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In this paper, the training sequence design for multiple-input multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) systems under the minimum mean square error (MMSE) criterion is addressed. The optimal training sequence for channel estimation in spatially correlated MIMO-OFDM systems was not known for an arbitrary signal-to-noise ratio (SNR). Only one class of training sequences was proposed in the literature in which the power allocation is given only for the extreme conditions of low and high SNRs. The current paper presents a necessary and sufficient condition for the optimal training sequence, and reformulates the training design problem as a convex optimization problem whose optimal solution is efficiently solved. In addition, tight upper bounds for MMSE and resulting low complexity iterative algorithms with the closed-form expression in iterations to find the optimum training sequence are derived. Simulation results confirm the superiority of the proposed design over the existing one in terms of both MSE estimation and BER performance. The proposed methods are also shown to be robust with respect to the spatial correlation mismatch at the transmitter.