Butterworth wavelet transforms derived from discrete interpolatory splines: recursive implementation

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 filter banks. These filter banks are related to Butterworth filters. The construction is performed in a “lifting” manner. The difference from the conventional lifting scheme is that the transforms of a signal are performed via recursive filtering with the use of IIR filters. These filters have linear phase property and the basic waveforms are symmetric. The filters allow fast cascade or parallel implementation. We present explicit formulas for construction of wavelets with arbitrary number of vanishing moments. In addition, these filters 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.