A computationally inexpensive method is presented for the recovery of spectra from measurements obtained with Hadamard transform spectrometers having nonideal masks. Normally, N measurements are required in order to recover an N-point spectrum; this method requires N + N0 measurements to be taken, where, typically, N0 ≤ 10. Once the additional measurements have been taken, only O( N[log2 N + 2]) arithmetic operations—mostly additions or subtractions—are needed in order to recover the spectrum; a conventional procedure requires O(2 N2) operations. Preliminary work for this method is minimal, requiring O( N) operations as opposed to O( N3) for a conventional procedure; this work needs to be done only once for a given spectrometer. The spectrum-estimate obtained is unbiased.
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.
The multiplexing inherent in the Hadamard transform (HT) spectrometer can result in an improved spectrum-estimate when the detector is the major source of noise. A spectrum-estimate may be further improved by taking into account any nonidealities in the system. In this paper, observations concerning the errors associated with such estimates are presented, with the use of results obtained from computer simulations. Three spectrum-recovery techniques for an HT spectrometer having a nonideal electro-optic mask are considered in terms of the mean-square error (MSE) associated with a given estimate. The discussion of the MSE is with respect to the input spectrum to be estimated, the detector noise, the transmittances of the nonideal mask, and the use of coaddition. Included is a review of the computational efficiency and the statistical bias of each method. The relative performances of the spectrum-recovery methods are presented with examples to help identify the sources of error for each of the techniques.
A spectrum-recovery method is presented which efficiently computes an optimal unbiased linear spectrum-estimate for measurements obtained with Hadamard transform (HT) spectrometers having nonideal masks. This method has the following advantages over other spectrum-recovery techniques: it is computationally efficient, it requires no additional measurements, and it computes an optimal spectrum-estimate. In the method presented, after the mask of the HT spectrometer has been characterized, approximately 3 N preliminary arithmetic operations are performed once for a given spectrometer, where N is both the number of spectral resolution-elements desired and the number of measurements required. Each spectrum-estimate to be recovered then requires only an additional O[ N(log2 N + 4)] arithmetic operations. In contrast, conventional methods for obtaining an optimal unbiased linear spectrum-estimate require O( N3) preliminary operations, and O(2 N2) operations during each spectrum-recovery.
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