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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.

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
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