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Channel characteristic-based physical layer authentication is one potential identity authentication scheme in wireless communication, such as used in a fog computing environment. While existing channel characteristic-based physical layer authentication schemes may be efficient when deployed in the conventional wireless network environment, they may be less efficient and practical for the industrial wireless communication environment due to the varying requirements. We observe that this is a topic that is understudied, and therefore in this paper, we review the constructions and performance of several commonly used test statistics and analyze their performance in typical industrial wireless networks using simulation experiments. The findings from the simulations show a number of limitations in existing channel characteristic-based physical layer authentication schemes. Therefore, we believe that it is a good idea to combine machine learning and multiple test statistics for identity authentication in future industrial wireless network deployment. Four machine learning methods prove that the scheme significantly improves the authentication accuracy and solves the challenge of choosing a threshold. [ABSTRACT FROM AUTHOR]