Fast filtering of noisy autoregressive signals

Autoregressive (AR) models are used in a wide variety of applications concerning the recovery of signals from noise-corrupted observations. In all real contexts of this kind also an additive broadband observation noise is present and the filtering of the observations is usually performed by means of standard Kalman filtering that requires a state space realization of the AR model to describe the observed process and the solution, at every step, of the Riccati equation. This paper proposes a faster filtering algorithm suitable for stationary processes and based on the decomposition of Toeplitz matrices described in [J. Rissanen, Algorithms for triangular decomposition of block Hankel and Toeplitz matrices with application to factoring positive matrix polynomials, Math. Comput. 27 (January 1973) 147–154] that operates directly on AR models. The computational complexity of the proposed algorithm increases only linearly with the order of the process.