Research On Sparse Recovery Algorithm And Its Application In Signal Processing

Sparse signal is widely existed in realistic world, how to recover sparse signalusing a small quantity of measurements, is called sparse recovery problem. Thisproblem, which is successfully applied to sparse channel estimation, radar signalprocessing, sparse component analysis and DOA estimation in array signal processingetc, is a prosperous area in the signal processing domain. In this thesis, severalalgorithms are proposed for sparse recovery problem and their applications in signalprocessing are researched. The main research work and contributions are detailed asfollows:1. Approximatel0norm minimization algorithm (AL0) and iterative approximatel0norm minimization algorithm (ISD-AL0) are proposed for noiseless singlemeasurement sparse recovery problem, and convergence property of AL0and ISD-AL0are analyzed. In the first algorithm, the problem of approximatel0norm minimization isestablished with the sparsity enforced byl0norm, which is approximated by arctanfunction, then this minimization problem is solved by fast fixed point iterative methodto recover sparse signal. ISD-AL0is the improved version of AL0. This algorithmestimates the partial support set from the existing results to improve performance. Bothabove proposed algorithms need fewer measurements than existing methods, whileneeding the low computational cost. Compared with AL0, ISD-AL0needs much fewermeasurements, but needs more computational cost.2. Expectation Minimization of approximatel0norm algorithm (EML0) androbust approximatel0norm minimization algorithm (RAL0) are proposed for noisysingle measurement sparse recovery problem, and RAL0is extended to complexnumber field for frequency spectrum estimation. The basic idea of EML0algorithm isthat sparse signal is recovered by minimizing the expectation of approximatel0norm.We simplify the expectation model by statistical character of noise and it can be solvedby steepest descent method. For the RAL0algorithm,l0norm is firstly approximatelyexpressed by arctan function. Then the model of sparse recovery problem in the presentof noise is constructed based on approximatel0norm, and this model is solved byquasi-Newton method to estimate sparse signal.The two proposed algorithms are morerobust to noise than existing algorithms, while needing fewer measurements and lowercomputational cost. Compared with RAL0, EML0has a little better performance, but itneeds to know statistical characteristics of noise.3. The Bayesian sparse recovery algorithm (FBSR) is proposed for noisy singlemeasurement sparse recovery problem, and FBSR is extended to complex number field for frequency spectrum estimation. For the FBSR algorithm, The Bayesian model,basedon the Laplace priors, is constructed, and a fast fixed point algorithm is established toestimate hyperparameter by maximizing the joint distribution. Compared with RAL0,FBSR is more robust to noisy and it does not require the user to make any difficultselection of parameters, but it needs more computational cost. Compare with classicalsparse Bayesian learning (SBL), FBSR needs fewer measurements and lesscomputational cost.4. Multiple measurements approximatel0norm minimization algorithm (M-AL0)is propose for noiseless multiple measurements sparse recovery problem, and Multiplemeasurements robust approximatel0norm minimization algorithm (M-AL0) isproposed for noisy multiple measurements sparse recovery problem. Besides, M-RAL0is extended to complex number field for DOA estimation. For the M-AL0algorithm,l2,0norm is approximated by arctan function, and multiple measurements sparserecovery problem is cast as multiple measurements approximatel2,0norm minimi-zation problem. then a fast fixed algorithm, derived by Lagrangian function, isestablished to recover joint sparse signal. For the M-RAL0algorithms, noisy multiplemeasurements approximatel2,0norm minimization problem is constructed by appro-ximatel2,0norm, and quasi-Newton method is used to solve this problem. M-AL0andM-RAL0need fewer measurements and less computational cost than the exsitedalgorthms. Compared with M-AL0, M-RAL0is more robust to noise.