Asymptotics of Daubechies Filters, Scaling Functions, and Wavelets

Abstract

We study the asymptotic form asp→ ∞ of the Daubechies orthogonal minimum phase filterhp[n], scaling function φp(t), and waveletwp(t). Kateb and Lemarié calculated the leading term in the phase of the frequency responseHp(ω). The infinite productφ̂p(ω) = ∏Hp(ω/2k) leads us to a problem in stationary phase, for an oscillatory integral with parametert. The leading terms change form with τ =t/pand we find three regions for φp(τ):[formula]The numbers τ0≃ 0.1817 and τ1≃ 0.3515 are known constants. The functionGand its integralG(−1)are independent ofp. Regions 1 and 2 are matched over the intervalp−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 responsehp[n]—a different functiong(ω) controls the three regions.The difficulty throughout is to estimate the phase.