Prior Information Based Compressive Sensing Theory And Its Application

Conventional approaches to sampling signals follow Shannon/Nyquist theorem: thesampling rate must be at least twice the maximum frequency present in the signal (theso-called Nyquist rate).Under the framework of Shannon/Nyquist theorem, People's hugedemand for information result in mass samples, which increase greatly the cost of store,communication and analysis. The recently proposed Compressive sensing (CS), utilizingthe prior information that signal always can be sparse represented in certain dictionaryand the incoherence measurements, can recover certain signals with far lower sample ratethan Nyquist rate. CS theory not only is a great breakthrough of traditional samplingtheory, but also provides a new way for the research of other scientifc felds. However,besides the sparsity, signals have other prior information. Exploiting and incorporatingefectively these information will signifcantly improve the performance of CS. In thisdissertation, the theory and applications of CS reconstruction with prior information areinvestigated systematically.The main contributions of this dissertation are summarized as follows:1. The basic principle and method of CS, including background, research signifcanceand current situation of the selected topic are reviewed. Three key factors of CS are alsointroduced. Especially the important conclusions of1minimization method for sparsesignal reconstruction are summarized.2. For the CS model with partially known support, a sufcient and necessary con-dition of Modifed-CS to recover sparse signal is established. Furthermore, a probabil-ity estimation method of recoverability is proposed. The fault-tolerance capability ofModifed-CS is investigated through deducing some probability inequalities that refectthe error's efect on recoverability, and the upper bound of the number of errors in thepartially known support set. We prove theoretically the efectiveness of p(0<p <1)minimization model with partially known support. At the same time, a CS-based methodfor OFDM rapidly time-varying channel estimation is proposed inspired of the idea of CSmodel with partially known support. We compare the proposed method with the otherclassical methods, including Least Square (LS) and Basis Pursuit (BP).3. For the signal that follows non-uniform sparse models, a sufcient and neces-sary condition of weighted1minimization method to recover a sparse signal is derived.The probability estimation method of recoverability is also generalized to this model.The optimal weights selection under the guidance of recovery probability is investigated, which has some advantage, such as exactness, no limitation for the class of measurementmatrix. A wavelet domain weighted1minimization method is proposed for the SARreconstruction. The proposed method does not need the sparsity information on eachblock of signal, and obtains the best results compare with some state of art CS methodsin the experiments.4. For the problem of optical power monitoring, a weighted-RSR method is derived.Compared with the conventional methods, the proposed weighted-RSR method has someadvantages that may be of particular interest for various applications. For example,precise measurement of TOF characteristics is not necessary; It can achieve <0.5dBmaccuracy within a wide range of operation conditions and the channel power diferencecan be as large as10dBm. Experimental results demonstrated that the proposed methodcan be used to accurately monitor the power of mixed10G/40GWDM channels with50GHz spacing and10dBm power diference via low-cost market available TOFs.5. The problem of CS reconstruction for sparse signal has investigated in the presenceof noise. At frst, we analyze and prove that in the aim of sparsity pattern recovery, theLASSO exists an upper bound relating with the power of noise for the tuning parameterh. If h exceeds this bound, the recovery error of LASSO will increase with h. However, toguarantee sparsity pattern recovery, a large h is always required in LASSO. This indicatesthat the selection of h may be a dilemma for simultaneous sparsity pattern recovery andapproximation recovery. To mitigate this problem, an extend LASSO method is proposed.Comparing to the LASSO, BPDN method, our method produces higher probability ofsparsity pattern recovery and better approximation recoveryFinally, the main results of the dissertation are concluded and some issues for futureresearch are proposed.