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Design of Maximally Flat and Monotonic FIR Filters Using the Bernstein Polynomial

  • Suhash Chandra Dutta RoyEmail author
Chapter

Abstract

This chapter presents a recently introduced technique for the design of Maximally flat (MAXFLAT) FIR filters by using the Bernstein polynomial, and reviews its applications in (i) establishing the equivalence between the earlier known methods, and (ii) formulating a matrix approach for determining the coefficients of MAXFLAT FIR filters efficiently. We also review a new, optimal design procedure for MAXFLAT filters and extend the method to generate monotonic FIR filters with arbitrary magnitude specifications, for which, presently, no method exists. Further, these concepts have been used here to design Quadrature Mirror Filters (QMF), with extremely low reconstruction error.

Keywords

Digital filters Maximally-flat FIR filters Monotonic FIR filters Quadrature mirror filters 

References

  1. 1.
    O. Herrmann, On the approximation problem in nonrecursive digital filter design. IEEE Trans. CT-18, 411–413 (1971)Google Scholar
  2. 2.
    J.A. Miller, Maximally flat nonrecursive digital filters. Electron. Lett. 8, 157–158 (1972)CrossRefGoogle Scholar
  3. 3.
    M.F. Fahmy, Maximally flat nonrecursive digital filters. Int. J. Circuit Theory Appl. 4, 311–313 (1976)CrossRefGoogle Scholar
  4. 4.
    J.F. Kaiser, Comments on maximally flat nonrecursive digital filters. Int. J. Circuit Theory Appl. 5, 103 (1977)CrossRefGoogle Scholar
  5. 5.
    R.W. Hamming, Digital Filters (Englewood Cliffs, New Jersey; Prentice-Hall, 1977)Google Scholar
  6. 6.
    J.F. Kaiser, W.A. Reed, Data smoothing using lowpass digital filters. Rev. Sci. lnstrum. 48, 1447–1457 (1977)CrossRefGoogle Scholar
  7. 7.
    C. Gumacos, Weighting coefficients for certain maximally flat nonrecursive digital filters. IEEE Trans. CAS-25, 234–235 (1978)Google Scholar
  8. 8.
    J.F. Kaiser, Design subroutine (MAXFLAT) for symmetric FIR low pass digital filters, with maximally-flat pass and stop bands, in Programs for Digital Signal Processing (IEEE Press, New York, 1979), pp. 5.3-1–5.3-6Google Scholar
  9. 9.
    M.U.A. Bromba, H. Zeigler, Explicit formula for filter function of maximally flat nonrccursive digital filters. Electron. Lett. 16, 905–906 (1980)MathSciNetCrossRefGoogle Scholar
  10. 10.
    B.C. Jinaga, S.C. Dutta Roy, Explicit formulas for the weighting coefficients of maximally flat nonrecursive digital filters. Proc. IEEE 72, 1092 (1984)Google Scholar
  11. 11.
    B.C. Jinaga, S.C. Dutta Roy, Coefficients of maximally-flat nonrecursive digital filters. Sig. Process. 7, 185–189 (1984)CrossRefGoogle Scholar
  12. 12.
    B.C. Jinaga, S.C. Dutta Roy, Explicit formula for the coefficients of maximally flat nonrecursive digital filter transfer function expressed in powers of cos ω. Proc. IEEE 73, 1135–1136 (1985)Google Scholar
  13. 13.
    P. Thajchayapong, M. Puangpool, S. Banjongjit, Maximally flat FIR filters with prescribed cut-off frequency. Electron. Lett. 16, 514–515 (1980)CrossRefGoogle Scholar
  14. 14.
    B.C. Jinaga, S.C. Dutta Roy, Coefficients of maximally flat low and high pass nonrecursive digital filters with specified cut-off frequency. Sig. Process. 9, 121–124 (1985)CrossRefGoogle Scholar
  15. 15.
    P.P. Vaidyanathan, Efficient and multiplierless design of FIR filters with very sharp cutoff via maximally flat building blocks. IEEE Trans. CAS-32, 236–244 (1985)Google Scholar
  16. 16.
    P.P. Vaidyanathan, Optimal design of linear phase FIR digital filters with very flat passband and equiripple stopbands. IEEE Trans. CAS-32, 904–907 (1985)Google Scholar
  17. 17.
    P.J. Davis, Interpolation and Approximation (Dover, New York, 1975)zbMATHGoogle Scholar
  18. 18.
    L.E. Bergeron, A maximally-flat filter design algorithm for Quadrature Mirror Filter (QMF), in IEEE International Conference on Acoustics, Speech, and Signal Processing, Washington, D.C. (1979), pp. 828–831Google Scholar
  19. 19.
    G.G. Lorentz, Bernstein Polynomial (Toronto 1953)Google Scholar
  20. 20.
    L.R. Rajagopal, S.C. Dutta Roy, Design of maximally-flat FIR filters using the Bernstein polynomial. IEEE Trans. CAS-34, 1587–1590 (1987)Google Scholar
  21. 21.
    L.R. Rajagopal, S.C. Dutta Roy, A matrix approach for the coefficients of maximally-flat FIR filters expressed in powers of cos ω. Electron. Lett. 18, 391–393 (1987)CrossRefGoogle Scholar
  22. 22.
    C.J. Greaves, G.A. Gagne, C.W. Bordner, Evaluation of integrals appearing in minimization problems of discrete-data systems. IEEE Trans. AC-11, 145–148 (1966)Google Scholar
  23. 23.
    E.I. Jury, W.C. Chen, Combinatorial Rules for some useful transformations. IEEE Trans. CT-10, 476–480 (1973)Google Scholar
  24. 24.
    N.K. Bose, Properties of the Q-matrix in bilinear transformation. Proc. IEEE 71, 1110–1111 (1983)CrossRefGoogle Scholar
  25. 25.
    A.V. Oppenheim, W.F.G. Mecklenbrauker, R.M. Mersereau, Variable cutoff linear phase digital filters. IEEE Trans. CAS-23, 199–203 (1976)Google Scholar
  26. 26.
    L.R. Rajagopal, S.C. Dutta Roy, A matrix approach for the coefficients of maximally-flat FIR filter transfer functions. IEEE Trans. ASSP, ASSP-36, 1914–1917 (1988)Google Scholar
  27. 27.
    L.R. Rajagopal, S.C. Dutta Roy, Design of optimal maximally flat FIR filters using the Bernstein polynomial, in International Symposium on Electronic Devices, Circuits and Systems, Kharagpur, India (1987)Google Scholar
  28. 28.
    S.C. Dutta Roy, L.R. Rajagopal, A new approach to the design of maximally-flat and monotonic FIR filters, in Indo-US workshop on Systems and Signal Processing, Bangalore (1988)Google Scholar
  29. 29.
    L.R. Rajagopal, S.C. Dutta Roy, Design of optimal maximally flat FIR filters with arbitrary magnitude specifications. IEEE Trans. ASSP, ASSP-37, 4, 512–218 (1989)Google Scholar
  30. 30.
    R.E. Crochiere, L.R. Rabiner, Multirate Digital Signal Processing (Englewood Cliffs, New Jersey, Prentice-Hall, 1983)CrossRefGoogle Scholar
  31. 31.
    S.M. Kay, S.L. Marple, Spectrum analysis—a modern perspective. Proc. IEEE 69, 1380–1419 (1981)CrossRefGoogle Scholar
  32. 32.
    J.D. Johnston, A filter family designed for use in quadrature mirror filter banks, in IEEE International Conference on Acoustics, Speech, and Signal Processing (1980), pp. 291–294Google Scholar
  33. 33.
    V.K. Jain, R.E. Crochiere, A novel approach to the design of analysis/synthesis filter banks, in IEEE International Conference on Acoustics, Speech, and Signal Processing, Boston (1983), pp. 228–231Google Scholar
  34. 34.
    L.C.W. Dixon, Nonlinear Optimization (The English Universities Press, London, 1972)Google Scholar
  35. 35.
    L.R. Rajagopal, S.C. Dutta Roy, A new class of monotone quadrature mirror filters, in Symposium on Signals, Systems and Sonars, vol. 88, Cochin, India (1988) Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Indian Institute of Technology DelhiNew DelhiIndia

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