On the capacity of the multiple-hop relay channel with linear relaying

The relay nodes with linear relaying transmit linear combination of their past received signals. The capacity of the multiple-hop Gaussian relay channel with linear relaying is derived, assuming that each node in the channel only communicates with its nearest neighbor nodes. The capacity is formulated as an optimization problem over the relaying matrices and the covariance matrix of the signals transmitted from 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. We also show that the solution to the “single-letter” optimization problem has the same form as the expression of the rate achieved by time-sharing amplify-and-forward (TSAF). In order to solve this equivalent problem, we give an iterative algorithm. Simulation results show that the achievable rate with TSAF is close to the capacity, if channel gain of one certain hop is smaller than that of all the other hops relatively.