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Most prior works on deep learning-based wireless device classification using radio frequency (RF) data apply off-the-shelf deep neural network (DNN) models, which were matured mainly for domains like vision and language. However, wireless RF data possesses unique characteristics that differentiate it from these other domains. For instance, RF data encompasses intermingled time and frequency features that are dictated by the underlying hardware and protocol configurations. In addition, wireless RF communication signals exhibit cyclostationarity due to repeated patterns (PHY pilots, frame prefixes, and so on) that these signals inherently contain. In this article, we begin by explaining and showing the unsuitability as well as limitations of existing DNN feature design approaches currently proposed to be used for wireless device classification. We then present novel feature design approaches that exploit the distinct structures of RF communication signals and the spectrum emissions caused by transmitter hardware impairments to custom-make DNN models suitable for classifying wireless devices using RF signal data. Our proposed DNN feature designs substantially improve classification robustness in terms of scalability, accuracy, signature anti-cloning, and insensitivity to environment perturbations. We end the article by presenting other feature design strategies that have great potential for providing further performance improvements of the DNN-based wireless device classification, and discuss the open research challenges related to these proposed strategies.