INFORMATION CHANGE THE WORLD

### International Journal of Information Technology and Computer Science(IJITCS)

ISSN: 2074-9007 (Print), ISSN: 2074-9015 (Online)

IJITCS Vol.4, No.12, Nov. 2012

#### Analogue Wavelet Transform Based the Solution of the Parabolic Equation

Full Text (PDF, 842KB), PP.1-20

#### Author(s)

Jean-Bosco Mugiraneza, Amritasu Sinha

#### Index Terms

Wavelet Transform, Morlet Wavelet, PDE, FFT, Power Spectral Density, Matlab, Parabolic Equation

#### Abstract

In this paper we have proved that the solution of parabolic equation and its Fast Fourier Transform generate continuous wavelet transforms. Indeed, we have solved the parabolic equation using PDETool, exported its solution and coefficients to Matlab workspace. We have then imported the solution from workspace to signal processing tool. We have sampled the imported solution with the sampling frequency of 8192Hz and applied the band pass filter with that frequency. The convolution of the sampled PDE solution with the impulse response of the band pass filter has generated wavelet transform. This algorithm computes the wavelet transform either directly of via Faster Fourier Transform. The computation of the FFT of the PDE solution has produced complex wavelet.

#### Cite This Paper

Jean-Bosco Mugiraneza, Amritasu Sinha,"Analogue Wavelet Transform Based the Solution of the Parabolic Equation", International Journal of Information Technology and Computer Science(IJITCS), vol.4, no.12, pp.1-20, 2012. DOI: 10.5815/ijitcs.2012.12.01

#### Reference

[1]M.Sifuzzaman, M.R.Islam and M. Z.Ali, Application of Wavelet Transform and Its Advantages Compared to Fourier Transform, Journal of Physical Sciences, Vol.13, 2009, 121-134

[2]Wells, R.O, Parametrizing Smooth Compactly Supported Wavelet Transform, American Mathematical Society, 338(2): 9019-931, 1993

[3]Strang, G. Wavelets and Dilatation Equations: A Brief Introduction. SIAM Review, 31:614-627, 1989

[4]R99942036, Term Paper, A Tutorial of the Mother Wavelet Transform,

[5]Timothy Herron and John Byrnes, Families of Orthogonal Differential Operators for Signal Processing, 2001

[6]Y. Lyubarskii, Frames in the Bargann Space of Entire Functions, Entire and Subharmonic Functions, 167-180, Adv. Soviet Mth., 11, Amer.Math.Soc, Providence, RI(1992)

[7]K.Seip, Density Theorems for Sampling and Interpolation in the Bargmann-Fork Space I, J.Reine Angew. Math.429, 91-106(1992)

[8]K.Seip, R. Wallstén, Density Theorems for Sampling and Interpolation in the Bargmann-Fork Space II, J.Reine Angew. Math.429, (1992), 107-113

[9]K.Seip, Beurling Type Density Theorems in the Unit Disc, Invent. Math. 113, 21-39, 1993

[10]Luis Daniel Abreu, Wavelet Frames, Bergmann Spaces and Fourier Transforms of Laguerre Functions, April 2007

[11]A.J.E.M Janssen, T. Strohmer, Hyperbolic Secants Yield Gabor Frames. Appl. Comput. Harmon. Anal.12, no 2, 259-267 (2002)

[12]A.J.E.M Janssen, Zak Transforms with Few Zeros and the Tie, in ‘Advances in Gabor Analysis’(H.G Feichtinger, T.trohmer, eds.), Boston , 2003, ppp.31-70

[13]J. Ramanathan and T. Steger, Incompleteness of Sparse Coherent States. Appl. Comput. Harmon.Anal.2, no 2, 148-153 (1995)

[14]Dorin Ervin Dutkay and Palle E.T. Jorgensen, Iterated Function Systems, Ruelle Operators, and Invariant Projective Measures, March 2008

[15]E.B. Postinikov and M.V. Lomonosov, Wavelet Transform and Diffusion Equations: Applications to the Processing of the “CASSINI” Space Craft Observations, 2008

[16]Haase M.A , Family of Complex Wavelets for the Characterization of Singularities//Paradigms of Complexity, Ed.M.M Novak.World Scientific, 2000, P.287-288.

[17]E.B. Postnikov, Time-Frequency Analysis of Non-Stationary Signals Using the Continuous Wavelet Transform Based on the Solving of PDE, XVIII Session of the Russian Acoustical Society, 2006

[18]M.Meléndez-Rodríguez, J. Silva-Martínez and R. Spencer, Efficient Circuit Implementation of Morlet Wavelets, Journal of Applied Research and Technology, Vol.1no1 April 2005.

[19]Stefan Goedecker, Wavelets and their Application for the Solution of Partial Differential Equations in Physics, Max-Planck Institute for Solid State Research, Stuttgart, Germany, May 20, 2009

[20]Amritasu Sinha and Jean Bosco Mugiraneza, Principles of Engineering Analysis, Alpha Science Int’l Ltd, Oxford, UK, 2012, ISBN: 978-1842657010