Gaussian two-hop relay channel with linear relaying

In this paper, the optimization problem of the two-hop relay channel with linear relaying is discussed. The expression of the capacity for the Gaussian two-hop relay channel with linear relaying is shown, which is an optimization problem over the relaying function and the covariance matrix of the signals transmitted by the source node. We show that the solution to the optimization problem is equivalent to a "single-letter" optimization problem, the solution to which has the same form as the expression of the rate achieved by time-sharing amplify and forward (TSAF). Taking a special case as example, we present that the achievable rate with TSAF and the maximal rate computed by exhaustively search are almost equal to each other at any SNR, which demonstrates the efficiency of TSAF.