Sampling and Recovery of Continuously-Defined Sparse Signals and Its Applications

Abstract

The common guideline for sampling continuously-defined signals has been provided by the Nyquist frequency for long. Recently it was clarified that even though signals in radar, echo, and sonar are wide-band with high Nyquist frequency, they can be sampled at extremely low frequency compared with the Nyquist frequency by taking the fact into account that such signals are sparse linear combinations of time-delayed versions of a transmitted (known) pulse. Such sampling scheme can also be applied to signals defined by piecewise polynomials or exponentials, in spite that they are not band-limited. In this article, we introduce a class of signals called signals with finite rate of innovation that covers not only the bandlimited signals but also aforementioned non band-limited signals, and review sampling and reconstruction schemes for those signals in noiseless and noisy scenarios. This is followed by the more stable approach based on maximum likelihood estimation, which is connected to the so-called structured low-rank estimation.We further briefly introduce an application of these techniques to image feature extraction.