稀疏重构算法及其在信号处理中的应用研究
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摘要
信号稀疏广泛存在现实世界中，利用信号的稀疏特征通过少量的测量值重构稀疏信号，称为稀疏重构问题。该问题已经成功应用于稀疏信道估计、雷达信号处理、稀疏分量分析以及阵列信号处理中的DOA估计等领域，是信号处理领域中一个新的研究热点。本文针对稀疏重构算法及其在信号处理中的应用展开了研究，主要工作概括如下：
1.针对不含噪单观测稀疏重构算法需要较多测量值和计算量较大的不足，提出了稳定收敛的近似l0范数最小化算法（SCAL0）和基于迭代支撑集检测的近似l0范数最小化算法（ISD-AL0），并分析了两种算法的收敛性。SCAL0算法利用了反正切函数近似l0范数，构造了快速收敛迭代格式近似l0范数表示信号的稀疏性，并建立近似l0范数的最小化问题，最后通过构造快速的不动点迭代格式求解该最优化问题以重构稀疏信号。ISD-AL0算法是对FCAL0算法的改进，提高了算法对少量测量值的适应能力。FCAL0算法和ISD-AL0算法需要较少的测量值，且所需计算量较小。相比AL0算法，ISD-AL0算法需要更少的测量值，但计算量有所增加。
2.针对单观测含噪稀疏重构算法对噪声适应能力较弱的不足，提出了近似l0范数期望值最小化算法（EML0）和快速收敛的稳健近似l0范数最小化算法（FCRAL0）。对于已知噪声方差的单观测稀疏重构问题提出了EML0算法，提高了对噪声的适应能力；对于未知噪声特性的单观测稀疏重构问题提出了FCRAL0算法，理论证明了该算法的收敛性和稳定性，并应用于解决非均匀样本的频谱估计问题。
3.针对现有概率类稀疏重构算法计算量较大的问题，提出了一种快速的贝叶斯稀疏重构算法（FBSR），并将该算法推广至复数域用于频谱估计。FBSR算法简化了基于拉普拉斯分布的贝叶斯模型，构造了快速的不动点迭代格式最大化联合概率估计超参数。FBSR算法减小了计算量，且具有较好的重构性能。与FCRAL0算法相比较，FBSR算法对噪声适应能力更强且不需要主观确定参数，但计算量有所增加。
4.针对不含噪和含噪的多观测稀疏重构问题，分别提出了多观测近似l0范数最小化算法（M-AL0）和多观测稳健近似l0范数最小化算法（M-RAL0），然后将M-AL0算法扩展至复数域，用于阵列信号处理中的DOA估计。M-AL0算法利用了反正切函数近似l2,0范数，通过拉格朗日函数构造了快速收敛的不动点迭代格式，并证明了算法的收敛性。M-RAL0算法将含噪多观测稀疏重构问题看成多目标规划问题，通过参数法将该问题转化为无约束最优化问题，然后采用拟牛顿法求解。M-AL0算法和M-RAL0算法重构联合稀疏信号时，需要较少的测量值数量且计算量较小。与M-AL0算法相比较，M-RAL0算法对噪声有较强的适应能力。
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.
1.针对不含噪单观测稀疏重构算法需要较多测量值和计算量较大的不足，提出了稳定收敛的近似l0范数最小化算法（SCAL0）和基于迭代支撑集检测的近似l0范数最小化算法（ISD-AL0），并分析了两种算法的收敛性。SCAL0算法利用了反正切函数近似l0范数，构造了快速收敛迭代格式近似l0范数表示信号的稀疏性，并建立近似l0范数的最小化问题，最后通过构造快速的不动点迭代格式求解该最优化问题以重构稀疏信号。ISD-AL0算法是对FCAL0算法的改进，提高了算法对少量测量值的适应能力。FCAL0算法和ISD-AL0算法需要较少的测量值，且所需计算量较小。相比AL0算法，ISD-AL0算法需要更少的测量值，但计算量有所增加。
2.针对单观测含噪稀疏重构算法对噪声适应能力较弱的不足，提出了近似l0范数期望值最小化算法（EML0）和快速收敛的稳健近似l0范数最小化算法（FCRAL0）。对于已知噪声方差的单观测稀疏重构问题提出了EML0算法，提高了对噪声的适应能力；对于未知噪声特性的单观测稀疏重构问题提出了FCRAL0算法，理论证明了该算法的收敛性和稳定性，并应用于解决非均匀样本的频谱估计问题。
3.针对现有概率类稀疏重构算法计算量较大的问题，提出了一种快速的贝叶斯稀疏重构算法（FBSR），并将该算法推广至复数域用于频谱估计。FBSR算法简化了基于拉普拉斯分布的贝叶斯模型，构造了快速的不动点迭代格式最大化联合概率估计超参数。FBSR算法减小了计算量，且具有较好的重构性能。与FCRAL0算法相比较，FBSR算法对噪声适应能力更强且不需要主观确定参数，但计算量有所增加。
4.针对不含噪和含噪的多观测稀疏重构问题，分别提出了多观测近似l0范数最小化算法（M-AL0）和多观测稳健近似l0范数最小化算法（M-RAL0），然后将M-AL0算法扩展至复数域，用于阵列信号处理中的DOA估计。M-AL0算法利用了反正切函数近似l2,0范数，通过拉格朗日函数构造了快速收敛的不动点迭代格式，并证明了算法的收敛性。M-RAL0算法将含噪多观测稀疏重构问题看成多目标规划问题，通过参数法将该问题转化为无约束最优化问题，然后采用拟牛顿法求解。M-AL0算法和M-RAL0算法重构联合稀疏信号时，需要较少的测量值数量且计算量较小。与M-AL0算法相比较，M-RAL0算法对噪声有较强的适应能力。
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.
引文
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