Optimal measurement techniques utilizing Hadamard transforms

A classic measurement problem-weighing n objects on a scale of limited absolute accuracy-is addressed, with a modern application of this problem-Hadamard-transform spectroscopy. For both of these measurement systems, optimal recovery of the desired measurements requires computation of a Hadamard transform. With the advent of digital-signal-processing methods, researchers soon realized that this transform could be performed most efficiently with a fast-Hadamard-transform (FHT) algorithm. However, in addition to introducing greater speed into these measurement systems, the FHT introduced some confusion because there are several different orderings or equivalence classes of Hadamard matrices and FHT´s that could be used. The goal of this paper is to relieve the confusion about the applicability of existing techniques for dealing with ordering problems by 1) clarifying under what conditions these techniques work, 2) explaining under what conditions they fail and why, and 3) presenting a new technique (requiring less memory and fewer computational steps) that never fails