Disjoint Sets of Daubechies Polynomial Roots for Generating Wavelet Filters with Extremal Properties Carl Taswell ∗ Computational Toolsmiths, Stanford, CA 94309-9925 Abstract A new set of wavelet filter families has been added to the systematized collection of Daubechies wavelets. This new set includes com- plex and real, orthogonal and biorthogonal, least and most disjoint families defined using con- straints derived from the principle of separably disjoint root sets in the complex z -domain. All of the new families are considered to be constraint selected without a search and without any eval- uation of filter properties such as time-domain regularity or frequency-domain selectivity. In contrast, the older families in the collection are considered to be search optimized for extremal properties. Some of the new families are demon- strated to be equivalent to some of the older families, thereby obviating the necessity for any search in their computation. 1 Introduction Daubechies wavelet filters with minimal length and maximal flatness can be readily computed via spectral factorization of a symmetric posi- tive polynomial [1]. All of the complex orthogo- nal, real orthogonal, and real biorthogonal fami- lies of the Daubechies class computable by spec- tral factorization have been studied experimen- tally in the systematized collection developed by Taswell [2, 3, 4, 5, 6] over a wide range of van- ishing moment numbers and filter lengths. In contrast, angular parameterization meth- ods have usually been demonstrated for wavelets * Email: taswell@toolsmiths.com; Tel/Fax: 650-323- 4336/5779. Technical Report CT-1998-08: original 7/15/98; revision 8/30/98. with only one vanishing moment (i.e., less than maximal flatness) and very short length filters [7, 8] with the exception of [9]. But the lat- ter only verified orthogonality and vanishing mo- ment numbers for the filters and did not attempt any search through the angular parametrization space for filters with desirable properties. These comments highlight the essential ques- tion in the development of an algorithm for the design of wavelet filters: How much computa- tional effort (measured by time, flops, and com- plexity of implementation) should be expended in the construction of a wavelet filter possess- ing which properties over which range of filter lengths? A basic assumption inherent in the sys- tematized collection of Daubechies wavelets [6] hypothesizes that the spectral factorization ap- proach affords the most economical generation of wavelet filters with the best variety and com- bination of properties over the widest range of filter lengths. The economy of the spectral factorization method in comparison with the angular param- eterization method is achieved by the reduced size of the search space for the filter root codes [6] relative to that for the filter coefficient an- gles [7]. In [6], conjectures were made regarding schemes to enhance the efficiency of the combina- torial search used in the design algorithm. This report investigates the next step in the develop- ment of an efficient algorithm: Can the search be completely eliminated? Section 2 clarifies the distinction between con- straint selected and search optimized filter fam- ilies, explains the principle underlying the least and most disjoint root sets, and defines the new filter families. Section 3 presents examples and