On a mean-square approximation problem (Corresp.)

Abstract

This correspondence is concerned with the problem of choosing an approximation for the random variable obtained by operating on a stochastic process <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> with a zero-memory non-linearity followed by a linear transformation. It is desired to approximate the nonlinearity with a simpler function and two different criteria are considered for selecting this approximation, viz., the mean-square error obtained before and after the linear transformation. A sufficient condition is presented-for an approximation to simultaneously minimize both error criteria; in addition, it is shown that if the two minima coincide for every linear transformation and every nonlinearity, this condition is also necessary. The condition appears as a restriction on the class of approximating functions and is related to the second-order distributions of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> . For several processes of interest, this restriction is satisfied when the approximating functions are polynomials, the most notable example being the Gaussian process.