基于阶梯波的周期非正弦电压精密测量——阶梯波研究之四
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摘要
经典傅里叶变换要求采样数据必须布满整个周期。以周期非正弦波分析为例,提出一种利用规则分布的部分采样数据仍可进行傅里叶变换的新方法。以阶梯波为参考的差分测量用于周期非正弦波,在每个阶梯波台阶上丢弃起始和结束部分的采样以克服过渡过程和吉布斯现象后,高频分量的傅里叶变换将产生明显偏差。为克服这种偏差,将傅里叶变换的基函数(三角函数)作同样的离散,组成相应的影响矩阵,其逆矩阵将能恢复准确结果。由于影响矩阵的阶次是最高谐波次数的2倍,一般情况下求逆十分困难,并需要较大存储空间。提出了使影响矩阵成为稀疏因而简化运算的条件。进一步讨论了基于阶梯波的差分采样所需要的限幅条件在周期非正弦电压下的表现及其应对方法。模拟和演示实验结果表明,该方法可极大减少计算量,仍具有较高准确度。最后探讨了该方法扩展到一般采样缺失情况下的可能性。
A novel method using the regular distributed partial data for the periodic signal analysis is proposed.Differential measurement of the non-sinusoidal voltage signal based on a staircase waveform is discussed,where the first and last parts of each step are discarded to overcome the transient effect and Gibbs phenomenon. In Fourier transformation of higher frequency components the significant deviations will be caused. When the base functions are divided with same mode,the relevant influence matrix will be derived; its inverse matrix can then recover the original Fourier coefficients.The key point here is to simplify calculation of the inverse matrix. The condition of this simplification is proposed in which the influence matrix becomes sparse. The limiting amplitude condition required in the differential sampling based on the staircase waveform is investigated. Simulations and demonstration experiments show an acceptable workload and a very high accuracy. This method can be applied to spectrum analysis when partial samples are regularly lost.
A novel method using the regular distributed partial data for the periodic signal analysis is proposed.Differential measurement of the non-sinusoidal voltage signal based on a staircase waveform is discussed,where the first and last parts of each step are discarded to overcome the transient effect and Gibbs phenomenon. In Fourier transformation of higher frequency components the significant deviations will be caused. When the base functions are divided with same mode,the relevant influence matrix will be derived; its inverse matrix can then recover the original Fourier coefficients.The key point here is to simplify calculation of the inverse matrix. The condition of this simplification is proposed in which the influence matrix becomes sparse. The limiting amplitude condition required in the differential sampling based on the staircase waveform is investigated. Simulations and demonstration experiments show an acceptable workload and a very high accuracy. This method can be applied to spectrum analysis when partial samples are regularly lost.
引文
[1]陆祖良,杨雁,黄璐,等.片段采样与阶梯波测量---阶梯波研究之二[J].计量学报,2019,40(1):31-39.Lu Z L,Yang Y,Huang L,et al.Piecewise sampling and measurement of staircase waveform[J].Acta Metrologia Sinica,2019,40(1):31-39.
[2]陆祖良,杨雁,黄璐,等.正弦波电压差分测量与多周期策略---阶梯波研究之三[J].计量学报,2019,40(2):319-328.Lu Z L,Yang Y,Huang L,et al.Differential Measurement of Sine Wave Voltage and Multi-period Strategy[J].Acta Metrologia Sinica,2019,40(2):319-328.
[3]Behr R,Palafox L,Ramm G,et al.Direct comparison of Josephson waveforms using an AC quantum voltmeter[J].IEEE Trans Instrum Meas,2007,56(2):235-238.
[4]Rufenacht A,Burroughs C J,Benz S P.Precision sampling measurements using AC programmable Josephson voltage standards[J].Rev Sci Instrum,2008,79(4):044704-1-044704-9.
[5]Rufenacht A,Burroughs C J,Dresselhaus P D,et al.Differential sampling measurement of a 7 V RMS sine wave with a programmable Josephson voltage standard[J].IEEE Trans Instrum Meas,2013,62(6):1587-1593.
[6]Schubert M,Starkloff M,Lee J,et al.An AC Josephson voltage standard up to the kilohertz range tested in a calibration laboratory[J].IEEE Trans Instrum Meas,2015,64(3):1620-1626.
[7]Arseneau R,Filipski P S,Zelle J J.An improved threephase digital recorder system for calibrating power instrumentation[J].IEEE Trans Instrum Meas,1997,46:399-402.
[8]Svensson S.Verification of calibration system for power quality instruments[J].IEEE Trans Instrum Meas,1998,47(5):1391-1394.
[9]Ihlenfeld W G K,Mohns E,Bachmair H,et al.Evaluation of the synchronous generation and sampling technique[J].IEEE Trans Instrum Meas,2003,52(2):371-374.
[10]Lu Z L,Wang L,Li M,et al.Harmonic power standard at NIM and its compensation algorithm[J].IEEE Trans Instrum Meas,2010,59(1):180-187.
[11]Marvasti F.Nonuniform Sampling,Theory and Practice[M].New York:Springer US,2001.
[12]Yen J L.On non-uniform sampling of bandwidth limited signals[J].IRE Trans Circuit Theory,1956,3:251-257.
[13]Kida T,Akagawa K.Sampling theorems using nonuniform sample points and interpolation functions with band-limited continuous spectra[J].Electron Commun Jpn,Part III:Fundam Electron Sci,1989,72(6):59-70.
[14]Strohmer T,Tanne J.Fast reconstruction algorithms for periodic nonuniform sampling with applications to time-interleaved ADCs[J].2007 IEEE Int Conf Acoustics,Speech and Signal Processing(ICASSP'07),2007,3(1):III-881-III-884.
[15]Adcock B,Gataric M,Hansen A C.Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples[J].Applied&Computational Harmonic Analysis,2017,42(3):508-535.
[16]Dunlop J,Phillips V J.Signal recovery from repetitive non-uniform sampling patterns[J].Radio Electron Eng,1974,44(9):491-503.
[17]Sommen P,Janse K.On the relationship between uniform and recurrent nonuniform discrete-time sampling schemes[J].IEEE Trans Signal Process,2008,56(10):5147-5156.
[18]Fjllbrant T.Method of data reduction of sampled speech signals by using nonuniform sampling and a timevariable digital filter[J].Electronics Letters,1977,13(11):334-335.
[19]Bonacci D,Lacaze B.Lowpass/bandpass signal reconstruction and digital filtering from nonuniform samples[C]//2015 IEEE Int Conf Acoustics,Speech and Signal Processing(ICASSP),3626-3630,2015.
[20]Razafinjatovo H N.Iterative reconstructions in irregular sampling with derivatives[J].J Fourier Anal Appl,2001,1(3):281-295.
[21]陆祖良,杨雁,黄璐,等.阶梯波性质再讨论---阶梯波研究之一[J].计量学报,2018,39(6):759-767.Lu Z L,Yang Y,Huang L,et al.Further discussion on characteristics of staircase waveform[J].Acta Metrologia Sinica,2018,39(6):759-767.
[22]Landau H J.Necessary density conditions for sampling and interpolation of certain entire functions[J].Acta Math,1967,117(7):37-52.
[23]Lu Z L.Discussion on new properties of staircase waveform through connection to an infinite series[J].IET Sci Meas Technol,2015,9(1):63-72.
[24]中国计量科学研究院.一种准确测量非正弦电压信号的方法及其系统:ZL 2014 1 0401895.X[P]2014.
[25]中国计量科学研究院.部分采样数据规则性缺失时的信号频谱分析方法及系统:2017109009804[P].2017.
[2]陆祖良,杨雁,黄璐,等.正弦波电压差分测量与多周期策略---阶梯波研究之三[J].计量学报,2019,40(2):319-328.Lu Z L,Yang Y,Huang L,et al.Differential Measurement of Sine Wave Voltage and Multi-period Strategy[J].Acta Metrologia Sinica,2019,40(2):319-328.
[3]Behr R,Palafox L,Ramm G,et al.Direct comparison of Josephson waveforms using an AC quantum voltmeter[J].IEEE Trans Instrum Meas,2007,56(2):235-238.
[4]Rufenacht A,Burroughs C J,Benz S P.Precision sampling measurements using AC programmable Josephson voltage standards[J].Rev Sci Instrum,2008,79(4):044704-1-044704-9.
[5]Rufenacht A,Burroughs C J,Dresselhaus P D,et al.Differential sampling measurement of a 7 V RMS sine wave with a programmable Josephson voltage standard[J].IEEE Trans Instrum Meas,2013,62(6):1587-1593.
[6]Schubert M,Starkloff M,Lee J,et al.An AC Josephson voltage standard up to the kilohertz range tested in a calibration laboratory[J].IEEE Trans Instrum Meas,2015,64(3):1620-1626.
[7]Arseneau R,Filipski P S,Zelle J J.An improved threephase digital recorder system for calibrating power instrumentation[J].IEEE Trans Instrum Meas,1997,46:399-402.
[8]Svensson S.Verification of calibration system for power quality instruments[J].IEEE Trans Instrum Meas,1998,47(5):1391-1394.
[9]Ihlenfeld W G K,Mohns E,Bachmair H,et al.Evaluation of the synchronous generation and sampling technique[J].IEEE Trans Instrum Meas,2003,52(2):371-374.
[10]Lu Z L,Wang L,Li M,et al.Harmonic power standard at NIM and its compensation algorithm[J].IEEE Trans Instrum Meas,2010,59(1):180-187.
[11]Marvasti F.Nonuniform Sampling,Theory and Practice[M].New York:Springer US,2001.
[12]Yen J L.On non-uniform sampling of bandwidth limited signals[J].IRE Trans Circuit Theory,1956,3:251-257.
[13]Kida T,Akagawa K.Sampling theorems using nonuniform sample points and interpolation functions with band-limited continuous spectra[J].Electron Commun Jpn,Part III:Fundam Electron Sci,1989,72(6):59-70.
[14]Strohmer T,Tanne J.Fast reconstruction algorithms for periodic nonuniform sampling with applications to time-interleaved ADCs[J].2007 IEEE Int Conf Acoustics,Speech and Signal Processing(ICASSP'07),2007,3(1):III-881-III-884.
[15]Adcock B,Gataric M,Hansen A C.Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples[J].Applied&Computational Harmonic Analysis,2017,42(3):508-535.
[16]Dunlop J,Phillips V J.Signal recovery from repetitive non-uniform sampling patterns[J].Radio Electron Eng,1974,44(9):491-503.
[17]Sommen P,Janse K.On the relationship between uniform and recurrent nonuniform discrete-time sampling schemes[J].IEEE Trans Signal Process,2008,56(10):5147-5156.
[18]Fjllbrant T.Method of data reduction of sampled speech signals by using nonuniform sampling and a timevariable digital filter[J].Electronics Letters,1977,13(11):334-335.
[19]Bonacci D,Lacaze B.Lowpass/bandpass signal reconstruction and digital filtering from nonuniform samples[C]//2015 IEEE Int Conf Acoustics,Speech and Signal Processing(ICASSP),3626-3630,2015.
[20]Razafinjatovo H N.Iterative reconstructions in irregular sampling with derivatives[J].J Fourier Anal Appl,2001,1(3):281-295.
[21]陆祖良,杨雁,黄璐,等.阶梯波性质再讨论---阶梯波研究之一[J].计量学报,2018,39(6):759-767.Lu Z L,Yang Y,Huang L,et al.Further discussion on characteristics of staircase waveform[J].Acta Metrologia Sinica,2018,39(6):759-767.
[22]Landau H J.Necessary density conditions for sampling and interpolation of certain entire functions[J].Acta Math,1967,117(7):37-52.
[23]Lu Z L.Discussion on new properties of staircase waveform through connection to an infinite series[J].IET Sci Meas Technol,2015,9(1):63-72.
[24]中国计量科学研究院.一种准确测量非正弦电压信号的方法及其系统:ZL 2014 1 0401895.X[P]2014.
[25]中国计量科学研究院.部分采样数据规则性缺失时的信号频谱分析方法及系统:2017109009804[P].2017.