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Wireless device classification techniques play a key role in promoting emerging wireless applications such as allowing spectrum regulatory agencies to enforce their access policies and enabling network administrators to control access and prevent impersonation attacks to their wireless networks. Leveraging spectrum distortions of transmitted RF signals, caused by transceiver hardware impairments created during manufacture and assembly stages, to provide device classification has been the focus of many recent works. These prior works essentially apply deep learning to extract features of the devices from their hardware-impaired signals and rely on feature variations across the devices to distinguish devices from one another. As technology advances, the manufacturing impairment variations across devices are becoming extremely insignificant, making these prior classification approaches inaccurate. This paper proposes a novel, deep learning-based technique that provides scalable and highly accurate classification of wireless devices, even when the devices exhibit insignificant variation across their hardware impairments and have the same hardware, protocol, and software configurations. The novelty of the proposed technique lies in leveraging both the {\em in-band} and {\em out-of-band} signal distortion information by oversampling the captured signals at the receiver and feeding IQ samples collected from the RF signals to a deep neural network for classification. Using a convolutional neural network (CNN) model, we show that our proposed technique, when applied to high-end, high-performance devices with minimally distorted hardware, doubles the device classification accuracy when compared to existing approaches.