  This paper investigates the downlink channel state information (CSI) sensing
in 5G heterogeneous networks composed of user equipments (UEs) with different
feedback capabilities. We aim to enhance the CSI accuracy of UEs only affording
the low-resolution Type-I codebook. While existing works have demonstrated that
the task can be accomplished by solving a phase retrieval (PR) formulation
based on the feedback of precoding matrix indicator (PMI) and channel quality
indicator (CQI), they need many feedback rounds. In this paper, we propose a
novel CSI sensing scheme that can significantly reduce the feedback overhead.
Our scheme involves a novel parameter dimension reduction design by exploiting
the spatial consistency of wireless channels among nearby UEs, and a
constrained PR (CPR) formulation that characterizes the feasible region of CSI
by the PMI information. To address the computational challenge due to the
non-convexity and the large number of constraints of CPR, we develop a
two-stage algorithm that firstly identifies and removes inactive constraints,
followed by a fast first-order algorithm. The study is further extended to
multi-carrier systems. Extensive tests over DeepMIMO and QuaDriGa datasets
showcase that our designs greatly outperform existing methods and achieve the
high-resolution Type-II codebook performance with a few rounds of feedback.
