FFT算法的优化及DSP的实现
摘要
信号处理方法是当前机械设备故障诊断中重要的技术基础之一,分析结果的精确程度是诊断成功与否的关键因素。研究频谱分析的校正方法和误差分析,提高谱分析方法的精度是当前主要的发展方向之一。
本文系统的评述了目前国内对FFT变换后谱校正的四种方法。第一种是对幅值谱进行校正的比值法,这种方法利用归一化后差值为1的两点窗谱函数比值,建立一个以校正频率为变量的方程,解出频率,再进行幅值和相位的校正。第二种能量重心校正法,从理论上推导了常用离散窗谱函数的能量重心就是坐标原点,由此得到了能量重心法校正频率和相位的公式。第三种离散频谱的相位差校正法—时域平移+改变窗长法,即第二段时域序列比第一段滞后L点,对这两段时域分别作N点和M点的FFT分析,利用对应峰值谱线的相位差进行频谱校正的方法。第四种方法是FFT+FT谱连续细化分析傅立叶变换,对FFT局部进行细化分析,可以得到某个主要频率成份频率、幅值、相位的精确值,极大的提高了频率分辨率。
仿真分析了比值校正法和双窗法。误差分析和仿真计算结果表明比值法解决了精确求FFT后幅度、频率和相位,为精确测量信号参数提供了一种有效手段;双窗法从减小栅栏效应带来的幅值估计误差的角度,设计了一个具有平顶窗谱的窗函数,分析对比了它和矩形窗函数的性能差异,在此基础上,提出了用矩形窗和平顶窗谱窗函数联合进行频谱分析的双窗函数频谱估计算法,可在不降低频谱频率分辨率的条件下提高其幅值估计精度。并在DSP平台实现了FFT幅度谱的优化。
Signal processing methods are the important technical foundation of current mechanical equipment fault diagnosis, the analysis results are the key factor in accuracy of the diagnosis. On the spectrum of correction methods and error analysis, improving the accuracy of spectral analysis method is one of the main direction of development.
This paper reviews the current system of internal transformation of the FFT spectrum correction after the four methods. The first is the Amplitude Spectrum of the Ratio Correction, this approach useing of a margin after a function of the ratio of two spectrum, the establishment of a correction to the frequency of variable equation of the frequency and then the amplitude and phase correction. The second is Energy Centrobaric Correction Method, this paper demonstrates that the energy centroid of discrete spectrum of frequently used windows is near the origin of coord -inate and deduces the correction formula of frequency and phase using Energy Centrobaric Correcting Method. The third is Phase Difference Correcting Method for Calibration, this method including time-domain translation and the change of window width,the second time-domain sequence lags behind the first one with L points,performing N point FFT analysis on the first sequence and M-point FFT analysis on the second one,and finally correcting spectrum by making use of the phase difference of two corresponding peak lines. The fourth is Zoom FFT+FT Spectrum by Fourier Transform,the method get the precise value of some main frequency,and thus greatly improve the frequency discrimation of FFT spectrum.
Error analysis and simulation results show that the Ratio Correction method for resolving the precise extent after the FFT, frequency and phase for the accurate measurement of signal parameters provides an effective means; Correction For The Amplitude Error In FFT By Two Window Funtions -A window function which has a flat top in frequency domain is designed to correct the amplitude error caused by Picket-Fence effect.The window function has a wide main lobe,which results in a loss of frequency resolution of the spectrum.To correct the amplitude error and still keep a high frequency resolution in the FFT spectrum,a new algorithm is proposed,which uses two window functions to do spectrum analysis.The one is a rectangular window function and the other is a window function that has a flat top inrequency domain.By combination of the two window functions,a spectrum can be reached which has both the frequency resolution of rectangular window function and the amplitude resolution of the window function that has a flat top in frequency domain. the implememtation of FFT algorithm optimization on the DSP is introuduced.
本文系统的评述了目前国内对FFT变换后谱校正的四种方法。第一种是对幅值谱进行校正的比值法,这种方法利用归一化后差值为1的两点窗谱函数比值,建立一个以校正频率为变量的方程,解出频率,再进行幅值和相位的校正。第二种能量重心校正法,从理论上推导了常用离散窗谱函数的能量重心就是坐标原点,由此得到了能量重心法校正频率和相位的公式。第三种离散频谱的相位差校正法—时域平移+改变窗长法,即第二段时域序列比第一段滞后L点,对这两段时域分别作N点和M点的FFT分析,利用对应峰值谱线的相位差进行频谱校正的方法。第四种方法是FFT+FT谱连续细化分析傅立叶变换,对FFT局部进行细化分析,可以得到某个主要频率成份频率、幅值、相位的精确值,极大的提高了频率分辨率。
仿真分析了比值校正法和双窗法。误差分析和仿真计算结果表明比值法解决了精确求FFT后幅度、频率和相位,为精确测量信号参数提供了一种有效手段;双窗法从减小栅栏效应带来的幅值估计误差的角度,设计了一个具有平顶窗谱的窗函数,分析对比了它和矩形窗函数的性能差异,在此基础上,提出了用矩形窗和平顶窗谱窗函数联合进行频谱分析的双窗函数频谱估计算法,可在不降低频谱频率分辨率的条件下提高其幅值估计精度。并在DSP平台实现了FFT幅度谱的优化。
Signal processing methods are the important technical foundation of current mechanical equipment fault diagnosis, the analysis results are the key factor in accuracy of the diagnosis. On the spectrum of correction methods and error analysis, improving the accuracy of spectral analysis method is one of the main direction of development.
This paper reviews the current system of internal transformation of the FFT spectrum correction after the four methods. The first is the Amplitude Spectrum of the Ratio Correction, this approach useing of a margin after a function of the ratio of two spectrum, the establishment of a correction to the frequency of variable equation of the frequency and then the amplitude and phase correction. The second is Energy Centrobaric Correction Method, this paper demonstrates that the energy centroid of discrete spectrum of frequently used windows is near the origin of coord -inate and deduces the correction formula of frequency and phase using Energy Centrobaric Correcting Method. The third is Phase Difference Correcting Method for Calibration, this method including time-domain translation and the change of window width,the second time-domain sequence lags behind the first one with L points,performing N point FFT analysis on the first sequence and M-point FFT analysis on the second one,and finally correcting spectrum by making use of the phase difference of two corresponding peak lines. The fourth is Zoom FFT+FT Spectrum by Fourier Transform,the method get the precise value of some main frequency,and thus greatly improve the frequency discrimation of FFT spectrum.
Error analysis and simulation results show that the Ratio Correction method for resolving the precise extent after the FFT, frequency and phase for the accurate measurement of signal parameters provides an effective means; Correction For The Amplitude Error In FFT By Two Window Funtions -A window function which has a flat top in frequency domain is designed to correct the amplitude error caused by Picket-Fence effect.The window function has a wide main lobe,which results in a loss of frequency resolution of the spectrum.To correct the amplitude error and still keep a high frequency resolution in the FFT spectrum,a new algorithm is proposed,which uses two window functions to do spectrum analysis.The one is a rectangular window function and the other is a window function that has a flat top inrequency domain.By combination of the two window functions,a spectrum can be reached which has both the frequency resolution of rectangular window function and the amplitude resolution of the window function that has a flat top in frequency domain. the implememtation of FFT algorithm optimization on the DSP is introuduced.
引文
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[2]周耀华,汪凯仁.数字信号处理.上海:复旦大学出版社.1992,132-138
[3]杨位钦,顾岚.时间序列分析与动态数据建模.北京:北京工业学院出版社.1986,183-194,195-217
[4]孙仲康.快速傅里叶变换及其应用.北京:人民邮电出版社.1982,240-243
[5]宋琼,朱长春,牛宝良.基2快速傅里叶变换引入的不确定度估算.计量学报[J].2004,25(3):281-283
[6]郑星亮等.数字信号处理中的窗效应及窗函数的应用原则.北京联合大学学报[J].1997,11(2):33-37
[7]刘进明,应怀樵.频率在FFT谱上位置对加窗泄露的影响.应用力学学报[J].1998,15(2):25-31
[8]缪贇,马瑶珠等.FFT谱上频率的位置对加窗泄漏影响的研究.浙江海洋学院学报.2001,20(2):123-126
[9]Xie Ming,Ding Kang.Correction For the Frequency,Amplitude and phase in FFT of Harmonic sigial.Mechanical System and Signal Processing[J].1996,10(2):211-221
[10]丁康,张晓飞.频谱校正理论的发展.振动工程学报[J].2000,13(1):14-22
[11]谢明,丁康.频谱分析的校正方法.振动工程学报[J].1994,7(2):172-179
[12]谢明,丁康.离散频谱分析的一种新校正方法.重庆大学报[J].1995,18(2):47-54
[13]Qu L,Liu X.The holospectrum:a new method for rotor surveillance and diagnosis.Mechanical System and Signal Porcessing.1989,3(3):255-267
[14]徐培民.改进的FFT线形谱分析方法中窗函数的选用.抚顺石油学院学报[J].1997,17(1):41-45
[15]徐培民.提高FFT谱分析精度的一种方法.抚顺石油学院学报[J].1996,16(4):81-84
[16]Thomas Grandke.Interpolation algorithms for discrete Fourier tran- sforms of weighted signals.IEEE Transactions on Instrum entation and Measurement.1983,32(2):350-355
[17]黄迪山.FFT相位误差分析及实用修正方法.振动工程学报[J].1994,7(2):185-189
[18]丁康,谢明等.离散频谱的幅值、相位和频率的新校正方法及误差分析.动态分析与测试技术[J].1996,14(1):10-29
[19]余佳兵,史铁林,杨叔子.窗谱校正方法的实用峰值搜索算法研究.振动工程学报[J].1997,10(2):12-16
[20]丁康,谢明.离散频谱三点卷积幅值修正法的误差分析.振动工程学报[J].1996,9(1):92-98
[21]陈孚奎,焦群英,高小榕.谱峰法的窗函数选择.中国农业大学学报[J].1997,2:21-27
[22]朱利民等.离散频谱多点卷积幅值修正法的理论分析.振动工程学报[J].1999,12(1)120-126
[23]Offelli C,Petri D.A frequency-doma in procedure for accurate real-time signal parameter measurement.IEEE Trans M.1990,39(2):363-368
[24]丁康,江利旗.离散频谱能量重心校正法.振动工程学报[J].2001,14(3):354-358
[25]谢明,丁康,莫克斌.频谱校正时谱线干涉的影响和判定方法.振动工程学报[J].1998,11(1):22-28
[26]刘渝.正弦波频率快速成估计方法.数据采集与处理[J].1998,13(1):7-11
[27]丁康,罗维等.提高扭振稳态信号处理精度和速度的方法研究与系统.振动、测试与诊断[J].1999,19(4):332-338
[28]刘进明,应怀憔.FFT谱连续细化分析的付里叶变换法.振动工程学报[J].1995,8(2):162-166
[29]陈奎孚,焦群英,高小蓉.提高FFT谱质量的一种新方法.振动、测试与诊断[J].1998,18(3):216-232
[30]徐敏强,张嘉钟,王日新.一种提高周期信号谱分析质量的方法.动态分析与测试技术[J].1995,3:10-15
[31]张文泉.窗函数.动态分析与测试技术[J].1991,2:8-13
[32]余佳兵,史铁林,杨叔子.窗谱校正方法的实用峰值搜索算法研究.振动工程学报[J].1997,10(2):12-16
[33]谢明,张晓飞,丁康.频谱分析中用于相位和频率校正的相位差校正法.振动工程学报[J].1999,12(4):454-459
[34]丁康,朱小勇.适用于加各种窗的一种离散频谱相位差校正法.电子学报[J].2001,29(7):987-989
[35]丁康,江利旗.离散频谱综合相位差校正法.振动工程学报[J].2002,15(1):114-116
[36]丁康,罗江凯,谢明.离散频谱时移相位差校正方法.应用数学与力学[J].2002,23(7):729-735
[37]丁康,朱小勇等.离散频谱综合相位差校正法.振动工程学报[J].2002,15(1):114-118
[38]丁康,钟舜聪,朱小勇.离散频谱相位差校正方法研究.电子学报[J].2003,31(1):142-145
[39]张学玲,唐毅.数字信号的加窗处理.计算机应用[J].2000,20:67-68
[40]陈孚奎等.矩形窗频谱的精确校正.动态测试与分析技术[J].1997,15(2):46-52
[41]陈奎孚,张森文,高小蓉.对称窗函数的幅值谱特性.农业工程学报[J].1999,15(1):36-40
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