1. bookVolume 31 (2021): Issue 4 (December 2021)
    Advanced Machine Learning Techniques in Data Analysis (special section, pp. 549-611), Maciej Kusy, Rafał Scherer, and Adam Krzyżak (Eds.)
Journal Details
First Published
05 Apr 2007
Publication timeframe
4 times per year
access type Open Access

On the statistical analysis of the harmonic signal autocorrelation function

Published Online: 30 Dec 2021
Page range: 729 - 744
Received: 31 May 2021
Accepted: 04 Oct 2021
Journal Details
First Published
05 Apr 2007
Publication timeframe
4 times per year

The article presents new tools for investigating the statistical properties of the harmonic signal autocorrelation function (ACF). These tools enable identification of the ACF estimator errors in measurements in which the triggering of the measurements is non-synchronized. This is important because in many measurement situations the initial phase of the measured signal is random. The developed tools enable testing the ACF estimator of a harmonic signal in the presence of Gaussian noise. These are the formulas on the basis of which the statistical properties of the estimator can be determined, including the bias, the variance and the mean squared error (MSE). For comparison, the article also presents the ACF statistical analysis tools used in the conditions of synchronized measurement triggering, known from the literature. Operation of the new tools is verified by simulation and experimental studies. The conducted research shows that differences between the MSE results obtained with the use of the developed formulas and those attained from simulations and experimental tests are not greater than 1 dB.


Akkucuk, U. (2019). Handbook of Research on Creating Sustainable Value in the Global Economy, IGI Global, Hershey.10.4018/978-1-7998-1196-1 Search in Google Scholar

Al-Qudsi, B., El-Shennawy, M., Joram, N. and Ellinger, F. (2017). Enhanced zero crossing frequency estimation for FMCW radar systems, Proceedings of the 13th Conference on PhD Research in Microelectronics and Electronics (PRIME), Taormina, Italy, pp. 53–56. Search in Google Scholar

Beck, M.S. and Plaskowski, A. (1987). Cross Correlation Flowmeters, Their Design and Application, Adam Hilger, Bristol. Search in Google Scholar

Bendat, J.S. and Piersol, A.G. (2010). Random Data: Analysis and Measurement Procedures, 4th Edition, Wiley, Hoboken.10.1002/9781118032428 Search in Google Scholar

Box, G.E.P., Jenkins, G. and Reinsel, G. (2008). Time Series Analysis: Forecasting and Control, 4th Edition, Wiley, Hoboken.10.1002/9781118619193 Search in Google Scholar

Broersen, P.M.T. (2006). Automatic Autocorrelation and Spectral Analysis, Springer-Verlag, London. Search in Google Scholar

Cao, Y., Wei, G. and Chen, F.-J. (2012). An exact analysis of modified covariance frequency estimation algorithm based on correlation of single-tone, Signal Processing 92(11): 2785–2790.10.1016/j.sigpro.2012.04.022 Search in Google Scholar

Chen, W.-K. (2004). The Electrical Engineering Handbook, Academic Press, Amsterdam. Search in Google Scholar

De Gooijer, J.G. and Anderson, O.D. (1985). Moments of the sampled space-time autocovariance and autocorrelation function, Biometrika 72(3): 689–693.10.1093/biomet/72.3.689 Search in Google Scholar

Elasmi-Ksibi, R., Besbes, H., López-Valcarce, R. and Cherif, S. (2010). Frequency estimation of real-valued single-tone in colored noise using multiple autocorrelation lags, Signal Processing 90(7): 2303–2307.10.1016/j.sigpro.2010.01.025 Search in Google Scholar

Grzegorzewski, P., Hryniewicz, O. and Romaniuk, M. (2020). Flexible resampling for fuzzy data, International Journal of Applied Mathematics and Computer Science 30(2): 281–297, DOI: 10.34768/amcs-2020-0022. Search in Google Scholar

Hague, D.A. and Buck, J.R. (2019). An experimental evaluation of the generalized sinusoidal frequency modulated waveform for active sonar systems, Journal of the Acoustical Society of America 145(6): 3741–3755.10.1121/1.5113581 Search in Google Scholar

Hanus, R. (2019). Time delay estimation of random signals using cross-correlation with Hilbert transform, Measurement 146: 792–799.10.1016/j.measurement.2019.07.014 Search in Google Scholar

Hołyst, R., Poniewierski, A. and Zhang, X. (2017). Analytical form of the autocorrelation function for the fluorescence correlation spectroscopy, Soft Matter 13(6): 1267–1275.10.1039/C6SM02643E Search in Google Scholar

Hussein, H.M., Terra, O., Hussein, H. and Medhat, M. (2020). Collinear versus non-collinear autocorrelation between femtosecond pulses for absolute distance measurement, Measurement 152: 107319.10.1016/j.measurement.2019.107319 Search in Google Scholar

Iqbal, S., Zang, X., Zhu, Y., Saad, H.M.A.A. and Zhao, J. (2015). Nonlinear time-series analysis of different human walking gaits, Proceedings of the 2015 IEEE International Conference on Electro/Information Technology, DeKalb, USA, pp. 25–30. Search in Google Scholar

JCGM (2008). Evaluation of measurement data. guide to the expression of uncertainty in measurement, Technical Report 100, Joint Committee for Guides in Metrology, Sévres. Search in Google Scholar

Kalvani, P.R., Jahangiri, A.R., Shapouri, S., Sari, A. and Jalili, Y.S. (2019). Multimode AFM analysis of aluminum-doped zinc oxide thin films sputtered under various substrate temperatures for optoelectronic applications, Radio Science 132: 106173.10.1016/j.spmi.2019.106173 Search in Google Scholar

Kogan, L. (1998). Correction functions for digital correlators with two and four quantization levels, Superlattices and Microstructures 33(5): 1289–1296.10.1029/98RS02202 Search in Google Scholar

Kowalczyk, A., Hanus, R. and Szlachta, A. (2016). Time delay measurement method using conditional averaging of the delayed signal module, Przegląd Elektrotechniczny 92(9): 279–282.10.15199/48.2016.09.66 Search in Google Scholar

Krajewski, M. (2018). Constructing an uncertainty budget for voltage rms measurement with a sampling voltmeter, Metrologia 55(1): 95–105.10.1088/1681-7575/aaa178 Search in Google Scholar

Lal-Jadziak, J. and Sienkowski, S. (2008). Models of bias of mean square value digital estimator for selected deterministic and random signals, Metrology and Measurement Systems 15(1): 55–67. Search in Google Scholar

Lal-Jadziak, J. and Sienkowski, S. (2009). Variance of random signal mean square value digital estimator, Metrology and Measurement Systems 16(2): 267–277. Search in Google Scholar

Martinez, M.A. and Ashrafi, A. (2018). Real-valued single-tone frequency estimation using half-length autocorrelation, Digital Signal Processing 83: 98–106.10.1016/j.dsp.2018.08.007 Search in Google Scholar

Menhaj, A., Assaad, J., Rouvaen, J.M., Heddebaut, M. and Bruneel, C. (1998). A new collision avoidance radar system using correlation receiver and compressed pulses, Measurement Science and Technology 9(2): 283–286.10.1088/0957-0233/9/2/017 Search in Google Scholar

Nielsen, E. and Rietveld, M.T. (2003). Observations of backscatter autocorrelation functions from 1.07-m ionospheric irregularities generated by the European incoherent scatter heater facility, Journal of Geophysical Research: Space Physics 108(A5): 1166. Search in Google Scholar

Peng, J. and Boscolo, S. (2016). Filter-based dispersion-managed versatile ultrafast fibre laser, Scientific Reports 6: 25995.10.1038/srep25995 Search in Google Scholar

Pfeifer, P.E. and Deutsch, S.J. (1981). Variance of the sample space-time autocorrelation function, Journal of the Royal Statistical Society B 43(1): 28–33.10.1111/j.2517-6161.1981.tb01144.x Search in Google Scholar

Rahman, M.M., Chowdhury, M.A. and Fattah, S.A. (2018). An efficient scheme for mental task classification utilizing reflection coefficients obtained from autocorrelation function of EEG signal, Brain Informatics 5: 1–12.10.1007/s40708-017-0073-7 Search in Google Scholar

Rice, F., Cowley, B., Moran, B. and Rice, M. (2001). Cramer–Rao lower bounds for QAM phase and frequency estimation, IEEE Transactions on Communications 49(9): 1582–1591.10.1109/26.950345 Search in Google Scholar

Roberts, P.P. (1997). Calculating quantization correction formulae for digital correlators with digital fringe rotation, Astronomy and Astrophysics Supplement Series 126(2): 379–383.10.1051/aas:1997271 Search in Google Scholar

Sienkowski, S. and Kawecka, E. (2013). Probabilistic properties of sinusoidal signal autocorrelation function, Przegląd Elektrotechniczny 89(11): 98–100. Search in Google Scholar

Sienkowski, S. and Krajewski, M. (2018). Simple, fast and accurate four-point estimators of sinusoidal signal frequency, Metrology and Measurement Systems 25(2): 359–376. Search in Google Scholar

Sienkowski, S. and Krajewski, M. (2020). Single-tone frequency estimation based on reformed covariance for half-length autocorrelation, Metrology and Measurement Systems 27(3): 473–493. Search in Google Scholar

Spiesberger, J.L. (1996). Identifying cross-correlation peaks due to multipaths with application to optimal passive localization of transient signals and tomographic mapping of the environment, Journal of the Acoustical Society of America 100(2): 910–918.10.1121/1.416250 Search in Google Scholar

Stodółka, J., Korzewa, L., Stodółka, W. and Gambal, J. (2017). The applicability of using parameters of the autocorrelation function in the assessment of human balance during quiet bipedal stance, Central European Journal of Sport Sciences and Medicine 17(1): 79–87.10.18276/cej.2017.1-10 Search in Google Scholar

Tagade, P.M. and Choi, H.-L. (2017). A dynamic bi-orthogonal field equation approach to efficient Bayesian inversion, International Journal of Applied Mathematics and Computer Science 27(2): 229–243, DOI: 10.1515/amcs-2017-0016.10.1515/amcs-2017-0016 Search in Google Scholar

Toth, L. and Kocsor, A. (2003). Harmonic alternatives to sine-wave speech, Proceedings of the 8th European Conference on Speech Communication and Technology, Geneva, Switzerland, pp. 2073–2076. Search in Google Scholar

Tu, Y.-Q. and Shen, Y.-L. (2017). Phase correction autocorrelation-based frequency estimation method for sinusoidal signal, Signal Processing 130: 183–189.10.1016/j.sigpro.2016.06.012 Search in Google Scholar

Uciński, D. (2000). Optimal selection of measurement locations for parameter estimation in distributed processes, International Journal of Applied Mathematics and Computer Science 10(2): 357–379. Search in Google Scholar

Vaseghi, S.V. (2008). Advanced Digital Signal Processing and Noise Reduction, 4th Edition, Wiley, Hoboken. Search in Google Scholar

Wang, K., Ding, J., Xia, Y., Liu, X., Hao, J. and Pei, W. (2018). Two high accuracy frequency estimation algorithms based on new autocorrelation-like function for noncircular/sinusoid signal, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences 101(7): 1065–1073.10.1587/transfun.E101.A.1065 Search in Google Scholar

Weber, R., Faye, C., Biraud, F. and Dansou, J. (1997). Spectral detector for interference time blanking using quantized correlator, Astronomy and Astrophysics Supplement Series 126(1): 161–167.10.1051/aas:1997257 Search in Google Scholar

Zeitler, R. (1997). Digital correlator for measuring the velocity of solid surfaces, IEEE Transactions on Instrumentation and Measurement 46(4): 803–806.10.1109/19.650777 Search in Google Scholar

Zhang, M., Yang, Q., Chen, X. and Zhang, A. (2018). A generalized correlation theorem for LTI systems, Journal of Physics: Conference Series 1169(1): 012018.10.1088/1742-6596/1169/1/012018 Search in Google Scholar

Recommended articles from Trend MD

Plan your remote conference with Sciendo