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On the Use of Fast Hadamard Transforms for Spectrum Recovery in Hadamard Transform Spectroscopy

The spectrum-recovery step in Hadamard transform spectroscopy is commonly implemented with a fast Hadamard transform (FHT). When the Hadamard or simplex matrix corresponding to the mask does not have the same ordering as the Hadamard matrix corresponding to the FHT, a modification is required. When the two Hadamard matrices are in the same equivalence class, this modification can be implemented as a permutation scheme. This paper investigates permutation schemes for this application. The investigation clarifies inaccurate claims about the applicability of existing methods; reveals a new, more efficient method; and leads to an extension that allows a permutation scheme to be applied to any Hadamard or simplex matrix in the appropriate equivalence class.