Adaptive Gabor transforms. (English) Zbl 1032.94001

Daubechies, Ingrid; Planchon, Fabrice

Appl. Comput. Harmon. Anal. 13, No. 1, 1-21 (2002).

Consider a family of Gaussian windows \(g^{\beta, \delta}(x) = \delta^{1/4} e^{i(\beta/2)x^2 - \delta x^2}\). In this paper the authors study generalized \((\beta, \delta)\)-Gabor tansforms, which are defined as \(G^{\beta, \delta} (f) (p,q) = \int g^{\bar \beta, \delta}_{p,q}(x)f(x) dx\), where \(g^{\beta, \delta}_{p,q}(x) = e^{ipx-pq/2}g^{\beta, \delta}(x-q)\). Based on these representations, the authors provide various adaptive Gabor transforms for analysis of one-dimensional signals by appropriately picking window functions from the family \(g^{\beta, \delta}\). The algorithms for choosing the best windows are computationally efficient, and multiple examples are provided.

Reviewer: Wojciech Czaj (Wroclaw)

MSC:

94A11	Application of orthogonal and other special functions
94A12	Signal theory (characterization, reconstruction, filtering, etc.)
42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems
42C15	General harmonic expansions, frames

Keywords:

Gabor transform; Wigner-Ville transform; adaptive time-frequency analysis

Software:

Time-frequency Toolbox

References:

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