基于EMD和BP网络联合的故障诊断技术
摘要
本文首先回顾了一些时频分析方法，分析其具有的特点以及存在的不足，然后介绍了一种新的分析非线性、非平稳信号的方法，并将它与以往的时频分析方法进行对比，指出其具有的优势。在此基础之上，对将它与神经网络联合进行故障诊断进行了研究。
以往的大多数时频分析方法均是对传统的傅里叶分析进行修改后得到的，因此，傅里叶分析所存在的问题，如用谐波分量来表达信号的中的突变，是一个时间区间上的积分平均，或者是由传统的频率定义所带来的问题，如时间分辨率和频率分辨率的矛盾性，各种时频分析方法或多或少都存在，提高某一参数指标的同时往往是以牺牲另一参数指标为代价的。
本文介绍了一种新的分析非线性、非平稳信号的有力工具-EMD方法，与传统的对信号做积分变换的方法不同的是，它是将信号中不同尺度的波动或趋势逐级分解开来，产生一系列具有不同特征尺度的数据序列，每一个序列称为内在模函数。而各个内在模函数包含并突出了原信号的局部特征信息，这样通过对各内在模函数进行分析，便能够良好的把握原信号中所含有的特征信息量。实验证明，EMD也适用于线性平稳信号，这时候他分解出来的内在模函数就代表了原信号的各频率分量。事实上作为一种只是提取信号特征量的方法，EMD方法常常是与其他各种方法联合起来对信号进行分析。
用神经网络进行故障诊断实质上是一个模式识别的问题。其重点与难点在于如何有效的提取信号中的特征量，即分类标准的找寻问题。EMD方法的提出为解决这一问题提供了新的途径。本文通过将原信号用EMD分解而成的内在模函数输入到BP网络中进行训练学习并进行故障信号的判别进行实验，验证了EMD方法与BP网络的联合不仅能够大大提高判别的准确率，而且大大缩短了BP网络的学习收敛时间。本文的最后还对减少输入的内在模函数数量，简化网络结构进行了研究，证明其可行性。
在本文的结尾，展望了EMD在故障诊断领域的应用前景，指出了EMD方法还存在的不足之处以及未来的发展方向。
In this thesis some Time-Frequency analyzing method are reviewed first, analyzing their advantages and disadvantages. Then a new analyzing nonlinear and non-stationary signal method is introduced. And a comparison between it and the other Time-Frequency analyzing method is done, pointing out the new method's advantages. On this basis, using it associated with BP neural network in fault diagnosis is investigated thoroughly.
Most of the former Time-Frequency analyzing methods are derived from the Fourier analysis. So they more or less have the problem exiting in Fourier analysis, for example: the discontinuity expressed with harmonic components, being integral mean of a time interval; and the problems derived from the conventional definition of frequency, that is, the contradiction between the time and frequency resolution. Improving the performance of some parameter is at the cost of sacrificing the other one.
A powerful tool for analyzing nonlinear and non-stationary signal which is called EMD (empirical mode decomposition) method, is introduced. Different from the traditional method in doing integral transformation to signal, it decompose signal into several IMFs (intrinsic mode function), which contain and extrude the local characteristic of signal. So the characteristic information of the original signal can be well held by analyzing the IMFs. The experimentation testifies that the EMD method applies to linear and stationary signal. At this circumstance, the decomposed IMFs are the frequency components. In fact, EMD is a method which extract characteristic from signal. So in most circumstances, it is combined with other method to analyse signal.
The essence of using neural network in fault diagnosis is a matter of pattern recognition. The emphasis and difficulty lie in how to extract characteristic of signal efficiently, that is, the problem of seeking classification criteria. The appearance of EMD method provides a new way to solve this problem. Fault diagnosis experimentation is carried out. In the experimentation the IMFs decomposed from
signal are input into a designed BP neural network to train the network. Through this experimentation it is verified that the method of EMD in conjunction with BP neural network can not only improve the distinguishing accuracy, but also reduce the learning time of BP neural network greatly. What's more, the study covers how to decrease the number of IMFs and simplify the structure of BP neural network. A experiment proves its feasibility.
At the end of the thesis, the application prospect of EMD in fault diagnosis is revealed. The disadvantages and development direction of EMD are pointed out.
以往的大多数时频分析方法均是对传统的傅里叶分析进行修改后得到的，因此，傅里叶分析所存在的问题，如用谐波分量来表达信号的中的突变，是一个时间区间上的积分平均，或者是由传统的频率定义所带来的问题，如时间分辨率和频率分辨率的矛盾性，各种时频分析方法或多或少都存在，提高某一参数指标的同时往往是以牺牲另一参数指标为代价的。
本文介绍了一种新的分析非线性、非平稳信号的有力工具-EMD方法，与传统的对信号做积分变换的方法不同的是，它是将信号中不同尺度的波动或趋势逐级分解开来，产生一系列具有不同特征尺度的数据序列，每一个序列称为内在模函数。而各个内在模函数包含并突出了原信号的局部特征信息，这样通过对各内在模函数进行分析，便能够良好的把握原信号中所含有的特征信息量。实验证明，EMD也适用于线性平稳信号，这时候他分解出来的内在模函数就代表了原信号的各频率分量。事实上作为一种只是提取信号特征量的方法，EMD方法常常是与其他各种方法联合起来对信号进行分析。
用神经网络进行故障诊断实质上是一个模式识别的问题。其重点与难点在于如何有效的提取信号中的特征量，即分类标准的找寻问题。EMD方法的提出为解决这一问题提供了新的途径。本文通过将原信号用EMD分解而成的内在模函数输入到BP网络中进行训练学习并进行故障信号的判别进行实验，验证了EMD方法与BP网络的联合不仅能够大大提高判别的准确率，而且大大缩短了BP网络的学习收敛时间。本文的最后还对减少输入的内在模函数数量，简化网络结构进行了研究，证明其可行性。
在本文的结尾，展望了EMD在故障诊断领域的应用前景，指出了EMD方法还存在的不足之处以及未来的发展方向。
In this thesis some Time-Frequency analyzing method are reviewed first, analyzing their advantages and disadvantages. Then a new analyzing nonlinear and non-stationary signal method is introduced. And a comparison between it and the other Time-Frequency analyzing method is done, pointing out the new method's advantages. On this basis, using it associated with BP neural network in fault diagnosis is investigated thoroughly.
Most of the former Time-Frequency analyzing methods are derived from the Fourier analysis. So they more or less have the problem exiting in Fourier analysis, for example: the discontinuity expressed with harmonic components, being integral mean of a time interval; and the problems derived from the conventional definition of frequency, that is, the contradiction between the time and frequency resolution. Improving the performance of some parameter is at the cost of sacrificing the other one.
A powerful tool for analyzing nonlinear and non-stationary signal which is called EMD (empirical mode decomposition) method, is introduced. Different from the traditional method in doing integral transformation to signal, it decompose signal into several IMFs (intrinsic mode function), which contain and extrude the local characteristic of signal. So the characteristic information of the original signal can be well held by analyzing the IMFs. The experimentation testifies that the EMD method applies to linear and stationary signal. At this circumstance, the decomposed IMFs are the frequency components. In fact, EMD is a method which extract characteristic from signal. So in most circumstances, it is combined with other method to analyse signal.
The essence of using neural network in fault diagnosis is a matter of pattern recognition. The emphasis and difficulty lie in how to extract characteristic of signal efficiently, that is, the problem of seeking classification criteria. The appearance of EMD method provides a new way to solve this problem. Fault diagnosis experimentation is carried out. In the experimentation the IMFs decomposed from
signal are input into a designed BP neural network to train the network. Through this experimentation it is verified that the method of EMD in conjunction with BP neural network can not only improve the distinguishing accuracy, but also reduce the learning time of BP neural network greatly. What's more, the study covers how to decrease the number of IMFs and simplify the structure of BP neural network. A experiment proves its feasibility.
At the end of the thesis, the application prospect of EMD in fault diagnosis is revealed. The disadvantages and development direction of EMD are pointed out.
引文
[1] 朱大奇，于盛林．基于知识的故障诊断综述．安徽工业大学学报．2002，第19卷，第3期．
[2] 轩建平，张华．基于动力模型的故障检测与诊断理论和方法综述．振动工程学报(增刊)，1998．4 7．
[3] 张萍，周东华．动态系统的故障诊断方法．控制理论与应用．2000，17(2)：153-158．
[4] Frank, R M.. Analytical and Qualitative Model Based Fault Diagnosis—A Survey and Some New Results. European J. Control, 5(2): 6-28
[5] 周东华，王桂增．第五讲故障诊断技术综述
[6] 张贤达，保铮．非平稳信号分析与处理．北京：国防工业出版社，1998，9
[7] [美]L．科恩．时-频分析：理论与应用．西安：西安交通大学出版社，1998，3
[8] 张贤达．现代信号处理．北京：清华大学出版社，1994．
[9] 徐胜旺，石海．时频分析方法初探．现代电子，1996，(1)：38-44
[10] 马维祯，殷瑞祥．子波分析与子波变换．广州：华南理工大学出版社，1996
[11] 王宏禹．非平稳随机信号分析与处理．北京：国防工业出版社，1999(2001．1重印)
[12] Norden E. Huang, Zhen Shen. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. A, 1998, 454(1971): 903-995
[13] Norden E. Huang, Zhen Shen, Long S R. A new view of nonlinear water waves: the Hilbert Spectrum. Ann Rev Fluid Mech, 1999, 31:417-457
[14] 邓拥军，王伟，钱成春等 EMD方法及Hilbert变换中边界问题的处理科学通报．2001，46(3)．-257-263
[15] 闻新，周露，熊晓英，神经网络的故障诊断技术，自动驾驶仪与红外技术．1999，(2).-6-11
[16] Liemg jing. Using Neural Networks and Fuzzy Theory in Failure Diagnosis Expert system[A] ICTD'95 1995;JINLIN China: 27 30
[17] Naidu S R. Use of Neural Networks for Sensor Failure detection is a control System[J] IEEE Control System Magazine 1990,49(2):225 231
[18] Sorsa T, Hei Kki N. Networks in process Fault Diagnosis[J] IEEE Transactions on system, Man and cybernetics, 1991,21(4):815 825
[19] 梁华．基于神经网络及模糊理论的机械磨损故障诊断模型与策略研究[D]．武汉：武汉汽车工业大学，1995．65～70．
[20] 任国全，郑海起，汪伟．发动机磨损诊断的神经网络技术研究．华北工学院测试技术学报．2002，第16卷，第1期．
[21] Vaidynathan R and Venkatadubramawan V. Process Fault detection and diagnosis Using Neural Networks-2. Dynamics processes[A]. Artificial Intelligence Annual Meeting[C].Chicago Ⅱ , NOV(1990)
[22] 张绪省，孙金玮，赵新民．用时频分析方法分析非平稳信号．宇航计测技术．1995，15(5)．-34-41．
[23] Peyrin F, Prost R. A Unified Definition for the Discrete-Time, Discrete-Frequency, and Discrete-Time-Frequency Wigner Distribution. IEEE Trans. Signal Processing, vol. 34, August, 1986
[24] 胡广书．数字信号处理—理论、算法与实现．北京：清华大学出版社，1997
[25] 秦前清，杨宗凯．实用小波分析．西安：西安电子科技大学出版社，1994
[26] Classen T A C M, Mecklenbrauler W F G. Phillips J Res..1980,355(3-6): 217-250,
276-300, 372-389.
[27] Rioul O, Flandrin. IEEE Trans. on SP 1992. SP-40(7): 1746-1757
[28] Grossmann A, et al. Wavelets, Time-frequency methods and phase space. Berlin: Springer, 1989, 68-98.
[29] Youhong Lu and Joel M. Morris. Gabor Expansion for Adaptive Echo Cancellation. IEEE Signal Processing Magazine, Mar. 1999:68-80
[30] 沈民奋，孙丽莎．现代随机信号与系统分析．北京：科学出版社，1998
[31] 陈进．信号处理在机械设备故障诊断中的应用(连载)：第二讲现代信号处理与特征提取(下)．振动与冲击，1999，18(4)：84-87
[32] Devore R A, Jawerth B, Lucier P J. Image compression through wavelet transform coding. IEEE Trans on Information Theory, 1992,38(2): 719-746
[33] Jerome g. Shapiro. Embedded Image Coding Using Zerotrees of Wavelets Coefficients. IEEE Trans on Signal Processing. 1993, 41(12): 3445-3462
[34] Mallat S G and Hwang W L. Singularity detection and processing with wavelets. IEEE Trans. Inform. Theory, 1992(38): 617-643
[35] S. Qian and D. Chen. Joint Time-Frequency Analysis. IEEE Signal Processing Mag., pp. 52-67, Mar. 1999
[36] Y. Zhao, L. E. Atlas, and R. J. Marks. The use of cone-shaped kernels for generalized time-frequency representation of non-stationary signals. IEEE Trans. Acoust., Speech, Signal Processing, 1990, 38(7):1084-1091
[37] Mallat S. A theory for multi-resolution signal decomposition: the wavelet representation. IEEE Trans. on PAMI, 1989, 11(7): 674-693
[38] Victor C. Chen, Hao Ling. Joint Time-Frequency Analysis for Radar signal and image processing. IEEE Signal Processing Mag., Mar. 1999:81-93
[39] Hlawatsch F, Boudreaux-Bartels G F. Linear and Quadratic Time-Frequency Signal Representation. IEEE Signal Processing Magazine, April, 1992
[40] 杨福生．小波变换的工程分析与应用．北京：科学出版社，1999
[41] 杨福生．带通信号的采样定理，信号处理，2(1)，58-61，1986
[42] 贾民平．机械故障诊断学的理论及其应用：第三讲信号的特征分析与特征量选择．江苏机械制造与自动化，1999，(3)：44-47
[43] 徐佩霞，孙功宪．小波分析与应用实例．合肥：中国科学技术大学出版社，1996
[44] (美)崔锦泰(CharlesK．Chui)著，程正兴译．小波分析导论．西安：西安交通大学出版社，1995
[45] 吴大正 主编．信号与线性系统分析．北京：高等教育出版社，1998(2000 重印)
[46] Peyrin F, Prost R. A Unified Definition for the Discrete-Time, Discrete-Frequency, and Discrete-Time-Frequency Wigner Distribution. IEEE Trans. Signal Processing, vol. 34, August, 1998
[47] 傅海威，傅君眉．小波变换用于Wigner-Ville分布交叉项的抑制．西安石油学院学报(自然科学版)，2001，16(5)：37-39
[48] 何世平，张梅军等．现代信号分析在变速箱故障诊断中的应用．振动与冲击，2000，19(4)：71-75
[49] Boachan. B. and Escadie. B. Wigner-Ville Analysis of Asymptotic Signal and Application. Signal Processing, 1985, B(8):315-327
[50] 袁小宏，屈梁生．机器振动诊断中信号处理方法的研究．2001，35(7)：714-718．
[51] Gonzalo R. Arce. Elimination of Interference Terms of the Discrete Wigner Distribution Using Nonlinear Filtering. IEEE Trans. Signal Processing, vol. 48, pp. 2321-2331, Aug. 2000.
[52] 邓拥军，王伟，钱成春等 EMD方法及Hilbert变换中边界问题的处理科学通报．2001，46(3)．-257-263
[53] 陈小平，于盛林等．基于遗传算法的三次样条函数拟合．数据采集与处理．Jun．2000，vol．15，pp．138-141
[54] 尚磊等．采用三次样条函数拟合身高百分值数曲线．第四军医大学学报．2001，22(16)：1525-1527。
[55] 洪东旭．对汽车油罐车容量检定中插值算法的研究．汕头科技，2001，41-45。
[56] Michael Unser. Splines: A Perfect Fit for Signal and Image Processing. IEEE Signal Processing Mag., pp. 22-38, Nov. 1999.
[57] 贾民平．机械故障诊断学的理论及其应用：第二讲故障诊断中的信号检测与处理．江苏机械制造与自动化，1999，(2)：45-48
[58] 胡寿松，周川等．基于小波奇异性的结构故障检测．应用科学学报，2000，18(3)：198-201
[59] 邱阿瑞．电机滚动轴承故障诊断的新方法．中小型电机，1996，23(5)：47-50
[60] 陈进．信号处理在机械设备故障诊断中的应用(连载)：第一讲基于傅里叶谱分析的诊断方法(下)．振动与冲击，1999，18(2)：89-93
[61] 王世龙等．滚动轴承摩擦副表面损伤的检测与诊断系统．农业机械学报，2001，32(3)：106-108．
[62] S. Mallat and W.L. Hang. "Singularity Detection and Processing with Wavelets". IEEE Trans on Information Theory. Vol. 38, NO. 2, pp. 617-643, Mar 1992
[63] 潘旭峰，谢波等．小波变换理论及其在机械故障诊断中的应用．振动与冲击，17(1)：14-19
[64] 刘献栋，李其汉．机械振动信号中突变信息检测的新方法研究．中国机械正程，2000，11(5)：541-543
[65] 张绪省等，信号包络提取方法—从希尔伯特变换到小波变换．电子科学学刊，1997，19(1)：120-123
[66] 赵学智，符红羽等．小波分析在信号包络提取中的应用．广东工业大学学报，2000，17(3)：36-39
[67] 朱跃松．设备故障诊断中的包络分析．金山油化纤，2000，(4)：27-30
[68] 李宝年，耿苏黎．滚动轴承包络法诊断的应用．林业机械与木工设备，1997，(1)：40-43
[69] 袁晓，虞厥邦，复解析小波变换与语音信号包络提取和分析．电子学报，1999，27(5)：142-144
[70] [美]A．V．奥本海姆R．W．谢弗著，黄建国 刘树堂译．离散时间信号处理．北京：科学出版社，1998．7
[71] 赵学智等．奇异性信号检测时小波基的选择．华南理工大学学报(自然科学版)2000，28(10)：75-80
[72] 鞠德航，林可祥 陈捷．信号检测理论导论．北京：教育科学出版社，1977：511页．
[73] Tewfik A H. Sinha D, Jorgensen P. On the optimal choice of a wavelel for signal representation. IEEE Trans on Information Theory, 1992, 38(2): 713-718
[74] Ahmed H T, Deepen S, Paul J. On the optimal choice of a wavelet for signal
representation. IEEE Trans on Information Theory, 1992, 38(2): 747-765
[75] Rioul O, Vetterli M. Wavelets and signal processing. IEEE SP Magazine, 1991:14-38
[76] 李光林等．基于小波变换的心电信号的分析与处理．浙江大学学报(自然科学版)，1998，32(1)：82-87
[77] Wong PW. et al. Space-frequency Localized Image Compression. IEEE Trans on Image Processing, 1994, 3(3)
[78] 林京，屈梁生．基于连续小波变换的奇异性检测与故障诊断．振动工程学报2000，13(4)：523-530
[79] Staszewski W J, Torlinson G R. Application of the Wavelet Transform to Fault Detection in a Spur Gear. Mechanical System and Signal Processing. 1994,8(3):289-307
[80] 沈松等．用小波变换识别机械故障中的通过振动．振动与冲击，1999，18(2)：1-4
[81] 宋京伟，佐山隼敏，铃木和彦，人工神经网络与故障诊断，Journal of East China Jiaotong University，Vol．11No．3,Sep．1994
[82] 王伟杰，赵学增，基于BP网络的故障诊断正向推理方法，VEHICLE ENGINE，No．4(Serial No．134) Aug．2001
[83] 张绪锦，谭剑波，韩江洪 基于BP神经网络的故障诊断方法系统工程理论与实践2002年06期
[84] Frank PM, etal. New developments using AI in fault diagnosis[J].Engineering Application of Artificial Intelligence, 1997,10(1):3-14.
[2] 轩建平，张华．基于动力模型的故障检测与诊断理论和方法综述．振动工程学报(增刊)，1998．4 7．
[3] 张萍，周东华．动态系统的故障诊断方法．控制理论与应用．2000，17(2)：153-158．
[4] Frank, R M.. Analytical and Qualitative Model Based Fault Diagnosis—A Survey and Some New Results. European J. Control, 5(2): 6-28
[5] 周东华，王桂增．第五讲故障诊断技术综述
[6] 张贤达，保铮．非平稳信号分析与处理．北京：国防工业出版社，1998，9
[7] [美]L．科恩．时-频分析：理论与应用．西安：西安交通大学出版社，1998，3
[8] 张贤达．现代信号处理．北京：清华大学出版社，1994．
[9] 徐胜旺，石海．时频分析方法初探．现代电子，1996，(1)：38-44
[10] 马维祯，殷瑞祥．子波分析与子波变换．广州：华南理工大学出版社，1996
[11] 王宏禹．非平稳随机信号分析与处理．北京：国防工业出版社，1999(2001．1重印)
[12] Norden E. Huang, Zhen Shen. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. A, 1998, 454(1971): 903-995
[13] Norden E. Huang, Zhen Shen, Long S R. A new view of nonlinear water waves: the Hilbert Spectrum. Ann Rev Fluid Mech, 1999, 31:417-457
[14] 邓拥军，王伟，钱成春等 EMD方法及Hilbert变换中边界问题的处理科学通报．2001，46(3)．-257-263
[15] 闻新，周露，熊晓英，神经网络的故障诊断技术，自动驾驶仪与红外技术．1999，(2).-6-11
[16] Liemg jing. Using Neural Networks and Fuzzy Theory in Failure Diagnosis Expert system[A] ICTD'95 1995;JINLIN China: 27 30
[17] Naidu S R. Use of Neural Networks for Sensor Failure detection is a control System[J] IEEE Control System Magazine 1990,49(2):225 231
[18] Sorsa T, Hei Kki N. Networks in process Fault Diagnosis[J] IEEE Transactions on system, Man and cybernetics, 1991,21(4):815 825
[19] 梁华．基于神经网络及模糊理论的机械磨损故障诊断模型与策略研究[D]．武汉：武汉汽车工业大学，1995．65～70．
[20] 任国全，郑海起，汪伟．发动机磨损诊断的神经网络技术研究．华北工学院测试技术学报．2002，第16卷，第1期．
[21] Vaidynathan R and Venkatadubramawan V. Process Fault detection and diagnosis Using Neural Networks-2. Dynamics processes[A]. Artificial Intelligence Annual Meeting[C].Chicago Ⅱ , NOV(1990)
[22] 张绪省，孙金玮，赵新民．用时频分析方法分析非平稳信号．宇航计测技术．1995，15(5)．-34-41．
[23] Peyrin F, Prost R. A Unified Definition for the Discrete-Time, Discrete-Frequency, and Discrete-Time-Frequency Wigner Distribution. IEEE Trans. Signal Processing, vol. 34, August, 1986
[24] 胡广书．数字信号处理—理论、算法与实现．北京：清华大学出版社，1997
[25] 秦前清，杨宗凯．实用小波分析．西安：西安电子科技大学出版社，1994
[26] Classen T A C M, Mecklenbrauler W F G. Phillips J Res..1980,355(3-6): 217-250,
276-300, 372-389.
[27] Rioul O, Flandrin. IEEE Trans. on SP 1992. SP-40(7): 1746-1757
[28] Grossmann A, et al. Wavelets, Time-frequency methods and phase space. Berlin: Springer, 1989, 68-98.
[29] Youhong Lu and Joel M. Morris. Gabor Expansion for Adaptive Echo Cancellation. IEEE Signal Processing Magazine, Mar. 1999:68-80
[30] 沈民奋，孙丽莎．现代随机信号与系统分析．北京：科学出版社，1998
[31] 陈进．信号处理在机械设备故障诊断中的应用(连载)：第二讲现代信号处理与特征提取(下)．振动与冲击，1999，18(4)：84-87
[32] Devore R A, Jawerth B, Lucier P J. Image compression through wavelet transform coding. IEEE Trans on Information Theory, 1992,38(2): 719-746
[33] Jerome g. Shapiro. Embedded Image Coding Using Zerotrees of Wavelets Coefficients. IEEE Trans on Signal Processing. 1993, 41(12): 3445-3462
[34] Mallat S G and Hwang W L. Singularity detection and processing with wavelets. IEEE Trans. Inform. Theory, 1992(38): 617-643
[35] S. Qian and D. Chen. Joint Time-Frequency Analysis. IEEE Signal Processing Mag., pp. 52-67, Mar. 1999
[36] Y. Zhao, L. E. Atlas, and R. J. Marks. The use of cone-shaped kernels for generalized time-frequency representation of non-stationary signals. IEEE Trans. Acoust., Speech, Signal Processing, 1990, 38(7):1084-1091
[37] Mallat S. A theory for multi-resolution signal decomposition: the wavelet representation. IEEE Trans. on PAMI, 1989, 11(7): 674-693
[38] Victor C. Chen, Hao Ling. Joint Time-Frequency Analysis for Radar signal and image processing. IEEE Signal Processing Mag., Mar. 1999:81-93
[39] Hlawatsch F, Boudreaux-Bartels G F. Linear and Quadratic Time-Frequency Signal Representation. IEEE Signal Processing Magazine, April, 1992
[40] 杨福生．小波变换的工程分析与应用．北京：科学出版社，1999
[41] 杨福生．带通信号的采样定理，信号处理，2(1)，58-61，1986
[42] 贾民平．机械故障诊断学的理论及其应用：第三讲信号的特征分析与特征量选择．江苏机械制造与自动化，1999，(3)：44-47
[43] 徐佩霞，孙功宪．小波分析与应用实例．合肥：中国科学技术大学出版社，1996
[44] (美)崔锦泰(CharlesK．Chui)著，程正兴译．小波分析导论．西安：西安交通大学出版社，1995
[45] 吴大正 主编．信号与线性系统分析．北京：高等教育出版社，1998(2000 重印)
[46] Peyrin F, Prost R. A Unified Definition for the Discrete-Time, Discrete-Frequency, and Discrete-Time-Frequency Wigner Distribution. IEEE Trans. Signal Processing, vol. 34, August, 1998
[47] 傅海威，傅君眉．小波变换用于Wigner-Ville分布交叉项的抑制．西安石油学院学报(自然科学版)，2001，16(5)：37-39
[48] 何世平，张梅军等．现代信号分析在变速箱故障诊断中的应用．振动与冲击，2000，19(4)：71-75
[49] Boachan. B. and Escadie. B. Wigner-Ville Analysis of Asymptotic Signal and Application. Signal Processing, 1985, B(8):315-327
[50] 袁小宏，屈梁生．机器振动诊断中信号处理方法的研究．2001，35(7)：714-718．
[51] Gonzalo R. Arce. Elimination of Interference Terms of the Discrete Wigner Distribution Using Nonlinear Filtering. IEEE Trans. Signal Processing, vol. 48, pp. 2321-2331, Aug. 2000.
[52] 邓拥军，王伟，钱成春等 EMD方法及Hilbert变换中边界问题的处理科学通报．2001，46(3)．-257-263
[53] 陈小平，于盛林等．基于遗传算法的三次样条函数拟合．数据采集与处理．Jun．2000，vol．15，pp．138-141
[54] 尚磊等．采用三次样条函数拟合身高百分值数曲线．第四军医大学学报．2001，22(16)：1525-1527。
[55] 洪东旭．对汽车油罐车容量检定中插值算法的研究．汕头科技，2001，41-45。
[56] Michael Unser. Splines: A Perfect Fit for Signal and Image Processing. IEEE Signal Processing Mag., pp. 22-38, Nov. 1999.
[57] 贾民平．机械故障诊断学的理论及其应用：第二讲故障诊断中的信号检测与处理．江苏机械制造与自动化，1999，(2)：45-48
[58] 胡寿松，周川等．基于小波奇异性的结构故障检测．应用科学学报，2000，18(3)：198-201
[59] 邱阿瑞．电机滚动轴承故障诊断的新方法．中小型电机，1996，23(5)：47-50
[60] 陈进．信号处理在机械设备故障诊断中的应用(连载)：第一讲基于傅里叶谱分析的诊断方法(下)．振动与冲击，1999，18(2)：89-93
[61] 王世龙等．滚动轴承摩擦副表面损伤的检测与诊断系统．农业机械学报，2001，32(3)：106-108．
[62] S. Mallat and W.L. Hang. "Singularity Detection and Processing with Wavelets". IEEE Trans on Information Theory. Vol. 38, NO. 2, pp. 617-643, Mar 1992
[63] 潘旭峰，谢波等．小波变换理论及其在机械故障诊断中的应用．振动与冲击，17(1)：14-19
[64] 刘献栋，李其汉．机械振动信号中突变信息检测的新方法研究．中国机械正程，2000，11(5)：541-543
[65] 张绪省等，信号包络提取方法—从希尔伯特变换到小波变换．电子科学学刊，1997，19(1)：120-123
[66] 赵学智，符红羽等．小波分析在信号包络提取中的应用．广东工业大学学报，2000，17(3)：36-39
[67] 朱跃松．设备故障诊断中的包络分析．金山油化纤，2000，(4)：27-30
[68] 李宝年，耿苏黎．滚动轴承包络法诊断的应用．林业机械与木工设备，1997，(1)：40-43
[69] 袁晓，虞厥邦，复解析小波变换与语音信号包络提取和分析．电子学报，1999，27(5)：142-144
[70] [美]A．V．奥本海姆R．W．谢弗著，黄建国 刘树堂译．离散时间信号处理．北京：科学出版社，1998．7
[71] 赵学智等．奇异性信号检测时小波基的选择．华南理工大学学报(自然科学版)2000，28(10)：75-80
[72] 鞠德航，林可祥 陈捷．信号检测理论导论．北京：教育科学出版社，1977：511页．
[73] Tewfik A H. Sinha D, Jorgensen P. On the optimal choice of a wavelel for signal representation. IEEE Trans on Information Theory, 1992, 38(2): 713-718
[74] Ahmed H T, Deepen S, Paul J. On the optimal choice of a wavelet for signal
representation. IEEE Trans on Information Theory, 1992, 38(2): 747-765
[75] Rioul O, Vetterli M. Wavelets and signal processing. IEEE SP Magazine, 1991:14-38
[76] 李光林等．基于小波变换的心电信号的分析与处理．浙江大学学报(自然科学版)，1998，32(1)：82-87
[77] Wong PW. et al. Space-frequency Localized Image Compression. IEEE Trans on Image Processing, 1994, 3(3)
[78] 林京，屈梁生．基于连续小波变换的奇异性检测与故障诊断．振动工程学报2000，13(4)：523-530
[79] Staszewski W J, Torlinson G R. Application of the Wavelet Transform to Fault Detection in a Spur Gear. Mechanical System and Signal Processing. 1994,8(3):289-307
[80] 沈松等．用小波变换识别机械故障中的通过振动．振动与冲击，1999，18(2)：1-4
[81] 宋京伟，佐山隼敏，铃木和彦，人工神经网络与故障诊断，Journal of East China Jiaotong University，Vol．11No．3,Sep．1994
[82] 王伟杰，赵学增，基于BP网络的故障诊断正向推理方法，VEHICLE ENGINE，No．4(Serial No．134) Aug．2001
[83] 张绪锦，谭剑波，韩江洪 基于BP神经网络的故障诊断方法系统工程理论与实践2002年06期
[84] Frank PM, etal. New developments using AI in fault diagnosis[J].Engineering Application of Artificial Intelligence, 1997,10(1):3-14.