  Many signal processing applications require estimation of time-varying sparse
signals, potentially with the knowledge of an imperfect dynamics model. In this
paper, we propose an algorithm for dynamic filtering of time-varying sparse
signals based on the sparse Bayesian learning (SBL) framework. The key idea
underlying the algorithm, termed SBL-DF, is the incorporation of a signal
prediction generated from a dynamics model and estimates of previous time steps
into the hyperpriors of the SBL probability model. The proposed algorithm is
robust to imperfect dynamics models (due to the propagation of dynamics
information through higher-order statistics), robust to certain undesirable
dictionary properties such as coherence (due to properties of the SBL
framework), allows the use of arbitrary dynamics models, and requires the
tuning of fewer parameters than many other dynamic filtering algorithms do. We
also extend the fast marginal likelihood SBL inference procedure to the
informative hyperprior setting to create a particularly efficient version of
the SBL-DF algorithm. Numerical simulations show that SBL-DF converges much
faster and to more accurate solutions than standard SBL and other dynamical
filtering algorithms. In particular, we show that SBL-DF outperforms state of
the art algorithms when the dictionary contains challenging structure such as
coherence and column scaling, as is found in many practical applications.
