Right-cyclic Hadamard coding schemes and fast Fourier transforms for use in computing spectrum estimates in Hadamard-transform spectrometry

Two computationally efficient spectrum-recovery schemes have been developed for use by Hadamard-transform spectrometers that have static and dynamic nonidealities in their encoding masks. When a right-cyclic Hadamard pattern is used to encode the mask, a fast Fourier transform (FFT) algorithm or a Trench algorithm can be used to obtain one of the matrix inverses, and an FFT can be used to compute the vector-matrix product, both of which are required in the spectrum-recovery process. In general, the number of mask elements is not an integer power of two, normally requiring that non-radix-2 FFT algorithms be used. However, with appropriate zero-padding of the vectors, radix-2 FFTs can be used to compute the required vector-matrix product. Various combinations of algorithms were used and the total computation times were compared in order to determine the most efficient means of computing the spectrum-estimate. The results indicate that the number of elements in the mask is an important factor in determining which combination of algorithms provides the most efficient recovery of the spectrum-estimate