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In this paper we address various geometric properties of frames, such as spark, coherence, restricted isometry property, and frame order statistics. These properties play crucial role in various signal processing problems, including compressive sensing, phase retrieval, and quantization. We focus on a particular case of structured frames, namely on Gabor frames, where frame vectors are time and frequency...
Gabor analysis is playing an important role in time-frequency analysis for the last 60 years. The fundamental concept of Gabor frames has become a focus also in the finite dimensional setting. Here, we study which sets of discrete time frequency shifts admit a proper choice of window vector, so that they generate an orthonormal Gabor basis in the underlying finite dimensional vector space. We fully...
We establish some Balian-Low type results for Gabor spaces, which concern with discontinuity in some periodization of Zφ, where Zφ is the Zak transform of the window function φ ∈ L2 (ℝd). The results are compared with the case for shift-invariant spaces where the Zak transform is replaced by Fourier transform.
Operator sampling considers the question of when operators of a given class can be distinguished by their action on a single probing signal. The fundamental result in this theory shows that the answer depends on the area of the support S of the so-called spreading function of the operator (i.e., the symplectic Fourier transform of its Kohn-Nirenberg symbol). |S| < 1 then identification is possible...
A Gabor space is a space generated by a discrete set of time-frequency shifted copies of a single window function. Starting from the question of whether a Gabor space contains additional time-frequency shifts of the window function we establish a new Balian-Low type result. This result extends (for example) the well established Amalgam Balian-Low Theorem in the one dimensional case. The Gabor spaces...
We consider phaseless measurements in the case when the measurement frame is a Gabor frame, that is, the frame coefficients are of the form of masked Fourier coefficients where the masks are time shifts of the Gabor window. This makes measurements meaningful for applications, but at the same time preserves the flexibility of the frame-theoretic approach. We are going to present a recovery algorithm...
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