Finite-Alphabet Noma For Two-User Uplink Channel

We consider the non-orthogonal multiple access (NOMA) design for a classical two-user multiple access channel (MAC) with finite-alphabet inputs. In contrast to the majority of existing NOMA schemes using continuous Gaussian distributed inputs, we consider practical quadrature amplitude modulation (QAM) constellations at both transmitters, whose sizes are not necessarily the same. By adjusting the scaling factors (i. e., instantaneous transmitting powers) of both users, we aim to maximize the minimum Euclidean distance of the received sum-constellation for a maximum likelihood (ML) receiver. The formulated problem is a mixed continuous-discrete optimization problem and in general it is non-trivial to resolve. By carefully examining the structure of the objective function, we discover that Farey sequence can be employed to tackle the formulated problem. However, the existing Farey sequence is not applicable when the constellation sizes of the two users are different. To address this challenge, we define a new type of Farey sequence, termed punched Farey sequence. Based on this new definition and its properties, we manage to attain a closed-form optimal solution to the original problem by first dividing the entire feasible region into a finite number of Farey intervals and then taking the maximum over all the subintervals. Finally, computer simulations are carried out to verify our theoretical analysis, and to demonstrate the advantages of the proposed NOMA over known orthogonal and non-orthogonal designs.