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**11 - 12**of**12**### FAST COMPRESSIVE PHASE RETRIEVAL FROM FOURIER MEASUREMENTS

"... This paper considers the problem of recovering a k-sparse, N-dimensional complex signal from Fourier magnitude measure-ments. It proposes a Fourier optics setup such that signal recovery up to a global phase factor is possible with very high probability whenever M & 4k log2(N/k) random Fourier i ..."

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This paper considers the problem of recovering a k-sparse, N-dimensional complex signal from Fourier magnitude measure-ments. It proposes a Fourier optics setup such that signal recovery up to a global phase factor is possible with very high probability whenever M & 4k log2(N/k) random Fourier intensity measurements are available. The proposed algorithm is comprised of two stages: An algebraic phase retrieval stage and a compressive sensing step subsequent to it. Simulation results are provided to demonstrate the applicability of the algorithm for noiseless and noisy scenarios. Index Terms — Phase retrieval, compressive sampling, Fourier measurements 1.

### STRUCTURED RANDOM MEASUREMENTS IN SIGNAL PROCESSING

"... Compressed sensing and its extensions have recently triggered interest in randomized signal acquisition. A key finding is that random measurements provide sparse signal reconstruction guarantees for efficient and stable algorithms with a minimal number of samples. While this was first shown for (uns ..."

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Compressed sensing and its extensions have recently triggered interest in randomized signal acquisition. A key finding is that random measurements provide sparse signal reconstruction guarantees for efficient and stable algorithms with a minimal number of samples. While this was first shown for (unstructured) Gaussian random measurement matrices, applications require certain structure of the measurements leading to structured random measurement matrices. Near optimal recovery guarantees for such structured measurements have been developed over the past years in a variety of contexts. This article surveys the theory in three scenarios: compressed sensing (sparse recovery), low rank matrix recovery, and phaseless estimation. The random measurement matrices to be considered include random partial Fourier matrices, partial random circulant matrices (subsampled convolutions), matrix completion, and phase estimation from magnitudes of Fourier type measurements. The article concludes with a brief discussion of the mathematical techniques for the analysis of such structured random measurements.