This article studies the problem of reconstructing spectrally sparse signals from a small random subset of time domain samples via low-rank Hankel matrix completion with the aid of prior information. By leveraging the low-rank structure of spectrally sparse signals in the lifting domain and the similarity between the signals and their prior information, we propose a convex method to recover the undersampled spectrally sparse signals. The proposed approach integrates the inner product of the desired signal and its prior information in the lift domain into vanilla Hankel matrix completion, which maximizes the correlation between the signals and their prior information. Theoretical analysis indicates that when the prior information is reliable, the proposed method has a better performance than vanilla Hankel matrix completion, which reduces the number of measurements by a logarithmic factor. We also develop an ADMM algorithm to solve the corresponding optimization problem. Numerical results are provided to verify the performance of proposed method and corresponding algorithm.

中文翻译：

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通过Hankel矩阵完成并具有先验信息的光谱稀疏信号恢复

本文研究了借助先验信息通过低秩汉克矩阵完成从时域样本的随机小子集重构频谱稀疏信号的问题。通过利用提升域中频谱稀疏信号的低秩结构以及信号与其先验信息之间的相似性，我们提出了一种凸方法来恢复欠采样的频谱稀疏信号。所提出的方法将期望信号的内积及其在提升域中的先验信息整合到香草汉克尔矩阵完成中，从而使信号与其先验信息之间的相关性最大化。理论分析表明，在先验信息可靠的情况下，所提出的方法具有比香草汉克尔矩阵完备性更好的性能，通过对数因子减少了测量次数。我们还开发了ADMM算法来解决相应的优化问题。数值结果验证了所提方法和相应算法的性能。