Identifying the modulation type of radio signals is challenging in both military and civilian applications such as radio monitoring and spectrum allocation. This has become more difficult as the number of signal types increases and the channel environment becomes more complex. Deep learning-based automatic modulation classification (AMC) methods have recently achieved state-of-the-art performance with massive amounts of data. However, existing models struggle to achieve the required level of accuracy, guarantee real-time performance at edge devices, and achieve higher classification performance on high-performance computing platforms when deployed on various platforms. In this paper, we present a family of AMC models based on communication domain knowledge for various computing platforms. The higher-order statistical properties of signals, customized data augmentation methods, and narrowband convolution kernels are the domain knowledge that is specifically employed to the AMC task and neural network backbone. We used separable convolution and depth-wise convolution with very few residual connections to create our lightweight model, which has only 4.61k parameters while maintaining accuracy. On the four different platforms, the classification accuracy and inference time outperformed those of the existing lightweight models. Meanwhile, we use the squeeze-and-excitation attention mechanism, channel shuffle module, and expert feature parallel branch to improve the classification accuracy. On the three most frequently used benchmark datasets, the high-accuracy models achieved state-of-the-art average accuracies of 64.63%, 67.22%, and 65.03%, respectively. Furthermore, we propose a generic framework for evaluating the complexity of deep learning models and use it to comprehensively assess the complexity strengths of the proposed models.