New Conditions for Sparse Phase Retrieval

We consider the problem of sparse phase retrieval, where a k-sparse signal x ∈ Rn (or C) is measured as y = |Ax|, where A ∈ R (or C respectively) is a measurement matrix and | · | is the element-wise absolute value. For a real signal and a real measurement matrix A, we show that m = 2k measurements are necessary and sufficient to recover x uniquely. For complex signal x ∈ Cn and A ∈ C, we show that m = 4k − 2 phaseless measurements are sufficient to recover x. It is known that the multiplying constant 4 in m = 4k − 2 cannot be improved.