The one-dimensional phase retrieval problem consists in the recovery of a complex-valued signal from its Fourier intensity. Due to the well-known ambiguousness of this problem, the determination of the original signal within the extensive solution set is challenging and can only be done under suitable a priori assumptions or additional information about the unknown signal. Depending on the application, one has sometimes access to further interference intensity measurements between the unknown signal and a reference signal. Beginning with the reconstruction in the discrete-time setting, we show that each signal can be uniquely recovered from its Fourier intensity and two further interference intensity measurements between the unknown signal and a modulation of the signal itself. Afterwards, we consider the continuous-time problem, where we obtain an equivalent result. Moreover, the unique recovery of a continuous-time signal can also be ensured by using interference intensity measurements with a known or an unknown reference which is unrelated to the unknown signal.
Results in Mathematics – Springer Journals
Published: Dec 9, 2016
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