一种同时稀疏逼近的多维谐波恢复方法
A Simultaneous Sparse Approximation Method for Multidimensional Harmonic Retrieval
摘要：
本文提出了一种基于稀疏估计的噪声中多维（$R$ -D）模态（谐波或阻尼）复信号参数估计方法。该问题被公式化为多个1-D信号的$R$同时稀疏逼近。为了得到一种方法，能够处理大尺寸的信号，同时保持足够的分辨率，多重网格字典细化技术相关联的同时稀疏逼近问题。在单$R$ -D模式的情况下，证明了该细化过程的收敛性。然后，对于一般的多模式$R$ -D的情况下，信号张量模型被分解，以便在迭代方案中分别处理每个模式。所提出的方法不需要关联步骤，因为估计的模式自动“配对”。我们还导出了模态$R$ -D信号参数的Cramér-Rao下界。在单$R$ -D模式的情况下，以紧凑的形式给出的表达式。最后，通过数值仿真验证了该方法的有效性.
英文摘要：
In this paper, a sparse-based method for the estimation of the parameters of multidimensional ($R$-D) modal (harmonic or damped) complex signals in noise is presented. The problem is formulated as $R$ simultaneous sparse approximations of multiple 1-D signals. To get a method able to handle large size signals while maintaining a sufficient resolution, a multigrid dictionary refinement technique is associated with the simultaneous sparse approximation problem. The refinement procedure is proved to converge in the single $R$-D mode case. Then, for the general multiple modes $R$-D case, the signal tensor model is decomposed in order to handle each mode separately in an iterative scheme. The proposed method does not require an association step since the estimated modes are automatically "paired". We also derive the Cramér-Rao lower bounds of the parameters of modal $R$-D signals. The expressions are given in compact form in the single $R$-D mode case. Finally, numerical simulations are conducted to demonstrate the effectiveness of the proposed method.
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