Automatic modulation classification (AMC) is crucial for the subsequent analysis and processing of modulated signals. In this paper, we propose an AMC method based on bispectrum and convolutional neural network (CNN) Alexnet model (Bispectrum-Alexnet) after studying the latest achievements in related fields. The proposed Bispectrum-Alexnet method feeds the CNN with amplitude spectrums of bispectrum (ASB) belonging to different signals, enabling it to learn and extract deep features automatically from those images, and finally obtain the ideal label. To present our work more clearly, firstly, we analyze the feasibility of using ASB for classification, especially focusing on the physical meaning of ASB. Secondly, an introduction to the complete algorithm implementation is given, including bispectrum estimation and CNN architecture. Finally, this method was designed to distinguish among 7 popular modulation signals (BPSK, 2ASK, 2FSK, 4FSK, 8FSK, LFM, OFDM) to validate the recognition performance. Simulation results show that the overall accuracy ratio can reach 97.7% when the symbolic signal to noise ratio is no less than 5dB.