  Optimal beamforming designs under imperfect successive interference
cancellation (SIC) decoding for a symbiotic network of non-orthogonal multiple
access (NOMA) primary users and a secondary ambient tag have been lacking. We
address that issue here. The primary base station (BS) serves NOMA users and a
passive tag simultaneously in this network. We develop two transmit beamforming
designs to meet the user and tag requirements while mitigating the effect of
imperfect SIC. Specifically, we design optimal BS transmit beamforming and
power allocation to either maximize the weighted sum rate of NOMA users and the
tag or minimize the BS transmit power under the minimum rate requirements while
satisfying the tag minimum energy requirement. Because both these problems are
non-convex, we propose algorithms using alternative optimization, fractional
programming, and semi-definite relaxation techniques. We also analyze their
computational complexity. Finally, we present extensive numerical results to
validate the proposed schemes and to show significant performance gains while
keeping the tag design intact. For example, the proposed digital beamforming
increases the harvested power and data rate by 2.16e3 % and 314.5 % compared to
random beamforming.
