Perturbations and radar in compressed sensing.

摘要

Compressed sensing is a radical new approach to signal processing where far fewer data measurements are collected than what is dictated by the classic Nyquist-Shannon sampling theory. This is followed at a later stage by an appropriate method to recover the original signal. The two most popular approaches are convex optimization and greedy algorithms. The success of compressed sensing relies on two critical phenomena. First, the signal of interest must be sparse under some basis or dictionary of waveforms. Fortunately, many signals in the real world naturally have this structure. Second, the sensing modality, or the system which the signal passes though, must have an incoherence property.;Information in the real world is always corrupted with noise. Previous studies in compressed sensing have analyzed the stability of recovery algorithms primarily in the presence of additive noise. We generalize this by introducing a completely perturbed model which allows for both additive as well as multiplicative noise. In this study we examine the behavior of a convex optimization program called Basis Pursuit, and a greedy algorithm called Compressive Sampling Matching Pursuit. Our results show that, under suitable conditions, the stability of the recovered signal is limited by the total noise level (additive and multiplicative) in the observation. This completely perturbed model, in particular, establishes a framework for analyzing real-world applications where one has to make assumptions about a system model. These errors manifest themselves as multiplicative noise.;In terms of real-world applications, our other contribution consists of a stylized compressed sensing radar system. Here we discretize the time-frequency plane into a fine grid in order to super-resolve targets. Assuming the number of targets is small, then we can transmit a sufficiently "incoherent" pulse and employ the techniques of compressed sensing to reconstruct the target scene. A theoretical upper bound on the sparsity is presented. Numerical simulations verify that even better performance can be achieved in practice. This novel compressed sensing approach offers the potential for better resolution over traditional radar which is limited by classical time-frequency uncertainty principles.