### A THEORETICAL ANALYSIS OF THE METHODS OF HARMONIC DECOMPOSITION

G. M. Molchan and W. I. Newman

Abstract

We present a theoretical study of the statistical properties and the resolution capability of Pisarenko's Harmonic Decomposition Method (MHD). The main result is as follows: The MHD resolves closely adjacent frequencies of low harmonics in white noise if and only if $N^{\gamma^*}\Delta \omega\gg 1$, where $\gamma^* =1/6,\;\Delta\omega$ is the radial frequency separation, and $N$ is number of observed values of signal. The least upper bound $\gamma^*$ of $\gamma$, such that the method resolves two harmonics $\Delta\omega=O(N^{-\gamma}$) apart, is a new and realistic characteristic describing the resolution capability of spectral methods in the case of small samples. The estimate $\gamma^*$ for the MHD is a first result in the general program to rank modern high resolution thechniques. Probably, the value $\gamma^*=1/6$ is a universal resolution characteristic of spectral methods based on a fixed number of signal correlations. At the same time optimal correlation methods, which do not use signal phase (like MHD), resolve frequencies under the condition $N^{\gamma^*}\Delta\omega\gg 1$, where $\gamma^*=7/6$. We analyze the limit distribution for MHD estimate of $\Delta\omega$ under the condition that $\Delta \omega=hN^{-\gamma},\;\gamma<1/6$, and find an unusual statistical effect. The errors of the MHD estimate $\hat h$ of $h$ for two harmonics with equal powers has approximately Gaussian distribution when $0<\gamma <1/8$ and $\chi_1^2$-distribution if $1/8<\gamma <1/6$. Statistical properties of $\hat {h}$ significantly change, depending on whether the harmonic powers are equal or different. Consequently, notwithstanding the accepted practice, a signal with equal harmonic amplitudes cannot be successfully used to test resolution capability of spectral methods. We also analyze MHD estimates of multiple sinusoids in the ordinary situation where the frequencies are fixed as $N\to\infty$. We show that MHD is highly sensitive to the geometry of all hidden frequencies.

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Computational Seismology, Vol. 1.