基于最优型确定性测量矩阵的振动信号数据压缩采集方法
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摘要
针对目前机械振动信号频带越来越宽,依据奈奎斯特采样定律进行数据采集时,会得到海量的振动数据,对存储、传输和处理带来困难的问题,结合压缩感知理论,提出了一种基于最优型确定性测量矩阵的振动信号数据压缩采集方法。该方法的核心内容为测量矩阵的设计:以托普利兹(Toeplitz)矩阵为基础,建立正交对称托普利兹(Orthogonal Symmetric Toeplitz, OST)测量矩阵模型;为了降低测量矩阵与稀疏基之间的互相干性,应用阈值迭代收缩算法对该矩阵进行迭代,得到改进的OST矩阵;为了提高OST矩阵自身列独立性,采用奇异值分解(Singular Value Decomposition, SVD)算法继续优化OST矩阵,从而得到最优型确定性矩阵。仿真实验结果显示,提出的最优型测量矩阵与Toeplitz矩阵、高斯矩阵及优化前的OST矩阵相比,在重建振动信号时,重建精度较高,并且易于工程实现。
Aiming at frequency bandwidth of mechanical vibration signals being getting wider and wider, when using Nyquist sampling theorem for data collection, a huge amount of vibration data are acquired to bring difficulties of storage, transmission and processing. Here, combining the compression sensing theory, a data compression collecting method for vibration signals based on optimal deterministic measurement matrix was proposed. The core of this method was to design a measurement matrix. Firstly, based on Toeplitz matrix, an orthogonal symmetric Toeplitz(OST) measurement matrix model was established. Secondly, in order to reduce the mutual coherence between the measurement matrix and sparse bases, the threshold iterative contraction algorithm was used for OST matrix iteration to get a modified OST matrix. Finally, to improve the column independence of the OST matrix, the singular value decomposition(SVD) algorithm was used to optimize the OST matrix to acquire the optimal deterministic measurement matrix. The simulation results showed that compared with Toeplitz matrix, Gaussian one and the OST one before optimization, the proposed optimal deterministic measurement matrix has higher reconstruction accuracy when reconstructing vibration signals, and it is easy to implement in engineering.
Aiming at frequency bandwidth of mechanical vibration signals being getting wider and wider, when using Nyquist sampling theorem for data collection, a huge amount of vibration data are acquired to bring difficulties of storage, transmission and processing. Here, combining the compression sensing theory, a data compression collecting method for vibration signals based on optimal deterministic measurement matrix was proposed. The core of this method was to design a measurement matrix. Firstly, based on Toeplitz matrix, an orthogonal symmetric Toeplitz(OST) measurement matrix model was established. Secondly, in order to reduce the mutual coherence between the measurement matrix and sparse bases, the threshold iterative contraction algorithm was used for OST matrix iteration to get a modified OST matrix. Finally, to improve the column independence of the OST matrix, the singular value decomposition(SVD) algorithm was used to optimize the OST matrix to acquire the optimal deterministic measurement matrix. The simulation results showed that compared with Toeplitz matrix, Gaussian one and the OST one before optimization, the proposed optimal deterministic measurement matrix has higher reconstruction accuracy when reconstructing vibration signals, and it is easy to implement in engineering.
引文
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[2] DONOHO D.Compressed sensing[J].IEEE Transaction on Information Theory,2006,52(4):1289-1306.
[3] CANDES E,ROMBERG J,TAO T.Robust uncertainty principles:exact signal reconstruction from highly incomplete frequency information[J].IEEE Transactions on Information Theory,2006,52(2):489-509.
[4] HAUPT J,BAJWA W U,RAZ G,et al.Toeplitz compr-essed sensing matrices with applications to sparse channel estimation[J].IEEE Transactions on Information Theory,2010,56(11):5862-5875.
[5] CANDES E,WAKIN M.An introduction to compressive sampling[J].IEEE Signal Processing Magazine,2008,25(2):21-30.
[6] ELAD M.Optimized projections for compressed sensing[J].IEEE Transactions on Signal Processing,2007,55(12):5695-5702.
[7] ABOLGHASEMI V,FERDOWSI S,SANEI S.A gradient-based alternating minimization approach for optimization of the measurement matrix in compressive sensing[J].Signal Processing,2012,92(4):999-1009.
[8] DUARTE-CARVAJALINO J M,SAPIRO G.Learning to sense sparse signals:simultaneous sensing matrix and sparsifying dictionary optimization[J].IEEE Transactions on Image Processing,2009,18(7):1395-1408.
[9] XU Jianping,PI Y,CAO Z.Optimized projection matrix for compressive sensing[J].Eurasip Journal on Advances in Signal Processing,2010,2010(1):560349.
[10] 刘记红,徐少坤,高勋章,等.基于随机卷积的压缩感知雷达成像[J].系统工程与电子技术,2011,33(7):1485-1490.LIU Jihong,XU Shaokun,GAO Xunzhang,et al.C-ompressed sensing radar imaging based on random c-onvolution[J].System Engineering and Electronics,2011,33(7):1485-1490.
[11] 孟庆微,黄建国,何成兵,等.采用时域测量矩阵的压缩感知稀疏信道估计方法[J].西安交通大学学报,2012,46(8):94-99.MENG Qingwei,HUANG Jianguo,HE Chengbing,et al.Compressed sensing sparse channel estimation using time domain measurement matrix[J].Journal of Xi’an Jiaotong University,2012,46(8):94-99.
[12] 王强,张培林,王怀光,等.基于稀疏分解的振动信号数据压缩算法[J].仪器仪表学报,2016,37(11):2497-2505.WANG Qiang,ZHANG Peilin,WANG Huaiguang,et al.Data compression algorithm of vibration signal based on sparse decomposition[J].Chinese Journal of Scientific Instrument,2016,37(11):2497-2505.
[13] TROPP J A,GILBERT A C.Singnal recovery fromrandom measurements via Orthogonal Matching Pursuit[J].IEEE Transaction on Information Theory,2008,53(12):4644-4666.
[14] LOPES M E.Estimating unknown sparsity in compressed sensing[C]// International Conference on Machine Learning.JMLR.org,2013:217-225.
[15] BOTTCHER A.Orthogonal symmetric toeplitz matrices[J].Complex Analysis & Operator Theory,2008,2(2):285-298.
[16] LI Kezhi,CONG Ling,LU Gan.Statistical restrictted isometry property of orthogonal symmetric Toeplitz matrices[C]// Information Theory Workshop,2009.IEEE,2009:183-187.
[17] 赵瑞珍,秦周,胡绍海.一种基于特征值分解的测量矩阵优化方法[J].信号处理,2012,28(5):653-658.ZHAO Ruizhen,QIN Zhou,HU Shaohai.An optimization method for measurement matrix based on eigenvalue decomposition[J].Signal Processing,2012,28(5):653-658.