Limiting performance of optimal linear filters

We study the lowest achievablemean square estimation error in two limiting optimal linear filtering problems. First, when the intensity of the process noise tends to zero, the lowest achievable mean square estimation error is a function of the unstable poles of the system. Second, when the intensity of the measurement noise tends to zero, the lowest achievable mean square estimation error is a function of the nonminimum phase zeros of the system. We link these results with Bode integral characterizations of performance limitations in linear filtering.