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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (5): 643-649.doi: 10.3969/j.issn.1005-3085.2015.05.002

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稀疏信号恢复问题解的个数

廖安平1,   杨  苗1,   谢家新1,    沈  坤2   

  1. 1- 湖南大学数学与计量经济学院，长沙 410082
    2- 湖南师范大学物理与信息科学学院，长沙 410081
  • 收稿日期:2014-07-01 接受日期:2014-12-27 出版日期:2015-10-15 发布日期:2015-12-15
  • 基金资助:
    国家自然科学基金 (11271117)；湖南省自然科学基金 (2015JJ6070).

Number of Solution for the Sparse Signal Recovery Problem

LIAO An-ping1,   YANG Miao1,   XIE Jia-xin1,   SHEN Kun2   

  1. 1- College of Mathematics and Econometrics, Hunan University, Changsha 410082
    2- College of Physics and Information Science, Hunan Normal University, Changsha 410081
  • Received:2014-07-01 Accepted:2014-12-27 Online:2015-10-15 Published:2015-12-15
  • Supported by:
    The National Natural Science Foundation of China (11271117); the Natural Science Foundation of Hunan Province (2015JJ6070).

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摘要： 稀疏信号恢复是信号处理研究领域中的重要问题，本文研究基于线性测量的稀疏信号恢复问题解的个数．在无噪测量下，采用组合分析方法，给出了稀疏信号恢复问题解的个数的一个上界，并通过构造一个特殊的线性测量矩阵，证明了该上界是最佳的．此外，如果测量矩阵还满足一定的条件，则该上界可减小．所得结果为特定情形下求解稀疏信号恢复问题提供了一种有限搜索的新思路．

关键词: 压缩感知, 稀疏信号恢复, 线性测量

Abstract:

This paper is concerned with the number of solution to sparse signal recovery problem based on linear measurements, which is an important problem in signal processing. In the noiseless measurement case, by taking advantage of the combinatorial analysis method, an upper bound is established for the number of solution to the sparse signal recovery problem, and by constructing a special linear measuring matrix, the best of the upper bound is proved as well. Moreover, if the measuring matrix satisfies some conditions, the upper bound could be improved. Based on these results, some new ideas of a finite search can be employed to solve the sparse signal recovery problem in some special cases.

Key words: compressed sensing, sparse signal recovery, linear measurement

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