  This paper focuses on the design of non-orthogonal multiple access (NOMA) in
a classical two-transmitter two-receiver Z-channel, wherein one transmitter
sends information to its intended receiver from the direct link while the other
transmitter sends information to both receivers from the direct and cross
links. Unlike most existing designs using (continuous) Gaussian input
distribution, we consider the practical finite-alphabet (i.e., discrete) inputs
by assuming that the widely-used quadrature amplitude modulation (QAM)
constellations are adopted by both transmitters. To balance the error
performance of two receivers, we apply the max-min fairness design criterion in
this paper. More specifically, we propose to jointly optimize the scaling
factors at both transmitters, which control the minimum Euclidean distance of
transmitting constellations, to maximize the smaller minimum Euclidean distance
of two resulting constellations at the receivers, subject to an individual
average power constraint at each transmitter. The formulated problem is a mixed
continuous-discrete optimization problem and is thus intractable in general. By
resorting to the Farey sequence, we manage to attain the closed-form expression
for the optimal solution to the formulated problem. This is achieved by
dividing the overall feasible region of the original optimization problem into
a finite number of sub-intervals and deriving the optimal solution in each
sub-interval. Through carefully observing the structure of the optimal
solutions in all sub-intervals, we obtain compact and closed-form expressions
for the optimal solutions to the original problem in three possible scenarios
defined by the relative strength of the cross link. Simulation studies are
provided to validate our analysis and demonstrate the merits of the proposed
design over existing orthogonal or non-orthogonal schemes.
