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A method proposed by Pickard for estimating the center frequency of the power spectrum of a Gaussian process is analyzed in terms of the mean and variance of the estimator. Through limiting and differentiation, this method consists essentially of counting zero-crossings with positive and negative weights so that our analysis of the center-frequency estimator sheds some light on the zero-crossing problem. A simple expression for the variance of the estimator is obtained by a novel application of Price's theorem for zero-memory nonlinear devices with Gaussian inputs. The effects of nonideal limiting and, hence, nonideal counting upon the mean and variance of the estimator are examined in detail. The mathematical techniques employed are felt to have potential usefulness for related problems in communication theory.