Recent advances have made it possible to create deep complex-valued neural networks. Despite this progress, many challenging learning tasks have yet to leverage the power of complex representations. Building on recent advances, we propose a new deep complex-valued method for signal retrieval and extraction in the frequency domain. As a case study, we perform audio source separation in the Fourier domain. Our new method takes advantage of the convolution theorem which states that the Fourier transform of two convolved signals is the elementwise product of their Fourier transforms. Our novel method is based on a complex-valued version of Feature-Wise Linear Modulation (FiLM) and serves as the keystone of our proposed signal extraction method. We also introduce a new and explicit amplitude and phase-aware loss, which is scale and time invariant, taking into account the complex-valued components of the spectrogram. Using the Wall Street Journal Dataset, we compared our phase-aware loss to several others that operate both in the time and frequency domains and demonstrate the effectiveness of our proposed signal extraction method and proposed loss.

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