#### Technical Report USC-SIPI-106

“On ESD's In System Identification”

by Ananthram Swami and Jerry M. Mendel

May 1987

In this note, we review several important properties of Elliptically Symmetric Distributions (ESD's). ESD's are generalizations of the Gaussian distribution; they can be represented as scale mixtures of the Gaussian, are closed under linear transformations and have conditional expectations that are linear on the conditioning variable. Several results in estimation theory, such as parameter estimation, Kalman filtering theory etc. hold for ESD's. In several cases, the null distributions of certain invariant test statistics remain unchanged from the Gaussian case. ESD's are also Bussgang processes. Finally, we show that the zeros of an ARMA system cannot be resolved in two cases: a) when the system is excited by an unobservable ESD process and b) when the output process is Bussgang.

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