摘要
压缩感知(CS:Compressed Sensing)理论作为一种新的信号采样理论框架,一经提出就受到各领域专家学者的广泛关注。该理论指出,如果信号是可压缩的,那么就可以以远低于奈奎斯特采样速率对信号进行采样并在接收端无失真的恢复,它实现了信号采样和压缩的同步进行,避免了大量冗余的采样数据。这一巨大优势,使得压缩感知理论有着广阔的应用前景。
本文从信号的稀疏表示、观测矩阵的设计、信号重构算法三个方面详细阐述了压缩感知理论。着重研究了信号的重构算法,对经典的算法进行了具体的描述和仿真对比。最后,结合无线信道的稀疏性,研究了压缩感知理论在信道估计中的应用。本文的贡献主要有以下两点：
(1)对光滑l0算法(SLO:Smoothed l0norm algorithm)进行了详细的推导,并从数学的角度分析了该算法在求解中存在的“锯齿现象”,此外,该算法求解过程中的步长选取是由人为经验确定的。这些因素导致目标函数逼近的最小值并不是最优的。针对此问题,文中提出一种基于步长优化和共轭梯度法的改进算法,并与原算法进行了仿真对比,比较两种算法的重构概率、精确度以及算法运行时间。
(2)对压缩感知的稀疏信道模型进行推导和分析,利用文中提出的改进算法对无线信道进行估计仿真,并与正交匹配追踪算法、正则化的正交匹配追踪算法、SLO算法以及传统的最小二乘估计法进行对比,分析了不同算法信道重构的精确度。从而说明压缩感知技术在信道估计中的优势和研究价值。
Compressed Sensing theory is a newly signal sampling framework, it has attracted a lot of experts and scholars from many fields. If the signal is compressible, the signal can be sampled with the rate far below the Nyquist frequency and reconstructed without distortion at the receiver. This new theory unites the signal sampling and compression and avoids the large number of redundant sampling data. This great advantage makes the Compressed Sensing theory has a great prospect in the future.
This thesis introduces the Compressed Sensing theory from the aspects of sparse representation of signal, the design of measurement matrix and reconstructed algorithms. It focuses on the signal recovery algorithms and has a detail description of several classical algorithms. Next, some simulation experiments carry out. Finally, combining with the sparse characteristics of the wireless channel, the thesis introduces the application of Compressed Sensing techniques to the problem of channel estimation. The main contribution of the thesis is as following two points.
(1) The thesis has a detail description of SLO (Smoothed l0norm algorithm) and analysis the "notched effect" in this method. In addition, the step in the process of solving is determined by experience. Which results in the approximation result is not optimal. In order to solve this problem, the thesis presents a new algorithm based on step optimization and the conjugate gradient method. Computer simulations confirm the effectiveness of the introduced algorithm comparisons with the existing methods in terms of run time, reconstructive probability and accuracy.
(2) The research includes analyzing the sparsity of the multipath wireless channel and deducing the system model. Providing some experimental results of the algorithm proposed in this thesis and its comparison with conventional SLO, orthogonal matching pursuit, regularized orthogonal matching pursuit, least square. From the results, we can see the advantages of the compressed channel sensing over tradition LS-based methods and the reconstructed quality of the proposed algorithm is better than other methods.
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