A subspace approach to estimation of autoregressive parameters from noisy measurements

This paper describes a method for estimating the parameters of an autoregressive (AR) process from a nite number of noisy measurements. The method uses a modi ed set of YuleWalker equations that lead to a quadratic eigenvalue problem which when solved, gives estimates of the AR parameters and the measurement noise variance. Version 2, submitted to IEEE Transactions on Signal Processing January 4, 1998 1This work was supported in part by a grant from the National Science Foundation (BCS-9308028) and a grant from the Whitaker Foundation. I. Background There are a number of applications where signals are modeled as autoregressive (AR) random processes. These include linear predictive coding (LPC) of speech [1], spectral estimation [2], biomedical signal processing [3], and time series forecasting [4]. When the signal to be modeled is observed in noise, the AR parameter estimates are biased and can produce misleading results. In this section, the problem of estimating AR parameters from noisy observations is formulated and some of the solutions that have been proposed are brie y described. In Sections II and III, a new method of estimating AR parameters is presented. This method is seen to be closely related to signal/noise subspace approaches used to estimate sinusoidal signal parameters. In Section IV the new method is compared with several well-known techniques and is seen to outperform them. A summary of the method is found in Section V. The following notation will be used throught: matrices are denoted by upper case letters (e.g., Ry; B), vectors are denoted by lower case letters (e.g., a; v); scalar elements of a sequence are denoted by lower case indexed letters (e.g., ry(n)), all other scalars are denoted by lower case greek letters (e.g., ). The p-order AR Random Process is de ned by x(n) = a(1)x(n 1) a(2)x(n 2) a(p)x(n p) + v(n) (1) where v(n) is white noise having variance 2 v and a(k); k = 1; : : : ; p are the AR parameters. Using the fact that the autocorrelation function, rx(k) of the AR process also satis es the autoregressive property rx(k) = p X i=1 a(i)rx(k i); k 1 (2) leads to the well-known Yule-Walker equations for the AR parameters 2 66664 rx(0) rx( (p 1)) .. . . . .. rx(p 1) rx(0) 3 77775 2 66664 a(1) .. a(p) 3 77775 = 2 66664 rx(1) .. rx(p) 3 77775 (3) Let y(n) = x(n) + z(n) (4)