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Computing the Poles of Autoregressive Models from the Reflection Coefficients 1

Summary: Computing the Poles of Autoregressive Models
from the Reflection Coefficients 1
Gregory S. Ammar 2 Daniela Calvetti 3 Lothar Reichel 4
A new approach to the computation of the poles of a stable autoregressive system
from the reflection coefficients is proposed. Equivalently, we compute the zeros of Szego
polynomials from the associated Schur parameters. The numerical method utilizes an
efficient algorithm for computing the (unimodular) zeros of a unitary Hessenberg matrix;
this step can be regarded as the computation of the poles of an associated lossless system.
These eigenvalues are then used as starting points for a continuation procedure for finding
the zeros of the desired polynomial. The procedure is efficient and parallelizable, and
may therefore be suitable for real­time applications.
1. Introduction
Autoregressive models are of fundamental importance in time series analysis and discrete­
time control. For example, in linear prediction, one is often given the autocorrelation
matrix of a wide sense stationary real­valued signal fx j g 1
. This matrix is a real
symmetric positive definite Toeplitz matrix
M n+1 = [¯ j \Gammak ] n


Source: Ammar, Greg - Department of Mathematical Sciences, Northern Illinois University


Collections: Mathematics