Convex Optimization Based Parameterized Sparse Estimation Theory And Its Application

Compressive sensing (CS) provides an efficient way to acquire and reconstruct sparse signals from a limited number of linear sub-Nyquist random measurements. As it achieves signal reconstruction using much fewer measurements than the number prescribed by the Nyquist theorem, there has been increasing attention to the issue of CS.Traditional sparse signal model merely assumes that the number of nonzero entreis is greatly smaller than the length of the signal, and other structral features and the measurment errores are not considered. This dissertation deals with the new problem in the CS applications. Parameterized sparse signal models are defined according to the new signal structures in applicactions. The research mainly focuses on the corresponding convex sparse signal estimation and its applications. The primary works are summarized as follows:1. Hierarchical sparse signal models are proposed according to different local distributions of nonzero elements of sparse signal. The new signal models include globally grouped-sparse signal, globally grouped-sparse and locally sparse signal, globally grouped-sparse and locally dense signal. Corresponding convex signal recovery methods and coherence analysis based sufficient conditions are given afterwards. For the corresponding signal model, the newly proposed signal recovery methods outperform traditional ones. This work further improves the performance of CS in applications, and enriches CS theory.2. A newly anti-loss CS system structure is proposed to deal with the errors in random sampling of CS. This work mainly focuses on the robust sparse signal estimation with measurement matrix uncertainty and amplitude component loss. When reconstructing the lossy signals, The newly proposed algorithms outperform the traditional ways. This work generalizes the CS theory to a more practical situation.3. To deal with the contradiction of high sampling rate and fast, robust and low-cost signal processing requirement, CS theory is applied to wideband spectrum sensing of cognitive radio. To discard the useless phase component and exploit the favorable structural information about fixed spectrum allocation, random autocorrelation measurement based structural compressive wideband spectrum sensing is introduced. The newly formed methods achieve wideband power spectrum estimation with much fewer measurements than Nyquist sampling. Therefore, a faster and more reliable spectrum monitoring can be provided for cognitive radio to perform dynamic spectrum access.4. High angle mismatch sensitivity and high sidelobe level are two major problems of the adaptive beamforming. To deal with them, a class of sparse beam pattern shaping constraints is discussed. It includes standard sparse constraint based beam pattern shaping, weighted sparse constraint based beam pattern shaping, mixed norm constraint based beam pattern shaping, total variation minimization based beam pattern shaping and mainlobe and sidelobe power ratio maximization based beam pattern shaping. Comparing to the traditional adaptive beamforming method, the proposed beamformers' robustness against high angle mismatch sensitivity and high sidelobe level are enhanced. This work further improves the performance of the adaptive beamforming in practice.