Zbl 1036.26004 |

[12] | Diethelm, K.; Ford, N. J.; Freed, A. D.; Luchko, Y.: Algorithms for the fractional calculus a selection of numerical methods, Computer methods in applied mechanics and engineering 194, No. 7, 743-773 (2005) · Zbl 1119.65352 |

[13] | Ferdi, Y.: Computation of fractional order derivative and integral via power series expansion and signal modelling, Nonlinear dynamics 46, No. 1, 1-15 (2006) · Zbl 1170.94311 |

[14] | Van Valkenburg, M. E.: Introduction to modern network synthesis, (1960) |

[15] | Manabe, S.: The non-integer integral and its application to control systems, Japan institute of electrical engineering 80, No. 860, 589-597 (1960) |

[16] | Manabe, S.: The noninteger integral and its application to control systems, English translation journal Japan 6, No. 34, 83-87 (1961) |

[17] | Carlson, G. E.; Halijak, C. A.: Approximation of a fractional capacitors (1/s)1/n by a regular Newton process, IEEE transactions on circuit theory 11, No. 2, 210-213 (1964) |

[18] | Oustaloup, A.: Fractional order sinusoidal oscillators: optimization and their use in highly linear modulation, IEEE transactions on circuits and systems 28, No. 10, 10-19 (1981) |

[19] | M. Axtel, M.E. Bise, Fractional calculus and applications in control systems, in: Proceedings of the IEEE National Aerospace and Electronics Conference, New York, 1990, pp. 563–566. |

[20] | Westerlund, S.: Dead matter has memory!, Physica scripta 43, No. 2, 174-179 (1991) |

[21] | Sugi, M.; Hirano, Y.; Saito, K.: Noninteger exponents in electronic circuits F matrix representation of the power law conductivity, IEICE transactions on fundamentals of electronics, communications and computer sciences 75-A, No. 6, 720-725 (1992) |

[22] | Matsuda, K.; Fujii, H.: H\(\infty \)‐optimized wave-absorbing control: analytical and experimental results, Journal of guidance, control and dynamics 16, No. 6, 1146-1153 (1993) · Zbl 0800.93313 |

[23] | Westerlund, S.; Ekstam, L.: Capacitor theory, IEEE transactions on dielectrics and electrical insulation 1, No. 5, 826-839 (1994) |

[24] | Sorimachi, K.; Nakagawa, M.: Basic characteristics of a fractance device, IEICE transactions on fundamentals of electronics, communications and computer sciences 6, No. 12, 1814-1818 (1998) |

[25] | Roy, S. C. Dutta: On the realization of constant–argument immittance or fractional operator, IEEE transactions on circuit theory 14, 264-274 (1967) |

[26] | Sugi, M.; Hirano, Y.; Miura, Y. F.; Saito, K.: Simulation of fractal immittance by analog circuits: an approach to the optimized circuits, IEICE transactions on fundamentals of electronics, communications and computer sciences 82, No. 8, 1627-1634 (1999) |

[27] | Petras, I.: The fractional order controllers methods for their synthesis and application, Journal of electrical engineering 50, No. 10, 284-288 (1999) |

[28] | Podlubny, I.: Fractional-order systems and \(PI{\lambda}\)D\({\mu}\) controllers, IEEE transactions on automatic control 44, No. 1, 208-214 (1999) · Zbl 1056.93542 |

[29] | Ortigueria, M. D.: Introduction to fractional linear systems I: Continuous time case, IEE Proceedings vision image and signal processing 147, No. 1, 62-70 (2000) |

[30] | G.W. Bohannan, Analog realization of a fractional control element revisited, in: IEEE CDC2002 Tutorial Workshop, Las Vegas, NE, USA, 27 October 2002. |

[31] | Hwang, C.; Leu, J. Fan; Tsay, S. -Y.: A note on time-domain simulation of feedback fractional-order systems, IEEE transactions on automatic control 47, No. 4, 625-631 (2002) · Zbl 1364.93772 |

[32] | Podlubny, I.; Petras, I.; Vinagre, B. M.; Oleary, P.; Dorcak, L.: Analogue realizations of fractional order controllers, Nonlinear dynamics 29, No. 2, 281-296 (2002) · Zbl 1041.93022 |

[33] | Poinot, T.; Trigeassou, J. C.: A method for modelling and simulation of fractional systems, Signal processing 83, No. 11, 2319-2333 (2003) · Zbl 1145.94372 |

[34] | W. Ahmad, R. Elkhazafi, Fractional-order passive low-pass filters, in: Proceedings of the 2003 10th IEEE International Conference on vol. 1, no. 1, 14–17 December 2003, pp. 160–163. |

[35] | Machado, J. A. T.: A probabilistic interpretation of the fractional order differentiation, Fractional calculus and applied analysis 6, No. 1, 73-80 (2003) · Zbl 1035.26010 |

[36] | Stanislavsky, A. A.: Twist of fractional oscillations, Physica A 354, No. 2, 101-110 (2005) |

[37] | Ortigueria, M.; Machado, J. A. T.; Da Costa, J. Sa: Which differeintegration?, IEE Proceedings on vision, image and signal processing 152, No. 6, 846-850 (2005) |

[38] | Pu, Y.; Yang, X.; Liao, K.; Zhou, J.; Zhang, N.; Zeng, Y.; Pu, X. X.: Structuring analog fractance circuit for 12 order fractional calculus, , 1039-1042 (2005) |

[39] | W. Jifeng, L. Yuanki, Frequency domain analysis and applications for fractional order control systems, in: 7th International Symposium on Measurement Technology and Intelligent Instruments, Journal of Physics Conference Series, vol. 13, 2005, pp. 268–275. |

[40] | R. Malti, M. Aoun, J. Sabatier, A. Oustaloup, Tutorial On system identification using fractional differentiation models, in: 14th IFAC Symposium on System Identification, Newcastle, Australia, 2006, pp. 606–611. |

[41] | Charef, A.: Analogue realisation of fractional order integrator differentiator and fractional \(PI{\lambda}\)D\({\mu}\) controller, IEE Proceedings of control theory applications 153, No. 6, 714-720 (2006) |

[42] | Yifei, P.; Xiao, Y.; Ke, L.; Jiliu, Z.; Ni, Z.; Xiaoxian, P.; Yi, Z.: A recursive two-circuits series analog fractance circuit for any order fractional calculus, Optical information processing, Proceedings of the SPIE 6027, No. Part 1, 509-519 (2006) |

[43] | Stanislavsky, A. A.: The peculiarity of self-excited oscillations in fractional systems, Acta physica polonica B 37, No. 2, 319-329 (2006) |

[44] | Machado, J. A. T.; Jesus, I. S.; Galhano, A.; Cunha, J. B.: Fractional order electromagnetics, Signal processing 86, No. 2, 2637-2644 (2006) · Zbl 1172.94454 |

[45] | Biswas, K.; Sen, S.; Dutta, P. K.: Realization of a constant phase element and its performance study in a differentiator circuit, IEEE transactions on circuits and systems 53, No. 9, 802-806 (2006) |

[46] | A. Djouambi, A. Charef, A.V. Besancon, Approximation and synthesis of non integer order systems, in: Workshop on Fractional Differentiation and its Applications Porto, Portugal, July 19–21, 2006. |

[47] | Vinagre, B. M.; Podlubny, I.; Hernndez, A.; Feliu, V.: Some approximations of fractional order operators used in control theory and applications, FCAA fractional calculus and applied analysis 3, No. 3, 231-248 (2000) · Zbl 1111.93302 |

[48] | Barbosa, R. S.; Machado, J. A. T.; Ferreira, I. M.: Tuning of PID controllers based on bode’s ideal transfer function, Nonlinear dynamics 38, 305-321 (2004) · Zbl 1134.93334 |

[49] | S. Ohhori, Realization of fractional order impedance by feedback control, in: Industrial Electronics Society, 2007, IECON 2007, 33rd Annual Conference of the IEEE, Taipei, 5–8 November 2007, pp. 299–304. |

[50] | Dork, L.; Terpk, J.; Petr, I.; Dorkov, F.: Electronic realization of the fractional-order systems, Acta montanistica slovaca ronk 12, No. 3, 231-237 (2007) |

[51] | Djouambi, A.; Charef, A.; Besancon, A. V.: Optimal approximation, simulation and analog realization of the fundamental fractional order transfer function, International journal of applied mathematics and computer science 17, No. 4, 455-462 (2007) · Zbl 1234.93049 |

[52] | Tofighi, A.; Pour, H. N.: \(\epsilon \) expansion and the fractional oscillator, Physica A 374, No. 1, 41-45 (2007) |

[53] | Ferreira, N. M. F.; Duarte, F. B.; Lima, M. F. M.; Marcos, M. G.; Machado, J. A. T.: Application of fractional calculus in the dynamical analysis and control of mechanical manipulators, Fractional calculus and applied analysis 11, No. 1, 91-113 (2008) · Zbl 1159.26303 |

[54] | Ortigueira, M. D.: An introduction to the fractional continuous-time linear systems: the 21st century systems, IEEE circuits and systems magazine 147, No. 1, 19-26 (2000) |

[55] | J.A.T. Machado, A.M.S. Galhano, A new method for approximating fractional derivatives: application in non-linear control, in: EnoC-2008, Saint Petersburg, Russia, 30th July, 2008, pp. 4–8. |

[56] | Jumarie, G.: From self-similarity to fractional derivative of non-differentiable functions via Mittag–Leffler function, Applied mathematical sciences 2, No. 40, 1949-1962 (2008) · Zbl 1175.26012 |

[57] | Krishna, B. T.; Reddy, K. V. V.S.; Kumari, S. Santha: Time domain response calculations of fractance device of order 12, Journal of active and passive electronic devices 3, No. 3, 355-367 (2008) |

[58] | B.T. Krishna, K.V.V.S. Reddy, Active and passive realization of fractance device of order 12, Journal of Active and Passive Electronic Components (2008) 5, Article ID 369421, doi:10.1155/2008/369421. |

[59] | Krishna, B. T.; Reddy, K. V. V.S.: Analysis of fractional order lowpass and highpass filters, Journal of electrical engineering 8, No. 1, 41-45 (2008) |

[60] | Radwan, A. G.; Soliman, A. M.; Elwakil, A. S.: First-order filters generalized to the fractional domain, Journal of circuits, systems, and computers 17, No. 1, 55-66 (2008) · Zbl 1191.94175 |

[61] | Proakis, J. G.; Manolakis, D. G.: Digital signal processing, principles, algorithms, and applications, (1999) |

[62] | Ifeachor, E. C.; Jervis, B. W.: Digital signal processing–a practical approach, (2004) |

[63] | A. Antoniou, Digital Filters–Analysis, Design and Applications, second ed., Tata McGraw Hill Edition, New Delhi, 2000. |

[64] | Al-Alaoui, M. A.: Novel approach to designing digital differentiators, IEEE electronic letters 28, No. 15, 1376-1378 (1992) |

[65] | Al-Alaoui, M. A.: Novel digital integrator and differentiator, Electronics letters 29, No. 4, 376-378 (1993) |

[66] | Bihan, J. L.: Novel class of digital integrators and differentiators, IEEE electronic letters 29, No. 11, 971-973 (1993) |

[67] | Al-Alaoui, M. A.: Novel IIR digital differentiator from simpson integration rule, IEEE transactions on circuits systems. I fundamental theory applications 41, No. 2, 186-187 (1994) · Zbl 0943.93501 |

[68] | Al-Alaoui, M. A.: Filling the gap between the bilinear and the backward difference transforms an interactive design approach, International journal of electrical engineering education 34, No. 4, 331-337 (1997) |

[69] | Al-Alaoui, M. A.: Novel stable higher order s to z transforms, IEEE transactions on circuits and systems–I 48, No. 11, 1326-1329 (2001) |

[70] | Al-Alaoui, M. A.: Al-alaoui operator and the \({\alpha}\) approximation for discretization of analog systems, Facta universitatis 19, No. 1, 143-146 (2006) |

[71] | Ngo, N. Q.: A new approach for the design of wideband digital integrator and differentiator, IEEE transactions on circuits and systems–II 53, No. 9, 936-940 (2006) |

[72] | Al-Alaoui, M. A.: Novel approach to analog-to-digital transforms, IEEE transactions on circuits and systems 54, No. 2, 338-351 (2007) · Zbl 1374.94692 |

[73] | Chassaing, R.: Digital signal processing and applications with the C6713 and C6416 DSK, (2005) |

[74] | Ortigueria, M. D.: Introduction to fractional linear systems part II: Discrete time case, IEE Proceedings vision, image and signal processing 147, No. 1, 71-78 (2000) |

[75] | Tseng, C. C.: Design of fractional order digital FIR differentiators, IEEE signal process letters 8, No. 3, 77-79 (2001) |

[76] | Ostalczyk, P.: Fundamental properties of the fractional order discrete time integrator, Signal processing 83, No. 11, 2367-2376 (2003) · Zbl 1145.94368 |

[77] | Chen, Y. Q.; Vinagre, B. M.: A new IIR-type digital fractional order differentiator, Signal processing 83, No. 11, 2359-2365 (2003) · Zbl 1145.93423 |

[78] | Vinagre, B. M.; Chen, Y. Q.; Petras, I.: Two direct tustin discretization methods for fractional order differentiator and integrator, Journal of franklin institute 340, No. 5, 349-362 (2003) · Zbl 1051.93031 |

[79] | Aoun, M.; Malti, R.; Levron, F.; Oustaloup, A.: Numerical simulations of fractional systems, Nonlinear dynamics 38, No. 1, 117-131 (2004) · Zbl 1134.65300 |

[80] | Barbosa, R. S.; Machado, J. A. T.; Silva, M. F.: Time domain design of fractional differ integrators using least-squares, Signal processing 86, No. 10, 2567-2581 (2006) · Zbl 1172.94364 |

[81] | Tseng, C. C.: Design of variable and adaptive fractional order FIR differentiators, Signal processing 86, No. 10, 2554-2566 (2006) · Zbl 1172.94495 |

[82] | Maione, G.: A digital noninteger order differentiator using Laguerre orthogonal sequences, International journal of intelligent control and systems 11, No. 2, 77-81 (2006) |

[83] | A. Djouambi, A. Charef, A.V. Besancon, Approximation and synthesis of non integer order systems, in: Workshop on Fractional Differentiation and its Applications Porto, Portugal, July 19–21, 2006. |

[84] | Machado, J. A. Tenreiro: Analysis and design of fractional-order digital control systems, SAMS journal of systems analysis, modelling and simulation 27, 107-122 (1997) · Zbl 0875.93154 |

[85] | Barbosa, R. S.; Machado, J. A. T.: Implementation of discrete-time fractional order controllers based on LS approximations, Acta polytechnica hungarica 3, No. 4, 5-22 (2006) |

[86] | Goutas, A.; Ferdi, Y.; Herbeuval, J. J.; Boudraa, M.; Ham, B. Bouche: Digital fractional order differentiation-based algorithm for P and T waves detection and delineation, Itbm 26, No. 2, 127-132 (2006) |

[87] | Tseng, C. C.: Design of FIR and IIR fractional order simpson digital integrator, Signal processing 87, No. 5, 1045-1057 (2007) · Zbl 1186.94343 |

[88] | Djouambi, A.; Charef, A.; Besancon, A. V.: Optimal approximation simulation and analog realization of the fundamental fractional order transfer function, International journal of applied mathematics and computer science 17, No. 4, 455-462 (2007) · Zbl 1234.93049 |

[89] | Mocak, J.; Janiga, I.; Rievaj, M.; Bustin, D.: The use of fractional differentiation or integration for signal improvement, Measurement science review 7, No. 5, 39-42 (2007) |

[90] | Krishna, B. T.; Reddy, K. V. V.S.: Design of digital differentiators and integrators of order 12, World journal of modelling and simulation 4, No. 3, 182-187 (2008) · Zbl 1154.26009 |

[91] | Krishna, B. T.; Reddy, K. V. V.S.: Design of fractional order digital differentiators and integrators using indirect discretization, Fractional calculus and applied analysis 11, No. 2, 143-151 (2008) · Zbl 1154.26009 |

[92] | Dobbs, S. E.; Schmitt, N. M.; Ozemek, H. S.: QRS detection by template matching using realtime correlation on a microcomputer, Journal of clinical engineering 9, No. 3, 197-212 (1984) |

[93] | Pan, J.; Tompkins, W. J.: A real-time QRS detection algorithm, IEEE transactions on biomedical engineering 32, No. 3, 230-236 (1985) |

[94] | Jesus, S.; Rix, H.: High resolution ECG analysis by an improved signal averaging method and comparison with beat-to-beat approach, Journal of biomedical engineering 10, No. 1, 25-32 (1988) |

[95] | Reddy, B. R. S.; Christenson, D. W.; Rowlandson, G. I.; Hammill, S. C.: High resolution ECG, Medical electronics 23, No. 2, 60-73 (1992) |

[96] | T. KF, C. KL, C. K, Detection of the QRS complex, P wave and T wave in electrocardiogram, in: First International Conference on Advances in Medical Signal and Information Processing Proceedings 2000, Bristol, UK, 2000, pp. 41–47. |

[97] | Kohler, B. U.; Hennig, C.; Orglmeister, R.: The principles of software QRS detection, IEEE engineering in medicine and biology magazine 21, No. 1, 42-57 (2002) |

[98] | Ferdi, Y.; Herbeuval, J. J.; Charef, A.; Boucheham, B.: R wave detection using fractional digital differentiation, Itbm 24, No. 5, 273-280 (2003) |

[99] | Jurko, S.; Rozinaj, G.: High resolution of the ECG signal by polynomial approximation, Radioengineering 15, No. 1, 32-37 (2006) |

[100] | Mathieu, B.; Melchior, P.; Oustaloup, A.; Ceyral, C.: Fractional differentiation for edge detection, Signal processing 83, No. 11, 2421-2432 (2003) · Zbl 1145.94309 |

[101] | Pu, Y. -F.: Apply fractional calculus to digital image processing, Journal of sichuan university (Engineering science edition) 39, No. 2, 124-132 (2007) |

[102] | J. Huading, P. Yifei, Fractional calculus method for enhancing digital image of bank slip, in: 2008 IEEE Congress on Image and Signal Processing held at Washington, USA, vol. 3, 2008, pp. 326–330. |

[103] | Yifei, P.; Weixing, W.; Jiliu, Z.: Fractional differential approach to detecting textural features of digital image and its fractional differential filter implementation, Science in China series E: information sciences 38, No. 2, 335-339 (2008) · Zbl 1147.68814 |

[104] | W. Yiyang, P. Yifei, Z. Jiliu, 1/2 order fractional differential tree type circuit of digital image, in: 2008 Congress on Image and Signal Processing, IEEE, vol. 3, 2008, pp. 331–334. |

[105] | A.C. Sparavigna, Fractional differentiation based image processing, arXiv:0910.2381v3[cs.CV], October 2009. |

[106] | J.A. Canny, Computational approach to edge detector, IEEE Transactions on PAMI (1986) 679–698. |

[107] | Davis, L. S.: Edge detection techniques, Computer graphics image process 4, 248-270 (1995) |

[108] | Gonzalez; Woods: Digital image processing, (2008) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.