The investigator develops a new mathematical framework for nonlinear sampling and recovery of signals with finite rate of innovation from theoretical viewpoint and in practical realization. The investigator works on several fundamental problems concerning the recovery of a signal from its noisy sampled data, the exploration of efficient algorithmic methods in the presence of substantial noise, and the application of the novel sampling methodology to engineering problems. This interdisciplinary project is based on the observations that signals in many engineering problems have finite rate of innovation and could be approximated by signals with sparse representations, and that the sampling process has strong neighbor dependency.
Sampling theory is one of the most basic and fascinating topics in mathematical science and in engineering sciences. In engineering applications, such as Global Positioning Systems (GPS), Ultra-wideband (UWB) ranging systems in communication, and mass spectrometry in medical diagnosis, noisy sampled data are obtained, real-time recovery is preferred, and very accurate restoration is crucial for meaningful justification. Standard Fourier approaches and conventional sampling techniques are inapplicable in such problems. The challenge resides in the requirement of rapid, accurate, and robust recovery. The investigator applies calculus for infinite matrices and introduces novel compressive techniques to tackle fundamental sampling problems. Success in the project could be both mathematically fundamental and technologically important, and has potential impact in the strategic areas of information technology and biotechnology.