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The problem of spectrum sensing in bandwidth-constrained cognitive sensor networks with quantised sensing observation is studied. In particular, the quantisation bit budget for achieving performance close to that of spectrum sensing with raw sensing observation is investigated. Both single-user spectrum sensing and collaborative spectrum sensing schemes are proposed. In single-user spectrum sensing, a sensor quantises its sensing observation and sends the quantised sensing observation to a processing point. The processing point performs spectrum sensing based on the reconstructed sensing observation. In collaborative spectrum sensing, each sensor sends a quantised sensing observation to a processing point and each processing point performs local spectrum sensing based on the reconstructed sensing observation. Then, each processing point sends a binary decision bit to a fusion centre. The fusion centre combines the decision bits by using a counting rule to generate the final spectrum sensing result. The detection probability and the false alarm probability are derived and simulated. The results show that there exists an optimal quantisation bit budget for which the detection probability and the false alarm probability are almost the same as those in spectrum sensing with raw sensing observation. Compared to existing spectrum sensing schemes with quantised sensing observation, the proposed schemes are more suitable for implementation in cognitive sensor networks.