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We propose a new spectrum-sensing technique based on the sample autocorrelation of the received signal. We assume that the received signal is oversampled and allow for frequency offset between the local oscillator and the carrier of the primary signal. We evaluate the performance of this algorithm for both additive white Gaussian noise (AWGN) and Rayleigh-fading channels and study its sensitivity to carrier frequency offset. Simulation results are presented to verify the accuracy of the approximation assumptions in our analysis. The performance of the proposed algorithm is also compared with those from the energy detector, the covariance detector, and the cyclic-autocorrelation detector. The results show that our algorithm outperforms the covariance detector and the cyclic autocorrelation detector. It also outperforms the energy detector in the presence of noise power uncertainty or in the case of unknown primary signal bandwidth. Finally, we investigate three diversity combining techniques, namely 1) equal gain combining, 2) selective combining and 3) equal gain correlation combining. Our simulations show that for detection probabilities of interest (e.g., > 0.9), a system with a four-branch diversity achieves a signal-to-noise ratio (SNR) gain of more than 5 dB over the no-diversity system that uses the same number of received signal samples.