Fast adaptive spectrum estimation: Bayesian approach and long AR models

Adaptive spectrum estimation is based on a local stationarity assumption for the studied process, and uses methods of the stationary case with data windows of reduced length. But conventional least squares methods and parsimony principle (for example Akaïke´s criterion) preclude use of long AR models necessary for a good spectral resolution. We develop here a bayesian adaptive spectrum estimation method using long AR models and normal prior distributions expressing a smoothness priors on the solution. This is now a classical approach to spectrum estimation. The main originality of our approach lies in the order choice and in the computation of the solution which is performed by a fast Kalman filter of the Chandrasekhar type (B-CAR), with a reduced complexity of O(p) per recursion, p being the model order. The likelihood of the regularizing factor which weights the smoothness priors is maximized to obtain the best data-dependent priors and is computed recursively as a by-product of our fast Kalman filter, which facilitates the determination of the hyperparameters. The method performances are illustrated by examples of adaptive spectrum estimation for simulated signals and Doppler signals.