Optimal Training Sequence for MIMO Wireless Systems in Colored Environments

In this paper, we design the training signal for a multi-input multi-output (MIMO) communication system in a colored medium. We assume that the known channel covariance matrix (CM) is a Kronecker product of a transmit channel CM and a receive channel CM. Similarly, the CM of the additive Gaussian noise is modeled by a Kronecker product of a temporal CM and a spatial CM. We maximize the differential entropy gained by receiver for a limited energy budget for training at the transmitter. Using, singular value decomposition of the involved CMs, we turn this problem into a convex optimization problem. We prove that the left and right singular vectors of the optimal training matrix are eigenvectors of the channel transmit CM and the noise temporal CM. In general case, this problem can be solved numerically using efficient methods. The impact of the optimal training is more significant in environments with larger eigenvalue spread. The expression of the optimal solution is interesting for some specific cases. For uncorrelated receive channel, the optimal training looks like water filling, i.e., more training power must be invested on the directions which have more impact. For high signal-to-noise ratios (SNRs), any orthogonal training is optimal; this means that if large amount of energy is available, it must be invested uniformly in all directions. In low SNR scenarios where low amount of energy is available for channel training, all the energy must be allocated to the best mode of channel (which has the highest ratio of the received channel variance to the received noise power).