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Optimal training design and channel estimation for spatially correlated multi-user multi-input multi-output with orthogonal frequency-division multiplexing (MIMO-OFDM) systems is still an open research topic of great interest. This study first applies tractable semi-definite programming (SDP) to obtain the optimal training signal for the general case of spatial channel correlations for multi-user MIMO-OFDM. In order to reduce the computational complexity of the SDP-based solution, an approximate solution in closed-form is then presented. For a special case of transmit correlations, an optimal solution in closed-form expression is also derived. Analytical and simulation results demonstrate the excellent performance of the proposed designs and their performance advantage over the existing equi-powered training designs.