Sparsity Constrained Joint Activity and Data Detection for Massive Access: A Difference-of-Norms Penalty Framework

Grant-free random access is a promising mechanism to support modern massive machine-type communications in which devices are sporadically active with small payloads. Under this random access, a unique challenge is the detection of device activity without the cooperation from devices. Furthermore, for only a few bits of data, it is more efficient to embed the data to the signature sequences so that the activity and data detection can be jointly carried out. However, compared with the vanilla device activity detection, joint activity and data detection has an extra discontinuous sparsity constraint, which makes the detection problem more challenging. In contrast to the prevalent way of first neglecting the discontinuous sparsity constraint and reinforcing it at the end, this paper proposes a novel approach to incorporate the discontinuous sparsity constraint into the optimization procedure. In particular, we first establish the equivalence between the discontinuous sparsity constraint and a continuous difference-of-norms (DN) form. Then, by introducing a DN penalty term in the objective function, an iterative DN penalty method with an increasing penalty weight is adopted. We prove theoretically that by solving each penalized problem to a stationary solution, the discontinuous sparsity constraint can be exactly satisfied when the penalty weight is sufficiently large, and the resulting solution is guaranteed to be at least a stationary point of the original problem. Due to the superior theoretical guarantee, simulation results demonstrate that the proposed method achieves around 10 times better detection performance than state-of-the-art approaches.