The Causes of Leakage Deviations from Sinc Function in DFT and Ways to Minimize Them
Keywords:aliasing, DFT, frequency estimation, leakage
The article describes new model for a solution to the instability problem of DFT spectra leakage for the incoherent real sinusoidal signal, which is a fundamental problem, for example when solving the most accurate estimation of its frequency in various types of measurement practical tasks. It shows that the main cause of leakage deviations from the sync function is aliasing, which, together with an undefined value of the input signal phase, causes a seemingly random deviation of leakage. This paper shows it by a new form of DFT spectrum expression. It consists mainly of the spectrum of the rectangular signal in the form of the function sinc, modulated to the frequency of the tested signal in baseband. To do this, analogous spectra from other neighbouring bands formed by the sampling effect are added as aliasing. In doing so, the effect of adding or subtracting the phase of the test signal is reflected, depending on the signal parameters. Based on the analysis of these effects, some ways to minimize or correct them are shown. On the other hand, the paper shows a simpler and substantially lesser effect of aliasing in the DFT spectrum for a complex sinusoidal signal.
OPPENHEIM, A.V. and SCHAFER, R.V. Discrete-Time Signal Processing. 3rd ed. London: Pearson, 2009. 1144 p. ISBN 978-0-13-198842-2.
LYONS, R.G. Understanding Digital Signal Processing. 2nd ed. New Jersey: Prentice Hall, 2004. 688 p. ISBN 978-0-13-108989-1.
SMITH, S.W. The Scientist and Engineer’s Guide to Digital Signal Processing. 2nd ed. San Diego: California Technical Publishing, 1999. 664 p. ISBN 978-0-96-601767-0.
ROBERTS, M.J. Signals and Systems: Analysis Using Transform Methods and MATLAB®. New York: McGraw Hill, 2003. 1072 p. ISBN 978-0-07-249942-1.
ZIELIŃSKI, T.P. and DUDA, K. Frequency and Damping Estimation Methods – an Overview. Metrology and Measurement Systems, 2011, vol. 18, no. 4, p. 505-528. https://doi.org/10.2478/v10178-011-005-y.
DJUKANOVIĆ, S. An Accurate Method for Frequency Estimation of a Real Sinusoid. IEEE Signal Processing Letters, 2016, vol. 23, no. 7, p. 915-918. https://doi.org/10.1109/LSP.2016.2564102.
LUO, J., HOU, S., LI, X., OUYANG, Q. and ZHANG, Y. Generalization of Interpolation DFT Algorithms and Frequency Estimators with high Image Component Interference Rejection. EURASIP Journal on Advances in Signal Processing, 2016, Article number 30. https://doi.org/10.1186/s13634-016-0330-6.
BELEGA, D., PETRI, D. and DALLET, D. Frequency Estimation of a Sinusoidal Signal via a Three-Point Interpolated DFT Method with High Image Component Interference Rejection Capability. Digital Signal Processing, 2014, vol. 24, p. 162-169. https://doi.org/10.1016/j.dsp.2013.09.014.
RADIL, T., RAMOS, P.M. and SERRA, A.C. New Spectrum Leakage Correction Algorithm for Frequency Estimation of Power System Signals. IEEE Transactions on Instrumentation and Measurement, 2009, vol. 58, no. 5, p. 1670-1679. https://doi.org/10.1109/TIM.2009.2014506.
CANDAN, C. A Method for Fine Resolution Frequency Estimation from Three DFT Samples. IEEE Signal Processing Letters, 2011, vol. 18, no. 6, p. 351-354. https://doi.org/10.1109/LSP.2011.2136378
CANDAN, C. Analysis and Further Improvement of Fine Resolution Frequency Estimation Method from Three DFT Samples. IEEE Signal Processing Letters, 2013, vol. 20, no. 9, p. 913-916. https://doi.org/10.1109/LSP.2013.2273616.
CANDAN, C. Fine Resolution Frequency Estimation from Three DFT Samples: Case of Windowed Data. Signal Processing, 2015, vol. 114, p. 245-250. https://doi.org/10.1016/j.sigpro.2015.03.009.
LIAO, J.-R. and LO, S. Analytical Solutions for Frequency Estimators by Interpolation of DFT Coefficients. Signal Processing, 2014, vol. 100, p. 93-100. https://doi.org/10.1016/j.sigpro.2014.01.012.
LIAO, J.-R. and CHEN, C.-M. Phase Correction of Discrete Fourier Transform Coefficients to Reduce Frequency Estimation Bias of Single Tone Complex Sinusoid. Signal Processing, 2014, vol. 94, p. 108-117. https://doi.org/10.1016/j.sigpro.2013.05.021.
XIANG, J., CUI, W. and SHEN, Q. Flexible and Accurate Frequency Estimation for Complex Sinusoid Signal by Interpolation Using DFT Samples. Chinese Journal of Electronics, 2018, vol. 27, no. 1, p. 109-114. https://doi.org/10.1049/cje.2017.09.019.
DUDA, K. and BARCZENTEWICZ, S. Interpolated DFT for sinα(x) Windows. IEEE Transactions on Instrumentation and Measurement, 2014, vol. 63, no. 4, p. 754-760. https://doi.org/10.1109/TIM.2013.2285795.
DUDA, K. DFT Interpolation Algorithm for Kaiser–Bessel and Dolph-Chebyshev Windows. IEEE Transactions on Instrumentation and Measurement, 2011, vol. 60, no. 3, p. 784-790. https://doi.org/10.1109/TIM.2010.2046594.
WERNER, K.J. and GERMAIN, F.G. Sinusoidal Parameter Estimation Using Quadratic Interpolation around Power-Scaled Magnitude Spectrum Peaks. Applied Sciences, 2016, vol. 6, no. 10, p. 1-22. https://doi.org/10.3390/app6100306.
AGREŽ, D. Weighted Multipoint Interpolated DFT to Improve Amplitude Estimation of Multifrequency Signal. IEEE Transactions on Instrumentation and Measurement, 2002, vol. 51, no. 2, p. 287-292. https://doi.org/10.1109/19.997826.
JACOBSEN, E. and KOOTSOOKOS, P. Fast, Accurate Frequency Estimators. IEEE Signal Processing Magazine, 2007, vol. 24, no. 3, p. 123-125. https://doi.org/10.1109/MSP.2007.361611.
CHEN, H., XU, F. an LI, J. A Frequency Estimator for Real Valued Sinusoidal Signals Using Three DFT Samples. In Proceedings of the 2018 International Conference on Radar (RADAR). Brisbane: IEEE, 2018. https://doi.org/10.1109/RADAR.2018.8557341.
How to Cite
Copyright (c) 2020 Advances in Military Technology
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Authors who publish with this journal agree to the following terms:
1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
Users can use, reuse and build upon the material published in the journal for any purpose, even commercially.