Large system analysis of iterative multiuser joint decoding with an uncertain number of users

We study iterative multiuser joint decoding in large randomly spread code division multiple access systems under the assumption that the number of users accessing the channel is unknown by the receiver. In particular, we focus on the factor graph representation and iterative algorithms based on belief propagation. We study a suboptimal iterative scheme that jointly detects the encoded data and the users' activity. By using the replica method from statistical physics, we analyze the performance of the iterative detector. Using density evolution, we provide a fixed-point equation of the overall iterative system where the probability messages depend on the users' activity. Finally, when the scaling between the log number of users and the block length is below a threshold, we show that in the large-system limit a simple structure on the users' codes yields a multiuser efficiency fixed-point equation that is equivalent to the case of all-active users with a system load scaled by the activity rate.