Systematic errors of the LIDFT method: analytical form and verification by a Monte Carlo method

• Autorzy: Borkowski J.
• Języki publikacji: PL

Abstrakty

EN

This paper derives analytical formulas for the systematic errors of the linear interpolated DFT (LIDFT) method when used to estimating multifrequency signal parameters and verifies this analysis using Monte-Carlo simulations. The analysis is performed on the version of the LIDFT method based on optimal approximation of the unit circle by a polygon using a pair of windows. The analytical formulas derived here take the systematic errors in the estimation of amplitude and frequency of component oscillations in the multifrequency signal as the sum of basic errors and the errors caused by each of the component oscillations. Additional formulas are also included to analyze particular quantities such as a signal consisting of two complex oscillations, and the analyses are verified using Monte-Carlo simulations.

Słowa kluczowe

EN

LIDFT; multifrequency signal; interpolated DFT; spectrum estimation; zero padding; unit circle; approximation by polygon

• Wydawca: Komitet Metrologii i Aparatury Naukowej PAN
• Czasopismo: Metrology and Measurement Systems
• Rocznik: 2012
• Tom: Vol. 19, nr 4
• Strony: 673--684
• Opis fizyczny: Bibliogr. 46 poz., wykr., wzory

Twórcy

autor

Borkowski J.

• Wroclaw University of Technology, Chair of Electronic and Photonic Metrology, Bolesława Prusa 53/55, 50-317 Wrocław, Poland,

Bibliografia

• [1] Marple, S.L. (1987). Digital Spectral Analysis with Applications, Prentice-Hall.
• [2] Kay, S.M. (1988). Modern Spectral Estimation: Theory and Application, Englewood Cliffs, Prentice-Hall.
• [3] Scharf, L.L. (1991). Statistical Signal Processing: Detection, Estimation and Time Series Analysis, Addison-Wesley.
• [4] Mitra, S.K., Kaiser, J.F. (1993). Handbook for Digital Signal Processing, Wiley.
• [5] Szmajda, M., Górecki, K., Mroczka, J. (2010). Gabor transform, spwvd, gabor-wigner transform and wavelet transform - tools for power quality monitoring. Metrol. Meas. Syst., 17(3), 383-396.
• [6] Mroczka, J., Szczuczyński, D. (2009). Inverse problems formulated in terms of first-kind Fredholm integral equations in indirect measurements. Metrol. Meas. Syst., 16(3), 333-357.
• [7] Zygarlicki, J., Zygarlicka, M., Mroczka, J., Latawiec, K.J. (2010). A reduced Prony's method in power-quality analysis-parameters selection. IEEE Trans. Power Del., 25(2), 979-986.
• [8] Zygarlicki, J., Mroczka, J. (2011). Short time algorithm of power waveforms fundamental harmonic estimation with use of prony’s methods. Metrol. Meas. Syst., 18(3), 371-378.
• [9] Zivanovic, M., Carlosena, A. (2001). Nonparametric Spectrum Interpolation Methods: A Comparative Study. IEEE Trans. Instrum. Meas., 50(5), 1127-1132.
• [10] Zivanovic, M., Carlosena, A. (2002). Extending the limits of resolution for narrow-band harmonic and modal analysis: a non-parametric approach. Measurement Science and Technology, 13(12), 2082-2089.
• [11] Aiello, M., Cataliotti, A., Nuccio, S. (2005). A Chirp-Z Transform-Based Synchronizer for Power System Measurements. IEEE Trans. Instrum. Meas., 54(3), 1025-1032.
• [12] Sarkar, I., Fam, A.T. (2006). The interlaced chirp Z transform. Signal Processing, 86, 2221-2232.
• [13] Duda, K., Borkowski, D., Bień, A. (2009). Computation of the network harmonic impedance with Chirp-Z transform. Metrol. Meas. Syst., 16(2), 299-312.
• [14] Makur, A., Mitra, S.K. (2001). Warped Discrete-Fourier Transform: Theory and Applications. IEEE Transactions on Circuits and Systems - I: Fundamental Theory and Applications, 48(9), 1086-1093.
• [15] Franz, S.F., Mitra, S.K., Doblinger, G. (2003). Frequency estimation using warped discrete Fourier transform. Signal Processing, 83, 1661-1671.
• [16] Venkataramanan, R., Prabhu, K.M.M. (2006). Estimation of frequency offset using warped discrete-Fourier transform. Signal Processing, 86, 250-256.
• [17] Rife, D.C., Vincent, G.A. (1970). Use of the Discrete Fourier Transform in the Measurement of Frequencies and Levels of Tones. Bell System Technical Journal, 49, 197-228.
• [18] Kamm, G.N. (1978). Computer Fourier-transform techniques for precise spectrum measurements of oscillatory data with application to the de Haas-van Alphen effect. J. of App. Phys., 49(12), 5951-5970.
• [19] Jain, V.K., Collins, W.L., Davis, D.C. (1979). High-Accuracy Analog Measurements via Interpolated FFT. IEEE Trans. Instrum. Meas., 28(2), 113-121.
• [20] Grandke, T. (1983). Interpolation Algorithms for Discrete Fourier Transforms of Weighted Signals. IEEE Transactions on Instrumentation and Measurement, 32, 350-355.
• [21] Andria, G., Savino, M., Trotta, A. (1989). Windows and Interpolation Algorithms to Improve Electrical Measurement Accuracy. IEEE Trans. Instrum. Meas., 38(4), 856-863.
• [22] Offelli, C., Petri, D. (1990). Interpolation Techniques for Real-Time Multifrequency waveform analysis. IEEE Trans. Instrum. Meas., 39(1), 106-111.
• [23] Schoukens, J., Pintelon, R., Van Hamme, H. (1992). The Interpolated Fast Fourier Transform: A Comparative Study. IEEE Trans. Instrum. Meas., 41(2), 226-232.
• [24] Quinn, B.G. (1994). Estimating Frequency by Interpolation Using Fourier Coefficients. IEEE Transactions on Signal Processing, 42(5), 1264-1268.
• [25] Quinn, B.G. (1997). Estimating of Frequency, Amplitude, and Phase from the DFT of a Time Series. IEEE Transactions on Signal Processing, 45(3), 814-817.
• [26] Macleod, M.D. (1998). Fast Nearly ML Estimation of the Parameters of Real or Complex Single Tones or Resolved Multiple Tones. IEEE Transactions on Signal Processing, 46(1), 141-148.
• [27] Sedlacek, M., Titera, M. (1998). Interpolations in frequency and time domains used in FFT spectrum analysis. Measurement, 23, 185-193.
• [28] Santamaria, I., Pantaleon, C., Ibanez, J. (2000). A Comparative Study of High-Accuracy Frequency Estimation Methods. Mechanical Systems and Signal Processing, 14(5), 819-834.
• [29] Borkowski, J. (2000). LIDFT - the DFT linear interpolation method. IEEE Trans. Instrum. Meas., 49(4), 741-745.
• [30] Borkowski, J., Mroczka, J. (2000). Application of the discrete Fourier transform linear interpolation method in the measurement of volume scattering function at small angle. Optical Eng., 39(6), 1576-1586.
• [31] Borkowski, J., Mroczka, J. (2002). Metrological analysis of the LIDFT method. IEEE Trans. Instrum. Meas., 51(1), 67-71.
• [32] Agrež, D. (2002). Weighted Multipoint Interpolated DFT to Improve Amplitude Estimation of Multifrequency Signal. IEEE Transactions on Instrumentation and Measurement, 51(2), 287-292.
• [33] Liguori, C., Paolillo, A. (2007). IFFTC-Based Procedure for Hidden Tone Detection. IEEE Trans. Instrum. Meas., 56(1), 133-139.
• [34] Belega, D., Dallet, D. (2008). Frequency estimation via weighted multipoint interpolated DFT. IET Science, Measurement and Technology, 2(1), 1-8.
• [35] Chen, K.F., Li, Y.F. (2008). Combining the Hanning windowed interpolated FFT in both directions. Computer Physics Communications, 178, 924-928.
• [36] Li, Y.F., Chen, K.F. (2008). Eliminating the picket fence effect of the fast Fourier transform. Computer Physics Communications, 178, 486-491.
• [37] Belega, D., Dallet, D. (2009). Multifrequency signal analysis by Interpolated DFT method with maximum sidelobe decay windows. Measurement, 42, 420-426.
• [38] Yang, X.Z., Li, H.Y., Chen, K.F. (2009). Optimally averaging the interpolated fast Fourier transform in both directions. IET Science, Measurement and Technology, 3(2), 137-147.
• [39] Chen, K.F., Jiang, J.T., Crowsen, S. (2009). Against the long-range spectral leakage of the cosine window family. Computer Physics Communications, 180, 904-911.
• [40] Chen, K.F., Mei, S.L. (2010). Composite Interpolated Fast Fourier Transform With the Hanning Window. IEEE Trans. Instrum. Meas., 59(6), 1571-1579.
• [41] Borkowski, J., Mroczka, J. (2010). LIDFT method with classic data windows and zero padding in multifrequency signal analysis. Measurement, 43, 1595-1602.
• [42] Borkowski, J. (2011). Methods of spectrum interpolation and the LIDFT method for estimation of multifrequency signal parameters. Publishing House of Wrocław University of Technology, Wrocław, Poland. (in Polish)
• [43] Duda, K. (2011). DFT Interpolation Algorithm for Kaiser-Bessel and Dolph-Chebyshev Windows. IEEE Trans. Instrum. Meas., 60(3), 784-790.
• [44] Belega, D., Dallet, D., Petri, D. (2012). Statistical description of the sine-wave frequency estimator provided by the interpolated DFT method. Measurement, 45(1), 109-117
• [45] Borkowski, J. (2011). Minimization of maximum errors in universal approximation of the unit circle by a polygon. Metrol. Meas. Syst., 18(3), 391-402.
• [46] Borkowski, J. (2011). Optimization of the unit circle approximation by a polygon. Electrical Review, 87(7), 95-99.