The integration of radar and communication systems in 6G networks has led to a significant challenge of spectrum congestion. To address this issue, we propose a deep-learning-based method for efficient waveform-based signal classification. Our method is designed to handle large and impaired radar and communication signals and is crucial for the implementation of resource-limited cognitive radio-enabled Internet of Things (CR-IoT) devices. We introduce waveform recognition network (WaveNet), a cost-efficient deep convolutional neural network that can aptly learn underlying radio features from time–frequency images transformed by a smooth pseudo Wigner–Ville distribution. WaveNet incorporates several innovative modules, including cost-efficient feature awareness, which integrates two well-designed structural blocks: 1) grouped-of-kernel-wise residual connections and 2) dual asymmetric channel attention. These enhancements significantly reduce network size without compromising classification accuracy. Based on various simulations experimented on an impaired signal data set containing eight radar and communication waveform types, the results demonstrate the effectiveness and robustness of WaveNet, achieving an overall classification accuracy of 92.02%. Compared to the current state-of-the-art deep models, WaveNet has the lowest architectural complexity, with a network size five times smaller, while still outperforming them by approximately 0.5%–1.69%. Consequently, WaveNet emerges as a valuable solution for waveform classification in integrated radar–communication 6G systems.