Diversity effects in the estimation of a Gauss-Markov process over a fading channel

A scalar first-order Gauss-Markov process is sent over d parallel i.i.d Rayleigh fading channels using analog transmission. With known channel realization at the decoder, the optimal MMSE estimator is the Kalman filter with random instantaneous expected estimation error variance. Statistical characteristics of the instantaneous expected 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 expected estimation error crosses a certain threshold. In this paper, we study the effect of channel diversity on the estimation error outage. We show that for a certain range of outage thresholds, the estimation error outage decreases with d-th power of the average channel SNR. 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.