Uniform observability and exponential convergence rate of the Kalman filter for the FIR deconvolution problem

In this paper we carry out a detailed stability analysis of the Kalman filt er for the time-varying finite impulse response (FIR) deconvolution problem , with white input and white noise. Although the exponential stability of t he Kalman filter is known from general results (since the system matrix is stable), the exponential convergence rate is unknown. The problem of determ ining the convergence rate is complicated by the singularity of the system matrix corresponding to the FIR response. A new definition of uniform obser vability is given which allows for singular system matrices, and a simple c haracterization of this property for the FIR system/channel is found. Under this observability condition, bounds on the exponential convergence rate o f the Kalman filter are obtained which show qualitatively the worst-case de pendence of the rate on the channel. (C) 2001 Elsevier Science B.V. All rig hts reserved.