Abstract
This paper considers a low-complexity Gaussian message passing (GMP) Multi-User Detection (MUD) scheme for a coded massive multiple-input multiple-output (MIMO) system with non-orthogonal multiple access (massive MIMO-NOMA), in which a base station with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{s}$ </tex-math></inline-formula> antennas serves <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{u}$ </tex-math></inline-formula> sources simultaneously in the same frequency. Both <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{u}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{s}$ </tex-math></inline-formula> are large numbers, and we consider the overloaded cases with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{u}>N_{s}$ </tex-math></inline-formula> . The GMP for MIMO-NOMA is a message passing algorithm operating on a fully-connected loopy factor graph, which is well understood to fail to converge due to the correlation problem. The GMP is attractive as its complexity order is only linearly dependent on the number of users, compared to the cubic complexity order of linear minimum mean square error (LMMSE) MUD. In this paper, we utilize the large-scale property of the system to simplify the convergence analysis of the GMP under the overloaded condition. We prove that the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">variances</i> of the GMP definitely converge to the mean square error (MSE) of the LMMSE multi-user detection. Second, the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">means</i> of the traditional GMP will fail to converge when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{u}/N_{s}< (\sqrt {2}-1)^{-2}\approx 5.83$ </tex-math></inline-formula> . Therefore, we propose and derive a new convergent GMP called scale-and-add GMP (SA-GMP), which always converges to the LMMSE multi-user detection performance for any <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{u}/N_{s}>1$ </tex-math></inline-formula> , and show that it has a faster convergence speed than the traditional GMP with the same complexity. Finally, the numerical results are provided to verify the validity and accuracy of the theoretical results presented.
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