In this paper, we consider a cognitive radio network in which energy constrained secondary users (SUs) can harvest energy from the randomly deployed power beacons (PBs). A new frame structure is proposed for the considered network. A wireless power transfer (WPT) model and a compressive spectrum sensing model are introduced. In the WPT model, a new WPT scheme is proposed, and the closed-form expressions for the power outage probability are derived. In compressive spectrum sensing model, two scenarios are considered: 1) Single SU, and 2) Multiple SUs. In the single SU scenario, in order to reduce the energy consumption at the SU, compressive sensing technique which enables sub-Nyquist sampling is utilized. In the multiple SUs scenario, cooperative spectrum sensing (CSS) is performed with adopting low-rank matrix completion technique to obtain the complete matrix at the fusion center. Throughput optimizations of the secondary network are formulated into two linear constrained problems, which aim to maximize the throughput of single SU and the CSS networks, respectively. Three methods are provided to obtain the maximal throughput of secondary network by optimizing the time slots allocation and the transmit power. Simulation results show that: 1) Multiple SUs scenario can achieve lower power outage probability than single SU scenario; and 2) The optimal throughput can be improved by implementing compressive spectrum sensing in the proposed frame structure design.