Achievable Rate for Multiple-hop Gaussian Relay Channel with Linear Relaying

In this paper, the optimization problem of the multiple-hop relay channel with linear relaying is discussed. The capacity for the multiple-hop Gaussian relay channel with linear relaying is derived, which can be formulated as an optimization problem over the relaying matrices and the covariance matrix of the signals transmitted at the source. It is proved that the solution to this optimization problem is equivalent to a "single-letter" optimization problem when some certain conditions are satisfied. And the solution to this "single-letter" optimization problem has the same form as the expression of the rate achieved by time-sharing amplify and forward (TSAF). Simulation results show that the achievable rate with linear relaying is close to the maxflow mm-cut capacity, if channel gain of one certain hop is smaller than that of all the other hops relatively.