Invariance property of gaussian signals: A new interpretation, extension and applications

This paper addresses the invariance property of Gaussian signals, originally derived by Bussgang, which characterizes the input/output moment relation of a hybrid nonlinear moment (HNM) estimator based on a zero-memory nonlinearity (ZMN) g(y). Some re-derivations of this property are reviewed, and an original, direct, and simple proof is presented (Appendix 1). The paper then derives a new interpretation of this property (Theorem 1) that shows a moment-sense equivalence between g(y) and a linear mapping h₁(y) whose coefficients a₀ and a₁ are completely characterized in terms of g(y) and are shown to be optimal in a mean square error (MSB) sense. A direct and very interesting byproduct of this interpretation is a simple linear relationship between the input and output of the HNM estimator involved. The property is then generalized (Theorem 2) to signals other than Gaussian, resulting in an infinite cumulant series expansion of the HNM estimator output, whose coefficients are all characterized in terms of g(y). Applications of Theorem 1 to some ZMNs commonly used in signal processing and control theory are presented that clearly illustrate the power and elegance of the invariance property. Finally, some conclusions are given.