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Revista de Matemática Teoría y Aplicaciones
versión impresa ISSN 1409-2433
Rev. Mat vol.22 no.1 San José ene./jun. 2015
Wavelets infinitamente oscilantes y un eficiente algoritmo de implementación basado en la FFT
Infinitely oscillating wavelets and a efficient implementation algorithm based on the FFT
Infinitely oscillating wavelets and a efficient implementation algorithm based on the FFT
Resumen
En este trabajo presentamos el diseño de una wavelet ortogonal, infinitamente oscilante, localizada en el tiempo con decaimiento 1=jtjn y de banda limitada. Su aplicación conduce a la descomposición de señales en ondas de frecuencia instantánea bien definida. Presentamos además el algoritmo de implementación para el análisis y síntesis basado en la Transformada Rápida de Fourier con la misma complejidad que el algoritmo de Mallat.
Palabras clave: wavelet tipo pasa-banda; algoritmo de Mallat; FFT; análisis de multirresolución; frecuencia instantánea.
Abstract
In this work we present the design of an orthogonal wavelet, infinitely oscillating, located in time with decay 1=jtjn and limited-band. Its application leads to the signal decomposition in waves of instantaneous, well defined frequency. We also present the implementation algorithm for the
analysis and synthesis based on the Fast Fourier Transform (FFT) with the same complexity as Mallat’s algorithm.
Keywords: pass-band wavelet; Mallat’s algorithm; FFT; multiresolution analysis; instantaneous frequency.
Mathematics Subject Classification: 42C40, 44A05.
Ver contenido en pdf.
Referencias
1 Donoho, D.L. (1995) “Nonlinear solution of Linear Inverse problems by Wavelet-Vaguelet decomposition”, Applied and Computational Harmonic Analysis 2(2): 101–126. [ Links ]
[2] Huang, N.E; Shen, Z.; Long, S.R.; Wu, M.C.; Shih, H.H.; Zheng, Q.; Yen, N.C.; Tung, C.C.; Liu, H.H. (1998) “The empirical mode decomposition and the Hilbert spectrum for non-stationary time series analysis”, Proc. R. Soc. Lond. A 454: 903–995. [ Links ]
[3] Jaffard, S.; Lashermes, B.; Abry, P. (2007) “Wavelet leaders in multifractal analysis”, in: T. Qian, M. Vai, X. Yuesheng (Esd.) Wavelet Analysis and Applications, Birkhäuser, Basel, Switzerland: 201–246. [ Links ]
[4] Li, L.C. (2010) “A new method of wavelet transform based on FFT for signal processing”, Second WRI Global Congres on Intelligent Systems, IEEE Computer Society: 203–206. [ Links ]
[5] Mallat, S. (2009) A Wavelet Tour of Signal Processing, The Sparse Way. Academic Press–Elsevier, Burlington MA. [ Links ]
[6] Meyer, Y. (1993) Wavelets, Algorithms and Applications. SIAM, Philadelphia PA. [ Links ]
[7] Meyer, Y. (2001) Oscillating Pattern in Image Processing and Nonlinear Evolution Equations. American Mathematical Society, Providence RI. [ Links ]
[8] Serrano, E.; Figliola, A. (2008) Littlewood-Paley spline wavelets: a simple and efficient tool for signal and image processing in industrial applications, Proceedings in Applied Mathematics and Mechanics (PAMM), Wiley InterScience, 7: 1040313–1040314. [ Links ]
[9] Serrano, E.; Fabio, M. (2010) “Diseño de funciones elementales combinando la transformada wavelet y la transformada de Hilbert”, UMA 2010, Tandil, Argentina. [ Links ]
[10] Serrano, E.; Fabio, M.; Aragón, A. (2011) “Caracterización de la frecuencia instantánea en señales tipo pasa-banda”, III MACI, Asociación Argentina de Matemática Aplicada, Computacional e Industrial. Bahía Blanca, Argentina. [ Links ]
[2] Huang, N.E; Shen, Z.; Long, S.R.; Wu, M.C.; Shih, H.H.; Zheng, Q.; Yen, N.C.; Tung, C.C.; Liu, H.H. (1998) “The empirical mode decomposition and the Hilbert spectrum for non-stationary time series analysis”, Proc. R. Soc. Lond. A 454: 903–995. [ Links ]
[3] Jaffard, S.; Lashermes, B.; Abry, P. (2007) “Wavelet leaders in multifractal analysis”, in: T. Qian, M. Vai, X. Yuesheng (Esd.) Wavelet Analysis and Applications, Birkhäuser, Basel, Switzerland: 201–246. [ Links ]
[4] Li, L.C. (2010) “A new method of wavelet transform based on FFT for signal processing”, Second WRI Global Congres on Intelligent Systems, IEEE Computer Society: 203–206. [ Links ]
[5] Mallat, S. (2009) A Wavelet Tour of Signal Processing, The Sparse Way. Academic Press–Elsevier, Burlington MA. [ Links ]
[6] Meyer, Y. (1993) Wavelets, Algorithms and Applications. SIAM, Philadelphia PA. [ Links ]
[7] Meyer, Y. (2001) Oscillating Pattern in Image Processing and Nonlinear Evolution Equations. American Mathematical Society, Providence RI. [ Links ]
[8] Serrano, E.; Figliola, A. (2008) Littlewood-Paley spline wavelets: a simple and efficient tool for signal and image processing in industrial applications, Proceedings in Applied Mathematics and Mechanics (PAMM), Wiley InterScience, 7: 1040313–1040314. [ Links ]
[9] Serrano, E.; Fabio, M. (2010) “Diseño de funciones elementales combinando la transformada wavelet y la transformada de Hilbert”, UMA 2010, Tandil, Argentina. [ Links ]
[10] Serrano, E.; Fabio, M.; Aragón, A. (2011) “Caracterización de la frecuencia instantánea en señales tipo pasa-banda”, III MACI, Asociación Argentina de Matemática Aplicada, Computacional e Industrial. Bahía Blanca, Argentina. [ Links ]
*Centro de Matemática Aplicada, Universidad de San Martín, Argentina. E-Mail: mfabio@unsam.edu.ar
†Centro de Matemática Aplicada, Universidad de San Martín y Escuela Superior Técnica del Ejército “General Manuel N. Savio”, I.E.S.E., Argentina. E-Mail: eserrano@unsam.edu.ar
Received: 5/Mar/2012; Revised: 3/Sep/2014; Accepted: 17/Oct/2014
