Ra. Dyer et al., RIGHT-CYCLIC HADAMARD CODING SCHEMES AND FAST FOURIER-TRANSFORMS FOR USE IN COMPUTING SPECTRUM ESTIMATES IN HADAMARD-TRANSFORM SPECTROMETRY, IEEE transactions on instrumentation and measurement, 45(5), 1996, pp. 860-864
Two computationally efficient spectrum-recovery schemes were recently
developed for use by Hadamard-transform spectrometers that have static
and dynamic nonidealities in their encoding masks, These methods make
use of a left-cyclic Hadamard encodement scheme and the ability to ex
press the left-cyclic W-D matrix in factored form as W-D = STD. The ma
trix W-D describes the dynamic characteristics of and the encodement s
cheme for the mask, This paper focuses on the use of a right-cyclic Ha
damard pattern to encode the mask and computationally efficient method
s that can be used to obtain the spectrum-estimate. The major advantag
e of right-cyclic over left-cyclic encodement schemes is due to the re
sulting right-cyclic nature of both W-D and W-D(-1) Fast algorithms, s
uch as a fast Fourier transform (FFT) or a Trench algorithm, that take
advantage of the right-cyclic nature of W-D can be used to obtain W-D
(-1) directly, In general, the number of mask elements is not an integ
er power of two, and non-radix-2 FFT's must be used to compute W-D(-1)
. Since W-D(-1) is right-cyclic, the vector-matrix product of W-D(-1)
and the measurement vector can be expressed as a circular correlation
and implemented indirectly via FFT's, With appropriate zero-padding of
the vectors, radix-2 FFT's can be used for this computation, Various
algorithms were used at each step in the overall computation of the sp
ectrum-estimate, and the total computation times are presented and com
pared, The size of the mask is important in determining which algorith
ms are the most efficient in recovering the spectrum-estimate.