Wirtinger Calculus-Based Expectation Propagation in Latent Variable Models Applied to Grant-Free NOMA

Grant-free non-orthogonal multiple access is an emerging communication paradigm, where devices transmit to an access point without explicit permission. However, unknown user activities add discrete latent variables on top of the usual hidden variables (accounting for channels and data), thus rendering exact inference - in the form a mixture of underlying distributions growing with time - intractable. Thus low-complexity user activity detection with reliable approximate distributed inference in the form of message-passing is relevant. We propose a new generic expectation propagation solution in the form of complex Gaussian distributions, that are consistently derived using a Wirtinger calculus-based second-order approximation. The proposed algorithm is then integrated with expectation propagation for other tasks (channel estimation, symbol detection and decoding) in a fully Bayesian inference setting. The main outcome is an excellent tradeoff between user missed detections and false alarms in a non-orthognal multiple access context, thus enabling further complexity reduction by gradually disabling message-passing for successfully identified inactive users. Numerical results demonstrate the superiority of the proposed method over similar approximations.