Spectral Analysis on the Canonical Autoregressive Decomposition.

摘要

Time series modeling as the sum of an autoregressive (AR) process and sinusoids is proposed. When the AR model order is infinite, it is called Canonical Autoregressive Decomposition (CARD) and is equivalent to the Wold decomposition. Maximum likelihood estimation of the sinusoidal and AR parameters is shown to require minimization with respect to only the unknown frequencies. Although the estimation problem is nonlinear in the sinusoidal amplitudes and AR parameters, it was reduced to a linear least squares problem by using a nonlinear parameter transformation. A general class of signals for which such parameter transformations are applicable, thereby reducing estimator complexity drastically, is derived. This class includes sinusoids as well as polynomials and polynomial-times-exponential signals. The ideas are based on the theory of invariant subspaces for linear operators. CARD serves as a powerful modeling tool in signal plus noise settings and therefore finds application in a large variety of statistical signal processing problems.