Robust Superimposed Training Designs for MIMO Relaying Systems under General Power Constraints

In this paper, we investigate the superimposed training matrix designs for the channel estimation of amplify-and-forward (AF) multiple-input multiple-output (MIMO) relaying systems under general power constraints. Furthermore, the imperfect channel and colored noise statistical information models with the corresponding nominal terms being Kronecker structure and the corresponding statistical errors belonging to the unitarily-invariant uncertainty sets are adopted. Based on the above analysis, the linear minimum mean-squared-error (LMMSE)-based robust training optimization problem is formulated, which is generally nonconvex and intractable. In order to effectively address this problem, we propose an iterative semidefinite programming (SDP) algorithm and two low-complexity upper bound optimization schemes. Particularly, for the proposed two upper bound optimization schemes, the diagonal structured optimal solutions of the relaxed robust training problems can be verified. Besides, the low-complexity iterative bisection search (IBS) can also be applied to derive the diagonal training matrix. Furthermore, we extend our work into the robust mutual information maximization of the AF MIMO relaying channel and demonstrate that all proposed robust training designs are still applicable. Finally, the numerical simulations illustrate the excellent performance of the proposed robust training designs in terms of the channel estimation MSE minimization and mutual information maximization.