Theory Research And Applications On Compressed Sensing With Non-ideal Model

Compressed sensing(CS) is an emerging research area which studies the problem of recovering a high dimensional sparse signal from its low dimensional linear samples. While CS can acquire a signal with a sub-Nyquist sampling rate, it requires a nonlinear signal reconstruction strategy which is computationally expensive compared to the simple linear reconstruction procedure in the traditional Nyquist sampling case. Given the exact information of the sensing system, existing results have shown that the signal of interest can be accurately recovered via convex relaxation(1? minimization) or other approaches provided the signal is sufficiently sparse. However, the sensing system may not be exactly known a priori, the signal recovery performance in the case of non-ideal CS model is in urgent need to be studid. Othe the other hand, in the fields of direction-finding and frequency estimation based on sparse representation, new approach to resolve the problem that the true directions-of-arrival/frequencies may not lie on the discretized sampling grid which is typically required.The main contributions of this thesis are listed as follows:1. The CS problem subject to a structured perturbation in the sensing matrix which has practical relevance is studied. Under mild conditions, we show that a sparse signal can be recovered by solving an 1? minimization problem and the recovery error is at most proportional to the measurement noise level, which is similar to the standard CS result. In the special noise free case, the recovery is exact provided that the signal is sufficiently sparse with respect to the perturbation level.2. The direction-finding algorithms based on off-grid sparse representation model are studied in-depth. Because of the limitations of model, the estimation performance of conventional methods based on sparse signal reconstruction can be highly deteriorated if the true directions of arrival are not on the preselected discretized grid. A new method based on a reformulated model for off-grid direction-finding is proposed. The reformulated model is based on the sparse spatial covariance model and the off-grid representation of the steering vector with Taylor expansion. According to this model, we present a new method alternating between a sparse recovery problem solved using the focal underdetermined system solver algorithm and a least squares problem to speed up the estimating process. In addition, we formulate the direction-finding problem as an array covariance matrix off-grid sparse representation model in a discretized grid, and relaxe the model as a convex problem. Thus, an alternating iterative estimator with grid matching is proposed. The proposed algorithm solves a series of basis pursuit denoising problems on a coarse grid for that problem, and revises the direction-finding results to achieve higher estimation accuracy. Numerical simulations demonstrate the superior performance of the proposed methods.3. We investigate the problem of estimating the K frequency components of a mixture of complex sinusoids in this work. Component frequencies of practical signals are not assumed to lie on a specified grid, but any values in the normalized frequency domain ?0,1?. To estimate the off-grid frequencies, we apply the first Taylor expansion to approximate the true unknown dictionary and establish a more accurate model for sparse approximation of practical complex sinusoids. Furthermore, we reformulate this model and develop a fast method alternating between a sparse recovery problem solved using the multi-measurement vector orthogonal matching pursuit algorithm and a least squares problem to speed up the estimating process. Numerical experiments demonstrate that the proposed method can achieve off-grid frequency estimation of complex sinusoids with low computational cost as well as high estimation accuracy.