, 44:61 | Cite as

A new design method for FIR notch filter using Fractional Derivative and swarm intelligence

  • A KUMAREmail author
  • S LEE
  • H-N LEE


In this paper, a new design method for the finite impulse response (FIR) notch filters using fractional derivative (FD) and swarm intelligence technique is presented. The design problem is constructed as a minimization of the magnitude response error w.r.t. filter coefficients. To acquire high accuracy of notch filter, fractional derivative (FD) is evaluated, and the required solution is computed using the Lagrange multiplier method. The fidelity parameters like passband error, notch bandwidth, and maximum passband ripple vary non-linearly with respect to FD values. Moreover, the tuning of appropriate FD value is computationally expensive. Therefore, modern heuristic methods, known as the constraint factor particle swarm optimization (CFI-PSO), which is inspired by swarm intelligence, is exploited to search the best values of FDs and number of FD required for the optimal solution. After an exhaustive analysis, it is affirmed that the use of two FDs results in 21% reduction in passband error, while notch bandwidth is slightly increased by 2% only. Also, it has been observed that, in the proposed methodology, at the most 66 iterations are required for convergence to optimum solution. To examine the performance of designed notch filter using the proposed method, it has been applied for removal of power line interference from an electrocardiography (ECG) signal, and the improvement in performance is affirmed.


Notch filter fractional derivative (FD) swarm intelligence 



This work was supported by the National Research Foundation (NRF) of Korea grant funded by the Korean government (MSIP) (NRF-2018R1A2A1A19018665).


  1. 1.
    Roy S C D, Jain S B and Kumar B 1997 Design of Digital FIR Notch Filters from Second Order IIR Prototype. IETE J. Res. 43(4): pp. 275–279CrossRefGoogle Scholar
  2. 2.
    Sharma I, Kuldeep B, Kumar A and Singh V K 2016 Performance of swarm based optimization techniques for designing digital FIR filter: A comparative study. Eng. Sci. Technol. Int. J. 19(3): 1564–1572CrossRefGoogle Scholar
  3. 3.
    Kumar A and Kuldeep B 2012 Design of M-channel cosine modulated filter bank using modified Exponential window. J. Franklin Inst. 349(3): 1304–1315MathSciNetCrossRefGoogle Scholar
  4. 4.
    Yu T H, Mitra, S K and Babic H 1990 Design of linear phase FIR notch filters. Sadhana 15(3): 133–155CrossRefGoogle Scholar
  5. 5.
    Hirano K, Nishimura S and Mitra S 1974 Design of digital notch filters. IEEE Trans. Circuits Syst. 21(4): 540–546CrossRefGoogle Scholar
  6. 6.
    Tseng, C C and Pei S C 1990 Design of an equiripple FIR notch filter using a multiple exchange algorithm. Signal Processing 75(3): 225–237CrossRefGoogle Scholar
  7. 7.
    Deshpande R, Jain S B and Kumar B 2008 Design of maximally flat linear phase FIR notch filter with controlled null width. Signal Processing 88(10): 2584–2592CrossRefGoogle Scholar
  8. 8.
    Tseng C C and Lee S L 2012 Digital image sharpening using fractional derivative and mach band effect. In: Procdings International Symposium on Circuits and Systems, IEEE, Seoul, South Korea, pp. 2765–2768Google Scholar
  9. 9.
    Mathieu B, Melchior P, Oustaloup A and Ceyral C 2003 Ceyral, Fractional differentiation for edge detection. Signal Processing 83(11): 2421–2432CrossRefGoogle Scholar
  10. 10.
    Ferdi Y, Herbeuval J P, Charef A and Boucheham B. 2003 R wave detection using fractional digital differentiation. ITBM-RBM. 24(5): 273–280CrossRefGoogle Scholar
  11. 11.
    Tseng C C and Lee S L 2012 Design of linear phase FIR filters using fractional derivative constraints. Signal Processing 92(5): 1317–1327CrossRefGoogle Scholar
  12. 12.
    Tseng C C and Lee S L 2013 Fractional Derivative Constrained Design of FIR Filter with Prescribed Magnitude and Phase Responses. In: Procdings of European Conference on Circuit Theory and Design, IEEE, Dresden, Germany, pp. 1–4Google Scholar
  13. 13.
    Tseng C C and Lee S.L 2010 Design of wideband fractional delay filters using derivative sampling method. IEEE Trans. Circuits Syst. I Regul. Pap. 57(8): 2087-2098Google Scholar
  14. 14.
    Tseng C C 2001 Design of fractional order digital FIR differentiators. IEEE Signal Process. Lett. 8(3): 77–79CrossRefGoogle Scholar
  15. 15.
    Tseng C C and Lee S L 2012 Designs of Fixed-Fractional-Delay Filters Using Fractional-Derivative Constraints. IEEE Trans. Circuits Syst. II Express Briefs. 59(10): 683–687Google Scholar
  16. 16.
    Baderia K, Kumar A and Singh G K 2015 Design of multi-channel filter bank using ABC optimized fractional derivative constraints. In: Prcodings of International Conference on Communications and Signal Processing, Melmaruvathur, India, pp. 0490–0494Google Scholar
  17. 17.
    Baderia K, Kumar A and Singh G K 2015 Hybrid method for designing digital FIR filters based on fractional derivative constraints. ISA Trans. 58: 493–508CrossRefGoogle Scholar
  18. 18.
    Kuldeep B, Singh V K, Kumar A and Singh G K 2015 Design of two-channel filter bank using nature inspired optimization based fractional derivative constraints. ISA Trans. 54: 101–116CrossRefGoogle Scholar
  19. 19.
    Kuldeep B, Kumar A and Singh G K 2015 Design of quadrature mirror filter bank using Lagrange multiplier method based on fractional derivative constraints. Eng. Sci. Technol. Int. J. 18(2): 235–243CrossRefGoogle Scholar
  20. 20.
    Kuldeep B, Kumar A and Singh G K 2015 Design of Multi-channel Cosine-Modulated Filter Bank Based on Fractional Derivative Constraints Using Cuckoo Search Algorithm. Circuits, Syst. Signal Process. 34(10): 3325–3351Google Scholar
  21. 21.
    Agrawal N, Kumar A and Bajaj V 2017 Design Method for Stable IIR Filters With Nearly Linear-Phase Response Based on Fractional Derivative and Swarm Intelligence IEEE Trans. Emerg. Top. Comput. Intell. 1(1): 464–477CrossRefGoogle Scholar
  22. 22.
    Charef A, Djouambi A and Idiou D 2014 Linear fractional order system identification using adjustable fractional order differentiator. IET Signal Process 8(4): 398–409CrossRefGoogle Scholar
  23. 23.
    Poli R, Kennedy J and Blackwell T 2007 Particle swarm optimization An overview. Swarm Intelligence 1(1): 33–57CrossRefGoogle Scholar
  24. 24.
    Ahirwal M K, Kumar A and Singh G K 2014 Adaptive filtering of EEG/ERP through noise cancellers using an improved PSO algorithm. Swarm Evol. Comput. 14: 76–91CrossRefGoogle Scholar
  25. 25.
    Karaboga D and Akay B 2009 A comparative study of Artificial Bee Colony algorithm. Appl. Math. Comput. 214(1): 108–132MathSciNetzbMATHGoogle Scholar
  26. 26.
    Rafi S M, Kumar A and Singh G K 2013 An improved particle swarm optimization method for multirate filter bank design. J. Franklin Inst. 350(4): 757–769MathSciNetCrossRefGoogle Scholar
  27. 27.
    Agrawal N, Kumar A, Bajaj V and Singh G K 2018 Design of Bandpass and Bandstop Infinite Impulse Response Filters using Fractional Derivative. IEEE Trans. Ind. Electron. 66(2): 1285–1295CrossRefGoogle Scholar
  28. 28.
    Dai H, Yin L and Li Y 2016 QRS residual removal in atrial activity signals extracted from single lead: a new perspective based on signal extrapolation. IET Signal Process. 10(9): 1169–1175CrossRefGoogle Scholar
  29. 29.
    Khamis H, Weiss R, Xie Y, Chang C W, Lovell N H and Redmond S J 2016 QRS Detection Algorithm for Telehealth Electrocardiogram Recordings IEEE Trans. Biomed. Eng. 63(7): 1377–1388Google Scholar
  30. 30.
    PhysioBank ATM, MIT-BIH arrhythmia ECG signal database, (n.d.).Google Scholar
  31. 31.
    Kumar R, Kumar A and Pandey R K 2013 Beta wavelet based ECG signal compression using lossless encoding with modified thresholding. Comput. Electr. Eng. 39(1): 130–140CrossRefGoogle Scholar
  32. 32.
    Kumar R, Kumar A and Singh G K 2016 Hybrid method based on singular value decomposition and embedded zero tree wavelet technique for ECG signal compression. Comput. Methods Programs Biomed. 129: 135–148CrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

    • 1
    Email author
    • 1
    • 2
  • S LEE
    • 3
  • H-N LEE
    • 3
  1. 1.PDPM Indian Institute of Information Technology, Design and Manufacturing JabalpurJabalpurIndia
  2. 2.Indian Institute of Technology RoorkeeRoorkeeIndia
  3. 3.School of Electrical Engineering and Computer ScienceGwangju Institute of Science and TechnologyGwangjuKorea

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