This paper considers the century-old phase retrieval problem of reconstructing a signal x in C^n from the amplitudes of its Fourier coefficients. To overcome the inherent ambiguity due to missing phase information, we create redundancy in the measurement process by distorting the signal multiple times, each time with a different mask. More specifically, we consider measurements of the form |<D_lf_k,x>| for k=1,2,...,n and l=1,2,...,L; where f_k^* are rows of the Fourier matrix and D_l are random diagonal matrices modeling the masks. We prove that the signal x can be recovered exactly (up to a global phase factor) from such measurements using a semi-definite program as long as the number of masks is at least on the order of (log n)^4. Numerical experiments complement our theoretical study, illustrating our approach and demonstrating its effectiveness.
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