A.A. L’vov – Dr.Sc.(Eng.), Professor, 

Department «Information and Communication Systems and Software Engineering», 

Yuri Gagarin State Technical University of Saratov

E-mail: alvova@mail.ru

A.A. Seranova – Post-graduate student, 

Department «Information and Communication Systems and Software Engineering», 

Yuri Gagarin State Technical University of Saratov

E-mail: seranova.anna@gmail.com

R.V. Ermakov – Ph.D.(Eng.), 

Yuri Gagarin State Technical University of Saratov

E-mail: roma-ermakov@yandex.ru

A.S. Muchkaev – Ph.D.(Eng.), 

Martec Corporation (Las-Vegas, USA)

E-mail: marsand@mail.ru

Due to the wide practical application, the problem of estimating the parameters of signal components from a finite number of noisy discrete measurements is still a relevant area of research. The solution of this problem is required when determining the parameters of the quality of electricity in industrial networks, in radiolocation, communication, metrology and other branches of science and technology. In the most difficult case, the frequency of the signal is unknown and should be evaluated along with the other parameters. There are many frequency estimation schemes, all of which can be classified as non-parametric and parametric methods. A brief overview of the methods used is given in the introduction of the presented work. This paper proposes a new method for estimating signal components from a finite number of noisy discrete measurements based on subfactoring data matrices and comparing it with one of the most popular methods described in IEEE-STD-1057, which represents algorithms for estimating the wave signal parameters from its noisy discrete time counts. The essence of the proposed method lies in the representation of the redundant model in the class of separable models depending on the input decomposition coefficients and the eigenvectors of Gram matrices. As a result, the search for values of unknown parameters minimizing the function of the residual is reduced to solving a linear system of equations for these decomposition coefficients. If the assumptions about the normality of errors are fulfilled, the resulting estimates of the unknown parameters are maximum likelihood estimates, that is, unbiased, consistent and asymptotically effective. An iterative procedure for solving the resulting non-linear equations is proposed. Mathematical modeling and application of this decomposition in practice showed good results. The IEEE-STD-1057 standard provides algorithms for estimating the parameters of a quasi-harmonic signal from its noisy discrete time samples. Estimation is carried out as a result of minimizing the sum of squared errors, that is, the difference between the observations and the values predicted by the model. The IEEE-STD-1057 algorithm provides accurate estimates in the presence of Gaussian and quantizing noise. The approaches described above were used in the processing of experimental data obtained in the study of AFC micromechanical gyroscopes. In the experiment, a reference rotary table was used, capable of making controlled movements in advance of a given law. The position of the platform with high accuracy was fixed by the built-in optical sensor of the angle of the turntable. As a result of processing the experimental data using the two methods described above, we obtained the estimates of the AFC and also the measurement errors. It was shown that the proposed method has an advantage over the methods outlined in the IEEE-STD-1057 standard, and the advantage of the proposed method is more noticeable with a low noise level. In addition to this advantage, the proposed method also has less computational complexity due to a reduction in the number of unknowns to be estimated.

1. Stoica R.P. Moses. Introduction to Spectral Analysis. Upper Saddle River. NJ: Prentice-Hall. 1997.
2. Tufts D.W., Kumaresan R. Estimation of Frequencies of Multiple Sinusoids: Making Linear Prediction Perform like Maximum Likelihood. Proc. of IEEE. September 1982. V. 70. № 9. P. 975−989.
3. Marpl-ml. S.L. Tsifrovoi spektralnyi analiz i ego prilozheniya. M.: Mir. 1990. (in Russian)
4. Roy R., Kailath T. ESPRIT Estimation of Signal Parameters via Rotational Invariance Techniques. IEEE Trans. Acoust. Speech. Signal Process. Jule 1989. V. 37. № 7. P. 984−995.
5. Hua Y., Sarkar T. Matrix pencil method for estimating parameters of exponentially damped. undamped sinusoids in noise. IEEE Trans. Acoust. Speech. Signal Process. May 1990. V. 38. № 4. P. 814−824.
6. Sarkar T., Pereira O. Using the Matrix Pencil Method to Estimate the Parameters of a Sum of Complex Exponentials. IEEE Anten. and Propag. Mag. February 1995. № 1. P. 48−55.
7. So H.C., Kit Wing Chan, Chan Y.T., Ho K.C. Linear Prediction Approach for Efficient Frequency Estimation of Multiple Real Sinusoids: Algorithms and Analyses. IEEE Trans. on Signal Processing. July 2005. V. 53. № 7. P. 2290−2305.
8. Linnik Yu.V. Metod naimenshikh kvadratov i osnovy teorii obrabotki nablyudenii. M.: GIFML. 1958. (in Russian)
9. Moutchkaev A.S., Kong S.-H., L’vov A.A. Parameter estimation of superimposed sinusoids by data matrix subfactorization: theory and algorithm. Conf. Proc. 2016 Int. Conf. on Actual Problems of Electron Devices Engineering. APEDE-2016. V. 2. P. 442−447.
10. Moutchkaev A.S., Kong S.-H., L’vov A.A. Parameter estimation of superimposed sinusoids by data matrix subfactorization: analysis and results. Conf. Proc. 2016 Int. Conf. on Actual Problems of Electron Devices Engineering. APEDE-2016. V. 2. P. 448−455.
11. L'vov A.A., Semenov K.V. A method of calibrating an automatic multiprobe measurement line. Measurement Techniques. 1999. V. 42. № 4. P. 357−365.
12. Girko V.L. Spektralnaya teoriya sluchainykh matrits. M.: Nauka. 1988. (in Russian)
13. Händel P. Properties of the IEEE-STD-1057 Four-Parameter Sine Wave Fit Algorithm. IEEE Trans. on Instrum. and Measur. December 2000. V. 49. № 6. P. 1189−1193.
14. Ermakov R.V., Popov A.N., Skripal E.N., Kalikhman D.M., Kondratov D.V., Lvov A.A. Metody i rezultaty ispytanii inertsialnykh datchikov, prednaznachennykh dlya ekspluatatsii na letatelnykh apparatakh vertoletnogo tipa. Sb. materialov XXIV Sankt-Peterburgskoi Mezhdunar. konf. po integrirovannym navigatsionnym sistemam. SPb.: Izd-vo TsNII «Elektropribor». 2017. S. 244−248. (in Russian)
15. Abakumov A.V., Gutsevich D.E., Ermakov R.V., Lvov A.A., Seranova A.A., Skripal E.N. Rezultaty ispytanii mikromekhanicheskikh inertsialnykh datchikov pri kombinirovannom vozdeistvii. Cb. tr. V Mezhdunar. yubileinoi nauch. konf. «Problemy upravleniya. obrabotki i peredachi informatsii» (UOPI-2017). Saratov: OOO SOP «Lodi». 2017. S. 506−511. (in Russian)
16. Ermakov R.V., Kondratov D.V., L'Vov A.A. et al. Methods for testing and test results of inertial sensors intended for operation in helicopter-type aircraft. 2017 24th Saint Petersburg Int. Conf. on Integrated Navigation Systems. ICINS 2017. Proc. 24. 2017. P. 335−338.
17. Ermakov R.V., Seranova A.A., L’vov A.A., Kalikhman D.M. Optimal Estimation of the Motion Parameters of a Precision Rotating Stand by Maximum Likelihood Method. Measurement Techniques. 2019. P. 1−8. https://doi.org/10.1007/s11018-019-01598-x.