Abstract
Mathematical methods are proposed for comparing the waveforms of signals with amplitudes distorted by unknown monotonic transformations and random noise. These methods are based on solving the problems of the best approximation of the signal under analysis to signals of a specified class and used in estimating the lag time of one fragment of signal with respect to another. Such problems arise, for example, in determining the direction of propagation of a sound wave in the atmosphere. The solution to this problem is based on the assumption that the conditions of the signal detection are different at different spatial points; as a result, the measured signals differ not only in their lag time, but also in nonlinear distortions, such that only the general waveform of the signal is preserved: if the amplitude of one signal is the result of a strictly monotonic transformation of the amplitude of another signal, then their waveforms are equivalent. In addition, the measurements are accompanied by an additive noise with an unknown dispersion.
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Acknowledgements
This work was supported by the Russian Foundation for Basic Research (projects nos. 17-07-00832 and 18-05-00576) and the program of the Presidium of Russian Academy of Science no. 56.
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Chulichkov, A.I., Kulichkov, S.N., Tsybulskaya, N.D. et al. Comparing Signal Waveforms and Their Use in Estimating Signal Lag Time. Pure Appl. Geophys. 176, 335–344 (2019). https://doi.org/10.1007/s00024-018-2056-x
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DOI: https://doi.org/10.1007/s00024-018-2056-x