Abstract: For scheduling-free non-orthogonal multiple access (NOMA) systems, the uplink channel estimation (CE) and active user detection (AUD) issues in multi-user transmission scenarios can be modeled as the reconstruction problem of compressed sensing. In view of the shortcomings of the existing compressed sensing algorithms, this paper proposes an improved Approximate Message Passing (AMP) algorithm-Threshold Adaptive Constrained Reweighted Approximate Message Passing (TA-CR-AMP) algorithm to combine Solve CE and AUD problems. The algorithm adds the operation of updating the sparse structure of the sparse signal to the AMP algorithm under a suitable iteration termination criterion, and on this basis introduces a constrained heavy weighting to the algorithm, and makes the threshold adaptively change. The simulation results show that, compared with the AMP algorithm, the algorithm proposed in this paper obtains more superior channel estimation and active user detection performance with lower complexity. In addition, the algorithm proposed in this paper has achieved similar performance to the more complex Expectation Maximization Bayesian Approximate Message Passing (EM-B-AMP) algorithm.