Application of the theory of orthogonal polynomials in two variables to a multi-gain equivalent linearization problem

It is shown that the multi-gain representation for a single-valued non-linearity with multiple inputs as developed by Somerville and Atherton may be regarded as an approximation problem involving orthogonal polynomials in two variables. Consider two stationary random processes, x(t) and y(t), possibly correlated, with a given second-order (zero-delay) joint probability density, p(x,y). If the input to a specified zero-memory non-linear device having the input/output characteristic v0(t) = f[vi(t)] is x(t) + y(t), the relevant polynomials satisfy orthonormality conditions over the xy-plane with respect to p(x, y) as weighting function. An inherent minimum property of these polynomials then allows the equivalent gains to be determined directly in terms of the expansion coefficients of f(x+y) with respect to the polynomials. When x and y are uncorrelated, the gains reduce to the values previously obtained by Somerville and Atherton. A further property of the polynomials is sufficient to prove that the zero-delay cross-correlation between the input and the error involved in the approximation is zero, and that this result remains true as the order of the approximation is increased.