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We address the problem of signal reconstruction from intensity measurements with respect to a measurement frame. This nonconvex inverse problem is known as phase retrieval. The case considered in this paper concerns phaseless measurements taken with respect to a Gabor frame. It arises naturally in many practical applications, such as diffraction imaging and speech recognition. We present a reconstruction algorithm that uses a nearly optimal number of phaseless time-frequency structured measurements and discuss its robustness in the case when the measurements are corrupted by noise. We show how geometric properties of the measurement frame are related to the robustness of the phaseless reconstruction. [ABSTRACT FROM AUTHOR]

Abstract: Classical and recent results on uncertainty principles for functions on finite Abelian groups relate the cardinality of the support of a function to the cardinality of the support of its Fourier transform. We obtain corresponding results relating the support sizes of functions and their short-time Fourier transforms. We use our findings to construct a class of equal norm tight Gabor frames that are maximally robust to erasures. Also, we discuss consequences of our findings to the theory of recovering and storing signals with sparse time–frequency representations. [Copyright &y& Elsevier]

Abstract: Finite dimensional matrices with more columns than rows have no left inverses while those with more rows than columns have no right inverses. We give generalizations of these simple facts to bi–infinite matrices. Our results are then used to obtain density results for p-frames of time–frequency molecules in modulation spaces and identifiability results for operators with bandlimited Kohn–Nirenberg symbols. [Copyright &y& Elsevier]

Abstract: Discrete Gabor multipliers are composed of rank one operators. We shall prove, in the case of rank one projection operators, that the generating operators for such multipliers are either Riesz bases (exact frames) or not frames for their closed linear spans. The same dichotomy conclusion is valid for general rank one operators under mild and natural conditions. This is relevant since discrete Gabor multipliers have an emerging role in communications, radar, and waveform design, where redundant frame decompositions are increasingly applicable. [Copyright &y& Elsevier]

We consider smoothness properties of the generator of a principal Gabor space on the real line which is invariant under some additional translation–modulation pair. We prove that if a Gabor system on a lattice with rational density is a Riesz basis for its closed linear span, and if the closed linear span, a Gabor space, has any additional translation–modulation invariance, then its generator cannot decay well in time and in frequency simultaneously. [ABSTRACT FROM AUTHOR]