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In this paper, we provide a performance analysis of cognitive radio systems employing energy detection based on power spectral density estimation. Specifically, we derive mathematical expressions for the probabilities of false alarm and missed detection when the periodogram estimate is used as a decision statistic. First, using the characteristic function of the quadratic form representation of the periodogram, we study the characteristics of the product of the sample covariance matrix of the observed vector and the symmetric matrix of the quadratic form. We have found that this product is Rank-1 with one non-zero eigenvalue and therefore a simplified expression for the cumulative density function (CDF) is obtained. Second, the CDF is exploited to derive accurate mathematical expressions for the probabilities of false alarm and missed detection in both Rayleigh and Rician fading channels. Also, for each case, we have derived the probability distribution function (PDF) of the corresponding eigenvalues. The results demonstrate that the probabilities of false alarm and missed detection are independent of the length of the observations when all the other parameters remain fixed. The results also reveal that the considered frequency-domain based energy detector is superior to its conventional time-domain counterpart as it provides fewer false alarms.