Walsh and other orthogonal representations of chemical signals

Includes bibliographical references (leaves 209-212)The piecewise-constant Walsh, Paley, and Haar systems are defined as examples of first-order basis functions. The orthogonal second-order basis functions are produced by integrating the first-order functions, then orthonormalizing them using the Gram-Schmidt procedure. The transform matrices associated with the first-order systems are examples of Hadamard-like matrices; such matrices can usually be formed from matrices of lower rank by the use of permuted Kronecker products. The nature of these products is such that a fast (O(Nlog₂N) operations) or a semi-fast transform may be accomplished by algorithms described within the paper. The transform matrices associated with the second-order basis systems are formed by calculating the vector-valued functions at all dyadic-rational points over [0,1). The second-order basis systems are of two types each, depending on whether the rectangular rule or Simpson's rule is used to evaluate the overlap (projection) integrals and vector norms. Because no fast or semi-fast transform algorithms are known yet for either form of the second-order basis vector transform matrices, expansion of 2 a data vector requires the point-by-point (0(N²) multiplications and additions) method. ..