V.A. Gaisky, P.V. Gaisky
Institute of Natural and Technical Systems, RF, Sevastopol, Lenin St., 28
Real natural processes have unlimited duration and spectrum. Real observations are discrete, have a finite duration, and in standard spectral analysis give a spectrum bounded on the top by Nyquist frequency equal to half the sampling frequency and with a resolution inverse to the duration of the implementation. Frequencies of process outside these boundaries are superimposed on the calculated as a result of aliasing. Their partial recovery with using modulation of the spectrum to the sampling procedures antialiasing, which involves thinning of the original series of samples with the formation of new ranks for number of divisors half of the original number of samples, computation of additional spectra, the number of which is equal to the sum of the factors of a number greater than the number in accordance with the well-known result in number theory about sum of factors is proposed.
The amplitudes of the upper frequencies of the overlay behind the Nyquist frequency up to a certain maximum frequency are found by the solution of a system of linear algebraic equations. The amplitudes of harmonics with periods higher than the duration of observations, i.e. the spectrum of lower frequencies are found applying anti-aliasing to the secondary spectrum. All together, it gives a wide-format amplitude spectrum for discrete implementation of processes and a spectral density function for the correlation function for random processes.
Inverse discrete Fourier transforms restore extended time implementations of processes and the correlation function.
Keywords: direct and inverse discrete Fourier transforms, eliasing, anti-leasing, primary spectrum, secondary spectrum.
To quote, follow the DOI link and use the Actions-Cite option or copy:[IEEE] V. A. Gaisky and P. V. Gaisky, “Wide spectral analysis of natural processes,” Monitoring systems of environment, no. 1, pp. 05–13, Mar. 2020.
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