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二进制信号的混沌压缩测量与重构

郭静波 李佳文

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二进制信号的混沌压缩测量与重构

郭静波, 李佳文

Chaotic compressive measurement and reconstruction of binary signals

Guo Jing-Bo, Li Jia-Wen
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  • 二进制信号的压缩感知问题对应超奈奎斯特信号系统中未编码的二进制符号的检测问题, 具有重要的研究意义. 已有的二进制信号压缩测量采用高斯随机矩阵, 信号重构采用经典的l1最小化方法. 本文利用混沌映射构造基于Cat序列的循环测量矩阵, 并提出一种针对二进制信号的全新的重构算法平滑函数逼近法. 文章构造的混沌循环测量矩阵兼具确定性和随机性的优点, 能够抵御低信令效率和低信噪比的影响, 取得更好的压缩测量效果. 文章提出的平滑函数逼近法利用非凸函数代替原问题不连续的目标函数, 将组合优化问题转化为具有等式约束的优化问题进行求解. 利用稀疏贝叶斯学习算法进一步修正误差, 得到更准确的重构信号. 在信道含有加性高斯白噪声的条件下对二进制信号进行了压缩测量与重构的数值仿真, 仿真结果表明:基于Cat 序列的循环测量矩阵的压缩测量效果明显优于传统的高斯随机矩阵; 平滑函数逼近法对二进制信号的重构性能明显优于经典的l1最小化方法.
    Compressive sensing of binary signals is corresponding to the problem of binary symbol detection in the faster-than-Nyquist signaling systems, which has significant research value. Traditional compressive measurement of a binary signal is based on Gaussian matrix, and l1 minimization is a classic algorithm for signal reconstruction. However, stochastic matrix such as the Gaussian matrix can hardly be realized by a digital circuit, and the reconstruction performance of l1 minimization is not well enough for binary signals. Thus, it is of great meaning to construct a new kind of measurement matrix as well as a better reconstruction algorithm for binary signals. This paper constructs a chaotic circulant measurement matrix based on Cat chaotic map (CCMM), and proposes a brand new algorithm for binary signal reconstructionsmooth function approximation method (SFAM). Chaotic sequence has characteristics of both internal certainty and external randomness, while a circulant matrix requires less elements and can be realized through fast Fourier transform. CCMM conbines the advantages of both chaotic sequence and circulant matrix, so that it not only satisfies the RIPless property required by the compressive measurement matrix because of external randomness, but also has the power to resist the effect of low signaling efficiency and low SNR due to the internal certainty. Moreover, the circle structure gives CCMM the potential to be digital realized in the future. In SFAM, we first use a non-convex function to approximate the original discontinuous objective function, in order to transfer the original combinatorial optimization problem into an optimization problem with equality constraints which can be solved much easier. Then we use the interior point method to solve this optimization problem. Furthermore, sparse Bayesian learning algorithm is used to correct the reconstruction error for a more accurate result. Compressive measurement and reconstruction of binary signals in additive Gaussian white noise channel are operated. Result of numerical experiments shows that CCMM is much better than the traditional Gaussian matrix for compressive measurement, especially in the condition of low signaling efficiency and low SNR, and SFAM is much better than l1 minimization for binary signal reconstruction. At the end of this paper, we explain the essential reason why CCMM performs better than the traditional Gaussian matrix, through calculating the autocorrelation function of compressive measurement vector in various conditions.
      通信作者: 郭静波, guojb@mail.tsinghua.edu.cn
    • 基金项目: 国家自然科学基金(批准号:51277100)资助的课题.
      Corresponding author: Guo Jing-Bo, guojb@mail.tsinghua.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 51277100).
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    Donoho D L 2006 IEEE Trans. Inform. Theory 52 1289

    [2]

    Cands E J 2008 Comptes. Rendus Math. 346 589

    [3]

    Cands E J, Wakin M B 2008 IEEE Sig. Proc. Mag. 25 21

    [4]

    Lustig M, Donoho D L, Pauly J M 2007 Magn Reson Med. 58 1182

    [5]

    Sen P, Darabi S 2011 IEEE Trans. Vis. Comput. Graphics. 17 487

    [6]

    Hrman M A, Strohmer T 2009 IEEE Trans. Signal Process. 57 2275

    [7]

    Hajela D 1990 IEEE Trans. Inf. Theory 36 289

    [8]

    Rusek F, Anderson J B 2009 IEEE Trans. Commun. 57 1329

    [9]

    Sugiura S 2013 IEEE Wireless Commun. Lett. 2 555

    [10]

    Yin W, Morgan S, Yang J F, Zhang Y 2010 Rice University CAAM Technical Report TR10-01

    [11]

    Yin W, Osher S, Xu Y Y 2012 Inverse Probl. Imag. 8 901

    [12]

    Cands E J, Plan Y 2011 IEEE Trans. Inform. Theory 57 7235

    [13]

    Guo J B, Wang R 2014 Acta Phys. Sin. 63 198402(in Chinese) [郭静波, 汪韧 2014 物理学报 63 198402]

    [14]

    Guo J B, Xu X Z, Shi Q H, Hu T H 2013 Acta Phys. Sin. 62 110508(in Chinese) [郭静波, 徐新智, 史启航, 胡铁华 2013 物理学报 62 110508]

    [15]

    Allen Y. Y, Zhou Z H, Arvind G B, S. Shankar S, Ma Y 2013 IEEE Trans. Image Process. 22 3234

    [16]

    Chen S S, Donoho D L, Saunders M A 1998 SIAM J Sci. Comput. 43 33

    [17]

    Lu C W, Liu X J, Fang G Y 2011 Acta Electronica Sinica 39 2204 (in Chinese) [卢策吾, 刘小军, 方广有 2011 电子学报 39 2204]

    [18]

    Chen G, Mao Y, Chui C K 2004 Chaos Soliton Fract. 21 749

    [19]

    Mohimani H, Babaie-Zadeh M, Jutten C 2009 IEEE Trans. Signal Process. 57 289

    [20]

    Wipf D P, Rao B D 2004 IEEE Trans. Signal Process. 52 2153

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出版历程
  • 收稿日期:  2015-04-17
  • 修回日期:  2015-05-23
  • 刊出日期:  2015-10-05

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