A sufficiency criteria for orthogonal QMF filters to ensure smooth wavelet decompositions

For the case of orthogonal QMF filters with maximum vanishing moments (MVM), there are only a finite number of N-point FIR filters that satisfy the constraint set, and all those solutions that are known result in continuous decompositions (N ≥ 4). Unfortunately, there are solutions to the non-maximum vanishing moment problem that result in wavelet decompositions that are highly irregular (i.e., discontinuous). This paper introduces a simple inequality constraint that can be used to quickly assure continuous wavelet decompositions for a non-maximum vanishing moments (non-MVM) solution. This is useful in schemes where the QMF filter is dynamically chosen (e.g., signal dependent compression). The sufficiency requirement developed is much easier to implement than the constraint on the Fourier transform of the filter developed previously. In addition, it can be easily extended to stricter regularity requirements (e.g., filters that are both continuous and differentiable).