In the edge computing, the communication between edge computing nodes and terminal devices is vulnerable to impersonation attacks, therefore a physical-layer impersonation attack detection algorithm based on Deep Q-Network (DQN) was proposed. Firstly, an impersonation attack model was built in the edge computing network, a hypothesis test based on the physical-layer Channel State Information (CSI) was established by the receiver, and the Euclidean distance between the currently measured CSI and the last recorded CSI was taken as the test statistics. Secondly, for the dynamic environment of edge computing, the DQN algorithm was used to adaptively select the optimal test threshold with the goal of maximizing the gain of the receiver. Finally, whether the current sender was an impersonation attacker was determined by comparing the statistics with the test threshold. The simulation results show that the Signal-to-Interference plus Noise Ratio (SINR) and channel gain ratio have certain effect on the performance of the detection algorithm, but when the relative change of channel gain is lower than 0.2, the false alarm rate, miss rate and average error rate of the algorithm are less than 5%. Therefore, the detection algorithm is adaptive to the dynamical environment of edge computing.