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Fast one-cycle frequency estimation of a single sinusoid in noise using downsampled linear prediction model

Krzysztof Duda Tomasz P. Zieliński
Metrology and Measurement Systems | 2021 | vol. 28 | No 4 | 661-672 | DOI: 10.24425/mms.2021.137701
Keywords frequency estimation linear prediction Prony method smart DFT
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Abstract

A new solution to the problem of frequency estimation of a single sinusoid embedded in the white Gaussian noise is presented. It exploits, approximately, only one signal cycle, and is based on the well-known 2nd order autoregressive difference equation into which a downsampling is introduced. The proposed method is a generalization of the linear prediction based Prony method for the case of a single undamped sinusoid. It is shown that, thanks to the proposed downsampling in the linear prediction signal model, the overall variance of the least squares solution of frequency estimation is decreased, when compared to the Prony method, and locally it is even close to the Cramér–Rao Lower Bound, which is a significant improvement. The frequency estimation variance of the proposed solution is comparable with, computationally more complex, the Matrix Pencil and the Steiglitz–McBride methods. It is shown that application of the proposed downsampling to the popular smart DFT frequency estimation method also significantly reduces the method variance and makes it even better than the least squares smart DFT. The noise immunity of the proposed solution is achieved simultaneously with the reduction of computational complexity at the cost of narrowing the range of measured frequencies, i.e. a sinusoidal signal must be sufficiently oversampled to apply the proposed downsampling in the autoregressive model. The case of 64 samples per period with downsampling up to 16, i.e. 1/4th of the cycle, is presented in detail, but other sampling scenarios, from 16 to 512 samples per period, are considered as well.
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Authors and Affiliations

Krzysztof Duda
1
Tomasz P. Zieliński
2
  1. AGH University of Science and Technology, Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering, Department of Measurement and Electronics, al. Mickiewicza 30, 30-059 Kraków, Poland
  2. AGH University of Science and Technology, Faculty of Computer Science, Electronics and Telecommunications, Institute of Telecommunications, al. Mickiewicza 30, 30-059 Kraków, Poland

Single-tone frequency estimation based on reformed covariance for half-length autocorrelation

Sergiusz Sienkowski Mariusz Krajewski
Metrology and Measurement Systems | 2020 | vol. 27 | No 3 | 473-493 | DOI: 10.24425/mms.2020.134590
Keywords frequency estimator sinusoidal signal autocorrelation function mean squared error
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Abstract

This paper presents a new simple and accurate frequency estimator of a sinusoidal signal based on the signal autocorrelation function (ACF). Such an estimator was termed as the reformed covariance for half-length autocorrelation (RC-HLA). The designed estimator was compared with frequency estimators well-known from the literature, such as the modified covariance for half-length autocorrelation (MC-HLA), reformed Pisarenko harmonic decomposition for half-length autocorrelation(RPHD-HLA), modified Pisarenko harmonic decomposition for half-length autocorrelation (MPHD-HLA), zero-crossing (ZC), and iterative interpolated DFT (IpDFT-IR) estimators. We determined the samples of the ACF of a sinusoidal signal disturbed by Gaussian noise (simulations studies) and the samples of the ACF of a sinusoidal voltage(experimental studies), calculated estimators based on the obtained samples, and computed the mean squared error(MSE) to compare the estimators. The errorswere juxtaposed with the Cramér–Rao lower bound (CRLB). The research results have shown that the proposed estimator is one of the most accurate, especially for SNR > 25 dB. Then the RC-HLA estimator errors are comparable to the MPHD-HLA estimator errors. However, the biggest advantage of the developed estimator is the ability to quickly and accurately determine the frequency based on samples collected from no more than five signal periods. In this case, the RC-HLA estimator is the most accurate of the estimators tested.

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Authors and Affiliations

Sergiusz Sienkowski
Mariusz Krajewski

Implementation of Goertzel-based frequency estimation for power quality monitoring in embedded measurement systems

Nuno M. Rodrigues Fernando M. Janeiro Pedro M. Ramos
Metrology and Measurement Systems | 2022 | vol. 29 | No 3 | 455-468 | DOI: 10.24425/mms.2022.142270
Keywords power quality frequency estimation non-integer Goertzel frequency interpolation embedded measurement systems
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Abstract

International standards from IEC and IEEE regulate power grid parameters such as theRMSvalue, frequency, harmonic and interharmonic distortion, unbalance or the presence of transients, that are important to assure the quality of distributed power. Standard IEC 61000-4-30 suggests the zero crossing algorithm for the measurement of the power grid frequency, but also states that different algorithms can be used. This paper proposes a new algorithm, the Fractional Interpolated Discrete Fourier Transform, FracIpDFT, to estimate the power grid frequency, suitable for implementation in resource limited embedded measurement systems. It is based on the non-integer Goertzel algorithm followed by interpolation at non-integer multiples of the DFT frequency resolution. The proposed algorithm is validated and its performance compared with other algorithms through numerical simulations. Implementation details of the FracIpDFT in an ARM Cortex M4 processor are presented along with frequency measurement results performed with the proposed algorithm in the developed system.
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Authors and Affiliations

Nuno M. Rodrigues
1
Fernando M. Janeiro
2
Pedro M. Ramos
1
  1. Instituto de Telecomunicações, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal
  2. Instituto de Telecomunicações, Universidade de Évora, 7000-671 Évora, Portugal

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