Conditional mutual information (CMI) analysis and precoding design for generally correlated wireless multi-hop multi-input multi-output (MIMO) channels are presented in this paper. Although some particular scenarios have been examined in existing publications, this paper investigates a generally correlated transmission system having spatially correlated channel, mutually correlated source symbols, and additive colored Gaussian noise (ACGN). First, without precoding techniques, we derive the optimized source symbol covariances upon mutual information maximization. Secondly, we apply a precoding technique and then design the precoder in two cases: maximizing the mutual information and minimizing the detection error. Since the optimal design for the end-to-end system cannot be analytically obtained in closed form due to the non-monotonic nature, we relax the optimization problem and attain sub-optimal designs in closed form. Simulation results show that without precoding, the average mutual information obtained by the asymptotic design is very close to the one obtained by the optimal design, while saving a huge computational complexity. When having the proposed precoding matrices, the end-to-end mutual information significantly increases while it does not require resources of the system such as transmission power or bandwidth
