Enhanced B-Wavelets via Mixed, Composite Packets

A modified B-wavelet construction with enhanced filter characteristics is considered. The design comprises a superposition of tessellated, integer dilated, “sister” wavelet functions. We here propose a cascaded filter-bank realization of this wavelet family together with some notable extensions. We prove that modifications of low-order members exist in the multiresolution subspace spanned by the half-translates of the original wavelets and, hence, that the resulting modified wavelet coefficients can be computed as convolutions of the undecimated original wavelet coefficients. Finite impulse response filters are thus designed and incorporated into a B-wavelet packet architecture such that the mainlobe-to-sidelobe ratio of the resulting wavelet filter characteristic is improved. This is achieved by designing the filters so that zeros are introduced near to the maxima of the harmonics. It is shown that the numbers of zeros can be balanced with the length of the corresponding filters by controlling the “modification order”. Several constructions are presented. We prove that two such constructions satisfy the perfect reconstruction property for all orders. The resulting modified wavelets preserve many of the properties of the original B-wavelets, such as differentiability, number of vanishing moments, symmetry, anti-symmetry, finite support, and the existence of a closed form expression.