  Although extensively investigated, the capacity of the two-hop half-duplex
(HD) relay channel is not fully understood. In particular, a capacity
expression which can be evaluated straightforwardly is not available and an
explicit coding scheme which achieves the capacity is not known either. In this
paper, we derive a new expression for the capacity of the two-hop HD relay
channel based on a simplified converse. Compared to previous results, this
capacity expression can be easily evaluated. Moreover, we propose an explicit
coding scheme which achieves the capacity. To achieve the capacity, the relay
does not only send information to the destination by transmitting
information-carrying symbols but also with the zero symbols resulting from the
relay's silence during reception. As examples, we compute the capacities of the
two-hop HD relay channel for the cases when the source-relay and
relay-destination links are both binary-symmetric channels (BSCs) and additive
white Gaussian noise (AWGN) channels, respectively, and numerically compare the
capacities with the rates achieved by conventional relaying where the relay
receives and transmits in a codeword-by-codeword fashion and switches between
reception and transmission in a strictly alternating manner. Our numerical
results show that the capacities of the two-hop HD relay channel for BSC and
AWGN links are significantly larger than the rates achieved with conventional
relaying.
