As the threats of small drones increase, not only the detection but also the classification of small drones has become important. Many recent studies have applied an approach to utilize the micro-Doppler signature (MDS) for the small drone classification by using frequency modulated continuous wave (FMCW) radars. In this letter, we propose a novel method to extract the MDS images of the small drones with the FMCW radar. Moreover, we propose a light convolutional neural network (CNN) whose structure is straightforward, and the number of parameters is quite small for fast classification. The proposed method contributes to increasing the classification accuracy by improving the quality of MDS images. We classified the small drones with the MDS images extracted by the conventional method and the proposed method through the proposed CNN. The experimental results showed that the total classification accuracy was increased by 10.00 % due to the proposed method. The total classification accuracy was recorded at 97.14 % with the proposed MDS extraction method and the proposed light CNN.