Calculation of signal spectrum by means of stochastic inversion

Abstract

The standard method of calculating the spectrum of a digital signal is based on the Fourier transform, which gives the amplitude and phase spectra at a set of equidistant frequencies from signal samples taken at equal intervals. In this paper a different method based on stochastic inversion is introduced. It does not imply a fixed sampling rate, and therefore it is useful in analysing geophysical signals which may be unequally sampled or may have missing data points. This could not be done by means of Fourier transform without preliminary interpolation. Another feature of the inversion method is that it allows unequal frequency steps in the spectrum, although this property is not needed in practice. The method has a close relation to methods based on least-squares fitting of sinusoidal functions to the signal. However, the number of frequency bins is not limited by the number of signal samples. In Fourier transform this can be achieved by means of additional zero-valued samples, but no such extra samples are used in this method. Finally, if the standard deviation of the samples is known, the method is also able to give error limits to the spectrum. This helps in recognising signal peaks in noisy spectra.