Abstract: For the channel estimation (CE) and the multiuser detection (MUD) with active user detection and data detection problems in the grant-free non-orthogonal multiple access (NOMA) system, this paper proposes a threshold aided-distributed weak selection stagewise adaptive matching pursuit (TA-DWSStAMP ) algorithm to jointly solve the CE and MUD problems in the multiple measurement vector-compressive sensing(MMV-CS) model. The algorithm terminates at precise iterations under the criterion and introduces a new identification parameter. When the identification parameter represents a large step, a variable step size method based on power function is designed. Simulation results show that, as compared to the existing algorithm, the proposed TA-DWSStAMP algorithm can obtain similar successful activity detection rate of the users, symbol error rate of the user data and the normalized mean squared error performance of the channel estimation. However, its computational complexity only accounts for about 10% of the existing algorithm.