Constraint-selected and search-optimized families of Daubechies wavelet filters computable by spectral factorization

Abstract

A unifying algorithm has been developed to systematize the collection of compact Daubechies wavelets computable by spectral factorization of a symmetric positive polynomial. This collection comprises all classes of real and complex orthogonal and biorthogonal wavelet filters with maximal flatness for their minimal length. The main algorithm incorporates spectral factorization of the Daubechies product filter into analysis and synthesis filters. The spectral factors are found for search-optimized families by examining a desired criterion over combinatorial subsets of roots indexed by binary codes, and for constraint-selected families by imposing sufficient constraints on the roots without any optimizing search for an extremal property. Daubechies wavelet filter families have been systematized to include those constraint-selected by the principle of separably disjoint roots, and those search-optimized for time-domain regularity, frequency-domain selectivity, time-frequency uncertainty, and phase nonlinearity. The latter criterion permits construction of the least and most asymmetric and least and most symmetric real and complex orthogonal filters. Biorthogonal symmetric spline and balanced-length filters with linear phase are also computable by these methods. This systematized collection has been developed in the context of a general framework enabling evaluation of the equivalence of constraint-selected and search-optimized families with respect to the filter coefficients and roots and their characteristics. Some of the constraint-selected families have been demonstrated to be equivalent to some of the search-optimized families, thereby obviating the necessity for any search in their computation.