Projeto de bancos de filtros wavelet quase-ortogonais simétricos e de suporte compacto baseado no processo de ortogonalização wavelet
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Data
2021-02-09
Autores
Gossler, Fabrício Ely
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Editor
Universidade Estadual Paulista (Unesp)
Resumo
A análise wavelet baseada em um banco de filtros de dois canais tem sido usada com frequência em várias aplicações. O algoritmo utilizado para tal propósito é a transformada wavelet rápida, que permite obter os coeficientes wavelet a partir da decompsição de um sinal de tempo discreto. Neste caso, o resultado dessa decomposição depende diretamente dos filtros wavelet escolhidos para tal análise. Geralmente, os filtros ortogonais e biortogonais, de suporte compacto, são os mais utilizados para esse fim, pois fornecem a propriedade de perfeita reconstrução. Entretanto, as propriedades de ortogonalidade e biortogonalidade podem implicar em distorções de fase e ganho, respectivamente. Dessa forma, utilizando-se tais bases, a análise wavelet de um sinal pode ser comprometida devido a essas distorções. Por meio das wavelets quase-ortogonais, as distorções de fase e ganho podem ser praticamente anuladas. Entretanto, para obter tal vantagem é preciso permitir que tais bases apresentem desvios de ortogonalidade, o que implica em erros no processo de reconstrução do sinal. Neste trabalho é proposto um novo método para projetar bancos de filtros wavelet quase-ortogonais, simétricos e de suporte compacto, sendo baseado na metodologia proposta para extrair filtros semi-conjugados a partir de uma wavelet não-ortogonal. Ao contrário das estratégias adotadas pelos projetos clássicos, o método proposto permite projetar filtros wavelet de acordo com as especificações desejadas no domínio da frequência, satisfazendo tolerâncias mínimas de desvios de ortogonalidade. Isso é possível por meio da função de Ely que pode ser vista como um filtro particular de Butterworth. As wavelets obtidas a partir dessa função podem ser caracterizadas como versões relaxadas da wavelet ideal, conhecida como wavelet de Shannon. Para os filtros wavelet obtidos, os desvios de ortogonalidade podem ser minimizados em função do aumento da ordem dos seus coeficientes. A fim de testar as wavelets propostas, três aplicações diferentes são exploradas neste trabalho. Tais aplicações envolvem os temas de monitoramento de integridade estrutural, compressão de imagens e classificação de distúrbios de tensão em sistemas elétricos de potência. Para essa última aplicação, é apresentado um novo método para construir vetores wavelet característicos utilizando a transformada wavelet contínua. Tal método implica melhores resultados quando comparado com outras técnicas utilizadas na literatura especializada. Os resultados obtidos pelas wavelets de Ely são significativos quando comparados com os resultados de outras wavelets clássicas.
Wavelet analysis based on a two-channel filter bank has been frequently used in several applications. The algorithm used for this purpose is the fast wavelet transform, which allows to obtain the wavelet coefficients from a discrete time signal decomposition. In this case, the result of this decomposition depends directly on the choices of wavelet filters for such analysis. Generally, orthogonal and biorthogonal compactly support wavelet filters are the most used for this purpose, since they provide the perfect reconstruction property. However, orthogonality and biortogonality properties can result in phase and gain distortions, respectively. Thus, using such bases, the signal analysis in the wavelet domain can be compromised due to these distortions. Through nearly-orthogonal wavelet filters, phase and gain distortions can be practically eliminated. However, to obtain such an advantage it is necessary to allow such bases to present orthogonality deviations, which imply errors in the signal reconstruction process. In this work, a new method is proposed to design symmetrical and compactly support nearly-orthogonal wavelet filter banks, which is based on the proposed methodology to extract semi-conjugate filters from a non-orthogonal wavelet. Unlike the classical design methods, the proposed method allows the design of wavelet filters according to the desired specifications in frequency domain, satisfying minimum orthogonality deviations tolerances. This is possible through the Ely function that can be seen as a particular Butterworth filter. The wavelets obtained from this function can be characterized as relaxed versions of the ideal wavelet, known as Shannon wavelet. For the obtained wavelet filters, the orthogonality deviations can be minimized due to the increase in their coefficients. In order to test the proposed wavelets, three different applications are explored in this work. Such applications involve the topics of structural health monitoring, image compression and voltage disturbances classification in electrical power systems. For this last application, a new method to construct wavelet feature vectors using the continuous wavelet transform is presented. Such a method implies better results when compared to other ones proposed in the specialized literature. Results obtained by Ely wavelets are significant when compared to the results of other classical wavelets.
Wavelet analysis based on a two-channel filter bank has been frequently used in several applications. The algorithm used for this purpose is the fast wavelet transform, which allows to obtain the wavelet coefficients from a discrete time signal decomposition. In this case, the result of this decomposition depends directly on the choices of wavelet filters for such analysis. Generally, orthogonal and biorthogonal compactly support wavelet filters are the most used for this purpose, since they provide the perfect reconstruction property. However, orthogonality and biortogonality properties can result in phase and gain distortions, respectively. Thus, using such bases, the signal analysis in the wavelet domain can be compromised due to these distortions. Through nearly-orthogonal wavelet filters, phase and gain distortions can be practically eliminated. However, to obtain such an advantage it is necessary to allow such bases to present orthogonality deviations, which imply errors in the signal reconstruction process. In this work, a new method is proposed to design symmetrical and compactly support nearly-orthogonal wavelet filter banks, which is based on the proposed methodology to extract semi-conjugate filters from a non-orthogonal wavelet. Unlike the classical design methods, the proposed method allows the design of wavelet filters according to the desired specifications in frequency domain, satisfying minimum orthogonality deviations tolerances. This is possible through the Ely function that can be seen as a particular Butterworth filter. The wavelets obtained from this function can be characterized as relaxed versions of the ideal wavelet, known as Shannon wavelet. For the obtained wavelet filters, the orthogonality deviations can be minimized due to the increase in their coefficients. In order to test the proposed wavelets, three different applications are explored in this work. Such applications involve the topics of structural health monitoring, image compression and voltage disturbances classification in electrical power systems. For this last application, a new method to construct wavelet feature vectors using the continuous wavelet transform is presented. Such a method implies better results when compared to other ones proposed in the specialized literature. Results obtained by Ely wavelets are significant when compared to the results of other classical wavelets.
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Palavras-chave
Banco de filtros wavelet, Filtros wavelet simétricos, Transformada wavelet rápida, Wavelets de suporte compacto, Wavelets quase-ortogonais, Wavelet filter bank, Symmetrical wavelet filters, Fast wavelet transform, Compactly support wavelets, Nearly-orthogonal wavelets