Multirate systems with shortest spline-wavelet filters

Motivated by the need of short FIR filters for perfect-reconstruction multirate systems, the main objective of this paper is to derive the shortest filters for such filter banks with M channels, for any integer M≥2, based on the M-dilated refinement sequence _pm of the mth order cardinal B-spline. By imposing the additional constraint of ℓth order sum rule on the M-dual low-pass sequence _am, the smoothness property of the M-dual scaling function, along with its corresponding analysis wavelets, is studied, and consequently yielding the ℓth order vanishing moment for each of the M-1 synthesis (spline) wavelets. Several illustrative examples and tables of the filter systems are also included in this paper.