Elliptically symmetric distributions

Elliptically symmetric distributions are second-order distributions with probability densities whose contours of equal height are ellipses. This class includes the Gaussian and sine-wave distributions and others which can be generated from certain first-order distributions. Members of this class have several desirable features for the description of the second-order statistics of the transformation of a random signal by an instantaneous nonlinear device. In particular, they are separable in Nuttall's sense, so that the output of the device may be described in terms of equivalent gain and distortion. These distributions can also simplify the evaluation of the output autovariance because of their similarity to the Gaussian distribution. For a certain class of functions, elliptically symmetric distributions yield averages which are simply proportional to those obtained with a Gaussian distribution. Furthermore, these distributions satisfy a relation analogous to Price's theorem for Gaussian distributions. Finally, a certain subclass of these distributions can be expanded in the series representation studied by Barrett and Lampard.