  Phase retrieval refers to recovering a signal from its Fourier magnitude.
This problem arises naturally in many scientific applications, such as
ultra-short laser pulse characterization and diffraction imaging.
Unfortunately, phase retrieval is ill-posed for almost all one-dimensional
signals. In order to characterize a laser pulse and overcome the ill-posedness,
it is common to use a technique called Frequency-Resolved Optical Gating
(FROG). In FROG, the measured data, referred to as FROG trace, is the Fourier
magnitude of the product of the underlying signal with several translated
versions of itself. The FROG trace results in a system of phaseless quartic
Fourier measurements. In this paper, we prove that it suffices to consider only
three translations of the signal to determine almost all bandlimited signals,
up to trivial ambiguities. In practice, one usually also has access to the
signal's Fourier magnitude. We show that in this case only two translations
suffice. Our results significantly improve upon earlier work.
