[Paper Review] Uplink Non-Orthogonal Multiple Access with Finite-Alphabet Inputs

Zheng Dong, He Chen+3 more|arXiv (Cornell University)|Sep 2, 2017

Advanced Wireless Communication Technologies51 references3 citations

TL;DR

This paper proposes a novel non-orthogonal multiple access (NOMA) design for a two-user uplink multiple access channel with finite-alphabet QAM inputs, optimizing power and phase scaling to maximize the minimum Euclidean distance of the received sum-constellation. By introducing a new 'punched Farey sequence' to solve a mixed continuous-discrete optimization problem, the method achieves a regular M₁²M₂²-QAM sum-constellation, enabling low-complexity ML detection and proving NOMA outperforms TDMA in minimum distance under total rate constraints.

ABSTRACT

This paper focuses on the non-orthogonal multiple access (NOMA) design for a classical two-user multiple access channel (MAC) with finite-alphabet inputs. We consider practical quadrature amplitude modulation (QAM) constellations at both transmitters, the sizes of which are assumed to be not necessarily identical. We propose to maximize the minimum Euclidean distance of the received sum-constellation with a maximum likelihood (ML) detector by adjusting the scaling factors (i.e., instantaneous transmitted powers and phases) of both users. The formulated problem is a mixed continuous-discrete optimization problem, which is nontrivial to resolve in general. By carefully observing the structure of the objective function, we discover that Farey sequence can be applied to tackle the formulated problem. However, the existing Farey sequence is not applicable when the constellation sizes of the two users are not the same. Motivated by this, we define a new type of Farey sequence, termed punched Farey sequence. Based on this, we manage to achieve 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 possible intervals. The resulting sum-constellation is proved to be a regular QAM constellation of a larger size. Moreover, the superiority of NOMA over time-division multiple access (TDMA) in terms of minimum Euclidean distance is rigorously proved. Furthermore, the optimal rate allocation among the two users is obtained in closed-form to further maximize the obtained minimum Euclidean distance of the received signal subject to a total rate constraint. Finally, simulation results are provided to verify our theoretical analysis and demonstrate the merits of the proposed NOMA over existing orthogonal and non-orthogonal designs.

Motivation & Objective

• To address the lack of practical NOMA designs using finite-alphabet inputs like QAM, rather than idealized Gaussian inputs.
• To maximize the minimum Euclidean distance of the received sum-constellation in a two-user uplink NOMA system with non-identical QAM constellation sizes.
• To develop a closed-form solution for power and phase scaling under total rate constraints, enabling optimal rate allocation.
• To prove that NOMA with finite-alphabet inputs outperforms time-division multiple access (TDMA) in terms of minimum distance.
• To design a practical, low-complexity detection scheme by ensuring the sum-constellation is a regular QAM constellation.

Proposed method

• Introduces a new mathematical construct called the 'punched Farey sequence' to handle mixed continuous-discrete optimization when QAM constellation sizes differ.
• Divides the feasible power and phase space into finite Farey intervals based on the new sequence to enable systematic optimization.
• Uses maximum likelihood (ML) detection by ensuring the sum-constellation is a regular QAM constellation of size M₁²M₂².
• Derives closed-form optimal power and phase scaling factors (w₁*, w₂*) based on channel gains and constellation sizes.
• Applies algebraic manipulation using channel gains |h₁|, |h₂| and QAM parameters M₁, M₂ to express the optimal solution in terms of normalized channel gains.
• Derives asymptotic solutions to reveal optimal rate allocation insights under high-SNR or extreme channel gain regimes.

Experimental results

Research questions

• RQ1 Can a closed-form solution be derived for NOMA power and phase optimization under finite-alphabet QAM inputs with non-identical constellation sizes?
• RQ2 How does the proposed NOMA design compare to TDMA in terms of minimum Euclidean distance for the received sum-constellation?
• RQ3 Can the sum-constellation be made a regular QAM constellation to allow low-complexity detection?
• RQ4 What is the optimal rate allocation between two users to maximize the minimum distance under a total rate constraint?
• RQ5 How do channel gain ratios and constellation sizes jointly affect the performance gain of NOMA over TDMA?

Key findings

• The proposed NOMA design achieves a regular M₁²M₂²-QAM sum-constellation, enabling simple quantization-based ML detection.
• The minimum Euclidean distance of the NOMA sum-constellation is proven to be strictly greater than that of TDMA under all channel gain ratios.
• The optimal power and phase scaling factors are derived in closed-form, depending on the ratio |h₂|/|h₁| and the constellation sizes M₁ and M₂.
• The minimum distance is maximized under a total rate constraint, with optimal rate allocation derived in closed-form.
• Asymptotic analysis reveals that when one user has a significantly stronger channel, the system should allocate more rate to the weaker user to maximize diversity gain.
• Simulation results confirm the theoretical analysis and demonstrate superior performance of the proposed NOMA over both orthogonal and non-orthogonal designs in terms of minimum distance and error rate.