Fundamental error bounds in state estimation: An information-theoretic analysis

This paper investigates fundamental performance bounds on estimation error for general state estimation systems that are not necessarily linear time invariant with noises that are not necessarily white Gaussian. In the analysis, concepts from information theory such as entropy play an instrumental role. We first propose an information-theoretic notion termed Gaussianity-whiteness to measure how Gaussian and white an asymptotically stationary stochastic process is. Subsequently, we derive lower bounds on estimation error variance which can be quantified explicitly by the Gaussianity-whiteness of the noises. Furthermore, the bounds are found to be tight in the particular case of a scalar linear time-invariant system with white Gaussian noises, as verified by the benchmark given by the renowned Kalman filter.