Robust Training Sequence Design for Correlated MIMO Channel Estimation

We study how to design a worst-case robust training sequence for multiple-input multiple-output (MIMO) channel estimation. We consider mean-squared error of channel estimates as the figure of merit which is a function of second-order statistics of the MIMO channel, i.e., channel covariance matrix, in order to optimize training sequences under a total power constraint. In practical applications, the channel covariance matrix is not known perfectly. Thus the main aspect of our design is to improve robustness of the training sequences against possible uncertainties in the available channel covariance matrix. Using a deterministic uncertainty model, we formulate a robust training sequence design as a minimax optimization problem where we take such imperfections into account. We investigate the robust design problem assuming the general case of an arbitrarily correlated MIMO channel and a non-empty compact convex uncertainty set. We prove that such a problem admits a globally optimal solution by exploiting the convex-concave structure of the objective function, and propose numerical algorithms to address the robust training design problem. We proceed the analysis by considering multiple-input single-output (MISO) channels and Kronecker structured MIMO channels along with unitarily-invariant uncertainty sets. For these scenarios, we show that the problem is diagonalized by the eigenvectors of the nominal covariance matrices so that the robust design is significantly simplified from a complex matrix-variable problem to a real vector-variable power allocation problem. For the MISO channel, we provide closed-form solutions for the robust training sequences with the uncertainty sets defined by the spectral norm and nuclear norm.