Stable Recovery from the Magnitude of Symmetrized Fourier Measurements

In this note we show that stable recovery of complex-valued signals $x\in {\mathbb{C}}^n$ up to global sign can be achieved from the magnitudes of $4n-1$ Fourier measurements when a certain “symmetrization and zero-padding” is performed before measurement ($4n-3$ is possible in certain cases). For real signals, symmetrization itself is linear and therefore our result is in this case a statement on uniform phase retrieval. Since complex conjugation is involved, such measurement procedure is not complex-linear but recovery is still possible from magnitudes of linear measurements on, for example, $(\mathrm{\Re }(x),\mathrm{\Im }(x))$.