  We study a special class of diamond channels which was introduced by Schein
in 2001. In this special class, each diamond channel consists of a transmitter,
a noisy relay, a noiseless relay and a receiver. We prove the capacity of this
class of diamond channels by providing an achievable scheme and a converse. The
capacity we show is strictly smaller than the cut-set bound. Our result also
shows the optimality of a combination of decode-and-forward (DAF) and
compress-and-forward (CAF) at the noisy relay node. This is the first example
where a combination of DAF and CAF is shown to be capacity achieving. Finally,
we note that there exists a duality between this diamond channel coding problem
and the Kaspi-Berger source coding problem.
