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Applied Optics

Applied Optics


  • Vol. 13, Iss. 11 — Nov. 1, 1974
  • pp: 2662–2668

Theoretical Comparison of Singly Multiplexed Hadamard Transform Spectrometers and Scanning Spectrometers

N. M. Larson, R. Crosmun, and Y. Talmi  »View Author Affiliations

Applied Optics, Vol. 13, Issue 11, pp. 2662-2668 (1974)

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We present a general mathematical framework within which the relative performance of singly multiplexed Hadamard transform spectrometers (HTS) and conventional scanning spectrometers (SS) may be compared. The theoretical multiplex advantage (Fellgett advantage) is calculated for spectrometers operating in two spectral regions. For the low energy region, i.e., infrared, the determined multiplex advantage F is (N/2)1/2 (N is the number of slots), in accordance with predictions given by Fellgett. For the high spectral energy region, i.e., uv-vis, F = ( x / 2 x ¯ ) 1 / 2, where x is the intensity of the spectal element sought and x ¯ is the average intensity produced across the whole spectrum. Our predictions are verified by computer simulation of various characteristic spectra. Based on these results, we arrive at some conclusions concerning the practical application of HTS systems.

© 1974 Optical Society of America

Original Manuscript: June 21, 1974
Published: November 1, 1974

N. M. Larson, R. Crosmun, and Y. Talmi, "Theoretical Comparison of Singly Multiplexed Hadamard Transform Spectrometers and Scanning Spectrometers," Appl. Opt. 13, 2662-2668 (1974)

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  16. δji is the Kronecker delta; δjk = 1 if i = j, δji = 0 if i ≠ j.
  17. Dummy variable j is substituted for dummy variable i in equations describing the SS mode, in order to facilitate comparison of SNR between the two modes of operation.
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  19. For accurate reproduction of Poisson distributions, α and β in Eqs. (29) and (31) must be equal to 1. We allow for the possibility of their assuming other values in order to maintain the highest degree of generality.
  20. Since our results are in good agreement with those described by Kahn (see Ref. 13) for the one type of noise he treated, we expect our conclusions concerning the use of HTS systems to also be applicable to Fourier transform systems.

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