  This paper considers precoding for multi-group multicasting with a common
message. The multiple antenna base station communicates with $K$ clusters, each
with $L$ users. There is a common message destined to all users and a private
multicast message for each cluster. We study the weighted sum rate (WSR)
maximization problem for two different schemes: (i) the base station transmits
the superposition of common and multicast messages, (ii) the base station
concatenates the multicast message vector with the common message. We also
formulate a second problem, weighted minimum mean square error (WMMSE)
minimization, and prove that WSR maximization and WMMSE minimization are
equivalent at the optimal solution. Inspired by the WMMSE problem, we suggest a
suboptimal algorithm, based on alternating optimization. We apply this
algorithm to the two transmission schemes, and understand that there is a
fundamental difference between the two. We compare the results with maximal
ratio transmission (MRT), and zero-forcing (ZF) precoding, and investigate the
effects of the number of base station antennas, the number of groups and the
number of users in a group. Finally, we study imperfect successive interference
cancellation (SIC) at the receivers and show that the first transmission scheme
is more robust.
