小波分析在光纤陀螺信号处理中的应用
摘要
作为惯性测量组合的基础测量单元,光纤陀螺的精度将决定惯性系统的精度,因而提高光纤陀螺的精度具有重要意义。光纤陀螺输出信号微弱,背景噪声强,由于噪声的干扰,光纤陀螺输出信号中包含着多种随机误差。本文的研究目的就是运用小波分析去噪的方法来减小陀螺信号中的随机误差。
小波分析作为一种最新的时-频分析工具,在时域和频域都有表征信号局部特征的能力,具有多分辨率分析的特点,很适合处理非平稳信号。利用信号和噪声在各尺度下的小波变换系数特性不同的特点,对它们进行分离,从而可以达到去除噪声恢复信号的目的。
由于可用于去噪的小波母函数是一个集合,在小波去噪的实际应用中采用哪一种小波函数才能得到最好的去噪效果,是本文的一个研究内容。小波去噪的另一个重要问题就是阈值的选取和量化问题,在去噪的过程中如何更有效地进行阈值选取和量化,使得在噪声被去除的同时尽可能的避免有用信号的丢失,是本文研究的另一个内容。同时,由于本文开放性研究的目的,还将研究原始信号长度与采样频率的变化,及小波消失距的变化对滤波效果的影响。
本文首先介绍了小波分析的基本理论和小波去噪的原理与方法。重点研究了小波阈值去噪方法,对不同小波基、不同阈值选取规则、不同阈值量化方式及不同信号长度与采样频率下小波的去噪效果进行了比较和分析,研究了小波的消失矩阶数对去噪效果的影响。根据仿真实验结果,得到了最优小波滤波方法,并研究了提升方案(Lifting Scheme)对于计算速度及滤波效果的影响。最后,根据将选出的最优小波方法应用到了目标光纤陀螺上。
As the basic measuring unit of inertial measurement units, the accuracy of Fiber Optic Gyroscope (FOG) will decide the inertia system’s precision. As a result, it is significant to improve the accuracy of FOG. Though, the signal of FOG is weak and the background noise is strong, so the output signal of FOG, which disturbed by the noises, contains many stochastic errors. The main purpose of this thesis is to reduce the stochastic errors in output signal of FOG by using wavelet analysis.
As a new tool for time-frequency analysis, wavelet analysis has excellent localizing quality in time domain and frequency domain simultaneously, and the characteristic of multi-resolution. For this reason, wavelet analysis is suitable for processing the non-stationary signals. By the differences between the wavelet transform coefficients, noise can be filtered from the initial signals and useful signals can be restored.
Because the wavelets are congregation, which one is better for denoising in the practical application is a research subject of this thesis. Another important problem of denoising by wavelet is the threshold selection and the threshold quantization method. How to select the proper threshold and the quantization method is another research subject of this thesis. In the meantime, as a result of the open ended purpose of this paper, the filtering effects with different length, sampling frequency in initial signals and different vanishing moments of wavelets will be researched.
Firstly, the basic theory of wavelet analysis, the principles and methods of wavelet denoising are introduced in this thesis. Wavelet threshold denoising method is the key research point. The denoising effect by using different wavelets, thresholding rules, quantization methods, length and sampling frequency of signal are studied respectively. The number of vanishing moments of the wavelets has an effect on the denoising results, which is researched in this thesis too. By the experiment results, the optimal wavelet method is obtained. Then, Lifting Scheme has been used to compare its effect on computing speed and denoising. At last, this optimal wavelet method is realized in the FOG.
小波分析作为一种最新的时-频分析工具,在时域和频域都有表征信号局部特征的能力,具有多分辨率分析的特点,很适合处理非平稳信号。利用信号和噪声在各尺度下的小波变换系数特性不同的特点,对它们进行分离,从而可以达到去除噪声恢复信号的目的。
由于可用于去噪的小波母函数是一个集合,在小波去噪的实际应用中采用哪一种小波函数才能得到最好的去噪效果,是本文的一个研究内容。小波去噪的另一个重要问题就是阈值的选取和量化问题,在去噪的过程中如何更有效地进行阈值选取和量化,使得在噪声被去除的同时尽可能的避免有用信号的丢失,是本文研究的另一个内容。同时,由于本文开放性研究的目的,还将研究原始信号长度与采样频率的变化,及小波消失距的变化对滤波效果的影响。
本文首先介绍了小波分析的基本理论和小波去噪的原理与方法。重点研究了小波阈值去噪方法,对不同小波基、不同阈值选取规则、不同阈值量化方式及不同信号长度与采样频率下小波的去噪效果进行了比较和分析,研究了小波的消失矩阶数对去噪效果的影响。根据仿真实验结果,得到了最优小波滤波方法,并研究了提升方案(Lifting Scheme)对于计算速度及滤波效果的影响。最后,根据将选出的最优小波方法应用到了目标光纤陀螺上。
As the basic measuring unit of inertial measurement units, the accuracy of Fiber Optic Gyroscope (FOG) will decide the inertia system’s precision. As a result, it is significant to improve the accuracy of FOG. Though, the signal of FOG is weak and the background noise is strong, so the output signal of FOG, which disturbed by the noises, contains many stochastic errors. The main purpose of this thesis is to reduce the stochastic errors in output signal of FOG by using wavelet analysis.
As a new tool for time-frequency analysis, wavelet analysis has excellent localizing quality in time domain and frequency domain simultaneously, and the characteristic of multi-resolution. For this reason, wavelet analysis is suitable for processing the non-stationary signals. By the differences between the wavelet transform coefficients, noise can be filtered from the initial signals and useful signals can be restored.
Because the wavelets are congregation, which one is better for denoising in the practical application is a research subject of this thesis. Another important problem of denoising by wavelet is the threshold selection and the threshold quantization method. How to select the proper threshold and the quantization method is another research subject of this thesis. In the meantime, as a result of the open ended purpose of this paper, the filtering effects with different length, sampling frequency in initial signals and different vanishing moments of wavelets will be researched.
Firstly, the basic theory of wavelet analysis, the principles and methods of wavelet denoising are introduced in this thesis. Wavelet threshold denoising method is the key research point. The denoising effect by using different wavelets, thresholding rules, quantization methods, length and sampling frequency of signal are studied respectively. The number of vanishing moments of the wavelets has an effect on the denoising results, which is researched in this thesis too. By the experiment results, the optimal wavelet method is obtained. Then, Lifting Scheme has been used to compare its effect on computing speed and denoising. At last, this optimal wavelet method is realized in the FOG.
引文
1金杰,王玉琴.光纤陀螺研究综述.光纤与电缆及其应用技术, 2003, 6: 4-7
2 R. B. Vaganov, B. G. Klevitskiy. Sagnac Effect in a Fiber-ring Interferometer. Radio Engineering and Electronic Physics. 1984, 29(3): 149-153, 170-172
3 R.M. Hiehn. The Sagnac Effect and the Chirality of Space Time. Proceeding of SPIE. 2007, 6644 66440L1-13
4 H. J. Arditty, H. C. Lefovre. Sagnac Effect in Fiber Gyroscopes. Optics Letters. 1981, 6: 401-403
5 Wang Ruyong, Zheng Yi, Yao Aiping. Generalized Sagnac effect. Physical review Letters. 2004, 93(14): 143901-1-143901-3
6 (法)H. C. Lefevre著,张桂才,王巍,译.光纤陀螺仪.国防工业出版社, 2002: 4~9, 28-30
7丁杨斌,申功勋,王缜,满顺强.小波分析在光纤陀螺信号处理中的工程应用.光学工程. 2007, 34(5): 121-122
8 (法)S. Mallat著,张力华等译.信号处理的小波导引.机械工业出版社,2002: 22-23, 94-107, 244-245, 166-172
9 (美)崔锦泰.小波分析导论.西安交通大学出版社,1995: 71-86, 91-99, 183-193
10 A. Boggess, F. J. Narcowich. A first course in Wavelets with Fourier Analysis. Pearson Education, Inc.2001: 196-201, 227-243
11高亚楠,陈家斌.基于小波分析的光纤陀螺信号处理.火力与指挥控制,2005, 30(5): 35-37
12张传斌,邓正隆.基于小波分析的捷联式惯导系统的初始对准方法研究.高技术通讯,2002: 87-89
13刘镇平,张春熹,王妍.小波分析在捷联惯导陀螺信号滤波中的应用.中国惯性技术学报,2005, 13(1): 77-81
14 R. Ulrich. Fiber-optic Rotation Sensing with Low Drift. Optics Letters. 1980, 5: 173-175
15 I. Daubechies. Ten Lectures on Wavelets. Capital City Press, 1992: 9-22, 72-73
16秦前清,杨宗凯.实用小波分析.西安电子科技大学出版社, 2002: 3-4
17 A. Graps. An Introduction to Wavelets. IEEE Computional Science and Engineering. 1995, 2(2): 50-61
18 J.G. Jones, G. H. Watson. Multiresolution Analysis Using Adaptive Wavelets. Proceedings of SPIE. 1994, 2242: 119-129
19 S. G. Mallat. Compact Multiresolution Representation: the Wavelet Model. Proceedings of the IEEE Computer Society Workshop on Computer Vision. Miami Beach, FL, USA,1987: 2-7
20 G. W. Wornell, A. V. Oppenhiem. Estimation of Fractal Signals from Noisy Measurements Using Wavelets. IEEE Transactions on Signal Processing. 1992, 40(3): 611-623
21 S. G. Mallat, S. Zhong. Characterization of Signals from Multiscale Edges. IEEE Trans. on PAMI, 1992.7, 14(7): 710-732
22 Yansun Xu, J. B. Weaver, D. M. Healy. Wavelet Transform Domain Filters: A Spatially Selective Noise Filtration Technique. IEEE Trans. on IP. 1994, 3(6): 217-237
23 D. L. Donoho, I. M. Johnstone. Ideal Spatial Adaptation via Wavelet Shrinkage. Biometrika, 1994.12, 81: 425-455
24 D. L. Donoho, I. M. Johnstone. Adapting to unknown Smoothness via Wavelet Shrinkage. Journal of American Stat. Assoc. 1995.12, 90: 1200-1224
25 I. M. Johnstone, B. W. Silverman. Wavelet Threshold Estimators for Data with Correlated Noise. J. R. Statist. Soc. Series B, 1997, vol. 59
26 R. D. Nowak. Optimal Signal Estimation Using cross Validation. IEEE Signal Processing Letters, 1997, 4(1): 23-25
27 Chen Xiyuan. Adaptive Filtering Based on the Wavelet Transform for FOG on the Moving Base. Lecture Notes in Computer Science. 2005, 3644: 447-455
28赵忠华,江红,张炎华.小波理论及其在光纤陀螺信号分析中的应用.上海交通大学学报. 2001, 34(1): 1598-1600
29袁瑞铭,孙枫,陈慧.光纤陀螺信号的小波滤波方法研究.哈尔滨工业大学学报. 2004, 36(9): 1235-1238
30张志君.基于光纤陀螺的寻北定向技术研究.中国科学院长春光学精密机械与物理研究所博士论文. 2005, 1: 28-31
31 M. M. Tehrani. Ring Laser Gyro Data Analysis with Cluster Sampling Technique. Proceedings of SPIE. 1983, 412: 207-220
32董长虹等. Matlab小波分析工具箱原理与应用.国防工业出版社. 2004: 2-29
33葛哲学,廖剑利,李浩明等.小波分析理论与MATLAB7实现.电子工业出版社. 2005: 3-10, 37-47, 51-56, 57-60, 61-80
34程正兴.小波分析与算法.西安交通大学出版社, 1998: 28-31
35胡昌华,张军波,夏军等.基于MATLAB的系统分析与设计—小波分析.西安电子科技大学出版社, 1999: 1-157, 201-232
36程正兴,杨守志,冯晓霞.小波分析的理论、算法、进展和应用.国防工业出版社, 2007: 38-56
37 I. Daubechies. Orthonomal Bases of Compactly Supported Wavelet. Comm. on Pure and Appl. Math, 41, 1989: 909-996
38 S. Mallat. A Theory of Multiresolution Signal Decomposition: The Wavelet Transform. IEEE Trans., PAMI-11(7), 1989: 674-693
39 [法]Meyer著,尤众译.小波与算子.世界图书出版公司, 1992: 37-42
40缪玲娟.小波分析在光纤陀螺信号滤波中的应用研究,宇航学报,2000, 21(1): 42-46
41高广春.第二代小波变换理论及其在信号和图像编码算法中的应用.浙江大学博士论文. 2004: 29-31
42钟广军.基于提升方法的图像压缩技术的研究.国防科学技术大学博士论文. 2001: 41-45
43 D. L. Donoho. De-noising by Soft-thresholding. IEEE Transaction on Information Theory. 1995, 41(3): 613-627
44 S. Ma11at. Singularity Detection and Processing with Wavelets. IEEE Transaction on Information Theory. 1992, 38(2): 617-643
45赵瑞珍.小波理论及其在图像、信号处理中的算法研究.西安电子科技大学博士学位论文. 2001: 50-51
46 D. L. Donoho, I. M. Johnstone. Threshold Selection for Wavelet Shrinkage of Noisy Data. Proceedings of the 16th Annual International Conference of the IEEE. 3-6 Nov., 1994, 16 (1): 24a-25a
47张传斌,王学孝,邓正隆.基于小波分析的FOG中1/ fν噪声的去除方法研究.系统工程与电子技术. 2002, 24(4): 64-66
2 R. B. Vaganov, B. G. Klevitskiy. Sagnac Effect in a Fiber-ring Interferometer. Radio Engineering and Electronic Physics. 1984, 29(3): 149-153, 170-172
3 R.M. Hiehn. The Sagnac Effect and the Chirality of Space Time. Proceeding of SPIE. 2007, 6644 66440L1-13
4 H. J. Arditty, H. C. Lefovre. Sagnac Effect in Fiber Gyroscopes. Optics Letters. 1981, 6: 401-403
5 Wang Ruyong, Zheng Yi, Yao Aiping. Generalized Sagnac effect. Physical review Letters. 2004, 93(14): 143901-1-143901-3
6 (法)H. C. Lefevre著,张桂才,王巍,译.光纤陀螺仪.国防工业出版社, 2002: 4~9, 28-30
7丁杨斌,申功勋,王缜,满顺强.小波分析在光纤陀螺信号处理中的工程应用.光学工程. 2007, 34(5): 121-122
8 (法)S. Mallat著,张力华等译.信号处理的小波导引.机械工业出版社,2002: 22-23, 94-107, 244-245, 166-172
9 (美)崔锦泰.小波分析导论.西安交通大学出版社,1995: 71-86, 91-99, 183-193
10 A. Boggess, F. J. Narcowich. A first course in Wavelets with Fourier Analysis. Pearson Education, Inc.2001: 196-201, 227-243
11高亚楠,陈家斌.基于小波分析的光纤陀螺信号处理.火力与指挥控制,2005, 30(5): 35-37
12张传斌,邓正隆.基于小波分析的捷联式惯导系统的初始对准方法研究.高技术通讯,2002: 87-89
13刘镇平,张春熹,王妍.小波分析在捷联惯导陀螺信号滤波中的应用.中国惯性技术学报,2005, 13(1): 77-81
14 R. Ulrich. Fiber-optic Rotation Sensing with Low Drift. Optics Letters. 1980, 5: 173-175
15 I. Daubechies. Ten Lectures on Wavelets. Capital City Press, 1992: 9-22, 72-73
16秦前清,杨宗凯.实用小波分析.西安电子科技大学出版社, 2002: 3-4
17 A. Graps. An Introduction to Wavelets. IEEE Computional Science and Engineering. 1995, 2(2): 50-61
18 J.G. Jones, G. H. Watson. Multiresolution Analysis Using Adaptive Wavelets. Proceedings of SPIE. 1994, 2242: 119-129
19 S. G. Mallat. Compact Multiresolution Representation: the Wavelet Model. Proceedings of the IEEE Computer Society Workshop on Computer Vision. Miami Beach, FL, USA,1987: 2-7
20 G. W. Wornell, A. V. Oppenhiem. Estimation of Fractal Signals from Noisy Measurements Using Wavelets. IEEE Transactions on Signal Processing. 1992, 40(3): 611-623
21 S. G. Mallat, S. Zhong. Characterization of Signals from Multiscale Edges. IEEE Trans. on PAMI, 1992.7, 14(7): 710-732
22 Yansun Xu, J. B. Weaver, D. M. Healy. Wavelet Transform Domain Filters: A Spatially Selective Noise Filtration Technique. IEEE Trans. on IP. 1994, 3(6): 217-237
23 D. L. Donoho, I. M. Johnstone. Ideal Spatial Adaptation via Wavelet Shrinkage. Biometrika, 1994.12, 81: 425-455
24 D. L. Donoho, I. M. Johnstone. Adapting to unknown Smoothness via Wavelet Shrinkage. Journal of American Stat. Assoc. 1995.12, 90: 1200-1224
25 I. M. Johnstone, B. W. Silverman. Wavelet Threshold Estimators for Data with Correlated Noise. J. R. Statist. Soc. Series B, 1997, vol. 59
26 R. D. Nowak. Optimal Signal Estimation Using cross Validation. IEEE Signal Processing Letters, 1997, 4(1): 23-25
27 Chen Xiyuan. Adaptive Filtering Based on the Wavelet Transform for FOG on the Moving Base. Lecture Notes in Computer Science. 2005, 3644: 447-455
28赵忠华,江红,张炎华.小波理论及其在光纤陀螺信号分析中的应用.上海交通大学学报. 2001, 34(1): 1598-1600
29袁瑞铭,孙枫,陈慧.光纤陀螺信号的小波滤波方法研究.哈尔滨工业大学学报. 2004, 36(9): 1235-1238
30张志君.基于光纤陀螺的寻北定向技术研究.中国科学院长春光学精密机械与物理研究所博士论文. 2005, 1: 28-31
31 M. M. Tehrani. Ring Laser Gyro Data Analysis with Cluster Sampling Technique. Proceedings of SPIE. 1983, 412: 207-220
32董长虹等. Matlab小波分析工具箱原理与应用.国防工业出版社. 2004: 2-29
33葛哲学,廖剑利,李浩明等.小波分析理论与MATLAB7实现.电子工业出版社. 2005: 3-10, 37-47, 51-56, 57-60, 61-80
34程正兴.小波分析与算法.西安交通大学出版社, 1998: 28-31
35胡昌华,张军波,夏军等.基于MATLAB的系统分析与设计—小波分析.西安电子科技大学出版社, 1999: 1-157, 201-232
36程正兴,杨守志,冯晓霞.小波分析的理论、算法、进展和应用.国防工业出版社, 2007: 38-56
37 I. Daubechies. Orthonomal Bases of Compactly Supported Wavelet. Comm. on Pure and Appl. Math, 41, 1989: 909-996
38 S. Mallat. A Theory of Multiresolution Signal Decomposition: The Wavelet Transform. IEEE Trans., PAMI-11(7), 1989: 674-693
39 [法]Meyer著,尤众译.小波与算子.世界图书出版公司, 1992: 37-42
40缪玲娟.小波分析在光纤陀螺信号滤波中的应用研究,宇航学报,2000, 21(1): 42-46
41高广春.第二代小波变换理论及其在信号和图像编码算法中的应用.浙江大学博士论文. 2004: 29-31
42钟广军.基于提升方法的图像压缩技术的研究.国防科学技术大学博士论文. 2001: 41-45
43 D. L. Donoho. De-noising by Soft-thresholding. IEEE Transaction on Information Theory. 1995, 41(3): 613-627
44 S. Ma11at. Singularity Detection and Processing with Wavelets. IEEE Transaction on Information Theory. 1992, 38(2): 617-643
45赵瑞珍.小波理论及其在图像、信号处理中的算法研究.西安电子科技大学博士学位论文. 2001: 50-51
46 D. L. Donoho, I. M. Johnstone. Threshold Selection for Wavelet Shrinkage of Noisy Data. Proceedings of the 16th Annual International Conference of the IEEE. 3-6 Nov., 1994, 16 (1): 24a-25a
47张传斌,王学孝,邓正隆.基于小波分析的FOG中1/ fν噪声的去除方法研究.系统工程与电子技术. 2002, 24(4): 64-66