Diversity Effects in Kalman Filtering Over Rayleigh Fading Channels

A scalar first-order Gauss-Markov signal is sent over d parallel i.i.d. Rayleigh fading channels using uncoded transmission. Assuming perfect channel estimation at the receiver, the optimal MMSE estimator is the Kalman filter with random instantaneous estimation error variance. Statistical characteristics of the instantaneous estimation error variance are studied to characterize the estimation quality. One such characteristic is the estimation error outage which probabilistically depicts how often the value of the instantaneous estimation error variance exceeds a certain threshold. In this paper, we study the effect of channel diversity on the estimation error outage probability. We show that for a certain range of outage thresholds, the estimation error outage probability decreases with inverse of d-th power of the average signal-to-noise ratio at the receiver side. This means that analogous to the context of digital communication over fading channels, our result implies a diversity order of d in estimating the input process.