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Let , the signal, be a complex Gaussian process corrupted by additive Gaussian noise . Observations on and are assumed to be available where is a smooth weighting function and . Using the Fourier transform of the samples of and , estimators are derived for estimating the mean frequency and spectral width of the unknown power spectrum of the unweighted signal process. The means and variances of these statistics are computed in general, and explicitly for nontrivial practical examples. Asymptotic formulas for the moment estimators as a function of the number of realizations, frequency resolution, signal-to-noise ratio and spectral width, and consistency of the estimators are some of the results that are discussed in detail.