On fast multivariable bilinear and Hadamard transforms
Applying the bilinear transformation to an n-variance polynomial, n>or=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. >