© 2020 IEEE.Automatic modulation classification (AMC) has been studied for more than a quarter of a century; however, it has been difficult to design a classifier that operates successfully under changing multipath fading conditions and other impairments. Recently, deep learning (DL)-based methods are adopted by AMC systems and major improvements are reported. In this paper, a novel convolutional neural network (CNN) classifier model is proposed to classify modulation classes in terms of their families, i.e., types. The proposed classifier is robust against realistic wireless channel impairments and in relation to that, when the data sets that are utilized for testing and evaluating the proposed methods are considered, it is seen that RadioML2016.10a is the main dataset utilized for testing and evaluation of the proposed methods. However, the channel effects incorporated in this dataset and some others may lack the appropriate modeling of the real-world conditions since it only considers two distributions for channel models for a single tap configuration. Therefore, in this paper, a more comprehensive dataset, named as HisarMod2019.1, is also introduced, considering real-life applicability. HisarMod2019.1 includes 26 modulation classes passing through the channels with 5 different fading types and several number of taps for classification. It is shown that the proposed model performs better than the existing models in terms of both accuracy and training time under more realistic conditions. Even more, surpassed their performance when the RadioML2016.10a dataset is utilized.