Multidimensional Systems and Signal Processing

, Volume 28, Issue 4, pp 1569–1587 | Cite as

Optimal design of 2-D FIR digital differentiator using \(L_1\)-norm based cuckoo-search algorithm

  • Apoorva Aggarwal
  • Manjeet Kumar
  • Tarun K. Rawat
  • D. K. Upadhyay
Article
First Online:
Received:
Revised:
Accepted:

Abstract

In this article, an optimal design of two-dimensional finite impulse response digital differentiators (2-D FIR-DD) with quadrantally odd symmetric impulse response is presented. The design problem of 2-D FIR-DD is formulated as an optimization problem based on the \(L_1\)-error fitness function. The novel error fitness function is based on the \(L_1\) norm which is unique and is liable to produce a flat response. This design methodology incorporates advantages of \(L_1\)-error approximating function and cuckoo-search algorithm (CSA) which is capable of attaining a global optimal solution. The optimized system coefficients are computed using \(L_1\)-CSA and performance is measured in terms of magnitude response, phase response, absolute magnitude error and elapsed time. Simulation results have been compared with other optimization algorithms such as real-coded genetic algorithm and particle swarm optimization and it is observed that \(L_1\)-CSA delivers optimal results for 2-D FIR-DD design problem. Further, performance of the \(L_1\)-CSA based 2-D FIR-DD design is evaluated in terms of absolute magnitude error and algorithm execution time to demonstrate their effect with variation in the control parameters of CSA.

Keywords

Finite impulse response Quadrantally odd symmetric 2-D differentiators \(L_1\)-norm Cuckoo-search algorithm 
This is a preview of subscription content, log in to check access.

References

  1. Aggarwal, A., Kumar, M., Rawat, T. K., & Upadhyay, D. K. (2016a). Optimal design of 2D FIR filters with quadrantally symmetric properties using fractional derivative constraints. Circuits, Systems, and Signal Processing, 35(6), 2213–2257.CrossRefMATHGoogle Scholar
  2. Aggarwal, A., Rawat, T. K., Kumar, M., & Upadhyay, D. K. (2016b). Design of optimal band-stop FIR filter using \(L_1\)-norm based RCGA. Ain Shams Engineering Journal. doi: 10.1016/j.asej.2015.11.022.
  3. Aggarwal, A., Rawat, T. K., Kumar, M., & Upadhyay, D. K. (2015). Optimal design of FIR high pass filter based on \(L_1\) error approximation using real coded genetic algorithm. Enggineeing Science and Technology, An International Journal, 18(4), 594–602.CrossRefGoogle Scholar
  4. Aggarwal, A., Rawat, T. K., & Upadhyay, D. K. (2016c). Design of optimal digital FIR filters using evolutionary and swarm optimization techniques. AEU-International Journal of Electronics and Communications, 70, 373–385.CrossRefGoogle Scholar
  5. Al-Alaoui, M. A. (2011). Class of digital integrators and differentiators. IET Signal Processing, 5(2), 251–260.CrossRefGoogle Scholar
  6. Bhattacharya, D., & Antoniou, A. (1999). Design of 2-D FIR filters by a feedback neural network. Multidimensional Systems and Signal Processing, 10(3), 319–330.CrossRefMATHGoogle Scholar
  7. Chu, C. H., Delp, E. J., & Buda, A. J. (1988). Detecting left ventricular endocardial and epicardial boundaries by digital two-dimensional echocardiography. IEEE Transactions on Medical Imaging, 7(2), 81–90.CrossRefGoogle Scholar
  8. Cichocki, A., & Amari, S. I. (2002). Adaptive blind signal and image processing: Learning algorithms and applications (Vol. 1). Hoboken: Wiley.CrossRefGoogle Scholar
  9. Dhabal, S., & Venkateswaran, P. (2015). A novel accelerated artificial bee colony algorithm for optimal design of two dimensional FIR filter. Multidimensional Systems and Signal Processing. doi: 10.1007/s11045-015-0352-5
  10. Gandomi, A. H., Yang, X. S., & Alavi, A. H. (2013). Cuckoo search algorithm: A metaheuristic approach to solve structural optimization problems. Engineering with Computers, 29(1), 17–35.CrossRefGoogle Scholar
  11. Grossmann, L. D., & Eldar, Y. C. (2007). An \(L_1\)-method for the design of linear-phase FIR digital filters. IEEE Transactions on Signal Processing, 55(11), 5253–5266.MathSciNetCrossRefGoogle Scholar
  12. Gupta, M., Jain, M., & Kumar, B. (2011). Recursive wideband digital integrator and differentiator. International Journal of Circuit Theory and Applications, 39, 775–782.CrossRefGoogle Scholar
  13. Gupta, M., Relan, B., Yadav, R., & Aggarwal, V. (2014). Wideband digital integrators and differentiators designed using particle swarm optimization. IET Signal Processing, 8(6), 668–679.CrossRefGoogle Scholar
  14. Gupta, M., & Yadav, R. (2014). New improved fractional order differentiator models based on optimized digital differentiators. The Scientific World Journal. doi: 10.1155/2014/741395.
  15. Hinamoto, T., & Doi, A. (1996). A technique for the design of 2-D recursive digital filters with guaranteed stability. Multidimensional Systems and Signal Processing, 7(2), 225–237.CrossRefMATHGoogle Scholar
  16. Jalloul, M. K., & Al-Alaoui, M. A. (2015). Design of recursive digital integrators and differentiators using particle swarm optimization. International Journal of Circuit Theory and Applications. doi: 10.1002/cta.2115.
  17. Kockanat, S., & Karaboga, N. (2015). The design approaches of two-dimensional digital filters based on meta-heuristic optimization algorithms: A review of the literature. Artificial Intelligence Review, 44(2), 265–287.Google Scholar
  18. Kumar, M., & Rawat, T. K. (2015a). Optimal design of FIR fractional order differentiator using cuckoo search algorithm. Expert Systems with Applications, 42, 3433–3449.CrossRefGoogle Scholar
  19. Kumar, M., & Rawat, T. K. (2015b). Optimal fractional delay-IIR filter design using cuckoo search algorithm. ISA Transactions, 59, 39–54.CrossRefGoogle Scholar
  20. Kumar, M., Rawat, T. K., Jain, A., Singh, A. A., & Mittal, A. (2015a). Design of digital differentiators using interior search algorithm. Procedia Computer Science, 57, 368–376.CrossRefGoogle Scholar
  21. Kumar, M., Rawat, T. K., Singh, A. A., Mittal, A., Jain, A. (2015b). Optimal design of wideband digital integrators using gravitational search algorithm. In International conference on international conference on computing, communication and automation (ICCCA), pp. 1314–1319.Google Scholar
  22. Lai, X. (2007). Design of smallest size two-dimensional linear-phase FIR filters with magnitude error constraint. Multidimensional Systems and Signal Processing, 18(4), 341–349.CrossRefMATHGoogle Scholar
  23. Lee, J. S. (1980). Digital image enhancement and noise filtering by use of local statistics. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2, 165–168.CrossRefGoogle Scholar
  24. Lin, Y. P., & Vaidyanathan, P. P. (1996). Theory and design of two-dimensional filter banks: A review. Multidimensional Systems and Signal Processing, 7(3–4), 263–330.CrossRefMATHGoogle Scholar
  25. Lu, W. S., & Hinamoto, T. (2011). Two-dimensional digital filters with sparse coefficients. Multidimensional Systems and Signal Processing, 22(1–3), 173–189.MathSciNetCrossRefMATHGoogle Scholar
  26. Pan, B., Xie, H., Guo, Z., & Hua, T. (2007). Full-field strain measurement using a two-dimensional Savitzky–Golay digital differentiator in digital image correlation. Optical Engineering, 46(3), 033601–033601.CrossRefGoogle Scholar
  27. Patwardhan, A. P., Patidar, R., & George, N. V. (2014). On a cuckoo search optimization approach towards feedback system identification. Digital Signal Processing, 32, 156–163.CrossRefGoogle Scholar
  28. Rawat, T. K. (2015). Digital signal processing (1st ed.). New Delhi: Oxford Publication.Google Scholar
  29. Shyu, J. J., Pei, S. C., Huang, Y. D., & Chen, Y. S. (2014). A new structure and design method for variable fractional-delay 2-D FIR digital filters. Multidimensional Systems and Signal Processing, 25(3), 511–529.CrossRefGoogle Scholar
  30. Tseng, C. C., & Lee, S. L. (2008). Design of digital differentiator using difference formula and richardson extrapolation. IET Signal Processing, 2(2), 177–188.MathSciNetCrossRefGoogle Scholar
  31. Tseng, C. C., & Lee, S. L. (2013). Designs of two-dimensional linear phase FIR filters using fractional derivative constraints. Signal Processing, 93(5), 1141–1151.CrossRefGoogle Scholar
  32. Tseng, C. C., & Lee, S. L. (2014). Designs of fractional derivative constrained 1-D and 2-D FIR filters in the complex domain. Signal Processing, 95, 111–125.CrossRefGoogle Scholar
  33. Tzafestas, S. G. (1986). Multidimensional systems, techniques and applications. New York: Marcel Dekker.MATHGoogle Scholar
  34. Tzeng, S. T. (2004). Genetic algorithm approach for designing 2-D FIR digital filters with 2-D symmetric properties. Signal Processing, 84(10), 1883–1893.CrossRefMATHGoogle Scholar
  35. Upadhyay, D. K., & Singh, R. K. (2011). Recursive wide band digital differentiators and integrators. Electronic Letters, 47(11), 647648.CrossRefGoogle Scholar
  36. Yang, X. S. (2014). Nature-inspired optimization algorithms. Amsterdam: Elsevier.MATHGoogle Scholar
  37. Yang, X. S., Cui, Z., Xiao, R., Gandomi, A. H., & Karamanoglu, M. (2013). Swarm intelligence and bio-inspired computation: Theory and applications (1st ed.). Amsterdam: Elsevier.Google Scholar
  38. Yang, X. S., Deb, S. (2009). Cuckoo search via lvy flights. In Proceeding of world congress on nature and biologically inspired computing, IEEE Publications, pp. 210–214.Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Apoorva Aggarwal
    • 1
  • Manjeet Kumar
    • 1
  • Tarun K. Rawat
    • 1
  • D. K. Upadhyay
    • 1
  1. 1.Department of Electronics and CommunicationNetaji Subhas Institute of TechnologyDelhiIndia

Personalised recommendations