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Automatic modulation classification (AMC) is an critical step to identify signal modulation types so as to enable more accurate demodulation in the non-cooperative scenarios. Convolutional neural network (CNN)-based AMC is believed as one of the most promising methods with great classification accuracy. However, the conventional CNN-based methods are lack of generality capabilities under time-varying signal-to-noise ratio (SNR) conditions, because these methods are merely trained on specific datasets and can only work at the corresponding condition. In this paper, a novel CNN-based generalized AMC method is proposed, and a more realistic scenario is considered, including white non-Gaussian noise and synchronization error. Its generalization capability stems from the mixed datasets under varying noise scenarios, and the CNN can extract common features from these datasets. Simulation results show that our proposed architecture can achieve higher robustness and generalization than the conventional ones.