The quality of wireless communication will directly affect the performance of federated learning (FL), so this paper analyze the influence of wireless communication on FL through symbol error rate (SER). In FL system, non-orthogonal multiple access (NOMA) can be used as the basic communication framework to reduce the communication congestion and interference caused by multiple users, which takes advantage of the superposition characteristics of wireless channels. The Minimum Mean Square Error (MMSE) based serial interference cancellation (SIC) technology is used to recover the gradient of each terminal node one by one at the receiving end. In this paper, the gradient parameters are quantized into multiple bits to retain more gradient information to the maximum extent and to improve the tolerance of transmission errors. On this basis, we designed the SER-based device selection mechanism (SER-DSM) to ensure that the learning performance is not affected by users with bad communication conditions, while accommodating as many users as possible to participate in the learning process, which is inclusive to a certain extent. The experiments show the influence of multi-bit quantization of gradient on FL and the necessity and superiority of the proposed SER-based device selection mechanism.