  This paper considers the problem of sampling and reconstruction of a
continuous-time sparse signal without assuming the knowledge of the sampling
instants or the sampling rate. This topic has its roots in the problem of
recovering multiple echoes of light from its low-pass filtered and
auto-correlated, time-domain measurements. Our work is closely related to the
topic of sparse phase retrieval and in this context, we discuss the advantage
of phase-free measurements. While this problem is ill-posed, cues based on
physical constraints allow for its appropriate regularization. We validate our
theory with experiments based on customized, optical time-of-flight imaging
sensors. What singles out our approach is that our sensing method allows for
temporal phase retrieval as opposed to the usual case of spatial phase
retrieval. Preliminary experiments and results demonstrate a compelling
capability of our phase-retrieval based imaging device.
