This paper observes that there is an issue of high frequencies missing in the discriminator of standard GAN, and we reveal it stems from downsampling layers employed in the network architecture. This issue makes the generator lack the incentive from the discriminator to learn high-frequency content of data, resulting in a significant spectrum discrepancy between generated images and real images. Since the Fourier transform is a bijective mapping, we argue that reducing this spectrum discrepancy would boost the performance of GANs. To this end, we introduce SSD-GAN, an enhancement of GANs to alleviate the spectral information loss in the discriminator. Specifically, we propose to embed a frequency-aware classifier into the discriminator to measure the realness of the input in both the spatial and spectral domains. With the enhanced discriminator, the generator of SSD-GAN is encouraged to learn high-frequency content of real data and generate exact details. The proposed method is general and can be easily integrated into most existing GANs framework without excessive cost. The effectiveness of SSD-GAN is validated on various network architectures, objective functions, and datasets. Code is available at https://github.com/cyq373/SSD-GAN.