An \(L_1\)-Method: Application to Digital Symmetric Type-II FIR Filter Design

  • Apoorva Aggarwal
  • Tarun K. Rawat
  • Manjeet Kumar
  • Dharmendra K. Upadhyay
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 384)


In this paper, the design of digital symmetric type-II linear-phase FIR low-pass (LP) and band-pass (BP) filter is formulated using the \(L_1\) optimality criterion. In order to obtain better filter performance we compute the optimal filter coefficients using the \(L_1\)-norm based fitness function. The use of \(L_1\) technique in digital filter design applications has the advantages of a flatter passband and high stopband attenuation over other gradient-based filter optimization methods. This technique is applied to optimally design type-II FIR filters. Simulations and statistical analysis have been performed for the 25th order LP and BP filters. It is observed, that the \(L_1\)-based filter results is an improved design in comparison with the filters obtained using the equiripple, least-square and window techniques.


Finite impulse response \(L_1\)-error criterion Stopband attenuation Least-square Window method 


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  1. 1.
    Mitra, S.K.: Digital Signal Processing: A Computer-based Approach. Tata Mc-Graw Hill (2008)Google Scholar
  2. 2.
    Rawat, T.K.: Digital Signal Processing, 1st edn. Oxford University Press (2014)Google Scholar
  3. 3.
    Vaidyanathan, P.P., Nquyen, T.Q.: Eigenfilters: A new approach to least squares FIR filter design and appli. IEEE Trans. Circuits Syst. 22, 943–953 (1975)CrossRefGoogle Scholar
  4. 4.
    Ramachandran, R.P., Sunder, S.: A unified and efficient least-squares design of linear-phase nonrecursive filters. Signal Proc. 36, 41–53 (1994)CrossRefGoogle Scholar
  5. 5.
    Parks, T.W., McClellan, J.H.: Chebyshev approximation for non-recursive digital filters with linear phase. IEEE Trans. Circuits Theory, 189–194 (1972)Google Scholar
  6. 6.
    Antoniou, A.: New improved method for the design of weighted chebyshev nonrecursive digital filters. IEEE Trans. Circuits Syst. CAS–30, 740–750 (1983)CrossRefGoogle Scholar
  7. 7.
    Kumar, M., Rawat, T.K.: Optimal design of FIR fractional order differentiator using cuckoo search algorithm. Expert Syst. with Appli. 42, 3433–3449 (2015)CrossRefGoogle Scholar
  8. 8.
    Grossmann, L.D., Eldar, Y.C.: The design of optimal \(L_I\) linear phase digital FIR filters. In: Proc. IEEE Int. Conf. Acoustics, Speech, Signal Process. (ICASSP), vol. 3, pp. 884–887 (2006)Google Scholar
  9. 9.
    Rice, J.R.: The Approximation of Functions, vol. I. Addison- Wesley, MA (1964)MATHGoogle Scholar
  10. 10.
    Chen, C.K., Lee, J.H.: Design of high-order digital differentiatiors using LI error criteria. IEEE Trans. Circuits Syst., Analog Digit. Signal Process. 42, 287–291 (1995)CrossRefGoogle Scholar
  11. 11.
    Yu, W.S., Fong, L.K., Chang, K.C.: An \(L_1\)-Approximation based method for synthesizing FIR details. IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. 39, 578–561 (1992)CrossRefGoogle Scholar
  12. 12.
    Aggarwal, A., Rawat, T.K., Kumar, M., Upadhyay, D.K.: Optimal design of FIR high pass filter based on \(L_1\) error approximation using real coded genetic algorithm. Eng. Sci. Tech., an. Int. J. (2015). doi: 10.1016/j.jestch.2015.04.004
  13. 13.
    Aggarwal, A., Kumar, M., Rawat, T.K.: \(L_1\) error criterion based optimal FIR filters. In: Annual IEEE India Conference (INDICON) (2014)Google Scholar
  14. 14.
    Watson, G.A.: An algorithm for linear \(L_1\) approximation of continuous functions. IMA J. Numer. Anal. 1, 157–167 (1981)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Grossmann, L.D., Eldar, Y.C.: An \(L_1\)-Method for the Design of Linear-Phase FIR Digital Filters. IEEE Trans. Signal Process. 55(11), 5253–5266 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Yarlagadda, R., Bednar, J.B., Watt, T.L.: Fast algorithms for \(l_p\) deconvolution. IEEE Trans. Acoust., Speech, Signal Process. 33, 174–184 (1985)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Apoorva Aggarwal
    • 1
  • Tarun K. Rawat
    • 1
  • Manjeet Kumar
    • 1
  • Dharmendra K. Upadhyay
    • 1
  1. 1.Department of Electronics and Communication EngineeringNetaji Subhas Institute of TechnologyDwarka, DelhiIndia

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