The Systematized Collection of Daubechies Wavelets
Carl Taswell · 1998
A single unifying algorithm has been developed to systematize the collection of compact Daubechies wavelets. This collection comprises all classes of real and complex orthogonal and biorthogonal wavelets with the maximal number K of vanishing moments for their finite length. Named and indexed families of wavelet filters were generated by spectral factorization of a product filter in which the optimal subset of roots was selected by a defining criterion within a combinatorial search of subsets meeting required constraints. Several new families have been defined some of which were demonstrated to be equivalent to families with roots selected solely by geometric criteria that do not require an optimizing search. Extensive experimental results are tabulated for 1 ≤ K ≤ 24 for each of the families and most of the filter characteristics defined in both time and frequency domains. For those families requiring optimization, a conjecture for K > 24 is provided for a search pattern that reduces the order of the computational complexity but permits attainment of the desired optimum. Technical Report CT-1998-06: original 5/22/98; revision 6/23/98. Carl Taswell, Computational Toolsmiths, Stanford, CA 94309-9925 Email: [email protected]; Tel: 650-323-4336, Fax: 650-323-5779 Technical Report CT-1998-06. 3 0 List of Symbols D, L in upper case calligraphic font for Daubechies parameter and Lagrange parameter; N , M , K, R in upper case math font for filter bank related parameters; d, p in lower case math font for degree and parity or power parameters; n, t, z in lower case math font for independent variables in discrete time, continous time, and complex z-domain; z = reiα for complex z with radius r, imaginary i, angle α; α, ω in lower case Greek font for angle and frequency in radians; P, A, S, F , G, H in upper case calligraphic font for polynomials (usually expressed as P(z) in terms of roots z); p, a, s, f , g, h in lower case bold font for vectors of coefficients; A, S, F, G, H in upper case bold font for matrices of coefficients; i, j, k, m, n in lower case math font for running indices in sums or products, or for subscript indices in vectors or matrices; ncq, nrd with number n in lower case math font and superscripts in roman font; Np, Na, Ns, n cq p , n cq a , n cq s various numbers with subscripts p, a, and s for product, analysis, and synthesis filters; γ, η, φ, ψ, τ , e, φ, ρ in lower case Greek fonts for gamma, eta, phi, psi, tau, varepsilon, varphi, rho; ς, υ, ν, e in lower case Greek bold font for varsigma, upsilon, nu, varepsilon; Φ, Ψ, Υ in upper case Greek fonts for Phi, Psi, Upsilon; 4 Taswell: Systematized Collection Daubechies Wavelets.