Asymptotics of Daubechies Filters, Scaling Functions, and Wavelets

Abstract

We study the asymptotic form as p → ∞ of the Daubechies orthogonal minimum phase filter hp[n], scaling function φp(t), and wavelet wp(t). Kateb and Lemarié calculated the leading term in the phase of the frequency response Hp(ω). The infinite product φ̂p(ω) = Π Hp(ω/2k) leads us to a problem in stationary phase, for an oscillatory integral with parameter t. The leading terms change form with τ = t/p and we find three regions for φp(τ): (1) An Airy function up to near τ0: 3√42π/p Ai(-3√42πp22(τ - τ0)) + o(p-1/3) (2) An oscillating region √2/πpG′(ωτ)cos[p(G(-1)(ω τ) - G(ωτ)ωτ) + π/4] + o(p-1/2) (3) A rapid decay after τ1: (1/pπ)(1/(τ - τ1))sin[p(G(-1)(π) - τπ)] + o(p-1) The numbers τ0 ≃ 0.1817 and τ1 ≃ 0.3515 are known constants. The function G and its integral G(-1) are independent of p. Regions 1 and 2 are matched over the interval p-2/3 ≪ τ - τ0 ≪ 1. The wavelets have a simpler asymptotic expression because the Airy wavefront is removed by the highpass filter. We also find the asymptotics of the impulse response hp[n] - a different function g(ω) controls the three regions. The difficulty throughout is to estimate the phase. © 1998 Academic Press.