Spectral analysis based on the canonical autoregressive decomposition

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 the canonical autoregressive decomposition 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 is reduced to a linear least-squares problem by using a nonlinear parameter transformation. Similar results are derived for AR processes in polynomial or polynomial-times-exponential signals. Applications include frequency estimation/transient analysis in unknown colored noise