The aim of this paper is to build up the theoretical framework for the
recovery of sparse signals from the magnitude of the measurement. We first
investigate the minimal number of measurements for the success of the recovery
of sparse signals without the phase information. We completely settle the
minimality question for the real case and give a lower bound for the complex
case. We then study the recovery performance of the β1β minimization. In
particular, we present the null space property which, to our knowledge, is the
first sufficient and necessary condition for the success of β1β
minimization for k-sparse phase retrievable.Comment: 14 page