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In this paper, we study the achievable rate of the training-based multi-input multi-output (MIMO) systems over time-varying flat fading channels modeled with the L-th order autoregressive, AR(L), process. Using the Bayesian Cramer-Rao lower bound (BCRB) to characterize the mean squared error of channel estimation, the achievable rate of the MIMO systems is investigated from the information-theoretical perspective. The optimum lengths for the training and the data blocks are determined to maximize the achievable rate. Besides, by modeling the time-varying wireless fading channel with a proper AR(L) model for the specific normalized Doppler frequency, fdTs, the aforementioned results can be extended to characterize the achievable rate in a more realistic time-varying wireless fading channel.