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Speech enhancement techniques that use a generative adversarial network (GAN) can effectively suppress noise while allowing models to be trained end-to-end. However, such techniques directly operate on time-domain waveforms, which are often highly-dimensional and require extensive computation. This paper proposes a novel GAN-based speech enhancement method, referred to as S-ForkGAN, that operates on log-power spectra rather than on time-domain speech waveforms, and uses a forked GAN structure to extract both speech and noise information. By operating on log-power spectra, one can seamlessly include conventional spectral subtraction techniques, and the parameter space typically has a lower dimension. The performance of S-ForkGAN is assessed for automatic speech recognition (ASR) using the TIMIT data set and a wide range of noise conditions. It is shown that S-ForkGAN outperforms existing GAN-based techniques and that it has a lower complexity.