Asymptotics of Daubechies Filters, Scaling Functions, and Wavelets

✍ Scribed by Jianhong Shen; Gilbert Strang

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 325 KB
• Volume: 5
• Category: Article
• ISSN: 1063-5203
• DOI: 10.1006/acha.1997.0234

✦ Synopsis

We study the asymptotic form as p r ϱ of the Daubechies orthogonal minimum phase filter h p [n], scaling function f p (t), and wavelet w p (t). Kateb and Lemarie ´calculated the leading term in the phase of the frequency response

leads us to a problem in stationary phase, for an oscillatory integral with parameter t. The leading terms change form with t Å t/p and we find three regions for f p (t):

(1) An Airy function up to near t 0 :

(3) A rapid decay after t 1 : (1/pp)(1/(t 0 t 1 ))sin[p(G ( 01) (p) 0 tp)] / o(p 01 ) The numbers t 0 á 0.1817 and t 1 á 0.3515 are known constants. The function G and its integral G ( 01) are independent of p. Regions 1 and 2 are matched over the interval p 02/3 Ӷ t 0 t 0 Ӷ 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 h p [n] -a different function g(v) controls the three regions.

The difficulty throughout is to estimate the phase.