Ph.D Student | Dvorkind Gregory |
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Subject | Generalized Sampling in the Presence of Nonlinearities: Theory and Methods |

Department | Department of Electrical Engineering |

Supervisor | Professor Yonina Eldar |

Full Thesis text |

Digital signal processing applications are mainly concerned with the ability to store and process discrete sets of numbers, which are related to continuous-time signals through an acquisition process. One major goal, which is at the heart of digital signal processing, is the ability to reconstruct continuous-time functions, by proper processing their available samples.

Traditional sampling problems assume ideal sampling in which the samples are point-wise evaluations of the signal. However, in practice, the samples are typically non-ideal. Furthermore, in many practical settings, the signal is first distorted by a non-linear mapping, before sampling takes place. Examples include nonlinearities of radiometric cameras, non-linear distortions caused by high power amplifiers, and nonlinear optical sampling systems.

In our work we develop a general framework for reconstructing a continuous-time signal which is distorted by a nonlinear mapping, and then sampled in a non-ideal manner. We begin by treating the case in which the mapping is linear and develop robust and consistent approximations of the input signal from the given non-ideal samples. We then consider the more difficult scenario in which the signal is also distorted by a memoryless nonlinearity.

In this setting, we first identify classes of problems in which it is possible to obtain, in a closed form, perfect and stable reconstruction of the input signal from the given samples. Next we treat the more general setting and suggest an iterative algorithm that recovers the input signal from its nonlinear and non-ideal

samples. Three alternative formulations of the algorithm are developed that provide different insight into the structure of the solution: A series of oblique projections, an approximated projections onto convex sets (POCS) method, and quasi-Newton iterations. Using classical analysis techniques for descent-based methods, and recent results of frame perturbation theory, we prove convergence of our algorithm to the input signal. We demonstrate our method by simulating a nonlinear optical sampling system, and explain the applicability of our theory to Wiener-Hammerstein analog to digital hybrid systems.