A theory of spectral analysis based on the characteristic property of a linear dynamic system

Abstract

We present a detailed description of a new method of spectral analysis named ‘Sompi’. The basic idea of this method originates in the physical concept of the characteristic property of the linear dynamic system that is described by a linear differential equation. The time series modelling in the Sompi method consists essentially of estimating the governing differential equation of the hypothetical linear dynamic system that has yielded the given time series data. Due to the equivalence of a linear differential equation and a linear difference equation [or an autoregressive (AR) equation], this method takes the form of the familiar AR method. However, our basic concept of the AR model and the exact formulation based on the maximum likelihood principle have led to a model estimation algorithm different from previous AR methods, and further, to spectral estimation with higher resolution and reliability. By the Sompi method, a time series is deconvoluted into a linear combination of coherent oscillations with amplitudes decaying (or growing) exponentially with time, and additional noise. In other words, it yields a line-shaped spectrum in complex frequency space, unlike the traditional harmonic decomposition in real frequency space, and is powerful for the analysis of the decaying characteristics, as well as the periods, of the oscillations. Also, the variances of the spectral estimates by the Sompi method can be given in simple formulae unlike most modern parametric methods. Although some practical problems still remain unresolved, the theory presented here will provide the theoretical prototype for a new discipline of physical spectral analysis.