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Sparse Model Uncertainties in Compressed Sensing with Application to Convolutions and Sporadic Communication

  • Peter JungEmail author
  • Philipp Walk
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

The success of the compressed sensing paradigm has shown that a substantial reduction in sampling and storage complexity can be achieved in certain linear and non-adaptive estimation problems. It is therefore an advisable strategy for noncoherent information retrieval in, for example, sporadic blind and semi-blind communication and sampling problems. But, the conventional model is not practical here since the compressible signals have to be estimated from samples taken solely on the output of an un-calibrated system which is unknown during measurement but often compressible. Conventionally, one has either to operate at suboptimal sampling rates or the recovery performance substantially suffers from the dominance of model mismatch. In this work we discuss such type of estimation problems and we focus on bilinear inverse problems. We link this problem to the recovery of low-rank and sparse matrices and establish stable low-dimensional embeddings of the uncalibrated receive signals whereby addressing also efficient communication-oriented methods like universal random demodulation. Exemplarily, we investigate in more detail sparse convolutions serving as a basic communication channel model. In using some recent results from additive combinatorics we show that such type of signals can be efficiently low-rate sampled by semi-blind methods. Finally, we present a further application of these results in the field of phase retrieval from intensity Fourier measurements.

Keywords

Sporadic Communication Sparse Convolution Fourier Measurements Semi-blind Methods Phase Retrieval Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The authors would like to thank the anonymous reviewers for their detailed and valuable comments. We also thank Holger Boche, David Gross, Richard Kueng, and Götz Pfander for their support and many helpful discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under grant JU 2795/2-1.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Technische Universität BerlinBerlinGermany
  2. 2.Technische Universität MünchenMünchenGermany

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