Archive




Volume 8, Issue 6, December 2019, Page: 100-105
Filter Banks from the Fibonacci Sequence
Fuxian Chen, Faculty of Mathematics and Informatics, South China Agricultural University, Guangzhou, China
Qiuhui Chen, Faculty of Mathematics and Informatics, South China Agricultural University, Guangzhou, China
Weibin Wu, Faculty of Engineer, South China Agricultural University, Guangzhou, China
Xiaoming Wang, Faculty of Engineer, South China Agricultural University, Guangzhou, China
Received: Oct. 21, 2019;       Accepted: Dec. 2, 2019;       Published: Dec. 31, 2019
DOI: 10.11648/j.pamj.20190806.12      View  322      Downloads  145
Abstract
Wavelet transform is an important quadratic representation in time-frequency domain of signals. The main advantage of wavelet transform is the time frequency localization as compared with the fourier transform. Due to the reason of dilation and translation operation acting the basic time-frequency atoms. Therefore a multi-resoloution analysis strategy is devoted to the construction of wavelet basis of L2(R), which also establishes a bridge between engineer and mathematics. The construction of wavelets is equivalent to the design of filter banks with complete reconstruction. In this note we investigate filter banks from the Fibonacci sequence. The draw back is that, the convergence z-transform is less than 1, hence it can not be used as filter. By adopting the Hadamard product of the Fibonacci sequence and a geometric sequence, a type of Fibonacci-based bi-orthogonal filter banks are constructed. This kind of filter banks are based two bricks: Bezout polynomials and the mask of the cardinal B-splines. These filters are essentially rational functions, which have potential applications in system identification and signal processing.
Keywords
Filter Banks, Bezout Polynomial, Wavelet, Symbol of B-spline
To cite this article
Fuxian Chen, Qiuhui Chen, Weibin Wu, Xiaoming Wang, Filter Banks from the Fibonacci Sequence, Pure and Applied Mathematics Journal. Vol. 8, No. 6, 2019, pp. 100-105. doi: 10.11648/j.pamj.20190806.12
Copyright
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[1]
A. Cohen, I. Daubechies, Non-separable bidimensional wavelet bases, Rev. Mat. Iberoamericana 9 (1993), 51-137.
[2]
A. Croisier, D. Esteban and C. Galand, Perfect channel splitting by use of interpolation/ decimation/tree decomposition techniques, in Int. Conf. on Info. Sciences and Systems, 443-446, Patras, Greece, August 1976.
[3]
I. Daubechies, Ten Lectures on Wavelets, CBMS, 61, SIAM, Philadelphia, 1992.
[4]
R. E. Edwards, Fourier Series, A Mordern Introduction, Volumn 2, GTM 64, Springer-Verlag New York,Inc., 1979.
[5]
C. Heil and D. Colella, Matrix refinement equation: existence and uniqueness, J. Fourier Anal. Appl. 2 (1996), 363-377.
[6]
J. Kovacevic and M. Vetterli, Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for Rn, IEEE Tran. on Information Theory 38(1992), 533-555.
[7]
S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 3rd edition, 2008.
[8]
T. Q. Nguyen and P. P. Vaidyanathan, Two-channel perfect reconstruction FIR QMF structure which yield linear phase FIR analysis and synthesis filters, IEEE Trans. Acoustics, Speech and Signal Proc. 37 (1989), 676-690.
[9]
M. J. Smith and T. P. Barnwell, A procedure for designing exact reconstruction filter banks for tree structured subband coders, in Proc. IEEE Int. Conf. Acoust. Speech and Signal Proc., San Diego, CA, March 1984.
[10]
P. P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice Hall, Englewood Cliffs, New Jersey, 1993.
[11]
P. Przemyslaw, H. James, A. Lindenmayer, Systems, Fractals, and Plants, Lecture Notes in Biomathematics, Springer-Verlag, ISBN 0387970924,1992.
[12]
Ahamad, Mohd Vasim, et al. An Improved Playfair Encryption Technique Using Fibonacci Series Generated Secret Key. International Journal of Engineering and Technology 7.4.5 (2018): 347-351.
[13]
Zhang, Yingge, et al. Multiplanar Fourier-transform phase-shifting digital holography with a generalized Fibonacci lens. Laser Physics Letters, 2019, 16.5: 055201.
[14]
Ghosh, Nabakumar. Fibonacci numbers in real life applications. Mugberia Gangadhar Mahavidyalaya, 1 (2018): 62-69.
[15]
Singh, P., et al. Fractal and periodical biological antennas: Hidden topologies in DNA, wasps and retina in the eye. Soft Computing Applications. Springer, Singapore, 2018. 113-130.
Browse journals by subject