G. M. Molchan


A model of a short signal is considered. It consists of two complex harmonics with unknown amplitudes and frequencies $\Delta\omega=O(T^{-\gamma})$ apart plus a complex white noise, where $T$ is a period of observation. The upper bound $\gamma^*$ of exponents $\gamma$, such that a spectral method resolves these harmonics, is considered as a new measure of resolution capability for small samples. The knowledge of $\gamma^*$ allows to rank modern high resolution thechniques originally developed for geophysical data proccesing. As a first step on this way, we find $\gamma^{**}={\rm sup}\gamma^*$, where supremum is taken over all spectral methods. For a generic signal $\gamma^{**}=5/4$. It reduces to 7/6 for harmonics with equal powers. In other words, $\gamma^*\le 7/6$ for spectral techniques, which are based on power spectrum and thus lose initial harmonic phases. This results are closely related to the old harmonic analysis problem that goes back to Lord Rayleigh. He defined, on the basis of nonoptimal Fourier method, a frequency resoltion condition in the absence of noise by relation: $| \omega _{1}- \omega _{2}| >2\pi T^{-1}$. However, the frequency resolution condition, even in the presence of noice, takes the form $| \omega _{1}- \omega _{2}| \gg T^{-\gamma^{**}}$. The likelihood of the signal in hand does not posess the property of local asimptotic normality for all $\gamma <\gamma^{**}$, therefore our results are not based on the general asymptotic theory of Le Cam-Hajek-Ibragimov-Khas'minsky.

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Computational Seismology, Vol. 1.