On the noise-compensated Yule-Walker equations

Recently a method of estimating the parameters of an AR(p) random process based on measurements corrupted by additive white noise was described. The method involves solving a matrix pencil, called the Noise-Compensated Yule-Walker (NCYW) equations, for the AR parameters and the variance of the measurement noise. In this correspondence we give a necessary and sufficient condition for there to exist a unique solution to the NCYW equations. submitted to IEEE Transactions on Signal Processing April 20, 2001 I. Background The pth-order AR (AR(p)) Random Process is given by x(n) = −a(1)x(n− 1)− a(2)x(n− 2)− · · · − a(p)x(n− p) + w(n) (1) where w(n) is white noise having variance σ2 w and a(k), k = 1, . . . , p are the AR parameters. We assume that x(n) is real. The autocorrelation function of the AR process, rx(k), also satisfies the autoregressive property, this leads to the well-known Yule-Walker equations for the AR parameters