Abstract: Cognitive Radios represent a powerful technology for improving the utilization of radio spectrum resources．Detection of low signal-to-noise ratio(SNR) signals with relaxed a priori information on the signal parameters is very important for cognitive radios．In this paper,an in-depth theoretical analysis is performed for the detector which can detect MFSK signals with noise based on first-order cyclic moment．For this detector,when the sampling length is limited,we derive a first-order cyclic moment amplitude probability density function formula for MFSK signals with noise and noise in the cycle frequency．Then this paper proposes a calculation method of this detector Receiver Operating Characteristic(ROC) curve．Theoretical analysis shows that with the increase of sampling length,the performance of detector based on first-order cyclic moment is improved obviously；under the same conditions,with MFSK signals dimension M increasing,the detector performance decline somewhat.

Key words: Cognitive radios,First-order cyclicmoment,M-ary frequency shift keying(MFSK)signal