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Monte Carlo uncertainty analysis of an ANN-based spectral analysis method

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Abstract

This work presents the uncertainty analysis of an artificial neural network (ANN)-based method, called multiharmonic ANN fitting method (MANNFM), which is able to obtain, at a metrological level, the spectrum of asynchronously sampled periodical signals. For sinusoidal and harmonic content signals, jitter and quantization noise contributions to uncertainty are considered in order to obtain amplitude and phase uncertainties using Monte Carlo method. The analysis performed identifies also both contributions to uncertainty for different parameters laboratory configurations. The analysis is performed simultaneously with our method and two others: discrete Fourier transform (DFT), for synchronously sampled signals, and multiharmonic sine-fitting method (MSFM), for asynchronously sampled signals, in order to compare them in terms of uncertainty. Regarding asynchronous methods, results show that MANNFM provides the same uncertainties than MSFM, with the advantage of a simpler implementation. Regarding asynchronous and synchronous methods comparison, results for sinusoidal signals show that MANNFM has the same uncertainty as DFT for amplitude and higher uncertainty for phase values; for signals with harmonic content, amplitude conclusions maintain but, regarding phase, both MANNFM and DFT uncertainties become closer as the frequency increases, which implies, in fact, that when synchronous sampling is not possible, spectrum analysis can be performed with asynchronous methods without incurring in uncertainty increases.

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Notes

  1. 1.

    The error of synchronism is due to the differences between generation and sampling time bases. Authors’ expertise (with real and simulated signals) has shown that this error affects only the convergence of the method to global or local minima, but not to the value of these minima. So, it will not be considered an uncertainty source. Nevertheless, it has been included in simulations just to perform the uncertainty evaluation in the more general conditions.

  2. 2.

    Brand names are used for purpose of identification. Such use does not imply endorsement by authors or assume that the equipment is the best available.

  3. 3.

    Only in the case of amplitude considering exclusively jitter, for both signals, very similar behaviours are appreciated. In the rest of the cases (included phase deviations considering exclusively jitter), different behaviours are observed.

  4. 4.

    For example, for amplitude standard deviation, supposing a starting point of 21 bits and 1311 samples and a resolution reduction from 21 to 18 bits, it can be mostly compensated if N = 15,457 is fixed. Nevertheless, if, in the same starting point, the reduction is from 21 to 16 bits, the uncertainty obtained with 16 bits at 15,457 samples is one order of magnitude higher than the initial one.

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Acknowledgements

This work was partially supported by the Universidad de Malaga - Campus de Excelencia Andalucia-Tech.

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Correspondence to José Ramón Salinas.

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Appendices

Appendix 1

Tables 10 and 11 show standard deviation results obtained for amplitude and phase of half-wave rectified harmonics for the case of a jitter value of 100 ns and 16 bits of quantization, 15,457 samples and different R values. In order to unify data as much as possible, both tables simultaneously present values obtained considering only jitter (j) and only quantization (q).

Table 10 Standard deviation of HWR harmonics deviations to amplitude nominal values (µV/V) in high number of samples points of Table 1, considering only: jitter (j), quantization (q)
Table 11 Standard deviation of HWR harmonics deviations to phase nominal values (µrad) in high number of samples points of Table 1, considering only: jitter (j), quantization (q)

Appendix 2

Tables 12 and 13 show standard deviation results obtained for amplitude and phase of half-wave rectified harmonics for the case of a jitter value of 100 ns and 16 bits of quantization, R fixed to 24 and different N values. In order to unify data as much as possible, both tables simultaneously present values obtained considering only jitter (j) and only quantization (q).

Table 12 Standard deviation of HWR harmonics deviations to amplitude nominal values (µV/V) in last three points of Table 1, considering: only jitter (j), only quantization (q)
Table 13 Standard deviation of HWR harmonics deviations to phase nominal values (µrad) in the last three points of Table 1, considering: only jitter (j), only quantization (q)

Appendix 3

Tables 14 and 15 show standard deviation results obtained for amplitude and phase of half-wave rectified harmonics for the case of a jitter value of 100 ns, R fixed to 24, different N values and different number of bits. Due to the amount of data to be included, values obtained considering only jitter (j) are presented.

Table 14 Standard deviation of HWR harmonics deviations to amplitude nominal values (µV/V) for Maxjit = 100 ns, Nbits and N values simulated, considering only jitter (j)
Table 15 Standard deviation of HWR harmonics deviations to phase nominal values (µrad) for Maxjit = 100 ns, Nbits and N values simulated, considering only jitter (j)

Tables 16 and 17 show standard deviation results obtained for amplitude and phase of half-wave rectified harmonics for the case of a jitter value of 100 ns, R fixed to 24, different N values and different number of bits. Due to the amount of data to be included, both tables present only values obtained considering only jitter (q).

Table 16 Standard deviation of HWR harmonics deviations to amplitude nominal values (µV/V) for Maxjit = 100 ns, Nbits and N values simulated, considering only quantization (q)
Table 17 Standard deviation of HWR harmonics deviations to phase nominal values (µrad) for Maxjit = 100 ns, Nbits and N values simulated, considering only quantization (q)

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Salinas, J.R., García-Lagos, F., Diaz de Aguilar, J. et al. Monte Carlo uncertainty analysis of an ANN-based spectral analysis method. Neural Comput & Applic 32, 351–368 (2020) doi:10.1007/s00521-019-04169-x

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Keywords

  • Sine-fitting methods
  • Spectral analysis
  • ADALINE
  • Digital measurement
  • Uncertainty
  • Monte Carlo