New Conditions for Sparse Phase Retrieval

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