Phase retrieval using a conjugate symmetric reference

It was recently shown that finding a compactly supported vector that solves the 1D Fourier phase retrieval problem in the least-squares sense can be computed in polynomial-time, although the solution is not unique. To resolve identifiability, we previously proposed adding a Kronecker delta reference with sufficiently large intensity to the signal before measuring its Fourier magnitude. In practice, however, it is difficult to add a reference that is both strong enough to meet the intensity requirement, and narrow enough to be considered a Kronecker delta after sampling. In this paper we propose a physically more accessible approach to correctly recover a signal from Fourier magnitude measurements, assuming we can (1) generate a reference that is conjugate symmetric (no specific requirement on the power), and (2) measure the Fourier intensity of both the desired signal alone and the signal plus the reference. Numerical simulations showcase the effectiveness of the proposed method in exact signal recovery, as well as noise robustness for certain choices of the references, which cannot be achieved by other methods under the same measuring settings.