Digital integrators (DIs) and digital differentiators (DDs) of second, third and fourth-order based on particle swarm optimisation (PSO) algorithm are presented. A modified particle swarm optimisation (MPSO) algorithm with reducing maximum velocity has been used to optimise the mean square error of the digital operators. Statistical and simulation results have been presented for comparing quality of optimal operators obtained by MPSO, genetic algorithm (GA), two variants of PSO and PSO-GA hybrid techniques. The results obtained for best solutions by the proposed algorithm are either superior or at par with the basic PSO variants and hybrid techniques. The proposed digital operators have also been simulated using MATLAB, and the results have been compared with that of existing DIs and DDs derived by different optimisation algorithms, to demonstrate the effectiveness of the use of proposed MPSO. The relative magnitude errors (dB) obtained for digital integrators and differentiators are as low as −40 and −35 dB, respectively, which are valid for almost the full band of normalised frequency.

References

1. 1)
  - 24. Eberhart, R.C., Shi, Y.: ‘Comparing inertia weights and constriction factors in particle swarm optimization’. Proc. Congress on Evolutionary Computation, July 2000, vol. 1, pp. 84–88.
2. 2)
  - S. Sunder , V. Ramachandran . Design of equiripple nonrecursive digital differentiators and Hilbert transformers using a weighted least-squares technique. IEEE Trans. Signal Process. , 2504 - 2509
3. 3)
  - 20. Carlisle, A., Dozier, G.: ‘An off-the shelf PSO’. Proc. Particle Swarm Optimization Workshop, 2001, pp. 1–6.
4. 4)
  - M.A. Al-Alaoui . Using fractional delay to control the magnitudes and phases of integrators and differentiators. IET Signal Process. , 2 , 107 - 119
5. 5)
  - 16. Upadhyay, D.K., Singh, R.K.: ‘Recursive wide band digital differentiators and integrators’, Electron. Lett., 2011, 47, (11), pp. 647–648 (doi: 10.1049/el.2011.0420).
6. 6)
  - N. Papamarkos , C. Chamzas . A new approach for the design of digital integrators. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. , 9 , 785 - 791
7. 7)
  - M.A. Al-Alaoui . Novel digital integrator and differentiator. Electron. Lett. , 376 - 378
8. 8)
  - 15. Chin-Chien, H., Wei-Yen, W., Chin-Yung, Y.: ‘Genetic algorithm-derived digital integrators and their applications in discretization of continuous systems’. Proc. CEC Congress on Evolutionary Computation. (USA), May 2002, 1, pp. 443–448.
9. 9)
  - 8. Pei, S.C., Wang, P.H.: ‘Closed-form design of maximally flat FIR Hilbert transformers, differentiators and fractional delayers by power series expansion’, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., 2001, 48, (4), pp. 389–398.
10. 10)
  - 1. Al-Alaoui, M.A.: ‘Class of digital integrators and differentiators’, IET Signal Process., 2011, 5, (2), pp. 251–260 (doi: 10.1049/iet-spr.2010.0107).
11. 11)
  - 29. Kao, Y.T., Zahara, E.: ‘A hybrid genetic algorithm and particle swarm optimization for multimodal functions’, Appl. Soft Comput., 2008, 8, pp. 849–857 (doi: 10.1016/j.asoc.2007.07.002).
12. 12)
  - 33. Steiglitz, K.: ‘Computer aided design of recursive digital filters’, IEEE Trans., 1970, AU-18, pp. 123–129.
13. 13)
  - 22. Valle, Y.D., Kumar, G., Mohagheghi, S., Hernandez, J.-C.: ‘Particle swarm optimization: basic concepts, variants and applications in power systems’, IEEE Trans. Evol. Comput., 2008, 12, (2), pp. 171–195.
14. 14)
  - 23. Eberhart, R.C., Shi, Y.: ‘Particle swarm optimization: developments, applications and resources’. Proc. Congress on Evolutionary Computation, May 2001, vol. 1, pp. 81–86.
15. 15)
  - N.Q. Ngo . A new approach for the design of wideband digital integrator and differentiator. IEEE Trans. Circuits Syst. II , 9 , 936 - 940
16. 16)
  - 28. Clerc, M., Kenney, J.: ‘The particle swarm-explosion, stability and convergence in a multidimensional complex space’, IEEE Trans. Evol. Comput., 2002, 6, (1), pp. 58–73.
17. 17)
  - 26. Tripathi, P.K., Bandyopadhyay, S., Pal, S.K.: ‘Multi-objective particle swarm optimization with time variant inertia and acceleration coefficients’, Inf. Sci., 2007, 177, pp. 5033–5049 (doi: 10.1016/j.ins.2007.06.018).
18. 18)
  - 17. Upadhyay, D.K.: ‘Class of recursive wide band digital differentiators and integrators’, Radio Eng. J., 2012, 21, (3), pp. 904–910.
19. 19)
  - M.A. Al-Alaoui . Novel stable higher order s-to-z transforms. IEEE Trans. Circuits Syst. I , 11 , 1326 - 1329
20. 20)
  - 11. Gupta, M., Jain, M., Kumar, B.: ‘Novel class of stable recursive digital integrators and differentiators’, IET Signal Process., 2009, 4, (5), pp. 560–566.
21. 21)
  - M.A. Al-Alaoui . Linear phase lowpass IIR digital differentiators. IEEE Trans. Signal Process. , 2 , 697 - 706
22. 22)
  - 18. Kennedy, J., Eberhart, R.: ‘Particle swarm optimization’. Proc. Fourth IEEE Int. Conf. Neural Networks, IEEE Service Centre, Perth, Australia, November 1995, vol. 4, pp. 1942–1948.
23. 23)
  - 14. Jain, M., Gupta, M., Jain, N.: ‘Linear phase second order recursive digital integrators and differentiators’, Radio Engineering Journal, 2012, 21, (2), pp. 712–717.
24. 24)
  - 19. Shi, Y., Eberhart, R.: ‘Parameter selection in particle swarm optimization’. Proc. Annual the Seventh Annual Conf. Evolutionary Programming VII, 1998, vol. 1447, pp. 591–600.
25. 25)
  - 32. Shi, X.H., Lu, Y.H., Zhou, C.G., Lee, H.P., Lin, W.Z., Liang, Y.C.: ‘Hybrid evolutionary algorithms based on PSO and GA’, Evol. Comput., CEC '03, 2003, 4, pp. 2393–2399.
26. 26)
  - D.K. Upadhyay . Recursive wideband digital differentiators. Electron. Lett. , 25 , 1661 - 1662
27. 27)
  - 25. Yufeng, L.: ‘Dynamic particle swarm optimization algorithm for resolution of overlapping chromatograms’. Fifth Int. Conf. Natural Computation, ICNC'09, August 2009, vol. 3, pp. 246–250.
28. 28)
  - M. Gupta , M. Jain , B. Kumar . Recursive wideband digital integrator and differentiator. Int. J. Circuit Theory Appl.
29. 29)
  - 9. Pie, S.C., Shyu, J.J.: ‘Design of FIR Hilbert transformers and differentiators by eigen filter’, IEEE Trans. Circuits Syst., 1988, CAS-35, (11), pp. 1457–1461.
30. 30)
  - 31. Thangaraj, R., Pant, M., Abraham, A., Bourvry, P.: ‘Particle swarm optimization: hybridization perspectives and experimental illustrations’, Appl. Math. Comput., 2011, 217, pp. 5208–5226 (doi: 10.1016/j.amc.2010.12.053).
31. 31)
  - 27. Rini, D.P., Shamsuddin, S.M., Yuhaniz, S.S.: ‘Particle swarm optimization: techniques, system and challenges’, Int. J. Comput. Appl., 2011, 14, (1), pp. 19–27.
32. 32)
  - 21. Shi, Y., Eberhart, R.C.: ‘Empirical study of particle swarm optimization’. Proc. IEEE Int. Conf. Evolution Computer, Washington DC, 1999, vol. 3, pp. 1945–1950.
33. 33)
  - 30. Duong, S.C., Kinjo, H., Uezato, E., Yamamoto, T.: ‘Particle swarm optimization with genetic recombination: a hybrid evolutionary algorithm’, Artif. Life Robot., 2010, 15, pp. 444–449 (doi: 10.1007/s10015-010-0846-z).

Wideband digital integrators and differentiators designed using particle swarm optimisation

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