Stable recovery from the magnitude of symmetrized fourier measurements

In this note we show that stable recovery of complex-valued signals x ϵ C
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up to a 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, (Re(x), Im(x)).