On fast multivariable bilinear and Hadamard transforms

Applying the bilinear transformation to an n-variance polynomial, n⩾1, one arrives at a rational function whose numerator is the transformed polynomial. Based on I.J. Good´s (1958) results and the symmetry properties of the single-dimensional bilinear transformation matrix, a fast implementation for computing the coefficients of such n-variable transformed polynomials is proposed. Letting Q denote the matrix relating the coefficients of the original polynomial to those of the transformed polynomials, it is shown that Sylvester-type Hadamard matrices coincide with the transformation matrices for the n-variable polynomials for which the highest degree of each variable is one. This new observation is used to derive W.R. Crowther and C.M. Rader´s (1966) formulation of the fast Hadamard transform