采用频域Prony方法估计信号重叠双分量
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摘要
给定加窗信号DFT峰值附近的5个点,应用汉宁窗的DTFT的性质,将表达式复杂的频率估计问题转化为简单的二元高次方程组,利用Prony方法将其进一步转化为差分方程的系数估计问题.将差分方程写成矩阵形式,利用数据矩阵的奇异值分解,得到判断插值点上重叠分量数目的方法、估计单分量和双分量的复频率的算法以及重叠分量数不小于3时的处理方法.在分量的复频率已知的情况下,分量的幅值和相位可以通过最小二乘法获得.当插值点仅包含单分量时,频域Prony方法和其他2种算法的估计误差十分接近;当插值点包含重叠双分量时,频域Prony方法能够同时得到2个分量的估计,但是其他2种算法失效.
Given five points around the peak of the windowed DFT values and based on the property of Hann window's DTFT,the estimation of complex frequencies was transformed to an equation group of higher degree with two unknowns.Solving the equation group was equivalent to estimating the coefficients of a difference equation according to Prony's method,The singular value decomposition method was applied by rewriting the difference equation into the matrix form,and the number of overlapped components on the interpolation points was evaluated.The algorithm to estimate single or double components was provided and easily extended to deal with three or more overlapped components.The amplitude and phase of components could be archived by the least squares method with frequencies estimated.When interpolation points contain only one component,all three methods have similar estimation results.when interpolation points contain two overlapped components,the proposed method provides accurate estimations,while the other two fail.
Given five points around the peak of the windowed DFT values and based on the property of Hann window's DTFT,the estimation of complex frequencies was transformed to an equation group of higher degree with two unknowns.Solving the equation group was equivalent to estimating the coefficients of a difference equation according to Prony's method,The singular value decomposition method was applied by rewriting the difference equation into the matrix form,and the number of overlapped components on the interpolation points was evaluated.The algorithm to estimate single or double components was provided and easily extended to deal with three or more overlapped components.The amplitude and phase of components could be archived by the least squares method with frequencies estimated.When interpolation points contain only one component,all three methods have similar estimation results.when interpolation points contain two overlapped components,the proposed method provides accurate estimations,while the other two fail.
引文
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[18]贾清泉,姚蕊,王宁,等.一种应用原子分解和加窗频移算法分析频率相近谐波/间谐波的方法[J].中国电机工程学报,2014,34(27):4605-4612.JIA Qing-quan,YAO Rui,WANG Ning,et al.An approach to detect harmonics/inter-harmonics with closing frequencies using atomic decomposition and windowed frequency shifting algorithm[J].Proceedings of the CSEE,2014,34(27):4605-4612.
[19]周菁菁,王彭,赵春宇,等.基于频率搜索的间谐波闪变检测方法[J].电力系统自动化,2014,38(7):60-65.ZHOU Jing-jing,WANG Peng,ZHAO Chun-yu,et al.A detection method of interharmonics-caused flicker based on frequency-searching[J].Automation of Electric Power Systems,2014,38(7):60-65.
[20]BELEGA D,PETRI D.Frequency estimation by two-or three-point interpolated Fourier algorithms based on cosine windows[J].Signal Processing,2015,117:115-125.
[21]DIAO R,MENG Q.An interpolation algorithm for discrete Fourier transforms of weighted damped sinusoidal signals[J].IEEE Transactions on Instrumentation and Measurement,2014,63(6):1505-1513.
[22]张贤达.矩阵分析与应用[M].北京:清华大学出版社,2004:17.
[2]ZYGARLICKI J,MROCZKA J.Variable-frequency Prony method in the analysis of electrical power quality[J].Metrology and Measurement Systems,2012,19(1):39-48.
[3]ZHOU Z,SO H C.Linear prediction approach to oversampling parameter estimation for multiple complex sinusoids[J].Signal Processing,2012,92(6):1458-1466.
[4]ZHOU Z,SO H C,CHRISTENSEN M G.Parametric modeling for damped sinusoids from multiple channels[J].IEEE Transactions on Signal Processing,2013,61(15):3895-3907.
[5]CHAKIR M,KAMWA I,LE HUY H.Extended C37.118.1PMU algorithms for joint tracking of fundamental and harmonic phasors in stressed power systems and microgrids[J].IEEE Transactions on Power Delivery,2014,29(3):1465-1480.
[6]RAY P K,SUBUDHI B.Ensemble-Kalman-filter-based power system harmonic estimation[J].IEEE Transactions on Instrumentation and Measurement,2012,61(12):3216-3224.
[7]BAGHERI A,MARDANEH M,RAJAEI A,et al.Detection of grid voltage fundamental and harmonic components using Kalman filter and generalized averaging method[J].IEEE Transactions on Power Electronics,2016,31(2):1064-1073.
[8]BORKOWSKI J,KANIA D,MROCZKA J.Interpolated-DFT-based fast and accurate frequency estimation for the control of power[J].IEEE Transactions on industrial Electronics,2014,61(12):7026-7034.
[9]CANDAN.Fine resolution frequency estimation from three DFT samples:case of windowed data[J].Signal Processing,2015,114:245-250.
[10]BELEGA D,DALLET D.Estimation of the multifrequency signal parameters by interpolated DFT method with maximum sidelobe decay[C]∥4th IEEE Workshop on.Intelligent Data Acquisition and Advanced Computing Systems:Technology and Applications.Dortmund:IEEE,2007:294-299.
[11]刘昊,王猛,王昌吉,等.一种实时精确估计电力谐波和间谐波参数的方法[J].电力系统自动化,2014,38(20):90-95.LIU Hao,WANG Meng,WANG Chang-ji,et al.A real-time accurate estimating method for electric power harmonics and inter-harmonics[J].Automation of Electric Power Systems,2014,38(20):90-95.
[12]贾清泉,杨晓雯,宋知用.一种电力系统谐波信号的加窗频移算法[J].中国电机工程学报,2014,34(10):1631-1640.JIA Qing-quan,YANG Xiao-wen,SONG Zhi-yong.A window frequency shift algorithm for power system harmonic analysis[J].Proceedings of the CSEE,2014,34(10):1631-1640.
[13]刘亚梅,惠锦,杨洪耕.电力系统谐波分析的多层DFT插值校正法[J].中国电机工程学报,2012,32(25):182-188.LIU Ya-mei,HUI Jin,YANG Hong-geng.Multilayer DFT interpolation correction approach for power system harmonic analysis[C]∥Proceedings of the CSEE.2012,32(25):182-188.
[14]蒋春芳,刘敏.基于双插值FFT算法的间谐波分析[J].电力系统保护与控制,2010(3):11-14.JIANG Chun-fang,LIU Min.Inter-harmonics analysis based on double interpolation FFT algorithm[J].Power System Protection and Control,2010(3):11-14.
[15]惠锦,杨洪耕.用于谐波/间谐波分析的奇数频点插值修正法[J].中国电机工程学报,2010,30(16):67-72.HUI Jin,YANG Hong-geng.An approach for harmonic/inter-harmonic analysis based on the odd points interpolation correction[J].Proceedings of the CSEE,2010,30(16):67-72.
[16]马晓春,刘旭东.一种电力谐波分析新算法[J].中国电机工程学报,2013,33(18):6-8.MA Xiao-chun,LIU Xu-dong.A novel algorithm for electric power system harmonic analysis[J].Proceedings of the CSEE,2013,33(18):6-8.
[17]曹健,林涛,徐遐龄,等.基于最小二乘法和时频原子变换的谐波/间谐波测量算法[J].电工技术学报,2011,26(10):1-7.CAO Jian,LIN Tao,XU Xia-ling,et al.Monitoring of power system harmonic/inter-harmonics based on least squares algorithm and time frequency transform[J].Transactions of China Electrotechnical Society,2011,26(10):1-7.
[18]贾清泉,姚蕊,王宁,等.一种应用原子分解和加窗频移算法分析频率相近谐波/间谐波的方法[J].中国电机工程学报,2014,34(27):4605-4612.JIA Qing-quan,YAO Rui,WANG Ning,et al.An approach to detect harmonics/inter-harmonics with closing frequencies using atomic decomposition and windowed frequency shifting algorithm[J].Proceedings of the CSEE,2014,34(27):4605-4612.
[19]周菁菁,王彭,赵春宇,等.基于频率搜索的间谐波闪变检测方法[J].电力系统自动化,2014,38(7):60-65.ZHOU Jing-jing,WANG Peng,ZHAO Chun-yu,et al.A detection method of interharmonics-caused flicker based on frequency-searching[J].Automation of Electric Power Systems,2014,38(7):60-65.
[20]BELEGA D,PETRI D.Frequency estimation by two-or three-point interpolated Fourier algorithms based on cosine windows[J].Signal Processing,2015,117:115-125.
[21]DIAO R,MENG Q.An interpolation algorithm for discrete Fourier transforms of weighted damped sinusoidal signals[J].IEEE Transactions on Instrumentation and Measurement,2014,63(6):1505-1513.
[22]张贤达.矩阵分析与应用[M].北京:清华大学出版社,2004:17.