In the paper we present a new family of biorthogonal wavelet transforms and the related library of biorthogonal symmetric waveforms. For the construction we used the interpolatory discrete splines which enabled us to design a library of perfect reconstruction ÿlter banks. These ÿlter banks are related to Butterworth ÿlters. The construction is performed in a "lifting" manner. The di erence from the conventional lifting scheme is that the transforms of a signal are performed via recursive ÿltering with the use of IIR ÿlters. These ÿlters have linear phase property and the basic waveforms are symmetric. The ÿlters allow fast cascade or parallel implementation. We present explicit formulas for construction of wavelets with arbitrary number of vanishing moments. In addition, these ÿlters yield perfect frequency resolution. The proposed scheme is based on interpolation and, as such, it involves only samples of signals and it does not require any use of quadrature formulas. ?
In this paper we consider equidistant discrete splines S( j ), j # Z, which may grow as O(| j | s ) as | j | Ä . Such splines are relevant for the purposes of digital signal processing. We give the definition of the discrete B-splines and describe their properties. Discrete splines are defined as linear combinations of shifts of the B-splines. We present a solution to the problem of discrete spline cardinal interpolation of the sequences of power growth and prove that the solution is unique within the class of discrete splines of a given order.
Academic Press
We consider equidistant discrete splines S(j), j ∈ Z, which may grow as O(|j| s ) as |j| → ∞. Such splines present a relevant tool for digital signal processing. The Zak transforms of Bsplines yield the integral representation of discrete splines. We define the wavelet space as a weak orthogonal complement of the coarse-grid space in the fine-grid space. We establish the integral representation of the elements of the wavelet space. We define and characterize the wavelets whose shifts form bases of the wavelet space. By this means we design a wide library of bases for the space of discrete-time signals of power growth construct multiscale representation of this space. We provide formulas for processing such the signals by discrete spline wavelets. Constructed bases are at the same time the Riesz bases for the space l 2 .
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