Structural Instantaneous Frequency Identification Based on the Fractional Fourier Transform

LU Lian; REN Weixin; WANG Shidong

doi:10.21656/1000-0887.420241

Volume 43 Issue 8

Aug. 2022

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2022 > 43(8): 825-834

LU Lian, REN Weixin, WANG Shidong. Structural Instantaneous Frequency Identification Based on the Fractional Fourier Transform[J]. Applied Mathematics and Mechanics, 2022, 43(8): 825-834. doi: 10.21656/1000-0887.420241

Citation:

PDF( 8192 KB)

Structural Instantaneous Frequency Identification Based on the Fractional Fourier Transform

doi: 10.21656/1000-0887.420241

LU Lian^1
,,
REN Weixin^{2
,
,},
WANG Shidong^1
,

1.
College of Civil Engineering, Hefei University of Technology, Hefei 230009, P.R.China
2.
Key Laboratory for Resilient Infrastructures of Coastal Cities, Ministry of Education, Shenzhen University, Shenzhen, Guangdong 518060, P.R.China

Received Date: 2021-08-12
Accepted Date: 2021-12-01
Rev Recd Date: 2021-11-25

Available Online: 2022-06-30

Publish Date: 2022-08-01

Abstract

Abstract

To identify the instantaneous frequencies of time-varying signals, the theoretical relationship between the frequency and rotational angle α in a signal was derived based on the definition of the fractional Fourier transform. Then the fractional Fourier transform was interpreted to be essentially an algorithm combining the ordinary Fourier transform with the dilation and translation window. A general expression of the signal instantaneous frequency in the fractional Fourier domain was thereafter formulated so that the structural instantaneous frequency can be extracted accordingly. The feasibility and reliability of the proposed method were verified with a simulated nonlinear frequency modulation signal and a numerical example of a 3DOF damped time-varying structure system. The results show that, the results of the proposed method are in good agreement with the theoretical values, and the method has a certain degree of anti-noise capability. Subsequently, the proposed method is applicable to the identification of the instantaneous frequencies of time-varying structures.
- fractional Fourier transform,
- non-stationary signal,
- instantaneous frequency,
- dilation and translation window,
- time-varying structure

FullText(HTML)

References(22)

References

[1]	赵雷, 刘宁国. 大跨度叠合梁斜拉桥多维多点非平稳随机地震响应分析[J]. 应用数学和力学, 2017, 38(1): 118-125 ZHAO Lei, LIU Ningguo. Non-stationary random seismic analysis of large-span composite beam cable-stayed bridges under multi-support and multi-dimentional earthquake excitations[J]. Applied Mathematics and Mechanics, 2017, 38(1): 118-125.(in Chinese)
[2]	HUANG N, SHEN S S P. Hilbert-Huang Transform and Its Applications[M]. Singapore: World Scientific, 2005.
[3]	FELDMAN M. Theoretical analysis and comparison of the Hilbert transform decomposition methods[J]. Mechanical Systems & Signal Processing, 2008, 22(3): 509-519.
[4]	BAI R F, LI B Z, CHENG Q Y. Wigner-Ville distribution associated with the linear canonical transform[J]. Journal of Applied Mathematics, 2012, 2012: 740161.
[5]	MATEO C, TALAVERA J A. Short-time Fourier transform with the window size fixed in the frequency domain[J]. Digital Signal Processing, 2018, 77: 13-21. doi: 10.1016/j.dsp.2017.11.003
[6]	SONG J, FAN H Y, YUAN H C. Wavelet transform of quantum chemical states[J]. International Journal of Quantum Chemistry, 2012, 112(11): 2343-2347.
[7]	陶然, 齐林, 王越. 分数阶Fourier变换的原理与应用[M]. 北京: 清华大学出版社, 2004. TAO Ran, QI Lin, WANG Yue. Principle and Application of Fractional Fourier Transform[M]. Beijing: Tsinghua University Press, 2004.(in Chinese)
[8]	FELDMAN M. Time-varying vibration decomposition and analysis based on the Hilbert transform[J]. Journal of Sound and Vibration, 2006, 295(3/5): 518-530.
[9]	续秀忠, 张志谊, 华宏星, 等. 结构时变模态参数辨识的时频分析方法[J]. 上海交通大学学报, 2003, 37(1): 122-126 doi: 10.3321/j.issn:1006-2467.2003.01.033 XU Xiuzhong, ZHANG Zhiyi, HUA Hongxing, et al. Time-varying modal parameter identification with time-frequency analysis methods[J]. Journal of Shanghai Jiaotong University, 2003, 37(1): 122-126.(in Chinese) doi: 10.3321/j.issn:1006-2467.2003.01.033
[10]	PACHORI R B, SIRCAR P. A new technique to reduce cross terms in the Wigner distribution[J]. Digital Signal Processing, 2007, 17(2): 466-474. doi: 10.1016/j.dsp.2006.10.004
[11]	CHEN G D, WANG Z C. A signal decomposition theorem with Hilbert transform and its application to narrowband time series with closely spaced frequency components[J]. Mechanical Systems and Signal Processing, 2012, 28: 258-279. doi: 10.1016/j.ymssp.2011.02.002
[12]	廖庆良. 基于短时傅立叶变换的结构振动模态的识别[D]. 硕士学位论文. 武汉: 华中科技大学, 2004. LIAO Qingliang. Identification of structural vibration mode based on short time Fourier transform[D]. Master Thesis. Wuhan: Huazhong University of Science and Technology, 2004.(in Chinese)
[13]	CHEN S L, LIU J J, LAI H C. Wavelet analysis for identification of damping ratios and natural frequencies[J]. Journal of Sound and Vibration, 2009, 323(1/2): 130-147.
[14]	KIJEWSKI T, KAREEM A. Wavelet transforms for system identification in civil engineering[J]. Computer-Aided Civil and Infrastructure Engineering, 2010, 18(5): 339-355.
[15]	WANG C, REN W X, WANG Z C, et al. Instantaneous frequency identification of time-varying structures by continuous wavelet transform[J]. Engineering Structures, 2013, 52: 17-25. doi: 10.1016/j.engstruct.2013.02.006
[16]	王超, 任伟新, 黄天立. 基于复小波变换的结构瞬时频率识别[J]. 振动工程学报, 2009, 22(5): 492-496 doi: 10.3969/j.issn.1004-4523.2009.05.009 WANG Chao, REN Weixin, HUANG Tianli. Structure instantaneous frequency identification based on complex wavelet transform[J]. Journal of Vibration Engineering, 2009, 22(5): 492-496.(in Chinese) doi: 10.3969/j.issn.1004-4523.2009.05.009
[17]	刘景良, 任伟新, 王超, 等. 基于最大坡度法提取非平稳信号小波脊线和瞬时频率[J]. 工程力学, 2018, 35(2): 30-37, 46 LIU Jingliang, REN Weixin, WANG Chao, et al. Wavelet ridgeline and instantaneous frequency extraction of nonstationary signal based on maximum slope method[J]. Engineering Mechanics, 2018, 35(2): 30-37, 46.(in Chinese)
[18]	陶然, 邓兵, 王越. 分数阶Fourier变换在信号处理领域的研究进展[J]. 中国科学(E辑): 信息科学, 2006, 36(2): 113-136 TAO Ran, DENG Bing, WANG Yue. Research progress of fractional Fourier transform in signal processing[J]. Science in China (Series E): Information Sciences, 2006, 36(2): 113-136.(in Chinese)
[19]	TAO R, MIAO H X, MA J M. Theory and methods for sampling in fractional domains[C]//2016 IEEE International Conference on Signal Processing, Communications and Computing (ICSPCC). Hong Kong, 2016.
[20]	MENG X Y, TAO R, WANG Y. Fractional Fourier domain analysis of decimation and interpolation[J]. Science in China (Series F): Information Sciences, 2007, 50(4): 521-538. doi: 10.1007/s11432-007-0040-7
[21]	SU X H, TAO R, KANG X J. Analysis and comparison of discrete fractional Fourier transforms[J]. Signal Processing, 2019, 160: 284-298. doi: 10.1016/j.sigpro.2019.01.019
[22]	赵兴浩, 邓兵, 陶然. 分数阶傅里叶变换数值计算中的量纲归一化[J]. 北京理工大学学报, 2005, 25(4): 360-364 doi: 10.3969/j.issn.1001-0645.2005.04.019 ZHAO Xinghao, DENG Bing, TAO Ran. Dimension normalization in numerical calculation of fractional Fourier transform[J]. Journal of Beijing Institute of Technology, 2005, 25(4): 360-364.(in Chinese) doi: 10.3969/j.issn.1001-0645.2005.04.019