Covariance sequence approximation for parametric spectrum modeling

Parametric methods of spectrum analysis are founded on finite-dimensional models for covariance sequences. Rational spectrum approximants for continuous spectra are based on autoregressive (AR), moving average (MA), or autoregressive moving average (ARMA) models for covariance sequences. Line spectrum approximants to discrete spectra are based on cosinusoidal models for covariance sequences. In this paper we make the point that a wide variety of spectrum types admit to modal analysis wherein the modes are characterized by amplitudes, frequencies, and damping factors. The associated modal decomposition is appropriate for both continuous and discrete components of the spectrum. The domain of attraction for the decomposition includes ARMA sequences, harmonically or nonharmonically related sinusoids, damped sinusoids, white noise, and linear combinations of these. The parametric spectrum analysis problem now becomes one of identifying mode parameters. This we achieve by solving two modified least squares problems. Numerical results are presented to illustrate the identification of mode parameters and corresponding spectra from finite records of perfect and estimated covariance sequences. The results for sinusoids and sinusoids in white noise are interpreted in terms of in-phase and quadrature effects attributable to the finite record length.