In this paper, a new two-step spectrum sensing scheme based on energy detection and maximum eigenvalue detection is proposed for cognitive radio. When the wideband frequency range is divided into multiple sub-channels, energy detection is firstly used for coarse sensing in each sub-channel to determine the existence of the primary user. As no primary user is decided, maximum eigenvalue detection is subsequently employed for fine sensing. For the proposed scheme, we analyze the probabilities of detection and false alarm and find the optimal thresholds for a given probability of false alarm. Moreover, the average sensing time of the proposed scheme is evaluated. We compare the performance of the proposed scheme with that of the schemes which employ only energy detection or maximum eigenvalue detection. For wireless microphone signal, simulation results show that the proposed scheme requires less sensing time than both energy detection and maximum eigenvalue detection under some conditions, while obtaining almost the same sensing performance with maximum eigenvalue detection.