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We propose a new cooperative spectrum sensing for cognitive radio (CR) systems that is designed to detect low-power primary users (PUs). In previous works, it has been assumed that all cognitive terminals (CTs) are within the no-talk zone of PU, which is the region that a CT should not use the band of PU when it lies within the region. However, when the PU transmits at low power, only some CTs will lie within the no-talk zone. It is therefore clear that low-power PUs cannot be protected adequately when conventional sensing schemes are used. To solve this problem, we propose a new scheme in which CTs perform sensing in a number of different stages. Through performance analysis and simulations, we show that our proposed scheme can detect low-power PUs with high accuracy, with the result that the PUs can be protected properly. We also show that the throughput of CR systems can be improved.