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In a cognitive radio (CR) network, cooperative spectrum sensing is a viable sensing technique to enhance spectral utilisation efficiency of secondary users (SUs) while ensuring the quality of service (QoS) of primary users (PUs). Intuitively, the more SUs are involved in sensing, the more sensing accuracy the CR can achieve, whereas the more sensing overhead the SUs consume, the less throughput the CR network can achieve. In this study, the authors investigate overhead-throughput trade-off over Rayleigh-fading channels in a cooperative CR network that consists of a number of the SUs employing energy detectors and a single decision fusion centre. Considering the trade-off, the authors prove that there is an optimal set of the sensing length and the number of SUs that maximise the throughput of an SU network. They further extend their analysis to a two-stage cooperative sensing mechanism where the second-stage fine sensing is triggered whenever any SU reports the presence of a PU after the first-stage detection. Numerical results showed that compared with the single-stage sensing, the two-stage sensing scheme achieves higher throughput via a reduction of the false alarm probability.