A new way to invert the sliding Fourier transform and its application to signal separation

In this paper we propose a simple method to invert the Fourier transform computed using zero-padding and sliding windows with delay of one point. Supposing that the window size is less than the number of frequencies, we show that the time-domain signal can be recovered multiplying the frequency-domain signal by a 2 × 2 matrix. Using this result, we propose a criterion to separate convolutive mixtures of signals in the frequency domain. The proposed approach has a reduced computational cost because only two frequency bins are considered.