Gabor Expansions of Signals: Computational Aspects and Open Questions

Feichtinger, Hans G.

doi:10.1007/978-3-030-05210-2_7

Cited by 4 publications

(2 citation statements)

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“…Hexagonal lattices are ubiquitous in Euclidean point configuration problems. In time-frequency analysis this is discussed in Feichtinger's survey [27]. An important conjecture by Strohmer and Beaver [58] expects the condition number of (deterministic) Gabor frames with Gaussian windows to be optimized by a hexagonal lattice.…”

Section: Theorem 1 All Local Minima Of H (Z) Are Zeros Of F(z) and No...mentioning

confidence: 99%

Local Maxima of White Noise Spectrograms and Gaussian Entire Functions

Abreu

2022

J Fourier Anal Appl

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We confirm Flandrin’s prediction for the expected average of local maxima of spectrograms of complex white noise with Gaussian windows (Gaussian spectrograms or, equivalently, modulus of weighted Gaussian Entire Functions), a consequence of the conjectured double honeycomb mean model for the network of zeros and local maxima, where the area of local maxima centered hexagons is three times larger than the area of zero centered hexagons. More precisely, we show that Gaussian spectrograms, normalized such that their expected density of zeros is 1, have an expected density of 5/3 critical points, among those 1/3 are local maxima, and 4/3 saddle points, and compute the distributions of ordinate values (heights) for spectrogram local extrema. This is done by first writing the spectrograms in terms of Gaussian Entire Functions (GEFs). The extrema are considered under the translation invariant derivative of the Fock space (which in this case coincides with the Chern connection from complex differential geometry). We also observe that the critical points of a GEF are precisely the zeros of a Gaussian random function in the first higher Landau level. We discuss natural extensions of these Gaussian random functions: Gaussian Weyl–Heisenberg functions and Gaussian bi-entire functions. The paper also reviews recent results on the applications of white noise spectrograms, connections between several developments, and is partially intended as a pedestrian introduction to the topic.

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Section: Theorem 1 All Local Minima Of H (Z) Are Zeros Of F(z) and No...mentioning

confidence: 99%

Local Maxima of White Noise Spectrograms and Gaussian Entire Functions

Abreu

2022

J Fourier Anal Appl

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show abstract

“…A reformulation in more general contexts, e.g. in the framework of Banach Gelfand triple setting, see [10], could be possible. In Section 5 we end the paper using the pseudo-differential form for the Gabor operator (0.5) to obtain a characterization for dual frames of {g h,k }.…”

Section: Introductionmentioning

confidence: 99%