基于指数再生窗Gabor框架的窄脉冲欠Nyquist采样与重构

陈鹏; 孟晨; 孙连峰; 王成; 杨森

doi:10.7498/aps.64.070701

摘要

基于Gabor框架的窄脉冲信号采样及重构效果已经得到验证, 其解决了有限新息率(finite rate of innovation, FRI)采样方法无法在波形未知的情况下重构出脉冲波形的问题.但是目前的Gabor框架采样系统的窗函数构造复杂且难以物理实现.本文将指数再生窗函数引入Gabor框架, 将窗函数序列调制部分简化为一阶巴特沃斯模拟滤波器, 构造了Gabor系数重构所需要的压缩感知(compressed sensing, CS)测量矩阵.为了使得测量矩阵满足信号精确重构所需的约束等距特性(restricted isometry property, RIP), 根据高阶指数样条函数能量聚集特性, 选择了最优的窗函数支撑宽度, 推导了信号重构所需的约束条件, 还对其鲁棒性进行了分析.本文通过仿真实验对上述分析进行了有效验证, 该系统可应用于测试仪器、状态监测、雷达及通信领域等多种背景下的窄脉冲信号采样与重构.

关键词:

Abstract

Sampling and reconstruction of short pulses based on Gabor frames have been proved to be effective, which overcome the difficulties that finite rate of innovation (FRI) sampling is unable to reconstruct the pulse streams without the prior information of waveforms. However, the windows sequences of sampling scheme based on Gabor frames proposed at present show complicated structure and are hard to realize physically. The exponential reproducing windows are then introduced in this paper and the windows sequences can be simplified as a first-order analog Butterworth filter. At the same time, the compressed sensing (CS) measurement matrix is constructed for the recovery of Gabor coefficients. In order to satisfy the restricted isometry property (RIP) of the measurement matrices for perfect signal reconstruction, we select appropriate windows for support according to the energy accumulation property. A restricted condition is deduced for perfecting the signal reconstruction and the system robustness is analyzed. By numerical simulations the above analysis is verified. This novel scheme can be used to implement short pulses sampling and reconstruction in the field of instrumentation, condition monitoring, radar and the communication.

Keywords:

作者及机构信息

1.
军械工程学院, 导弹工程系, 石家庄 050003;

2.
国家纳米科学中心, 北京 100190

基金项目: 国家自然科学基金(批准号: 61372039)资助的课题.

Authors and contacts

1.
Department of Missile Engineering, Mechanical Engineering College, Shijiazhuang 050003, China;

2.
National Center for Nanoscience and Technology, Beijing 100190, China

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61372039).

参考文献

[1]	Fang S, Wu W C, Ying K, Guo H 2013 Acta Phys. Sin. 62 048702 (in Chinese) [方晟, 吴文川, 应葵, 郭华 2013 62 048702]
[2]	Ning F L, He B J, Wei J 2013 Acta Phys. Sin. 62 174212 (in Chinese) [宁方立, 何碧静, 韦娟 2013 62 174212]
[3]	Zhang J C, Fu N, Qiao L Y, Peng X Y 2014 Acta Phys. Sin. 63 030701 (in Chinese) [张京超, 付宁, 乔立岩, 彭喜元 2014 63 030701]
[4]	Omer Bar-Ilan, Eldar Y C 2014 IEEE Trans. Signal Processing 62 1796
[5]	Herman M A, Strohmer T 2009 IEEE Trans. Signal Processing 57 2275
[6]	Razzaque M A, Bleakley C, Dobson S 2013 ACM Transactions on Sensor Networks 10 5
[7]	Mishali M, Eldar Y C, Dounaevsky O 2011 IET circuits, devices & systems 5 8
[8]	Tropp J A, Laska J N, Duarte M F 2010 IEEE Trans. Inf. Theory 56 520
[9]	Michaeli T, Eldar Y C 2012 Signal Processing, IEEE Transactions on. 60 1121
[10]	Urigiien J A, Eldar Y C, Dragotti P L 2012 Compressed Sensing: Theory and Applications (Cambridge, U.K.: Cambridge Univ. Press) p148
[11]	Matusiak E 2012 IEEE Ttransactions on Signal Processing 60 1134
[12]	Kloos T, Stöckler J 2013 Journal of Approximation Theory 184 209
[13]	Jeffrey D B, Michael C, David H, Yirong J 2013 IEEE Trans. Signal Processing 62 1694
[14]	Xie Z P, Chen S C 2013 Journal of Computer Research and Development 49 580 (in Chinese) [谢志鹏, 陈松灿 2013 计算机研究与发展 49 580]
[15]	Feichtinger H G 1981 Monatshefte fr Mathematik 92 269
[16]	Mishali M, Eldar Y C 2009 Information Theory Workshop 2009.IEEE
[17]	Daubenchies I 1992 Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics) p97
[18]	Unser M, Blu T 2005 IEEE Trans. Signal Processing 53 1425
[19]	Qu C W, He Y, Liu W H, Li N 2009 Frames Theory and Applications (Beijing: National Defense Industy Press) p152 (in Chinese) [曲长文, 何友, 刘卫华, 李楠 2009 框架理论及应用(北京: 国防工业出版社) 第152页]
[20]	Tropp J A, Laska J N, Duarte M F, Romberg J K, Baraniuk R G 2010 IEEE Trans Inf. Theory 56 520
[21]	Xu Z Q 2012 Scientia Sinica (Mathematica) 42 865 (in Chinese) [许志强中国科学:数学 42 865]
[22]	Haupt J, Nowak R 2010 Proc. 44th Annual Conf. on Information Sciences and Systems Princeton, NJ, March 2010
[23]	Xu G W, Xu Z Q 2013 arXiv:1301.0373
[24]	Rudelson M, Vershynin R 2008 Communications on Pure and Applied Mathematics 61 1025

施引文献

[1]	Fang S, Wu W C, Ying K, Guo H 2013 Acta Phys. Sin. 62 048702 (in Chinese) [方晟, 吴文川, 应葵, 郭华 2013 62 048702]
[2]	Ning F L, He B J, Wei J 2013 Acta Phys. Sin. 62 174212 (in Chinese) [宁方立, 何碧静, 韦娟 2013 62 174212]
[3]	Zhang J C, Fu N, Qiao L Y, Peng X Y 2014 Acta Phys. Sin. 63 030701 (in Chinese) [张京超, 付宁, 乔立岩, 彭喜元 2014 63 030701]
[4]	Omer Bar-Ilan, Eldar Y C 2014 IEEE Trans. Signal Processing 62 1796
[5]	Herman M A, Strohmer T 2009 IEEE Trans. Signal Processing 57 2275
[6]	Razzaque M A, Bleakley C, Dobson S 2013 ACM Transactions on Sensor Networks 10 5
[7]	Mishali M, Eldar Y C, Dounaevsky O 2011 IET circuits, devices & systems 5 8
[8]	Tropp J A, Laska J N, Duarte M F 2010 IEEE Trans. Inf. Theory 56 520
[9]	Michaeli T, Eldar Y C 2012 Signal Processing, IEEE Transactions on. 60 1121
[10]	Urigiien J A, Eldar Y C, Dragotti P L 2012 Compressed Sensing: Theory and Applications (Cambridge, U.K.: Cambridge Univ. Press) p148
[11]	Matusiak E 2012 IEEE Ttransactions on Signal Processing 60 1134
[12]	Kloos T, Stöckler J 2013 Journal of Approximation Theory 184 209
[13]	Jeffrey D B, Michael C, David H, Yirong J 2013 IEEE Trans. Signal Processing 62 1694
[14]	Xie Z P, Chen S C 2013 Journal of Computer Research and Development 49 580 (in Chinese) [谢志鹏, 陈松灿 2013 计算机研究与发展 49 580]
[15]	Feichtinger H G 1981 Monatshefte fr Mathematik 92 269
[16]	Mishali M, Eldar Y C 2009 Information Theory Workshop 2009.IEEE
[17]	Daubenchies I 1992 Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics) p97
[18]	Unser M, Blu T 2005 IEEE Trans. Signal Processing 53 1425
[19]	Qu C W, He Y, Liu W H, Li N 2009 Frames Theory and Applications (Beijing: National Defense Industy Press) p152 (in Chinese) [曲长文, 何友, 刘卫华, 李楠 2009 框架理论及应用(北京: 国防工业出版社) 第152页]
[20]	Tropp J A, Laska J N, Duarte M F, Romberg J K, Baraniuk R G 2010 IEEE Trans Inf. Theory 56 520
[21]	Xu Z Q 2012 Scientia Sinica (Mathematica) 42 865 (in Chinese) [许志强中国科学:数学 42 865]
[22]	Haupt J, Nowak R 2010 Proc. 44th Annual Conf. on Information Sciences and Systems Princeton, NJ, March 2010
[23]	Xu G W, Xu Z Q 2013 arXiv:1301.0373
[24]	Rudelson M, Vershynin R 2008 Communications on Pure and Applied Mathematics 61 1025

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