  This paper considers a low-complexity Gaussian Message Passing Iterative
Detection (GMPID) algorithm for massive Multiuser Multiple-Input
Multiple-Output (MU-MIMO) system, in which a base station with $M$ antennas
serves $K$ Gaussian sources simultaneously. Both $K$ and $M$ are very large
numbers, and we consider the cases that $K<M$. The GMPID is a low-complexity
message passing algorithm based on a fully connected loopy graph, which is well
understood to be not convergent in some cases. As it is hard to analyse the
GMPID directly, the large-scale property of the massive MU-MIMO is used to
simplify the analysis. Firstly, we prove that the variances of the GMPID
definitely converge to the mean square error of Minimum Mean Square Error
(MMSE) detection. Secondly, we propose two sufficient conditions that the means
of the GMPID converge to those of the MMSE detection. However, the means of
GMPID may not converge when $ K/M\geq (\sqrt{2}-1)^2$. Therefore, a new
convergent GMPID called SA-GMPID (scale-and-add GMPID) , which converges to the
MMSE detection in mean and variance for any $K<M$ and has a faster convergence
speed than the GMPID, but has no higher complexity than the GMPID, is proposed.
Finally, numerical results are provided to verify the validity and accuracy of
the theoretical results.
