l1范数约束下的自适应滤波算法及其在稀疏系统辨识中的应用

项目名称： l1范数约束下的自适应滤波算法及其在稀疏系统辨识中的应用

项目编号： No.60872087

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 轻工业、手工业

项目作者： 谷源涛

作者单位： 清华大学

项目金额： 25万元

中文摘要： 针对广义稀疏系统辨识问题和稀疏信号重建理论、方法及应用进行了深入研究。(1)将稀疏性范数引入自适应滤波算法的代价函数，建立了稀疏约束辨识的算法框架，提出了以l_0-LMS为代表的一类自适应算法，促进了稀疏辨识领域的深入发展。(2)提出一种理论分析非线性自适应算法性能的思路，推导出l_0-LMS的稳态表达式和收敛条件，讨论了瞬态收敛过程，从根本上解释了该算法优于传统算法原因；这种理论分析思路适用于一大类非线性自适应算法，发展了非线性自适应理论。(3)提出"稀疏信号重建等价于稀疏系统辨识"，将自适应滤波结构和l_0-LMS算法应用于压缩采样信号恢复；提出ZAP算法解决一般稀疏信号重建问题，严格证明其收敛性并理论分析性能，若干指标优于同类算法。(4)首次全面分析了经典OMP算法在系统噪声和观测噪声并存情况下，精确或部分恢复支撑集的充分条件，导出重建误差；严格证明了一大类贪婪算法在两种噪声并存时具有接近最优的重建性能；实现了一般噪声下分析贪婪算法的从无到有的突破，为压缩采样的实用化提供了理论支持。(5)对基于压缩采样的模数转换系统和分布式视频编码进行了探索性研究。研究成果达到并超过了预期目标。

中文关键词： 稀疏系统辨识；稀疏信号重建；l_0/l_1范数约束；自适应滤波；压缩采样

英文摘要： In this project, general sparse system identification problem and sparse signal reconstruction problem, which includes its theory, algorithm and applications, are researched in-depth. (1) A framework of sparse constraint LMS algorithms is established to improve the performance of sparse system identification by adding the sparse penalty to the cost function. The proposed framework promotes the further development in the field of sparse system identification. (2) A method of theoretical analysis on the nonlinear adaptive algorithm is employed. Expressions for steady-state mean square deviation of l_0-LMS, which is a representative algorithm of the proposed framework, are derived and discussed with respect to algorithm parameters and system sparsity. The reason that the proposed algorithm is superior to the traditional ones is explained by analyzing the instantaneous behavior. Furthermore, this kind of theoretical analysis method can be extended to other nonlinear adaptive algorithms, which develops the nonlinear adaptive theory. (3) Based on the methodological similarity between sparse signal reconstruction and system identification, the adaptive filtering framework and l_0-LMS algorithm are introduced to solve the compressive sensing signal reconstruction problem. A Zero-point Attracting Projection (ZAP) algorithm is also adopted. In addition, the convergence of ZAP is proven and the theoretical analysis of its performance is deduced. Several experiments show that the proposed ZAP algorithm has the best performance among state of the arts. (4) Performance of Orthogonal Matching Pursuit (OMP) under general perturbations from both system noise and measurement noise is studied. The sufficient condition for exact or partial recovery of the support set and the reconstruction error are provided as well. Besides, we also proves that greedy pursuits with replacement have near-oracle recovery performance. The results of this part make a breakthrough for the analysis of greedy algorithm and provide a theoretical support for the practical applications of CS. (5) We designs new schemes for both analog to digital converter (ADC) and distributed video coding (DVC) based on CS. The proposed schemes perform better than the traditional ones. The achievement in this project meets and exceeds the expectation of the plan.

英文关键词： sparse system identification; sparse signal recovery; l_0 norm or l_1 norm constraint; adaptive filtering; compressive sampling