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Trellis-Based Postprocessing for Short Delay Measurement Using NDFT

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Abstract

For digital measurement of short delays between fast analogues signals can be used phase shift method with non-uniform sampling and non-uniform Fourier transform (NDFT), but if the signals are short, non-uniform sampling causes random spurious peaks in the spectrum. In case we have sequence of spectrograms with slow frequency change, correct peaks could be found using evaluation of such sequence. Our new Trellis-based postprocessing selects several greatest peaks from each spectrogram in the sequence and interprets them as nodes of an acyclic directed graph. These nodes are weighted and correct ones are found by graph algorithm. Experiments showed that our method provides correct results in cases when ordinary peak detection is not able to distinguish correct and spurious peaks.

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Notes

  1. Phase shift computed using Fourier transform is in range \(\left\langle { - \pi , + \pi } \right\rangle \).

  2. According to [9], the bandwidth of analogue inputs of common AD converters is typically 4 to 8 times higher than half of the maximal (periodic) sampling frequency of the AD converter.

  3. Change of the frequency fp and of the random sampling instants causes different cross-interference in every burst; therefore, spurious peaks appear at the different positions.

  4. We also tried p = 1/2.

  5. Asterisk (*) in indexes (e.g. \({{P}_{{{i,}^{*}}}}\) or \({{w}_{{{0,}^{*}}}}\)) is used as a placeholder saying any valid value.

  6. It would be formally more proper to weight the nodes by \(\sqrt[p]{{{{w}_{{i + 1,\,k}}}}}\), but for purposes of our method, it is enough to use \({{w}_{{i + 1,\,k}}}\) since for any a < b < c; a, b, c > 0 holds ap < bp < cp ; p > 1.

  7. There could be more non-converging paths, but it is improbable due to random distribution of the most of the nodes.

  8. It would be formally more appropriate to use \({{w}_{c}} = \sqrt[p]{{k{{{(\lambda )}}^{p}}}}\), but we must use the same norm as we use for path weighting, i.e. \({{w}_{c}} = k{{(\lambda )}^{p}}\).

  9. Generator of pseudorandom numbers was also initiated to the same state.

  10. In the spectrogram, there is no local maximum for useful frequency fp , or it is to small.

  11. In case of increasing N, take care about ratio of λ and B/2N.

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  1. Department of Computer Science and Engineering, University of West Bohemia, Univerzitní 8, 30100, Pilsen, Czech Republic

    Karel Dudáček &  Karel Dudáček

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  1. Karel Dudáček
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  2. Karel Dudáček
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Correspondence to Karel Dudáček or Karel Dudáček.

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Karel Dudáček, Karel Dudáček Trellis-Based Postprocessing for Short Delay Measurement Using NDFT. Aut. Control Comp. Sci. 53, 270–280 (2019). https://doi.org/10.3103/S0146411619030039

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