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In this paper, we investigate the optimality of training signals for linear minimum mean square error (LMMSE) channel estimation in multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) with frequency-selective fading channels. This is a very challenging problem due to its mathematical intractability and has not been analytically solved in the literature. Using the Lagrange multiplier method, we derive the optimality conditions for training signal design. Important findings revealed on optimal training signals are twofold: (i) the energies of the training signals on each subcarrier are equal, and (ii) on each subcarrier, the training signals transmitted from the different antennas are orthogonal and of equal energy. We verify that our results are in line with the design principles that have been derived in single-carrier MIMO systems. Two types of optimal training signal examples that satisfy the optimality conditions are presented for practical implementations in MIMO-OFDM systems. Simulation results show that the training signals based on the optimality conditions outperform other non-optimal training signals in terms of channel estimation performance.

On the optimality of training signals for MMSE channel estimation in MIMO-OFDM systems