Easy-to-compute parameterizations of all wavelet filters: input-output and state-space
Offers a novel, systematic parameterization for rational wavelet filters, which may benefit signal processing and control theory researchers working with wavelet-based methods.
This work provides an easy-to-compute parameterization of all rational wavelet filters using linear shift-invariant dynamical systems, enabling a step-by-step construction of state-space models. The parameterization is based on a factorization involving elementary wavelet filters and unitary matrices.
We here use notions from the theory linear shift-invariant dynamical systems to provide an easy-to-compute characterization of all rational wavelet filters. For a given N bigger or equql to 2, the number of inputs, the construction is based on a factorization to an elementary wavelet filter along with of m elementary unitary matrices. We shall call this m the index of the filter. It turns out that the resulting wavelet filter is of McMillan degree $N((N-1)/2+m). Rational wavelet filters bounded at infinity, admit state space realization. The above input-output parameterization is exploited for a step-by-step construction (where in each the index m is increased by one) of state space model of wavelet filters.