Variance expressions for four spectral-centroid estimators of a complex Gaussian process

Variance expressions are derived for four unbiased centroid estimators of an asymmetrical spectrum of the complex envelope of a narrow-band Gaussian process. These estimators are obtained by cross-correlating the undistorted or hard-clipped component and the time derivative of the undistorted or hard-dipped quadrature component of the complex envelope. When both components are hard-clipped, the variance expression of the centroid estimator is a generalization of Blachman's zero-crossing analysis. The variance of the centroid estimator obtained from finite records of the undistorted components is shown to be smaller than that of the other three estimators when
m_{2} > 5.32\mu_{1}^{2}
, where
\mu_{1}
and
m_{2}
are the first simple and second central spectral moments respectively. However, when
m_{2} < 5.32\mu_{1}^{2}
, the variance of the centroid estimate obtained from one hard-clipped component is shown to be smallest. Furthermore, the dependence of the variances on both the magnitude and direction of spectral asymmetry is shown to be different for the four centroid estimators considered.