Spectrum sensing is a key technology of cognitive radio. The goal of spectrum sensing in this context is to achieve a high detection probability in order to avoid causing interference to the primary licensed users. Among the methods available for spectrum sensing, energy detection has been widely applied in hardware implementations. However, since previous work has been based on the traditional FFT, the side leakage problem makes it difficult to detect a primary user adjacent to a strong signal. In this paper, we propose an all-phase FFT based wideband spectrum sensing processor. Simulation results show that it can increase the probability of detection of a primary user that is adjacent in frequency to a strong signal. It can also improve the receiver operating characteristic curve. Hardware implementation requirements are also given.