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This paper proposes simple moment based spectrum sensing algorithm for cognitive radio networks in a flat fading channel. It is assumed that the transmitted signal samples are binary (quadrature) phase-shift keying BPSK (QPSK), Mary quadrature amplitude modulation (QAM) or continuous uniformly distributed random variables and the noise samples are independent and identically distributed circularly symmetric complex Gaussian random variables all with unknown (imperfect) variance. Under these assumptions, we propose a simple test statistics employing a ratio of fourth and second moments. For this statistics, we provide analytical expressions for both probability of false alarm (Pf) and probability of detection (Pd) in an additive white Gaussian noise (AWGN) channel. We confirm the theoretical expressions by computer simulation. Furthermore, under noise variance uncertainty, simulation results demonstrate that the proposed moment based detector gives better detection performance compared to that of energy detector in AWGN and Rayleigh fading channels.