Detection of cyclostationary primary user (PU) signals in white Gaussian noise for cognitive radio systems is considered based on looking for a cycle frequency at a particular time lag in the cyclic autocorrelation function (CAF) of the noisy PU signal. We explicitly exploit the knowledge that under the null hypothesis of PU signal absent, the measurements originate from white Gaussian noise with possibly unknown variance. Our formulation allows us to computationally simplify the spectrum sensing detector, obviating the need for estimating an unwieldy covariance matrix needed in prior works. We consider both single and multiple antenna receivers. A performance analysis of the proposed detector is carried out. Supporting simulation examples are provided using an OFDM PU signal and they verify our performance analysis and also show that our approaches either outperform or are at least as good as existing approaches while being computationally much cheaper.