Jump to ContentJump to Main Navigation
Show Summary Details

Metrology and Measurement Systems

The Journal of Committee on Metrology and Scientific Instrumentation of Polish Academy of Sciences

4 Issues per year


IMPACT FACTOR increased in 2015: 1.140

SCImago Journal Rank (SJR) 2015: 0.554
Source Normalized Impact per Paper (SNIP) 2015: 1.363
Impact per Publication (IPP) 2015: 1.260

Open Access
Online
ISSN
2300-1941
See all formats and pricing

Frequency and Damping Estimation Methods - An Overview

Tomasz Zieliński
  • Department of Telecommunications, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Kraków, Poland
/ Krzysztof Duda
  • Department of Measurement and Instrumentation, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Kraków, Poland
Published Online: 2011-12-25 | DOI: https://doi.org/10.2478/v10178-011-0051-y

Frequency and Damping Estimation Methods - An Overview

This overview paper presents and compares different methods traditionally used for estimating damped sinusoid parameters. Firstly, direct nonlinear least squares fitting the signal model in the time and frequency domains are described. Next, possible applications of the Hilbert transform for signal demodulation are presented. Then, a wide range of autoregressive modelling methods, valid for damped sinusoids, are discussed, in which frequency and damping are estimated from calculated signal linear self-prediction coefficients. These methods aim at solving, directly or using least squares, a matrix linear equation in which signal or its autocorrelation function samples are used. The Prony, Steiglitz-McBride, Kumaresan-Tufts, Total Least Squares, Matrix Pencil, Yule-Walker and Pisarenko methods are taken into account. Finally, the interpolated discrete Fourier transform is presented with examples of Bertocco, Yoshida, and Agrež algorithms. The Matlab codes of all the discussed methods are given. The second part of the paper presents simulation results, compared with the Cramér-Rao lower bound and commented. All tested methods are compared with respect to their accuracy (systematic errors), noise robustness, required signal length, and computational complexity.

Keywords: damped sinusoids; frequency estimation; damping estimation; linear prediction; subspace methods; interpolated DFT

  • http://en.wikipedia.org/wiki/Oscillation#Electrical

  • Sedlacek, M., Stoudek, Z. (2011). Active power measurements - an overview and comparison of DSP algorithms by noncoherent sampling. Metrol. Meas. Syst., 18(2), 173-184.

  • Ramos, P. M., Janeiro, F. M., Radil, T. (2010). Comparative analysis of three algorithms for two-channel common frequency sinewave parameter estimation: ellipse fit, seven parameter sine fit and spectral sinc fit. Metrol. Meas. Syst., 17(2), 255-270.

  • Source codes of all Matlab programs tested in this paper: http://kt.agh.edu.pl/~tzielin/papers/M&MS-2011/

  • Pintelon, R., Schoukens, J. (2001). System Identification: A Frequency Domain Approach. IEEE Press, Piscataway (USA).

  • Magalas, L. B. (2006). Determination of the logarithmic decrement in mechanical spectroscopy. Solid State Phenomena, 115, 7-14.

  • Duda, K., Zieliński, T. P., Magalas, L. B., Majewski, M. (2011). DFT-based Estimation of Damped Oscillation Parameters in Low-frequency Mechanical Spectroscopy. IEEE Trans. Instrum. Meas., 60(11), 3608-3618.

  • Radil, T., Ramos, P. M., Serra, A. C. (2009). New Spectrum Leakage Correction Algorithm for Frequency Estimation of Power System Signals. IEEE Trans. Instrum. Meas., 58(5), 1670-1679.

  • Duda, K. (2011). Fourier-Based Estimation of Line Spectra. AGH Publishing, Kraków. (in Polish)

  • Andria, G., Savino, M., Trotta, A. (1989). Windows and interpolation algorithms to improve electrical measurement accuracy. IEEE Trans. Instrum. Meas., 38, 856-863.

  • Duda, K. (2011). DFT Interpolation Algorithm for Kaiser-Bessel and Dolph-Chebyshev Windows. IEEE Trans. Instrum. Meas., 60(3), 784-790.

  • Oppenheim, A. V., Willsky, A. S., Nawab, S. H. (1997). Signals & Systems. Prentice Hall.

  • Oppenheim, A. V., Schafer, R. W., Buck, J. R. (1999). Discrete-Time Signal Processing. Prentice-Hall.

  • Poularikas, A. D., Seely, S. (1985). Signals and Systems. Boston, PWS Engineering.

  • Agneni, A., Balis-Crema, L. (Jan., 1989). Damping measurements from truncated signals via Hilbert transform. Mechanical Systems and Signal Processing, 3(1), 1-13.

  • Laila, D. S., Larsson, M., Pal, B. C., Korba, P. (2009). Nonlinear damping computation and envelope detection using Hilbert transform and its application to power systems wide area monitoring. IEEE Power & Energy Society General Meeting, 1-7.

  • Magalas, L. B., Malinowski, T. (2003). Measurement Techniques for Logarithmic Decrement. Solid State Phenomena, 89, 247-258.

  • Messina, A. R. et al. (2006). Interpretation and Visualization of Wide-Area PMU Measurements Using Hilbert Analysis. IEEE Trans. Power Systems, 21(4), 1763-1771. [Crossref]

  • Shin, K., Hammond, J. K. (2007). Fundamentals of Signal Processing for Sound and Vibration. Wiley.

  • Zieliński, T. P. (2005, 2007, 2009). Digital Signal Processing: From Theory To Applications, WKL.

  • Proakis, J. G., Manolakis, D. G. (1992). Digital Signal Processing: Principles, Algorithms, Applications. Macmillan.

  • Golub, G. G., Van Loan, Ch.F. (1996). Matrix Computation.3rd ed. Johns Hopkins Univ. Press.

  • Steiglitz, K., McBride, L. E. (1965). A technique for identification of linear systems. IEEE Trans. Automatic Control, 10, 461-464.

  • McClellan, J. H., Lee, D. (1991). Exact Equivalence of the Steiglitz-McBride Iteration and IQLM. IEEE Trans. Signal Processing, 39(2), 509-512. [Crossref]

  • Moon, T. K., Stirling, W. C. (1999). Mathematical Methods and Algorithms for Signal Processing. Prentice Hall.

  • Kumaresan, R., Tufts, D. W. (1982). Estimating the parameters of exponentially damped sinusoids and pole-zero modeling in noise. IEEE Trans. Acoust. Speech Signal Processing, ASSP-30, 837-840.

  • Van Beek, J. D. (2007). Software from http://matnmr.sourceforge.net/. matNMR: a flexible toolbox for processing, analyzing and visualizing magnetic resonance data in Matlab. J. Magn. Res, 187, 19-26.

  • Rahman, M. A., Yu, K. B. (1987). Total least squares approach for frequency estimation using linear prediction. IEEE Trans. Acoustics. Speech Signal Processing, 35(10), 1440-1454. [Crossref]

  • Hua, Y., Sarkar, T. K. (1990). Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoid in noise. IEEE Trans. Acoustics. Speech Signal Processing, 38(5), 814-824. [Crossref]

  • Sarkar, T. K., Pereira, O. (1995). Using the Matrix Pencil Method to Estimate the Parameters of a Sum of Complex Exponentials. IEEE Antennas and Propagation Magazine, 37(1), 48-55. [Crossref]

  • Li, Y., Ray Liu, K. J., Razavilar, J. (1997). A Parameter estimation Scheme for Damped Sinusoidal Signals Based on Low-Rank Hankel Approximation. IEEE Trans. Signal Process, 45(2), 481-486. [Crossref]

  • Razavilar, J., Li, Y., Ray Liu, K. J. (1998). A structured low-rank matrix pencil for spectral estimation and system identification. Signal Processing (Elsevier), 65, 363-372. [Crossref]

  • Ruiz, D. P., Carrion, M. C., Gallego, A., Medouri, A. (1995). Parameter Estimation of Exponentially Damped Sinusoids Using a Higher Order Correlation-Based Approach. IEEE Trans. on Signal Processing, 43(11), 2665-2677. [Crossref]

  • Allu, G. K. (2003). Estimating the parameters of exponentially damped sinusoids in noise. University of Rhode Island, Technical Report http://www.ele.uri.edu/~gopi/report.pdf

  • Kay, S. M. (1988). Modern Spectral Estimation: Theory and Applications, Prentice Hall.

  • Kay, S. M., Marple, S. L. (1981). Spectrum Analysis - A Modern Perspective. Proc. of IEEE, 69(11), 1380-1419. [Crossref]

  • Marple, S. L. (1987). Digital Spectral Analysis with Applications. Englewood Cliffs, Prentice Hall.

  • Hayes, M. H. (1996). Statistical Digital Signal Processing and Modeling. New York, Wiley.

  • Lobos, T., Leonowicz, Z., Rezmer, J., Schegner, P. (2006). High-Resolution Spectrum-Estimation Methods for Signal Analysis in Power Systems. IEEE Trans. Instrum. Meas., 55(1), 219-225.

  • Cooley, J. W., Tukey, J. W. (1965). An Algorithm for the Machine Computation of Complex Fourier Series. Mathematics of Computation, 19, 297-301. [Crossref]

  • Jacobsen, E., Lyons, R. (2003). The sliding DFT. IEEE Signal Processing Mag., 20(2), 74-80.

  • Duda, K. (2010). Accurate, Guaranteed-Stable, Sliding DFT. IEEE Signal Processing Mag., 124-127.

  • Borkowski, D., Bien, A. (2009). Improvement of accuracy of power system frequency analysis by coherent resampling. IEEE Trans. Power Delivery, 24(2), 1004-1013. [Crossref]

  • Harris, F. J. (1978). On the use of windows for harmonic analysis with the discrete Fourier transform. Proc. IEEE, 66, 51-83. [Crossref]

  • Bertocco, M., Offeli, C., Petri, D. (1994). Analysis of damped sinusoidal signals via a frequency-domain interpolation algorithm. IEEE Trans. Instrum. Meas., 43(2), 245-250.

  • Yoshida, Y. I., Sugai, T., Tani, S., Motegi, M., Minamida, K., Hayakawa, H. (1981). Automation of internal friction measurement apparatus of inverted torsion pendulum type. J. Phys. E: Sci. Instrum., 14, 1201-1206. [Crossref]

  • Jain, V. K., Collins, W. L., Davis, D. C. (1979). High-Accuracy Analog Measurements via Interpolated FFT. IEEE Trans. Instrum. Meas., Im-28(2), 113-122.

  • Grandke, T. (1983). Interpolation Algorithms for Discrete Fourier Transforms of Weighted Signals. IEEE Trans. Instrum. Meas., Im-32(2), 350-355.

  • Agrež, D. (2002). Weighted Multipoint Interpolated DFT to Improve Amplitude Estimation of Multifrequency Signal. IEEE Trans. Instrum. Meas., 51, 287-292.

  • Offelli, C., Petri, D. (1990). Interpolation Techniques for Real-Time Multifrequency Waveform Analysis. IEEE Trans. Instrum. Meas., 39(1), 106-111.

  • Borkowski, J. (2000). LIDFT—The DFT Linear Interpolation Method. IEEE Trans. Instrum. Meas., 49(4), 741-745.

  • Borkowski, J., Mroczka, J. (2002). Metrological Analysis of the LIDFT Method. IEEE Trans. Instrum. Meas., 51(1), 67-71.

  • Agrež, D. (2009). A frequency domain procedure for estimation of the exponentially damped sinusoids. International Instrumentation and Measurement Technology Conference.

  • Kay, S. M. (1993). Fundamentals of Statistical Signal Processing: Estimation Theory. Englewood Cliffs, NJ: Prentice-Hall.

  • Yao, Y., Pandit, S. M. (1995). Cramér-Rao lower bounds for a damped sinusoidal process. IEEE Trans. Signal Process., 43(4), 878-885.

  • Duda, K. (2011). Tracking performance of digital sinusoidal signals using adaptive filters, Electrical review, (1), 140-143. (in Polish)

About the article


Published Online: 2011-12-25

Published in Print: 2011-01-01


Citation Information: Metrology and Measurement Systems, ISSN (Print) 0860-8229, DOI: https://doi.org/10.2478/v10178-011-0051-y. Export Citation

This content is open access.

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[2]
Xin Liu, Yongfeng Ren, Chengqun Chu, and Wei Fang
Metrology and Measurement Systems, 2015, Volume 22, Number 3
[3]
Jiufei Luo, Zhijiang Xie, and Ming Xie
Digital Signal Processing, 2015, Volume 41, Page 118

Comments (0)

Please log in or register to comment.
Log in