Circuits, Systems, and Signal Processing

, Volume 37, Issue 8, pp 3295–3312 | Cite as

Design of Time–Frequency-Localized Two-Band Orthogonal Wavelet Filter Banks

  • Dinesh BhatiEmail author
  • Ram Bilas Pachori
  • Manish Sharma
  • Vikram M. Gadre


In this paper, we design time–frequency-localized two-band orthogonal wavelet filter banks using convex semidefinite programming (SDP). The sum of the time variance and frequency variance of the filter is used to formulate a real symmetric positive definite matrix for joint time–frequency localization of filters. Time–frequency-localized orthogonal low-pass filter with specified length and regularity order is designed. For nonmaximally regular two-band filter banks of length twenty, it is found that, as we increase the regularity order, the solution of the SDP converges to the filters with time–frequency product (TFP) almost same as the Daubechies maximally regular filter of length twenty. Unlike the class of Daubechies maximally regular minimum phase wavelet filter banks, a rank minimization algorithm in a SDP is employed to obtain mixed-phase low-pass filters with TFP of the filters as well as the scaling and wavelet function better than the equivalent two-band Daubechies filter bank.


Two-band filter bank Orthogonal wavelet Semidefinite programming Uncertainty principle Time–frequency localization 


  1. 1.
    D. Bhati, M. Sharma, R.B. Pachori, S.S. Nair, V.M. Gadre, Design of time–frequency optimal three-band wavelet filter banks with unit sobolev regularity using frequency domain sampling. Circuits Syst. Signal Process. 35(12), 4501–4531 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    D. Bhati, R.B. Pachori, V.M. Gadre, A novel approach for time–frequency localization of scaling functions and design of three-band biorthogonal linear phase wavelet filter banks. Digit. Signal Process. 69, 309–322 (2017a)CrossRefGoogle Scholar
  3. 3.
    D. Bhati, M. Sharma, R.B. Pachori, V.M. Gadre, Time–frequency localized three-band biorthogonal wavelet filter bank using semidefinite relaxation and nonlinear least squares with epileptic seizure EEG signal classification. Digit. Signal Process. 62, 259–273 (2017b)CrossRefGoogle Scholar
  4. 4.
    S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004)CrossRefzbMATHGoogle Scholar
  5. 5.
    L. Cohen, Time–frequency distributions—a review. Proc. IEEE 77(7), 941–981 (1989)CrossRefGoogle Scholar
  6. 6.
    A. Cohen, I. Daubechies, Orthonormal bases of compactly supported wavelets III. Better frequency resolution. SIAM J. Math. Anal. 24(2), 520–527 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    I. Daubechies, Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41, 909–996 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    L. Debnath, F.A. Shah, Wavelet Transforms and Their Applications (Springer, Berlin, 2002)CrossRefzbMATHGoogle Scholar
  9. 9.
    M. Dedeoglu, Y. Alp, O. Arikan, FIR filter design by convex optimization using directed iterative rank refinement algorithm. IEEE Trans. Signal Process. 64, 2209–2219 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    D. Gabor, Theory of communication. Proc. Inst. Electr. Eng. 93(26), 429–441 (1946)Google Scholar
  11. 11.
    A. Grossmann, J. Morlet, Decomposition of hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal. 15(4), 723–736 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. Haar, Zur theorie der orthogonalen funktionensysteme. Math. Ann. 69(3), 331–371 (1910)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    R.A. Haddad, A.N. Akansu, A. Benyassine, Time–frequency localization in transforms, subbands, and wavelets: a critical review. Opt. Eng. 32(7), 1411–1429 (1993)CrossRefGoogle Scholar
  14. 14.
    R. Ishii, K. Furukawa, The uncertainty principle in discrete signals. IEEE Trans. Circuits Syst. 33(10), 1032–1034 (1986)CrossRefzbMATHGoogle Scholar
  15. 15.
    R. Kolte, P. Patwardhan, V.M. Gadre, A class of time-frequency product optimized biorthogonal wavelet filter banks. InProceedings of the Sixteenth National Conference on Communications (2010)Google Scholar
  16. 16.
    J.E. Littlewood, R.E. Paley, Theorems on Fourier series and power series. J. Lond. Math. Soc. 1(3), 230–233 (1931)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Z.-Q. Luo, W.-K. Ma, A.M.-C. So, Y. Ye, S. Zhang, Semidefinite relaxation of quadratic optimization problems. IEEE Signal Process. Mag. 27(3), 20 (2010)CrossRefGoogle Scholar
  18. 18.
    S. Mallat, A theory of multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11, 647–693 (1989)CrossRefzbMATHGoogle Scholar
  19. 19.
    Y. Meyer, Orthonormal wavelets. in Wavelets, ed. by J.M. Combes, A. Grossmann, P. Tchamitchian (Springer, Berlin, 1989), pp. 21–37Google Scholar
  20. 20.
    J.M. Pak, C.K. Ahn, Y.S. Shmaliy, P. Shi, M.T. Lim, Switching extensible fir filter bank for adaptive horizon state estimation with application. IEEE Trans. Control Syst. Technol. 24(3), 1052–1058 (2016)CrossRefGoogle Scholar
  21. 21.
    A. Papoulis, Signal Analysis, vol. 2 (McGraw-Hill, New York, 1977)zbMATHGoogle Scholar
  22. 22.
    R. Parhizkar, Y. Barbotin, M. Vetterli, Sequences with minimal time–frequency uncertainty. Appl. Comput. Harmonic Anal. 38(3), 452–468 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    B.D. Patil, P.G. Patwardhan, V.M. Gadre, On the design of FIR wavelet filter banks using factorization of a halfband polynomial. IEEE Signal Process. Lett. 15, 485–488 (2008)CrossRefGoogle Scholar
  24. 24.
    M. Sharma, R. Kolte, P. Patwardhan, V.M. Gadre, Time-frequency localization optimized biorthogonal wavelets, in International Conference on Signal Processing and Communications (IISc, Bangalore, India, 2010), pp. 1–5Google Scholar
  25. 25.
    M. Sharma, V.M. Gadre, S. Porwal, An eigenfilter-based approach to the design of time-frequency localization optimized two-channel linear phase biorthogonal filter banks. Circuits Syst. Signal Process. 34(3), 931–959 (2015)CrossRefGoogle Scholar
  26. 26.
    M. Sharma, D. Bhati, S. Pillai, R.B. Pachori, V.M. Gadre, Design of time-frequency localized filter banks: transforming non-convex problem into convex via semidefinite relaxation technique. Circuits Syst. Signal Process. 35(10), 3716–3733 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    D. Slepian, H.O. Pollak, Prolate spheroidal wave function, Fourier analysis and uncertainty—I. Bell Syst. Tech. J. 40(1), 43–63 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    P. Steffen, P.N. Heller, R.A. Gopinath, C.S. Burrus, Theory of regular \(M\)-band wavelet bases. IEEE Trans. Signal Process. 41(12), 3497–3511 (1993)CrossRefzbMATHGoogle Scholar
  29. 29.
    G. Strang, Eigenvalues of (\(\downarrow \)2)h and convergence of the cascade algorithm. IEEE Trans. Signal Process. 44(2), 233–238 (1996)CrossRefGoogle Scholar
  30. 30.
    D.B. Tay, Balanced-uncertainty optimized wavelet filters with prescribed regularity, in International Symposium on Circuits and Systems, Scottsdale, Arizona (1999), pp. 532–535Google Scholar
  31. 31.
    P.P. Vaidyanathan, Multirate Systems and Filter Banks (Pearson Education, Delhi, 1993)zbMATHGoogle Scholar
  32. 32.
    Y. Venkatesh, K.S. Raja, V.G. Sagar, On bandlimited signals with minimal space/time-bandwidth product, in IEEE International Conference on Multimedia and Expo, Taipei, Taiwan (2004), pp. 1911–1914Google Scholar
  33. 33.
    M. Vetterli, J. Kovačević, V.K. Goyal, Signal Processing: Foundations (2012)Google Scholar
  34. 34.
    M. Vetterli, J. Kovacevic, Wavelets and Subband Coding (Prentice Hall, Englewood Cliffs, 1995)zbMATHGoogle Scholar
  35. 35.
    H. Xie, J.M. Morris, Design of orthonormal wavelets with better time–frequency resolution. Proc. SPIE 2242, 878–887 (1994)CrossRefGoogle Scholar
  36. 36.
    J.-K. Zhang, T.N. Davidson, K.M. Wong, Efficient design of orthonormal wavelet bases for signal representation. IEEE Trans. Signal Process. 52(7), 1983–1996 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Dinesh Bhati
    • 1
    Email author
  • Ram Bilas Pachori
    • 2
  • Manish Sharma
    • 3
  • Vikram M. Gadre
    • 4
  1. 1.Department of Electronics EngineeringAcropolis Institute of Technology and ResearchIndoreIndia
  2. 2.Discipline of Electrical EngineeringIndian Institute of Technology IndoreIndoreIndia
  3. 3.Department of Electrical EngineeringInstitute of Infrastructure Technology Research and ManagementAhmadabadIndia
  4. 4.Department of Electrical EngineeringIndian Institute of Technology BombayMumbaiIndia

Personalised recommendations