Bk. Harms et al., OPTIMAL MEASUREMENT TECHNIQUES UTILIZING HADAMARD TRANSFORMS, IEEE transactions on instrumentation and measurement, 43(3), 1994, pp. 397-402
A classic measurement problem-weighing n objects on a scale of limited
absolute accuracy-is addressed, with a modern application of this pro
blem-Hadamard-transform spectroscopy. For both of these measurement sy
stems, optimal recovery of the desired measurements requires computati
on of a Hadamard transform. With the advent of digital-signal-processi
ng methods, researchers soon realized that this transform could be per
formed most efficiently with a fast-Hadamard-transform (FHT) algorithm
. However, in addition to introducing greater speed into these measure
ment systems, the FHT introduced some confusion because there are seve
ral different orderings or equivalence classes of Hadamard matrices an
d FHT's that could be used. The goal of this paper is to relieve the c
onfusion about the applicability of existing techniques for dealing wi
th ordering problems by 1) clarifying under what conditions these tech
niques work, 2) explaining under what conditions they fail and why, an
d 3) presenting a new technique (requiring less memory and fewer compu
tational steps) that never fails.