On Signal Reconstruction from Fourier Magnitude

In this paper, three new algorithms for signal reconstruction from spectral magnitude are presented. The first algorithm reconstructs a signal from its discrete Fourier transform (DFT) magnitude and half of its samples using the decimation-in-time FFT algorithm, which results in a closed-form solution for the unknown data. The second is based on localized Fourier transform magnitudes and a single spatial sample to fully reconstruct an image. The process reconstructs equal-sized image blocks, until the entire image is restored, using the spatial sample as initial data. The third scheme is a modification of the Gerchberg-Saxton iterative approach to image reconstruction. From the separability property of the 2D DFT, an intermediate Fourier domain arises, and the reconstruction algorithm may impose twice as many Fourier magnitude constraints, compared to the conventional approach. The three algorithms are analyzed, and simulation results in the presence of measurement errors are presented.