In this paper, we consider cognitive radio networks in which energy constrained secondary users (SUs) can harvest energy from the randomly deployed power beacons. A new frame structure is proposed for the considered networks. In the considered network, a wireless power transfer model is proposed, and the closed-form expressions for the power outage probability are derived. In addition, in order to reduce the energy consumption at SUs, sub-Nyquist sampling are performed at SUs. Subsequently, compressive sensing and matrix completion techniques are invoked to recover the original signals at the fusion center by utilizing the sparsity property of spectral signals. Throughput optimizations of the secondary networks are formulated into two linear constrained problems, which aim to maximize the throughput of a single SU and the whole cooperative network, respectively. Three methods are provided to obtain the maximal throughput of secondary networks by optimizing the time slots allocation and the transmit power. Simulation results show that the maximum throughput can be improved by implementing compressive spectrum sensing in the proposed frame structure design.