Parametrizations of all wavelet filters: Input-output and state-space

We here use notions from the theory linear shift-invariant dynamical systems to provide an explicit characterization, both practical and computable, of all rational wavelet filters. For a given N, (N ≥ 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(1/2 (N − 1) + m).

Moreover, beyond the parameters N and m, one confine the spectrum of the filters to lie in an open disk of radius ρ (stable filters mean ρ ∈ [0, 1] and for FIR take ρ = 0). Then all filters can be described by a convex set of parameters ([0, π) × [0, 2π)^2(N−1) × [0, ρ))^m.

Rational wavelet filters bounded at infinity, admit state space realization. The above input-output parametrization is exploited for a step-bystep construction (where in each, the index m is increased by one) of state space model of wavelet filters.