Abstract

Multivariate optical computing (MOC) is an instrumentation design concept for optically demultiplexing the spectroscopic signals in radiometric measurements. The advantages of optically demultiplexing are improved precision, optical throughput, improved reliability, and reduced cost of instrumentation. Conceptually, the instrument implements a multivariate regression vector whose dot product with the spectrum yields a single value related to a spectroscopically active physical property of interest. Instrumentation designs for implementing MOC are diverse, and there has been no systematic comparison of the performance of these designs. This report develops a general expression for comparing the precision of the different instrumentation designs of MOC. Additionally, an expression is given for the transition from low- to high-signal-limited performance of MOC instrumentation. These two general expressions are applied to the traditional multivariate analysis and five examples of MOC.

© 2004 Optical Society of America

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References

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  1. M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, M. L. Myrick, “Multivariate optical computation for predictive spectroscopy,” Anal. Chem. 70, 73–82 (1998).
    [CrossRef]
  2. A. M. C. Prakash, C. M. Stellman, K. S. Booksh, “Optical regression: a method for improving quantitative precision of multivariate prediction with single channel spectrometers,” Chemom. Intell. Lab. Syst. 46, 265–274 (1999).
    [CrossRef]
  3. D. E. Battey, J. B. Slater, R. Wludyka, H. Owen, D. M. Pallister, M. D. Morris, “Axial transmissive F/1.8 imaging Raman spectrograph with volume-phase holographic filter and grating,” Appl. Spectrosc. 47, 1913–1919 (1993).
    [CrossRef]
  4. E. N. Lewis, P. J. Treado, I. W. Levin, “A miniaturized, no-moving-parts Raman spectrometer,” Appl. Spectrosc. 47, 539–543 (1993).
    [CrossRef]
  5. R. A. DeVerse, R. M. Hammaker, W. G. Fateley, “Realization of the Hadamard multiplex advantage using a programmable optical mask in a dispersive flat-field near-infrared spectrometer,” Appl. Spectrosc. 54, 1751–1758 (2000).
    [CrossRef]
  6. T. Diehl, W. Ehrfeld, M. Lacher, T. Zetterer, “Electrostatically operated micromirrors for Hadamard transform spectrometer,” IEEE J. Sel. Top. Quantum Electron. 5, 106–110 (1999).
    [CrossRef]
  7. R. A. DeVerse, R. M. Hammaker, W. G. Fateley, J. A. Graham, J. D. Tate, “Spectrometry and imaging using a digital micromirror array,” Am. Lab. 30, 112S–120S (1998).
  8. A. G. Ryabenko, G. G. Kasparov, “Numerical study of a pattern recognition multispectral system with optimal spectral splitting,” Pattern Recogn. Image Anal. 1, 347–354 (1991).
  9. S. E. Bialkowski, “Species discrimination and quantitative estimation using incoherent linear optical signal processing of emission signals,” Anal. Chem. 58, 2561–2563 (1986).
    [CrossRef]
  10. O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
    [CrossRef]
  11. M. L. Myrick, S. Soyemi, H. Li, L. Zhang, D. Eastwood “Spectral tolerance determination for multivariate optical element design,” Fresenius J. Anal. Chem. 369, 351–355 (2001).
    [CrossRef] [PubMed]
  12. N. M. Faber, L. M. C. Buydens, G. Kateman, “Aspects of pseudorank estimation methods based on the eigenvalues of principal component analysis of random matrices,” Chemom. Intell. Lab. Syst. 25, 203–226 (1994).
    [CrossRef]
  13. K. Booksh, B. R. Kowalski, “Error analysis of the generalized rank annihilation method,” J. Chemom. 8, 45–63 (1994).
    [CrossRef]
  14. L. A. Goodman, S. J. Haberman, “The analysis of nonadditivity in two-way analysis of variance,” J. Am. Stat. Assoc. 85, 139–145 (1990).
    [CrossRef]
  15. S. D. Hodges, P. G. Moore, “Data uncertainties and least squares regression,” Appl. Stat. 21, 185–195 (1972).
    [CrossRef]
  16. K. R. Beebe, R. J. Pell, M. B. Seasholtz, Chemometrics: a Practical Guide (Wiley, New York, 1998).
  17. H. Martens, T. Naes, Multivariate Calibration (Wiley, Chichester, England, 1989).
  18. O. O. Soyemi, F. G. Haibach, P. J. Gemperline, M. L. Myrick, “Nonlinear optimization algorithm for multivariate optical element design,” Appl. Spectrosc. 56, 477–487 (2002).
    [CrossRef]

2002 (1)

2001 (2)

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[CrossRef]

M. L. Myrick, S. Soyemi, H. Li, L. Zhang, D. Eastwood “Spectral tolerance determination for multivariate optical element design,” Fresenius J. Anal. Chem. 369, 351–355 (2001).
[CrossRef] [PubMed]

2000 (1)

1999 (2)

T. Diehl, W. Ehrfeld, M. Lacher, T. Zetterer, “Electrostatically operated micromirrors for Hadamard transform spectrometer,” IEEE J. Sel. Top. Quantum Electron. 5, 106–110 (1999).
[CrossRef]

A. M. C. Prakash, C. M. Stellman, K. S. Booksh, “Optical regression: a method for improving quantitative precision of multivariate prediction with single channel spectrometers,” Chemom. Intell. Lab. Syst. 46, 265–274 (1999).
[CrossRef]

1998 (2)

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, M. L. Myrick, “Multivariate optical computation for predictive spectroscopy,” Anal. Chem. 70, 73–82 (1998).
[CrossRef]

R. A. DeVerse, R. M. Hammaker, W. G. Fateley, J. A. Graham, J. D. Tate, “Spectrometry and imaging using a digital micromirror array,” Am. Lab. 30, 112S–120S (1998).

1994 (2)

N. M. Faber, L. M. C. Buydens, G. Kateman, “Aspects of pseudorank estimation methods based on the eigenvalues of principal component analysis of random matrices,” Chemom. Intell. Lab. Syst. 25, 203–226 (1994).
[CrossRef]

K. Booksh, B. R. Kowalski, “Error analysis of the generalized rank annihilation method,” J. Chemom. 8, 45–63 (1994).
[CrossRef]

1993 (2)

1991 (1)

A. G. Ryabenko, G. G. Kasparov, “Numerical study of a pattern recognition multispectral system with optimal spectral splitting,” Pattern Recogn. Image Anal. 1, 347–354 (1991).

1990 (1)

L. A. Goodman, S. J. Haberman, “The analysis of nonadditivity in two-way analysis of variance,” J. Am. Stat. Assoc. 85, 139–145 (1990).
[CrossRef]

1986 (1)

S. E. Bialkowski, “Species discrimination and quantitative estimation using incoherent linear optical signal processing of emission signals,” Anal. Chem. 58, 2561–2563 (1986).
[CrossRef]

1972 (1)

S. D. Hodges, P. G. Moore, “Data uncertainties and least squares regression,” Appl. Stat. 21, 185–195 (1972).
[CrossRef]

Aust, J. F.

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, M. L. Myrick, “Multivariate optical computation for predictive spectroscopy,” Anal. Chem. 70, 73–82 (1998).
[CrossRef]

Battey, D. E.

Beebe, K. R.

K. R. Beebe, R. J. Pell, M. B. Seasholtz, Chemometrics: a Practical Guide (Wiley, New York, 1998).

Bialkowski, S. E.

S. E. Bialkowski, “Species discrimination and quantitative estimation using incoherent linear optical signal processing of emission signals,” Anal. Chem. 58, 2561–2563 (1986).
[CrossRef]

Booksh, K.

K. Booksh, B. R. Kowalski, “Error analysis of the generalized rank annihilation method,” J. Chemom. 8, 45–63 (1994).
[CrossRef]

Booksh, K. S.

A. M. C. Prakash, C. M. Stellman, K. S. Booksh, “Optical regression: a method for improving quantitative precision of multivariate prediction with single channel spectrometers,” Chemom. Intell. Lab. Syst. 46, 265–274 (1999).
[CrossRef]

Buydens, L. M. C.

N. M. Faber, L. M. C. Buydens, G. Kateman, “Aspects of pseudorank estimation methods based on the eigenvalues of principal component analysis of random matrices,” Chemom. Intell. Lab. Syst. 25, 203–226 (1994).
[CrossRef]

DeVerse, R. A.

R. A. DeVerse, R. M. Hammaker, W. G. Fateley, “Realization of the Hadamard multiplex advantage using a programmable optical mask in a dispersive flat-field near-infrared spectrometer,” Appl. Spectrosc. 54, 1751–1758 (2000).
[CrossRef]

R. A. DeVerse, R. M. Hammaker, W. G. Fateley, J. A. Graham, J. D. Tate, “Spectrometry and imaging using a digital micromirror array,” Am. Lab. 30, 112S–120S (1998).

Diehl, T.

T. Diehl, W. Ehrfeld, M. Lacher, T. Zetterer, “Electrostatically operated micromirrors for Hadamard transform spectrometer,” IEEE J. Sel. Top. Quantum Electron. 5, 106–110 (1999).
[CrossRef]

Dobrowolski, J. A.

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, M. L. Myrick, “Multivariate optical computation for predictive spectroscopy,” Anal. Chem. 70, 73–82 (1998).
[CrossRef]

Eastwood, D.

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[CrossRef]

M. L. Myrick, S. Soyemi, H. Li, L. Zhang, D. Eastwood “Spectral tolerance determination for multivariate optical element design,” Fresenius J. Anal. Chem. 369, 351–355 (2001).
[CrossRef] [PubMed]

Ehrfeld, W.

T. Diehl, W. Ehrfeld, M. Lacher, T. Zetterer, “Electrostatically operated micromirrors for Hadamard transform spectrometer,” IEEE J. Sel. Top. Quantum Electron. 5, 106–110 (1999).
[CrossRef]

Faber, N. M.

N. M. Faber, L. M. C. Buydens, G. Kateman, “Aspects of pseudorank estimation methods based on the eigenvalues of principal component analysis of random matrices,” Chemom. Intell. Lab. Syst. 25, 203–226 (1994).
[CrossRef]

Fateley, W. G.

R. A. DeVerse, R. M. Hammaker, W. G. Fateley, “Realization of the Hadamard multiplex advantage using a programmable optical mask in a dispersive flat-field near-infrared spectrometer,” Appl. Spectrosc. 54, 1751–1758 (2000).
[CrossRef]

R. A. DeVerse, R. M. Hammaker, W. G. Fateley, J. A. Graham, J. D. Tate, “Spectrometry and imaging using a digital micromirror array,” Am. Lab. 30, 112S–120S (1998).

Gemperline, P.

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[CrossRef]

Gemperline, P. J.

Goodman, L. A.

L. A. Goodman, S. J. Haberman, “The analysis of nonadditivity in two-way analysis of variance,” J. Am. Stat. Assoc. 85, 139–145 (1990).
[CrossRef]

Graham, J. A.

R. A. DeVerse, R. M. Hammaker, W. G. Fateley, J. A. Graham, J. D. Tate, “Spectrometry and imaging using a digital micromirror array,” Am. Lab. 30, 112S–120S (1998).

Haberman, S. J.

L. A. Goodman, S. J. Haberman, “The analysis of nonadditivity in two-way analysis of variance,” J. Am. Stat. Assoc. 85, 139–145 (1990).
[CrossRef]

Haibach, F. G.

Hammaker, R. M.

R. A. DeVerse, R. M. Hammaker, W. G. Fateley, “Realization of the Hadamard multiplex advantage using a programmable optical mask in a dispersive flat-field near-infrared spectrometer,” Appl. Spectrosc. 54, 1751–1758 (2000).
[CrossRef]

R. A. DeVerse, R. M. Hammaker, W. G. Fateley, J. A. Graham, J. D. Tate, “Spectrometry and imaging using a digital micromirror array,” Am. Lab. 30, 112S–120S (1998).

Hodges, S. D.

S. D. Hodges, P. G. Moore, “Data uncertainties and least squares regression,” Appl. Stat. 21, 185–195 (1972).
[CrossRef]

Karunamuni, J.

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[CrossRef]

Kasparov, G. G.

A. G. Ryabenko, G. G. Kasparov, “Numerical study of a pattern recognition multispectral system with optimal spectral splitting,” Pattern Recogn. Image Anal. 1, 347–354 (1991).

Kateman, G.

N. M. Faber, L. M. C. Buydens, G. Kateman, “Aspects of pseudorank estimation methods based on the eigenvalues of principal component analysis of random matrices,” Chemom. Intell. Lab. Syst. 25, 203–226 (1994).
[CrossRef]

Kowalski, B. R.

K. Booksh, B. R. Kowalski, “Error analysis of the generalized rank annihilation method,” J. Chemom. 8, 45–63 (1994).
[CrossRef]

Lacher, M.

T. Diehl, W. Ehrfeld, M. Lacher, T. Zetterer, “Electrostatically operated micromirrors for Hadamard transform spectrometer,” IEEE J. Sel. Top. Quantum Electron. 5, 106–110 (1999).
[CrossRef]

Levin, I. W.

Lewis, E. N.

Li, H.

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[CrossRef]

M. L. Myrick, S. Soyemi, H. Li, L. Zhang, D. Eastwood “Spectral tolerance determination for multivariate optical element design,” Fresenius J. Anal. Chem. 369, 351–355 (2001).
[CrossRef] [PubMed]

Martens, H.

H. Martens, T. Naes, Multivariate Calibration (Wiley, Chichester, England, 1989).

Moore, P. G.

S. D. Hodges, P. G. Moore, “Data uncertainties and least squares regression,” Appl. Stat. 21, 185–195 (1972).
[CrossRef]

Morris, M. D.

Myrick, M. L.

O. O. Soyemi, F. G. Haibach, P. J. Gemperline, M. L. Myrick, “Nonlinear optimization algorithm for multivariate optical element design,” Appl. Spectrosc. 56, 477–487 (2002).
[CrossRef]

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[CrossRef]

M. L. Myrick, S. Soyemi, H. Li, L. Zhang, D. Eastwood “Spectral tolerance determination for multivariate optical element design,” Fresenius J. Anal. Chem. 369, 351–355 (2001).
[CrossRef] [PubMed]

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, M. L. Myrick, “Multivariate optical computation for predictive spectroscopy,” Anal. Chem. 70, 73–82 (1998).
[CrossRef]

Naes, T.

H. Martens, T. Naes, Multivariate Calibration (Wiley, Chichester, England, 1989).

Nelson, M. P.

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, M. L. Myrick, “Multivariate optical computation for predictive spectroscopy,” Anal. Chem. 70, 73–82 (1998).
[CrossRef]

Owen, H.

Pallister, D. M.

Pell, R. J.

K. R. Beebe, R. J. Pell, M. B. Seasholtz, Chemometrics: a Practical Guide (Wiley, New York, 1998).

Prakash, A. M. C.

A. M. C. Prakash, C. M. Stellman, K. S. Booksh, “Optical regression: a method for improving quantitative precision of multivariate prediction with single channel spectrometers,” Chemom. Intell. Lab. Syst. 46, 265–274 (1999).
[CrossRef]

Ryabenko, A. G.

A. G. Ryabenko, G. G. Kasparov, “Numerical study of a pattern recognition multispectral system with optimal spectral splitting,” Pattern Recogn. Image Anal. 1, 347–354 (1991).

Seasholtz, M. B.

K. R. Beebe, R. J. Pell, M. B. Seasholtz, Chemometrics: a Practical Guide (Wiley, New York, 1998).

Slater, J. B.

Soyemi, O.

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[CrossRef]

Soyemi, O. O.

Soyemi, S.

M. L. Myrick, S. Soyemi, H. Li, L. Zhang, D. Eastwood “Spectral tolerance determination for multivariate optical element design,” Fresenius J. Anal. Chem. 369, 351–355 (2001).
[CrossRef] [PubMed]

Stellman, C. M.

A. M. C. Prakash, C. M. Stellman, K. S. Booksh, “Optical regression: a method for improving quantitative precision of multivariate prediction with single channel spectrometers,” Chemom. Intell. Lab. Syst. 46, 265–274 (1999).
[CrossRef]

Synowicki, R. A.

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[CrossRef]

Tate, J. D.

R. A. DeVerse, R. M. Hammaker, W. G. Fateley, J. A. Graham, J. D. Tate, “Spectrometry and imaging using a digital micromirror array,” Am. Lab. 30, 112S–120S (1998).

Treado, P. J.

Verly, P. G.

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, M. L. Myrick, “Multivariate optical computation for predictive spectroscopy,” Anal. Chem. 70, 73–82 (1998).
[CrossRef]

Wludyka, R.

Zetterer, T.

T. Diehl, W. Ehrfeld, M. Lacher, T. Zetterer, “Electrostatically operated micromirrors for Hadamard transform spectrometer,” IEEE J. Sel. Top. Quantum Electron. 5, 106–110 (1999).
[CrossRef]

Zhang, L.

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[CrossRef]

M. L. Myrick, S. Soyemi, H. Li, L. Zhang, D. Eastwood “Spectral tolerance determination for multivariate optical element design,” Fresenius J. Anal. Chem. 369, 351–355 (2001).
[CrossRef] [PubMed]

Am. Lab. (1)

R. A. DeVerse, R. M. Hammaker, W. G. Fateley, J. A. Graham, J. D. Tate, “Spectrometry and imaging using a digital micromirror array,” Am. Lab. 30, 112S–120S (1998).

Anal. Chem. (3)

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, M. L. Myrick, “Multivariate optical computation for predictive spectroscopy,” Anal. Chem. 70, 73–82 (1998).
[CrossRef]

S. E. Bialkowski, “Species discrimination and quantitative estimation using incoherent linear optical signal processing of emission signals,” Anal. Chem. 58, 2561–2563 (1986).
[CrossRef]

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, M. L. Myrick, “Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy,” Anal. Chem. 73, 1069–1079 (2001).
[CrossRef]

Appl. Spectrosc. (4)

Appl. Stat. (1)

S. D. Hodges, P. G. Moore, “Data uncertainties and least squares regression,” Appl. Stat. 21, 185–195 (1972).
[CrossRef]

Chemom. Intell. Lab. Syst. (2)

N. M. Faber, L. M. C. Buydens, G. Kateman, “Aspects of pseudorank estimation methods based on the eigenvalues of principal component analysis of random matrices,” Chemom. Intell. Lab. Syst. 25, 203–226 (1994).
[CrossRef]

A. M. C. Prakash, C. M. Stellman, K. S. Booksh, “Optical regression: a method for improving quantitative precision of multivariate prediction with single channel spectrometers,” Chemom. Intell. Lab. Syst. 46, 265–274 (1999).
[CrossRef]

Fresenius J. Anal. Chem. (1)

M. L. Myrick, S. Soyemi, H. Li, L. Zhang, D. Eastwood “Spectral tolerance determination for multivariate optical element design,” Fresenius J. Anal. Chem. 369, 351–355 (2001).
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

T. Diehl, W. Ehrfeld, M. Lacher, T. Zetterer, “Electrostatically operated micromirrors for Hadamard transform spectrometer,” IEEE J. Sel. Top. Quantum Electron. 5, 106–110 (1999).
[CrossRef]

J. Am. Stat. Assoc. (1)

L. A. Goodman, S. J. Haberman, “The analysis of nonadditivity in two-way analysis of variance,” J. Am. Stat. Assoc. 85, 139–145 (1990).
[CrossRef]

J. Chemom. (1)

K. Booksh, B. R. Kowalski, “Error analysis of the generalized rank annihilation method,” J. Chemom. 8, 45–63 (1994).
[CrossRef]

Pattern Recogn. Image Anal. (1)

A. G. Ryabenko, G. G. Kasparov, “Numerical study of a pattern recognition multispectral system with optimal spectral splitting,” Pattern Recogn. Image Anal. 1, 347–354 (1991).

Other (2)

K. R. Beebe, R. J. Pell, M. B. Seasholtz, Chemometrics: a Practical Guide (Wiley, New York, 1998).

H. Martens, T. Naes, Multivariate Calibration (Wiley, Chichester, England, 1989).

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Figures (1)

Fig. 1
Fig. 1

Schematic flow diagrams for the specific measurement methods analyzed in the text. Light from a sample is the input to the system in all cases, and the output of all measurements is an estimation of the direct product of the light spectrum with the loading vector b. A. Diagram of the common scanning monochromator-digital evaluation method. A wavelength selector selects discrete wavelengths sequentially until a complete measurement of the sample spectrum has been obtained. The spectrum is then combined digitally with b. B. Diagram of the common multichannel array-digital evaluation method. After the light of the sample is dispersed, discrete sampling in parallel is used to produce a spectrum. C. Diagram of the SMOC method. A wavelength selector directs light to an integrating detector with sampling times dictated by b. D. Diagram of the PN MOC method. Integrating detectors sum all wavelengths in parallel with fractional efficiencies dictated by b, after which the difference in the measurements for positive and negative values of b is determined. E. Diagram of the MOE and BMOE methods. A beam splitter with weighting dictated by b is used, with parallel measurement of the polychromatic transmitted and reflected light. After integration, the difference between transmitted and reflected intensities is determined. d.p., direct product.

Equations (48)

Equations on this page are rendered with MathJax. Learn more.

ŷ=xTb=j=1J xjbj.
xj=τjxj,
xj=ξjX¯,
X¯=j=1Jx¯j.
τj=f¯T.
b˜=bbξ¯,
bξ¯j=1J bj2ξ¯j21/2.
bξ¯g¯j=1JX¯j=g¯TX¯f¯.
ŷ=g¯j=1J b˜jξj.
Dk=TX¯ j=1J wjkϕjξj,
ŷ=γk=1K κkDk,
ŷ=γk=1K κkk,
σŷ2=γ2k=1K κk2D¯k.
k=1K κk2D¯kk=1K κk2=σ02.
Dj=TX¯J ξj.
ŷSMDE=g¯JTX¯j=1J b˜jD¯j.
σSMDE2= g¯2J2T2X¯2j=1J b˜j2D¯j,= g¯2JTX¯j=1Jξ¯jb˜j2,
PSMDE=σ02Jj=1J b˜j2j=1J b˜j2ξ¯j=σ02J2bj=±c=σ02Jξ¯kbj=±δjkc.
Dj=TX¯ξj.
ŷMADE=g¯TX¯j=1J b˜jDj.
σMADE2= g¯2TX¯j=1Jξ¯jb˜j2.
PMADE=PSMDE/J.
D+=TX¯m=1J |b˜m|j=1J δb,+b˜jξj, D-=-TX¯m=1J |b˜m|j=1J δb,-b˜jξj.
y˜SMOC=g¯m=1J |b˜m|TX¯D+-D-.
σSMOC2= g¯2JTX¯j=1Jξ¯j|b˜j||b˜j|.
1σSMOC2σSMDE21J.
PSMOC=2σ02j=1J |b˜j|j=1J |b˜j|ξ¯j=2PSMDEJbj=±c,±δjkc.
D+=TX¯c λiλf t+λξλdλTX¯ j=1J tj+ξj, D-TX¯ j=1J tj-ξj,
tj+-tj-ξ¯j  Δjξ¯j=μb˜jξ¯j, J=1J Δj2ξ¯j2 = μ2, b˜j = Δjm=1J Δj2ξ¯j21/2=ΔjΔξ¯Δ˜j.
ŷPN=g¯Δξ¯TX¯TX¯ j=1J Δjξj=g¯Δξ¯TX¯TX¯ j=1J tj+ξj -TX¯j=1J tj-ξj=g¯Δξ¯TX¯D+-D-.
σPN2=g¯2j=1J|Δj|ξ¯jTX¯ j=1J Δj2ξ¯j2=g¯20|Δλ|ξ¯λdλTX¯c0Δλ2ξ¯λ2dλ.
σPN2σMADE2=j=1Jξ¯j|Δj|j=1Jξ¯jΔj2|b˜|maxj=1Jξ¯j|b˜j|j=1Jξ¯jb˜j2.
|Δj||b˜j||b˜|max,
PPN=2σ02j=1J |Δj|ξ¯j2J PMADEbj=±c2PMADEbj=δjkc.
Dt=TX¯c λiλf tλξjλdλTX¯ j=1J tjξj,DrTX¯ j=1J rjξj.
tj-rjΔj,b˜j=ΔjΔξ¯=Δ˜j.
ŷMOE=g¯Δξ¯TX¯TX¯ J=1J Δjξj=g¯Δξ¯TX¯Dt-Dr.
σMOE2=g¯2TX¯j=1J Δj2ξ¯j2=g¯2TX¯c0 Δj2ξ¯j2dλ.
σMOE2σMADE2=1j=1Jξ¯jΔj2  |b˜|max2j=1Jξ¯jb˜j2=1 Δj=±1=1ξ¯kΔj=±δjk.
PMOE=2σ02=2PMADEJbj=±c=2PMADE ξk(bj=±δjkc
αttj-αrrjΔj,b˜j=Δ˜j=ΔjΔξ¯.
ŷBMOE=g¯Δξ¯TX¯αtDt-αrDr,
σBMOE2=g¯2TX¯ j=1JΔj2ξ¯j2αt2j=1J tjξ¯j+αr2j=1J rjξ¯j=g¯2TX¯c0Δλ2ξ¯λ2dλ0αt2tλ+αr2rλξ¯λdλ.
X¯=FNAj=0J ηjLjPjψ¯j =FNA0 ηλLλPλψ¯λdλ,
FNA=1-NAdet1/2-11-NAinc1/2-1,
ψ¯j=Δλhc λjΦ¯j,dψ¯dλ=λΦ¯λhc,
X¯c=1hc1-1-NAdet1-1-NAinc0 ηλLλPλ×Φ¯λλdλ,
Φ¯j=t¯jΦj,

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