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Deep-learning-based automatic modulation recognition techniques have been extensively explored for wireless communication systems, because of their strong feature extraction and classification abilities of deep neural networks. Although the high recognition accuracy and low alarms, they also raise concerns about complexity and interpretability, which can affect the practical deployments. In this paper, we propose a randomly perturbation convolutional kernel activation mapping (RPCKAM) strategy to explain modulation recognition networks, and the RPCKAM-based filter pruning method to compress modulation recognition networks, combining the reasoning and interpretation of decisions made by modulation recognition networks with the compression of network models. The RPCKAM is used to evaluate the importance of convolutional kernels, and then the importance ranking is used to provide a basis for network pruning. Moreover, the effect of pruning is verified with VGG16 and ResNet34 as the baseline networks. The experimental results reflect that the RPCKAM-based convolutional kernel pruning can compress the model on a large scale while maintaining high accuracy, and can even improve the modulation recognition accuracy within a certain compression range. Specifically, the accuracy improvement is close to 1% at most in the VGG16, and it is also close to 0.5% at most in the ResNet34. Furthermore, by analyzing the performance changes before and after model compression, it is found that the compression effect of ResNet34 is better than that of VGG16, and the performance of the RPCKAM-based pruning method outperforms that of the Grad-CAM-based pruning method. [ABSTRACT FROM AUTHOR]