Phase retrieval is the process of algorithmically finding solutions to the phase problem. Given a complex spectrum ${\displaystyle F(k)}$, of amplitude ${\displaystyle |F(k)|}$, and phase ${\displaystyle \psi (k)}$:

where x is an M-dimensional spatial coordinate and k is an M-dimensional spatial frequency coordinate. Phase retrieval consists of finding the phase that satisfies a set of constraints for a measured amplitude. Important applications of phase retrieval include X-ray crystallography, transmission electron microscopy and coherent diffractive imaging, for which ${\displaystyle M=2}$. Uniqueness theorems for both 1-D and 2-D cases of the phase retrieval problem, including the phaseless 1-D inverse scattering problem, were proven by Klibanov and his collaborators (see References).

Here we consider 1-D discrete Fourier transform (DFT) phase retrieval problem. The DFT of a complex signal ${\displaystyle f[n]}$ is given by

${\displaystyle F[k]=\sum _{n=0}^{N-1}f[n]e^{-j2\pi {\frac {kn}{N}},}=|F[k]|\cdot e^{j\psi [k]}\quad k=0,1,\ldots ,N-1}$,

and the oversampled DFT of ${\displaystyle x}$ is given by

${\displaystyle F[k]=\sum _{n=0}^{N-1}f[n]e^{-j2\pi {\frac {kn}{M}},}\quad k=0,1,\ldots ,M-1}$,

where ${\displaystyle M>N}$.

Since the DFT operator is bijective, this is equivalent to recovering the phase ${\displaystyle \psi [k]}$. It is common recovering a signal from its autocorrelation sequence instead of its Fourier magnitude. That is, denote by ${\displaystyle {\hat {f}}}$ the vector ${\displaystyle f}$ after padding with ${\displaystyle N-1}$ zeros. The autocorrelation sequence of ${\displaystyle {\hat {f}}}$ is then defined as