Reconstruction of a Signal from the Real Part of Its Discrete Fourier Transform [Tips & Tricks]
Stephen So, Kuldip K. Paliwal · IEEE Signal Processing Magazine · 2018
In this tutorial, we present a procedure for reconstructing a complex-valued, discrete-time signal from only partial Fourier transform (FT) information, more specifically, the real part of its discrete FT (RDFT). By applying a delay, coupled with appropriate zero-padding to ensure a sufficiently dense sampling of the frequency axis, we show that any signal can be reconstructed perfectly from the RDFT alone. The presented procedure can, in the case of a densely sampled DFT, provide a reduction in the computational complexity of analysis-modification-synthesis-based speech processing methods that independently process the real and imaginary (RI) parts temporally. Furthermore, the perfect reconstruction property of this method implies that the RDFT alone captures all of the information about the signal, which suggests that it may be a potentially useful frequency-derived domain for the processing of speech signals.