Probabilistic Reconstruction in Compressed Sensing: Algorithms, Phase Diagrams, and Threshold Achieving Matrices

Compressed sensing is a signal processing method that acquires data directly in a compressed form. This allows one to make less measurements than what was considered necessary to record a signal, enabling faster or more precise measurement protocols in a wide range of applications. Using an interdisciplinary approach, we have recently proposed in [arXiv:1109.4424] a strategy that allows compressed sensing to be performed at acquisition rates approaching to the theoretical optimal limits. In this paper, we give a more thorough presentation of our approach, and introduce many new results. We present the probabilistic approach to reconstruction and discuss its optimality and robustness. We detail the derivation of the message passing algorithm for reconstruction and expectation max- imization learning of signal-model parameters. We further develop the asymptotic analysis of the corresponding phase diagrams with and without measurement noise, for different distribution of signals, and discuss the best possible reconstruction performances regardless of the algorithm. We also present new efficient seeding matrices, test them on synthetic data and analyze their performance asymptotically.Comment: 42 pages, 37 figures, 3 appendixe

Forward Error Correction for Optical Transponders

Forward error correction is an essential technique required in almost all communication systems to guarantee reliable data transmission close to the theoretical limits. In this chapter, we discuss the state-of-the-art forward error correction (FEC) schemes for fiber-optic communications. Following a historical overview of the evolution of FEC schemes, we first introduce the fundamental theoretical limits of common communication channel models and show how to compute them. These limits provide the reader with guidelines for comparing different FEC codes under various assumptions. We then provide a brief introduction to the general basic concepts of FEC, followed by an in-depth introduction to the main classes of codes for soft decision decoding and hard decision decoding. We include a wide range of performance curves, compare the different schemes, and give the reader guidelines on which FEC scheme to use. We also introduce the main techniques to combine coding and higher-order modulation (coded modulation), including constellation shaping. Finally, we include a guide on how to evaluate the performance of FEC in transmission experiments. We conclude the chapter with an overview of the properties of some state-of-the-art FEC schemes used in optical communications and an outlook