Notes on poles and convergence rate of autoregressive model

Abstract

The convergence rate of the power spectral density function and the properties of the poles of the AR model as an approximate representation of an ARMA(2,1) process are studied with full use of numerical analysis of specific examples. Attention is focused on the case where the ARMA(2,1) model transfer function in z ^-1 (time shift operator) has two real system poles 1/r₁, 1/r₂ and the system zero 1/b in the z ^-1-plane (complex-plane) with the relationship . When the AR model fitting is applied for the theoretical covariance sequence of the process based on the Levinson-Durbin Algorithm, (1) the convergence rate of power spectrum of the AR model is asymptotically proportional to b, (2) the pole 1/r₁ is transferred into the AR model as a real pole, however, the information about 1/r₂ merges into all poles of the AR model, and (3) the zero 1/b is equivalently represented in the AR model by a set of poles almost equally spaced on a circle centered at the origin in the complex plane. For the qualitative analysis of the AR model, an AR model type approximation (TAR model) of the ARMA model has been defined by truncating the AR model of infinite order which is equivalent to the ARMA model, and it has been shown that the TAR model is a useful intermediate to bridge over an ARMA model and its approximation, AR model.