On asymptotic convergence of the dual filters associated with two families of biorthogonal wavelets

The asymptotic behavior of the dual filters associated with biorthogonal spline wavelet (BSW) systems and general biorthogonal Coifman wavelet (GBCW) systems are studied. As the order of the wavelet systems approaches infinity, the magnitude responses of the dual filters in the BSW systems either diverge or converge to some nonideal frequency responses. However, the synthesis filters in the GBCW systems converge to an ideal halfband lowpass filter without exhibiting any Gibbs-like phenomenon, and a subclass of the analysis filters also converge to an ideal halfband lowpass filter but with a one-sided Gibbs-like behavior. The two approximations of the ideal lowpass filter by the filter associated with a Daubechies orthonormal wavelet and by the synthesis filter in a GBCW system of the same order are compared. Such a study of the asymptotic behaviors of wavelet systems provides insightful characterization of these systems and systematic assessment and global comparison of different wavelet systems.