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 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.