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The objective of this paper is twofold. The first part provides further insight in the statistical properties of the Welch power spectrum estimator. A major drawback of the Welch method reported in the literature is that the variance is not a monotonic decreasing function of the fraction of overlap. Selecting the optimal fraction of overlap, which minimizes the variance, is in general difficult since it depends on the window used. We show that the explanation for the nonmonotonic behavior of the variance, as reported in the literature, does not hold. In the second part, this extra insight allows one to eliminate the nonmonotonic behavior of the variance for the Welch power spectrum estimator (PSE) by introducing a small modification to the Welch method. The main contributions of this paper are providing extra insight in the statistical properties of the Welch PSE; modifying the Welch PSE to circular overlap-the variance is a monotonically decreasing function of the fraction of overlap, making the method more user friendly; and an extra reduction of variance with respect to the Welch PSE without introducing systematic errors-this reduction in variance is significant for a small number of data records only.