TOEPLITZ MATRICES AND TIME SERIES ANALYSIS

摘要

The analysis and control of mechanical and fluid systems require the capacity to analyze the associated time series. Covariance and third order cumulant matrices, [Nikias, Petropulu, 1993] are examples of Toeplitz matrices, which commonly arise, in the numerical study of stationary or shift invariant time series. The higher order spectral algorithms, as for example TOR, involving Toeplitz matrices, suppress Gaussian noise and detect phase coupling which may be present in nonlinear dynamical systems, [Raghuveer and Nikias, 1986]. There exists a large body of theoretical and computational research on Toeplitz matrices and forms. [Grenander and Szegoe, 1985], [Parter, 1986], [Tyrtshnikov, 1992], [Iohvidov, 1982]. However studies of low rank Toeplitz matrices are sparse. The following is an examination of the singular values of symmetric, (nxn), Toeplitz matrices, T(n, r), with trigonometric series elements, as a function of n. The T(n, r) matrices are of constant rank and are circulant for specific values of n. For such values it is shown that each coefficient of a given term of the trigonometric series is simply related to the corresponding singular values of T(n, r). The frequency of oscillation of a singular value pair trajectory is shown to relate directly to the frequency of the associated term in the trigonometric series. The separation of singular value pair trajectories and therefore of the frequency components of the trigonometric series is shown to be a function of the amplitude coefficients of corresponding terms in the trigonometric series. It is also noted that for nearly equal frequency components the oscillations of differences of singular value pairs of either component display a modulation with a frequency of one half of the frequency difference of the associated frequency components. The case of equal amplitude coefficients is examined. A number of numerical results support the analysis.