Title

On Fourier Transforms Of Wavelet Packets

Keywords

Fourier transform; Multi-resolution analysis; Quadrature mirror filter; Wavelet packets

Abstract

This paper deals with the Fourier transform ω̂n of wavelet packets ωn ∈ L2(ℝ) relative to the scaling function φ = ω0. Included there are proofs of the following statements: (i) ω̂n (0) = 0 for all n ∈ ℕ. (ii) ω̂n(4nkπ) = 0 for all k ∈ ℤ, n = 2i for some j ∈ ℕ0, provided |ψ̂|, |m0| are continuous. (iii) |ω̂n(ξ)|2 = ∑s=02r-1 |ω̂2r n+s(2r ξ)|2 for r ∈ ℕ. (iv) ∑j=1∞ ∑s=02r-1 ∑k∈ℤ|ω̂n(2j+r(ξ + 2kπ))|2 = 1 for a.a. ξ ∈ ℝ where r = 1, 2,⋯, j. Moreover, several theorems including a result on quadrature mirror filter are proved by using the Fourier transform of wavelet packets. © Heldermann Verlag.

Publication Date

12-1-2001

Publication Title

Zeitschrift fur Analysis und ihre Anwendung

Volume

20

Issue

3

Number of Pages

579-588

Document Type

Article

Personal Identifier

scopus

Socpus ID

26444529051 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/26444529051

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