Frames proved to be a powerful tool in many areas of applied mathematics, computer science, and engineering. The “quality” of a frame is often reflected in the properties of the associated synthesis matrix. For this reason, investigation of various geometric properties of (random) matrices plays a crucial role in different signal processing problems, such as compressive sensing, phase retrieval, and quantization. Such properties are sufficiently well-studied for random Gaussian matrices with independent entries. Moreover, it is often the case that Gaussian matrices have properties optimal for applications with high probability. At the same time, the concrete application for which a signal processing problem is studied usually dictates the structure of the frame used to represent a signal. This motivates the study of properties of structured application relevant frames and their synthesis matrices.
In this talk we address various geometric properties of matrices that are related to frame theory, including optimal frame bounds and frame order statistics. We focus on a particular case of time-frequency structured matrices associated to Gabor frames, where matrix columns are time and frequency shifts of a random window. We show that geometric properties of such matrices are close to optimum, which is usually demonstrated by Gaussian matrices with independent entries.
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