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The one-dimensional phase retrieval problem consists in the recovery of acomplex-valued signal from its Fourier intensity. Due to the well-knownambiguousness of this problem, the determination of the original signal withinthe extensive solution set is challenging and can only be done under suitable apriori assumption or additional information about the unknown signal. Dependingon the application, one has sometimes access to further interferencemeasurements between the unknown signal and a reference signal. Beginning withthe reconstruction in the discrete-time setting, we show that each signal canbe uniquely recovered from its Fourier intensity and two further interferencemeasurements between the unknown signal and a modulation of the signal itself.Afterwards, we consider the continuous-time problem, where we obtain anequivalent result. Moreover, the unique recovery of a continuous-time signalcan also be ensured by using interference measurements with a known or anunknown reference which is unrelated to the unknown signal.

One-dimensional phase retrieval with additional interference measurements