  This paper considers a multi-antenna multicast system with programmable
metasurface (PMS) based transmitter. Taking into account of the
finite-resolution phase shifts of PMSs, a novel beam training approach is
proposed, which achieves comparable performance as the exhaustive beam
searching method but with much lower time overhead. Then, a closed-form
expression for the achievable multicast rate is presented, which is valid for
arbitrary system configurations. In addition, for certain asymptotic scenario,
simple approximated expressions for the multicase rate are derived. Closed-form
solutions are obtained for the optimal power allocation scheme, and it is shown
that equal power allocation is optimal when the pilot power or the number of
reflecting elements is sufficiently large. However, it is desirable to allocate
more power to weaker users when there are a large number of RF chains. The
analytical findings indicate that, with large pilot power, the multicast rate
is determined by the weakest user. Also, increasing the number of radio
frequency (RF) chains or reflecting elements can significantly improve the
multicast rate, and as the phase shift number becomes larger, the multicast
rate improves first and gradually converges to a limit. Moreover, increasing
the number of users would significantly degrade the multicast rate, but this
rate loss can be compensated by implementing a large number of reflecting
elements.
