Adaptive Gabor transforms. (English) Zbl 1032.94001

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.


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
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