The authors propose novel spectrum sensing and utilisation schemes for cognitive radio networks using the chirp-z transform. To improve the spectral efficiency, a dispersed chirp-z transform is introduced with the energy detection method. The dispersed chirp-z transform enables one to analyse the dispersed frequency spectrum of any frequency range of interest. The analysis is first focused on the sensing part, including the derivations of closed-form expressions for the optimal detection thresholds minimising the total error rate over additive white Gaussian noise (AWGN), Rayleigh, Rician and log-normally distributed fading channels. Then, the performance analysis of the proposed system is investigated for efficient spectrum utilisation over AWGN and fading channels by presenting the receiver operating characteristics. Conventional and segmented chirp-z transform-based spectrum utilisation techniques are also introduced for performance comparison purposes. Finally, the theoretical framework and the optimal threshold derivations through simulations are verified. The analyses reveal that the proposed schemes have a considerable potential to improve the non-cooperative spectrum sensing and utilisation performance.