This paper deals with the challenging problem of spectrum sensing in cognitive radio. We consider a stochastic system model where the the Primary User (PU) transmits a periodic signal over fading channels. The effect of frequency offsets due to oscillator mismatch, and Doppler offset is studied. We show that for this case the Likelihood Ratio Test (LRT) cannot be evaluated poitnwise. We present a novel approach to approximate the marginilisation of the frequency offset using a single point estimate. This is obtained via a low complexity Constrained Adaptive Notch Filter (CANF) to estimate the frequency offset. Performance is evaluated via numerical simulations and it is shown that the proposed spectrum sensing scheme can achieve the same performance as the near-optimal scheme, that is based on a bank of matched filters, using only a fraction of the complexity required.