Circuits, Systems, and Signal Processing

, Volume 37, Issue 3, pp 965–983 | Cite as

Nonlinear Sequence Transformation-Based Continuous-Time Wavelet Filter Approximation



Time-domain synthesis is used to design continuous-time filters when their time-domain response for a given excitation is known. A case in point is the design of a continuous-time wavelet filter which requires a chosen wavelet as the filter’s impulse response. A key step in the design of a continuous-time wavelet filter is the approximation of its transfer function to a proper rational function. This approximation problem is addressed using methods that can be broadly categorized as closed-form methods and numerical optimization methods. While optimization methods are very effective, closed-form solutions are attractive because they are easy to design. In this work, we propose variants of nonlinear sequence transformation (a closed-form method) that obtain proper rational approximations of wavelet functions which are stable and also perform better in terms of mean square error when compared to those obtained by other closed-form methods. It is shown that the model-order reduction of the proposed variants leads to either similar or better performance compared to \(L_2\) optimization method. These variants are also shown to act as good starting points for optimization methods that use local search routines.


Wavelet filter Nonlinear sequence transformation Impulse response Rational approximation 


  1. 1.
    R. Bhattacharya, D. Roy, S. Bhowmick, Rational interpolation using Levin–Weniger transforms. Comput. Phys. Commun. 101(3), 213–222 (1997)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    A.J. Casson, An analog circuit approximation of the discrete wavelet transform for ultra low power signal processing in wearable sensor nodes. Sensors 15(12), 31914–31929 (2015)CrossRefGoogle Scholar
  3. 3.
    A.J. Casson, E. Rodriguez-Villegas, A 60 pw \( g_m\)c continuous wavelet transform circuit for portable EEG systems. IEEE J. Solid State Circuits 46(6), 1406–1415 (2011)CrossRefGoogle Scholar
  4. 4.
    A.J. Casson, E. Rodriguez-Villegas, Nanowatt multi-scale continuous wavelet transform chip. Electron. Lett. 50(3), 153–154 (2014)CrossRefGoogle Scholar
  5. 5.
    A.J. Casson, E.Rodriguez Villegas, in An Inverse Filter Realisation of a Single Scale Inverse Continuous Wavelet Transform, IEEE International Symposium on Circuits and Systems, ISCAS 2008 (IEEE, 2008), pp. 904–907Google Scholar
  6. 6.
    A.J. Casson, D.C. Yates, S. Patel, E. Rodriguez-Villegas, in An Analogue Bandpass Filter Realisation of the Continuous Wavelet Transform, Engineering in Medicine and Biology Society, 2007. EMBS 2007. 29th Annual International Conference of the IEEE, (IEEE, 2007) pp. 1850–1854Google Scholar
  7. 7.
    C.K. Chui, J. Wang, A cardinal spline approach to wavelets. Proceed. Am. Math. Soc. 113(3), 785–793 (1991)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    I.M. Filanovsky, P.N. Matkhanov, Synthesis of time delay networks approximating the pulse response described by an integer power of a sinusoid over its semi-period. Analog Integr. Circuits Signal Process. 28(1), 83–90 (2001)CrossRefGoogle Scholar
  9. 9.
    S.A.P. Haddad, W.A. Serdijn, et al., in Mapping the Wavelet Transform onto Silicon: The Dynamic Translinear Approach, International Symposium on Circuits and Systems 2002. ISCAS 2002, vol. 5 (IEEE, 2002) pp. V–621Google Scholar
  10. 10.
    S.A.P. Haddad, R. Houben, W.A. Serdijn, et al., in Analog Wavelet Transform Employing Dynamic Translinear Circuits for Cardiac Signal Characterization, IEEE International Symposium on Circuits and Systems, 2003. ISCAS 2003, vol. 1 (IEEE, 2003) pp. I–121Google Scholar
  11. 11.
    S.A.P. Haddad, N. Verwaal, R. Houben, W.A. Serdijn, et al. in Optimized Dynamic Translinear Implementation of the Gaussian Wavelet Transform, IEEE International Symposium on Circuits and Systems, 2004. ISCAS 2004, vol. 1 (IEEE, 2004) pp. I–145Google Scholar
  12. 12.
    S.A.P. Haddad, S. Bagga, W.A. Serdijn, Log-domain wavelet bases. IEEE Trans. Circuits Syst. I Regul. Pap. 52(10), 2023–2032 (2005a)CrossRefGoogle Scholar
  13. 13.
    S.A.P. Haddad, J.M.H. Karel, R.L.M. Peeters, R.L. Westra, W.A. Serdijn, et al., in Analog complex wavelet filters, IEEE International Symposium on Circuits and Systems, 2005. ISCAS 2005, (IEEE, 2005b) pp. 3287–3290Google Scholar
  14. 14.
    S.A.P. Haddad, W.A. Serdijn, Ultra Low-Power Biomedical Signal Processing: An Analog Wavelet Filter Approach for Pacemakers (Springer, Berlin, 2009)CrossRefGoogle Scholar
  15. 15.
    H. Kamada, N. Aoshima, in Analog Gabor Transform Filter with complex First Order System, SICE’97. Proceedings of the 36th SICE Annual Conference. International Session Papers, (IEEE, 1997) pp.925–930Google Scholar
  16. 16.
    J.M.H. Karel, R.L.M. Peeters, R.L. Westra, S.A.P. Haddad, W.A. Serdijn, in An L 2-Based Approach for Wavelet Approximation, 44th IEEE Conference on Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC’05, (IEEE, 2005a) pp. 7882–7887Google Scholar
  17. 17.
    J.M.H. Karel, R.L.M. Peeters, R.L. Westra, S.A.P. Haddad, W.A. Serdijn, in Wavelet Approximation for Implementation in Dynamic Translinear Circuits, Proceedings of 16th IFAC World Congress (IFAC WC05), Prague, Czech republic, vol. 8 (2005b)Google Scholar
  18. 18.
    J.M.H. Karel, S.A.P. Haddad, S. Hiseni, R.L. Westra, W. Serdijn, R.L.M. Peeters et al., Implementing wavelets in continuous-time analog circuits with dynamic range optimization. IEEE Trans. Circuits Syst. I Regul. Pap. 59(2), 229–242 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    G. Makkena, K.N. Abhilash, M.B. Srinivas, in Gaussian filter approximation using Levin’s transformation for implementation in analog domain, IEEE Asia Pacific Conference on Postgraduate Research in Microelectronics and Electronics, IEEE PrimeAsia, 2013, (IEEE, 2013) pp. 204–207Google Scholar
  20. 20.
    Stephane Mallat, A Wavelet Tour of Signal Processing: The Sparse Way (Academic Press, Cambridge, 2008)MATHGoogle Scholar
  21. 21.
    J.P. Marmorat, M. Olivi, RARL2: a Matlab based software for \(H^2\) rational approximation (2004)Google Scholar
  22. 22.
    Y.J. Min, H.K. Kim, Y.R. Kang, G.S. Kim, J. Park, S.W. Kim, Design of wavelet-based ECG detector for implantable cardiac pacemakers. IEEE Trans. Biomed. Circuits Systems 7(4), 426–436 (2013)CrossRefGoogle Scholar
  23. 23.
    K. Ogata, Y. Yang, Modern Control Engineering (Prentice-Hall, Englewood Cliffs, 1970)Google Scholar
  24. 24.
    L. Pernebo, L.M. Silverman, Model reduction via balanced state space representations. IEEE Trans. Autom. Control 27(2), 382–387 (1982)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    D. Roy, R. Bhattacharya, S. Bhowmick, Rational approximants using Levin–Weniger transforms. Comput. Phys. Commun. 93, 159–178 (1996)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    M. Tuckwell, C. Papavassiliou, An analog Gabor transform using sub-threshold 180-nm CMOS devices. IEEE Trans. Circuits Syst. I Regul. Pap. 56(12), 2597–2608 (2009)MathSciNetCrossRefGoogle Scholar
  27. 27.
    M. Unser, T. Blu, Cardinal exponential splines: part I theory and filtering algorithms. IEEE Trans. Signal Process. 53(4), 1425–1438 (2005)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    M.A. Unser, in Ten Good Reasons for Using Spline Wavelets, Optical Science, Engineering and Instrumentation’97, International Society for Optics and Photonics (1997) pp. 422–431Google Scholar
  29. 29.
    M. Vucic, G. Molnar, Time-domain synthesis of continuous-time systems based on second-order cone programming. IEEE Trans. Circuits Syst. I Regul. Pap. 55(10), 3110–3118 (2008). doi: 10.1109/TCSI.2008.925379
  30. 30.
    E.J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Comput. Phys. Rep. 10(5), 189–371 (1989)CrossRefGoogle Scholar
  31. 31.
    E.J. Weniger, Irregular input data in convergence acceleration and summation processes: general considerations and some special Gaussian hypergeometric series as model problems. Comput. Phys. Commun. 133(2), 202–228 (2001)CrossRefMATHGoogle Scholar
  32. 32.
    E.J. Weniger, Mathematical properties of a new Levin-type sequence transformation introduced by Čıžek, Zamastil, and Skála. I. Algebraic theory. J. Math. Phys. 45(3), 1209–1246 (2004)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    H. Xu, S. Jain, J. Song, T. Kamgaing, Y.S. Mekonnen, Acceleration of spectral domain immitance approach for generalized multilayered shielded microstrips using the Levins transformation. IEEE Antennas Wirel. Propag. Lett. 14, 92–95 (2015)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Electrical EngineeringBirla Institute of Technology and Science - PilaniHyderabadIndia

Personalised recommendations