  In many signal processing problems arising in practical applications, we wish
to reconstruct an unknown signal from its phaseless measurements with respect
to a frame. This inverse problem is known as the phase retrieval problem. For
each particular application, the set of relevant measurement frames is
determined by the problem at hand, which motivates the study of phase retrieval
for structured, application-relevant frames. In this paper, we focus on one
class of such frames that appear naturally in diffraction imaging,
ptychography, and audio processing, namely, multi-window Gabor frames. We study
the question of injectivity of the phase retrieval problem with these
measurement frames in the finite-dimensional setup and propose an explicit
construction of an infinite family of phase retrievable multi-window Gabor
frames. We show that phase retrievability for the constructed frames can be
achieved with a much smaller number of phaseless measurements compared to the
previous results for this type of measurement frames. Additionally, we show
that the sufficient for reconstruction number of phaseless measurements depends
on the dimension of the signal space, and not on the ambient dimension of the
problem.
