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Locating zeros of polynomials associated with Daubechies orthogonal wavelets

In the last decade, Daubechies orthogonal wavelets have been successfully used and proved their practicality in many signal processing paradigms. The construction of these wavelets via two channel perfect reconstruction filter bank requires the identification of necessary conditions that the coefficients of the filters and the roots of binomial polynomials associated with them should exhibit. In this paper, the low pass and high pass filters that generate a Daubechies mother wavelet are considered. From these filters, a new set of high pass and low pass filters are derived by using the "Alternating Flip" techniques. The new set of filters maintain perfect reconstruction status of an input signal, thus allowing the construction of a new mother wavelet and a new scaling function that are reflective to those of the originals. Illustration are given and new reflective wavelets are derived. Also, a subclass of polynomials is derived from this construction process by considering the ratios of consecutive binomial polynomials' coefficients. A mathematical proof of the residency of the roots of this class of polynomials inside the unit circle is presented along with an illustration for db6, a member of the Daubechies orthogonal wavelets family. The Kakeya-Enestrom theorem is discussed along with some of its generalizations. Finally, a λ-dependent difference among the coefficients of the new set of polynomials is examined and optimized locations of the roots are derived.