Abstract

Recent theoretical investigations have shown important radiometric disadvantages of interferential multiplexing in Fourier transform spectrometry that apparently can be applied even to coded aperture spectrometers. We have reexamined the methods of noninterferential multiplexing in order to assess their signal-to-noise ratio (SNR) performance, relying on a theoretical modeling of the multiplexed signals. We are able to show that quite similar SNR and radiometric disadvantages affect multiplex dispersive spectrometry. The effect of noise on spectral estimations is discussed.

© 2010 Optical Society of America

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References

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    [CrossRef] [PubMed]
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  5. J. R. Kimmel, O. K. Yoon, I. A. Zuleta, O. Trapp, and R. N. Zare, “Peak height precision in Hadamard transform time-of-flight mass spectra,” J. Am. Soc. Mass Spectrom. 16, 1117–1130 (2005).
    [CrossRef]
  6. R. A. DeVerse, R. M. Hammaker, and W. G. Fateley, “Hadamard transform Raman imagery with a digital micro-mirror array,” Vib. Spectrosc. 19, 177–186 (1999).
    [CrossRef]
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    [CrossRef] [PubMed]
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  13. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill1991).
  14. P. B. Fellgett, “Conclusions on multiplex methods,” J. Phys. (Paris) Colloq. 28, 165–171 (1967).
    [CrossRef]
  15. P. B. Fellgett, “A propos de la théorie du spectromètre interférentiel multiplex,” J. Phys. Radium 19, 187–191 (1958).
    [CrossRef]
  16. P. B. Fellgett, “The nature and origin of multiplex Fourier spectrometry,” Notes Rec. R. Soc. 60, 91–93 (2006).
    [CrossRef]
  17. L. Streeter, G. R. Burling-Claridge, M. J. Cree, and R. Künnemeyer, “Optical full Hadamard matrix multiplexing and noise effects,” Appl. Opt. 48, 2078–2085 (2009).
    [CrossRef] [PubMed]
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2010 (1)

2009 (1)

2007 (1)

2006 (3)

2005 (1)

J. R. Kimmel, O. K. Yoon, I. A. Zuleta, O. Trapp, and R. N. Zare, “Peak height precision in Hadamard transform time-of-flight mass spectra,” J. Am. Soc. Mass Spectrom. 16, 1117–1130 (2005).
[CrossRef]

2002 (1)

1999 (1)

R. A. DeVerse, R. M. Hammaker, and W. G. Fateley, “Hadamard transform Raman imagery with a digital micro-mirror array,” Vib. Spectrosc. 19, 177–186 (1999).
[CrossRef]

1992 (1)

R. Damaschini, “Binary-encoding image based on original Hadamard matrices,” Opt. Commun. 90, 218–220 (1992).
[CrossRef]

1970 (1)

1969 (1)

1967 (1)

P. B. Fellgett, “Conclusions on multiplex methods,” J. Phys. (Paris) Colloq. 28, 165–171 (1967).
[CrossRef]

1958 (1)

P. B. Fellgett, “A propos de la théorie du spectromètre interférentiel multiplex,” J. Phys. Radium 19, 187–191 (1958).
[CrossRef]

1949 (1)

Barducci, A.

Brady, D. J.

Burling-Claridge, G. R.

Cree, M. J.

Damaschini, R.

R. Damaschini, “Binary-encoding image based on original Hadamard matrices,” Opt. Commun. 90, 218–220 (1992).
[CrossRef]

Decker, J. A.

J. A. Decker, Jr., “Hadamard transform spectroscopy,” in Spectrometric Techniques, G.A.Vanasse ed. (Academic, 1977).

DeVerse, R. A.

R. A. DeVerse, R. M. Hammaker, and W. G. Fateley, “Hadamard transform Raman imagery with a digital micro-mirror array,” Vib. Spectrosc. 19, 177–186 (1999).
[CrossRef]

Fateley, W. G.

R. A. DeVerse, R. M. Hammaker, and W. G. Fateley, “Hadamard transform Raman imagery with a digital micro-mirror array,” Vib. Spectrosc. 19, 177–186 (1999).
[CrossRef]

Fellgett, P. B.

P. B. Fellgett, “The nature and origin of multiplex Fourier spectrometry,” Notes Rec. R. Soc. 60, 91–93 (2006).
[CrossRef]

P. B. Fellgett, “Conclusions on multiplex methods,” J. Phys. (Paris) Colloq. 28, 165–171 (1967).
[CrossRef]

P. B. Fellgett, “A propos de la théorie du spectromètre interférentiel multiplex,” J. Phys. Radium 19, 187–191 (1958).
[CrossRef]

P. B. Fellgett, “The multiplex advantage,” Ph.D. dissertation (University of Cambridge, 1951).

Fernandez, C.

Fine, T.

Gehm, M. E.

Golay, M. J. E.

Guenther, B. D.

Guzzi, D.

Hammaker, R. M.

R. A. DeVerse, R. M. Hammaker, and W. G. Fateley, “Hadamard transform Raman imagery with a digital micro-mirror array,” Vib. Spectrosc. 19, 177–186 (1999).
[CrossRef]

Harwit, M.

Kimmel, J. R.

J. R. Kimmel, O. K. Yoon, I. A. Zuleta, O. Trapp, and R. N. Zare, “Peak height precision in Hadamard transform time-of-flight mass spectra,” J. Am. Soc. Mass Spectrom. 16, 1117–1130 (2005).
[CrossRef]

Künnemeyer, R.

Lastri, C.

Marcoionni, P.

McCain, S. T.

Nardino, V.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill1991).

Phillips, P. G.

Pippi, I.

Pitsianis, N. P.

Potuluri, P.

Sloane, N. J.

Sloane, N. J. A.

Streeter, L.

Sullivan, M. E.

Trapp, O.

J. R. Kimmel, O. K. Yoon, I. A. Zuleta, O. Trapp, and R. N. Zare, “Peak height precision in Hadamard transform time-of-flight mass spectra,” J. Am. Soc. Mass Spectrom. 16, 1117–1130 (2005).
[CrossRef]

Wang, Y.

Yoon, O. K.

J. R. Kimmel, O. K. Yoon, I. A. Zuleta, O. Trapp, and R. N. Zare, “Peak height precision in Hadamard transform time-of-flight mass spectra,” J. Am. Soc. Mass Spectrom. 16, 1117–1130 (2005).
[CrossRef]

Zare, R. N.

J. R. Kimmel, O. K. Yoon, I. A. Zuleta, O. Trapp, and R. N. Zare, “Peak height precision in Hadamard transform time-of-flight mass spectra,” J. Am. Soc. Mass Spectrom. 16, 1117–1130 (2005).
[CrossRef]

Zuleta, I. A.

J. R. Kimmel, O. K. Yoon, I. A. Zuleta, O. Trapp, and R. N. Zare, “Peak height precision in Hadamard transform time-of-flight mass spectra,” J. Am. Soc. Mass Spectrom. 16, 1117–1130 (2005).
[CrossRef]

Appl. Opt. (4)

Appl. Spectrosc. (1)

J. Am. Soc. Mass Spectrom. (1)

J. R. Kimmel, O. K. Yoon, I. A. Zuleta, O. Trapp, and R. N. Zare, “Peak height precision in Hadamard transform time-of-flight mass spectra,” J. Am. Soc. Mass Spectrom. 16, 1117–1130 (2005).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. (Paris) Colloq. (1)

P. B. Fellgett, “Conclusions on multiplex methods,” J. Phys. (Paris) Colloq. 28, 165–171 (1967).
[CrossRef]

J. Phys. Radium (1)

P. B. Fellgett, “A propos de la théorie du spectromètre interférentiel multiplex,” J. Phys. Radium 19, 187–191 (1958).
[CrossRef]

Notes Rec. R. Soc. (1)

P. B. Fellgett, “The nature and origin of multiplex Fourier spectrometry,” Notes Rec. R. Soc. 60, 91–93 (2006).
[CrossRef]

Opt. Commun. (1)

R. Damaschini, “Binary-encoding image based on original Hadamard matrices,” Opt. Commun. 90, 218–220 (1992).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Vib. Spectrosc. (1)

R. A. DeVerse, R. M. Hammaker, and W. G. Fateley, “Hadamard transform Raman imagery with a digital micro-mirror array,” Vib. Spectrosc. 19, 177–186 (1999).
[CrossRef]

Other (3)

P. B. Fellgett, “The multiplex advantage,” Ph.D. dissertation (University of Cambridge, 1951).

J. A. Decker, Jr., “Hadamard transform spectroscopy,” in Spectrometric Techniques, G.A.Vanasse ed. (Academic, 1977).

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill1991).

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Equations (27)

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I ( x ) = { 0 + i ( λ ) F ( x , λ ) d λ + N ( x )     x D x 0 elsewhere ,
i ( λ ) = 0 + I ( x ) B ( x , λ ) d x     λ D λ ,
0 + F ( x , λ ) B ( x , ν ) d x = δ ( λ ν )     λ D λ , 0 + F ( x , λ ) B ( ξ , λ ) d λ = δ ( x ξ )     x D x ,
e ( x , λ ) x = 0 + e ( x , λ ) d x = δ ( λ ) , e ( x , λ ) λ = 0 + e ( x , λ ) d λ = δ ( x ) ,
F ( x , λ ) = F 0 e ( x , λ ) + f ( x , λ ) , B ( x , λ ) = B 0 e ( x , λ ) + b ( x , λ ) ,
0 F ( x , λ ) 1         λ D λ ,     x D x .
F 0 B 0 = 1.
F 0 = 1 2 , B 0 = 2 , f ( x , λ ) = 1 2 , b ( x , λ ) = 0.
0 + s 2 ( λ ) d λ = 0 + s ( λ ) [ 0 + S ( x ) e ( x , λ ) d x ] d λ = 0 + S ( x ) [ 0 + s ( λ ) e ( x , λ ) d λ ] d x = 0 + S 2 ( x ) d x .
0 + [ F 0 e ( x , ν ) + f ( x , ν ) ] [ B 0 e ( x , λ ) + b ( x , λ ) ] d x = F 0 B 0 δ ( λ ν ) + B 0 e ( x , λ ) f ( x , ν ) x + F 0 e ( x , ν ) b ( x , λ ) x + f ( x , ν ) b ( x , λ ) x = δ ( λ ν )     λ D λ ,
B 0 e ( x , λ ) f ( x , ν ) x = 0     λ D λ , F 0 e ( x , ν ) b ( x , λ ) x = 0     λ D λ , f ( x , ν ) b ( x , λ ) x = 0     λ D λ .
0 + [ F 0 e ( x , ν ) + f ( x , ν ) ] [ B 0 e ( x , λ ) + b ( x , λ ) ] d x = F 0 B 0 δ ( λ ν ) + B 0 f 0 δ ( λ ) , B 0 e ( x , λ ) f ( x , ν ) x = B 0 f 0 δ ( λ ) , F 0 e ( x , ν ) b ( x , λ ) x = f ( x , ν ) b ( x , λ ) x = 0 ,
I ( x ) = F 0 e ( x , λ ) i ( λ ) λ + f 0 i ( λ ) λ .
I 2 ( x ) x i 2 ( λ ) λ , I 2 ( x ) x = S 2 ( x ) x + U 2 ( x ) .
S 2 ( x ) x = F 0 2 D x [ e ( x , λ ) i ( λ ) λ ] 2 d x = F 0 2 i 2 ( λ ) λ ,
I 2 ( x ) x U 2 ( x ) x S 2 ( x ) x , D x [ f 0 i ( λ ) λ ] 2 d x F 0 2 D x [ e ( x , λ ) i ( λ ) λ ] 2 d x .
γ i ( λ ) 2 = i 2 ( λ ) λ μ ( D λ ) [ i ( λ ) λ μ ( D λ ) ] 2 i ( λ ) λ 2 = μ ( D λ ) i 2 ( λ ) λ μ 2 ( D λ ) γ i ( λ ) 2 .
D x I 2 ( x ) d x D x f 0 2 i ( λ ) λ 2 d x = f 0 2 μ ( D x ) i ( λ ) λ 2 = f 0 2 μ ( D x ) [ μ ( D λ ) i 2 ( λ ) λ μ 2 ( D λ ) γ i ( λ ) 2 ] ,
S 2 ( x ) x = F 0 2 i 2 ( λ ) λ F 0 2 D x I 2 ( x ) d x + f 0 2 μ ( D x ) μ 2 ( D λ ) γ i ( λ ) 2 f 0 2 μ ( D x ) μ ( D λ ) .
H n n ( λ , ν ) = E { 0 + N ( x ) e ( x , λ ) d x 0 + N ( ξ ) e ( ξ , ν ) d ξ } = 0 + 0 + E { N ( x ) N ( ξ ) } e ( x , λ ) e ( ξ , ν ) d ξ d x = 0 + 0 + N 0 δ ( x ξ ) e ( x , λ ) e ( ξ , ν ) d ξ d x = N 0 δ ( λ ν ) .
SNR datagram 2 = 0 + S 2 ( x ) d x 0 + N 2 ( x ) d x = 0 + s 2 ( λ ) d λ 0 + n 2 ( λ ) d λ = SNR spectrum 2 .
d σ Φ ( λ ) = λ i ( λ ) c h { F 0 e ( x , λ ) + f 0 } d λ ,
σ I = 0 + c h λ i ( λ ) { F 0 e ( x , λ ) + f 0 } d λ .
SNR eff   max ( x ) = F 0 i ( λ ) e ( x , λ ) λ c h λ i ( λ ) { F 0 e ( x , λ ) + f 0 } λ F 0 i ( λ ) e ( x , λ ) λ c h f 0 i ( λ ) λ λ .
SNR ¯ eff   max 2 = SNR eff   max 2 ( x ) x μ ( D x ) F 0 2 i ( λ ) e ( x , λ ) λ 2 x c h f 0 μ ( D x ) i ( λ ) λ λ = F 0 2 i 2 ( λ ) λ c h f 0 μ ( D x ) i ( λ ) λ λ ,
SNR ¯ nonmult   max 2 i 2 ( λ ) λ / μ ( D λ ) c h i ( λ ) λ λ / μ ( D λ ) = i 2 ( λ ) λ c h i ( λ ) λ λ .
{ SNR multiplexing SNR nonmult μ ( D x ) n phot n det SNR multiplexing SNR nonmult n phot n det ,

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