AM  Vol.3 No.7 , July 2012
On the Zeros of Daubechies Orthogonal and Biorthogonal Wavelets
Author(s) Jalal Karam*
ABSTRACT
In the last decade, Daubechies’ wavelets have been successfully used in many signal processing paradigms. The construction of these wavelets via two channel perfect reconstruction filter bank requires the identification of necessary conditions that the coefficients of the filters and the roots of binomial polynomials associated with them should exhibit. In this paper, orthogonal and Biorthogonal Daubechies families of wavelets are considered and their filters are derived. In particular, the Biorthogonal wavelets Bior3.5, Bior3.9 and Bior6.8 are examined and the zeros distribution of their polynomials associated filters are located. We also examine the locations of these zeros of the filters associated with the two orthogonal wavelets db6 and db8.

Cite this paper
J. Karam, "On the Zeros of Daubechies Orthogonal and Biorthogonal Wavelets," Applied Mathematics, Vol. 3 No. 7, 2012, pp. 778-787. doi: 10.4236/am.2012.37116.
References
[1]   G. Strang, and T. Nguyen, “Wavelets and Filter Banks,” Wellesley-Cambridge Press, Wellesley, 1996.

[2]   I. Daubechies, “Ten Lectures on Wavelets,” SIAM, Philadelphia, 1992.

[3]   S. Kakeya, “On the Limits of the Roots of an Algebraic Equation with Positive Coefficients,” Tohoku Mathematical Journal, Vol. 2, 1912, pp. 140-142.

[4]   J. Karam, “On the Kakeya-Enestrom Theorem,” MSc. Thesis, Dalhousie University, Halifax, 1995.

[5]   J. Karam, “Connecting Daubechies Wavelets with the Kakeya-Enestrom Theorem,” International Journal of Applied Mathematics, Vol. 14, No. 2, 2003, pp. 109-124.

[6]   J. Karam, “On the Roots of Daubechies Polynomials,” International Journal of Applied Mathematics, Vol. 20, No. 8, 2007, pp. 1069-1076.

[7]   M. Vetterli, “Wavelets and Filter Banks: Theory and Design,” IEEE Transactions on Signal Processing, Vol. 40, No. 9, 1992, pp. 2207-2232. doi:10.1109/78.157221

[8]   M. Vetterli, and J. Kovacevic, “Wavelets and Suband Coding,” Prentice Hall, Englewood Cliffs, 1995.

[9]   M. Misiti, Y. Misiti, G. Oppenheim and J. Poggi, “Matlab Wavelet Tool Box,” 1997.

[10]   J. Karam, “A Comprehensive Approach for Speech Related Multimedia Applications,” WSEAS Transactions on Signal Processing, Vol. 6, No. 1, 2010, pp. 12-21.

[11]   J. Karam, “Radial Basis Functions With Wavelet Packets For Recognizing Arabic Speech,” The 9th WSEAS International Conference on Circuits, Systems, Electronics, Control and Signal Processing, Athens, December 2010, pp. 34-39.

[12]   J. Karam, “On the Distribution of Zeros for Daubechies Orthogonal Wavelets and Associated Polynomials,” 15th WSEAS International Conference on Applied Mathematics, Athens, 29-31 December 2010, pp. 101-105.

[13]   C. A. Muresan, “Comparative Methods for the Polynomial Isolation,” Proceedings of the 13th WSEAS International Conference on Computers, 2009, pp. 634-638.

 
 
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