  This paper presents a two-stage online phase reconstruction framework using
causal deep neural networks (DNNs). Phase reconstruction is a task of
recovering phase of the short-time Fourier transform (STFT) coefficients only
from the corresponding magnitude. However, phase is sensitive to waveform
shifts and not easy to estimate from the magnitude even with a DNN. To overcome
this problem, we propose to use DNNs for estimating differences of phase
between adjacent time-frequency bins. We show that convolutional neural
networks are suitable for phase difference estimation, according to the
theoretical relation between partial derivatives of STFT phase and magnitude.
The estimated phase differences are used for reconstructing phase by solving a
weighted least squares problem in a frame-by-frame manner. In contrast to
existing DNN-based phase reconstruction methods, the proposed framework is
causal and does not require any iterative procedure. The experiments showed
that the proposed method outperforms existing online methods and a DNN-based
method for phase reconstruction.
