Research And Application Of High Precision Frequency Estimation Method Based On Symmetric DFT

Frequency is an important signal parameter,and it is widely used in various industries.For example,it is used in mechanical structure fault diagnosis,sonar and radar detection,spectroscopy,seismic location positioning,and celestial mechanics research.In the frequency-based fault diagnosis of beam structures,the more accurate the frequency estimation is,the more precise the crack location and depth estimation are.The signal excitation method based on resonance can effectively excite vibration.The extracted natural frequencies can be used for crack location and depth estimation.Affected by the spectrum leakage effect,none of the existing frequency estimation methods can accurately estimate the two components whose frequency distance is less than three bins.Fourier transform（FT）is widely used in signal parameter analysis.In practical application,discrete Fourier transform（DFT）carries the digital implementation of FT.DFT has two main forms:ordinary DFT（ODFT）and symmetric DFT（SDFT）.Currently,almost all signal analysis methods are based on ODFT.Compared with ODFT,SDFT has more FT properties,such as symmetry,integration,and interpolation.After analysis and comparison,this thesis concludes that SDFT is more suitable as the discrete form of FT than ODFT.In order to solve the shortcomings of frequency estimation,this thesis studies SDFT.It is found that even-number SDFT is not strictly symmetrical to zero,and this thesis improves it.In order to solve the estimation problem of two close specified frequency components,this thesis proposes two frequency estimation methods successively.The first one is the sidelobe method.When the frequency distance between two equal amplitude components is greater than 1.5 bins,the maximum estimation error is about 0.07 bins.The second method is the phase difference method based on SDFT（also called numerical method）.The simulation experiments show this method significantly improves the estimation accuracy of the two close specified frequency components,and this method has better anti-noise performance.When the frequency distance between two equal amplitude components is greater than 0.5 bins,the maximum estimation error is about 0.04 bins.Modal decomposition can decompose complex signals into single-tone signals,and it provides a new way to extract the natural frequencies of cracked beams.Assuming that the frequency resolution is 1Hz,when the frequency to be estimated is not very small,the minimum estimation error of the existing method reaches 10^-6Hz.A real single tone has two frequency components（a positive component and a negative component）,so the above numerical method can estimate the instantaneous frequency of a real single tone.However,the numerical method has disadvantages such as long calculation time,occasional failure,sensitivity to DC,and inability to calculate instantaneous amplitude.Based on the integral property of SDFT,a high precision calculation method of instantaneous frequency,amplitude,and phase is constructed.This method does not have the four shortcomings of the numerical method.When estimating the frequency of a real single tone with this method,the estimation error is about 10^-11Hz.When analyzing an AM and FM signal with this method,the instantaneous frequency and amplitude fluctuate heavily,but the instantaneous phase error is very small.The instantaneous frequency can be re-estimated by differentiating the instantaneous phase.Simulation experiments show that the re-estimated instantaneous frequency has extremely high accuracy.Finally,in order to verify the proposed frequency estimation method,this thesis combines the video motion magnification method,modal signal extraction method,and frequency estimation method to extract the first three mode frequencies of a beam with an open crack.The results show that this approach has high reliability,and the maximum relative error between the estimated frequency and the theoretical value is about 3.66%.