Abstract

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/2x¯)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

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References

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  1. H. J. Babrov, R. N. Tourin, Appl. Opt. 7, 2171 (1968).
    [CrossRef] [PubMed]
  2. R. E. Benn, W. S. Foote, C. T. Chase, J. Opt. Soc. Am. 39, 529 (1949).
    [CrossRef]
  3. R. E. Santini, M. J. Milano, H. L. Pardue, D. W. Margerum, Anal. Chem. 44, 826 (1972).
    [CrossRef] [PubMed]
  4. G. Horlick, E. G. Coddington, Anal. Chem. 45, 1490 (1973).
    [CrossRef]
  5. J. E. Carnes, Optical Spectra, page 27, March (1974).
  6. J. Adams, B. W. Manley, Phillips Tech. Rev. 28, 156 (1967).
  7. P. Fellgett, J. Phys. Radium 19, 187 (1958).
    [CrossRef]
  8. John A. Decker, Martin O. Harwit, Appl. Opt. 7, 2205 (1968).
    [CrossRef] [PubMed]
  9. E. D. Nelson, M. L. Fredman, J. Opt. Soc. Am. 60, 1664 (1970).
    [CrossRef]
  10. R. N. Ibbett, D. Aspinall, J. F. Grainger, Appl. Opt. 7, 1089 (1968).
    [CrossRef] [PubMed]
  11. F. W. Plankey, J. D. Winefordner, Paper No. 200, Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy, 4–8 March (1974).
  12. C. J. Oliver, E. R. Pike, Appl. Opt. 13, 158 (1974).
    [CrossRef] [PubMed]
  13. F. D. Kahn, Astrophys. J. 129, 518 (1959).
    [CrossRef]
  14. A. S. Filler, J. Opt. Soc. Am. 63, 589 (1973).
    [CrossRef]
  15. Martin O. Harwit, John A. Decker, in Progress in Optics (North Holland Publishing Co., Amsterdam, 1974), Vol. 12.
    [CrossRef]
  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.
  18. W. K. Pratt, J. Kane, H. C. Andrews, Proc. IEEE 57, 58 (1969).
    [CrossRef]
  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.

1974 (1)

1973 (2)

G. Horlick, E. G. Coddington, Anal. Chem. 45, 1490 (1973).
[CrossRef]

A. S. Filler, J. Opt. Soc. Am. 63, 589 (1973).
[CrossRef]

1972 (1)

R. E. Santini, M. J. Milano, H. L. Pardue, D. W. Margerum, Anal. Chem. 44, 826 (1972).
[CrossRef] [PubMed]

1970 (1)

1969 (1)

W. K. Pratt, J. Kane, H. C. Andrews, Proc. IEEE 57, 58 (1969).
[CrossRef]

1968 (3)

1967 (1)

J. Adams, B. W. Manley, Phillips Tech. Rev. 28, 156 (1967).

1959 (1)

F. D. Kahn, Astrophys. J. 129, 518 (1959).
[CrossRef]

1958 (1)

P. Fellgett, J. Phys. Radium 19, 187 (1958).
[CrossRef]

1949 (1)

Adams, J.

J. Adams, B. W. Manley, Phillips Tech. Rev. 28, 156 (1967).

Andrews, H. C.

W. K. Pratt, J. Kane, H. C. Andrews, Proc. IEEE 57, 58 (1969).
[CrossRef]

Aspinall, D.

Babrov, H. J.

Benn, R. E.

Carnes, J. E.

J. E. Carnes, Optical Spectra, page 27, March (1974).

Chase, C. T.

Coddington, E. G.

G. Horlick, E. G. Coddington, Anal. Chem. 45, 1490 (1973).
[CrossRef]

Decker, John A.

John A. Decker, Martin O. Harwit, Appl. Opt. 7, 2205 (1968).
[CrossRef] [PubMed]

Martin O. Harwit, John A. Decker, in Progress in Optics (North Holland Publishing Co., Amsterdam, 1974), Vol. 12.
[CrossRef]

Fellgett, P.

P. Fellgett, J. Phys. Radium 19, 187 (1958).
[CrossRef]

Filler, A. S.

Foote, W. S.

Fredman, M. L.

Grainger, J. F.

Harwit, Martin O.

John A. Decker, Martin O. Harwit, Appl. Opt. 7, 2205 (1968).
[CrossRef] [PubMed]

Martin O. Harwit, John A. Decker, in Progress in Optics (North Holland Publishing Co., Amsterdam, 1974), Vol. 12.
[CrossRef]

Horlick, G.

G. Horlick, E. G. Coddington, Anal. Chem. 45, 1490 (1973).
[CrossRef]

Ibbett, R. N.

Kahn, F. D.

F. D. Kahn, Astrophys. J. 129, 518 (1959).
[CrossRef]

Kane, J.

W. K. Pratt, J. Kane, H. C. Andrews, Proc. IEEE 57, 58 (1969).
[CrossRef]

Manley, B. W.

J. Adams, B. W. Manley, Phillips Tech. Rev. 28, 156 (1967).

Margerum, D. W.

R. E. Santini, M. J. Milano, H. L. Pardue, D. W. Margerum, Anal. Chem. 44, 826 (1972).
[CrossRef] [PubMed]

Milano, M. J.

R. E. Santini, M. J. Milano, H. L. Pardue, D. W. Margerum, Anal. Chem. 44, 826 (1972).
[CrossRef] [PubMed]

Nelson, E. D.

Oliver, C. J.

Pardue, H. L.

R. E. Santini, M. J. Milano, H. L. Pardue, D. W. Margerum, Anal. Chem. 44, 826 (1972).
[CrossRef] [PubMed]

Pike, E. R.

Plankey, F. W.

F. W. Plankey, J. D. Winefordner, Paper No. 200, Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy, 4–8 March (1974).

Pratt, W. K.

W. K. Pratt, J. Kane, H. C. Andrews, Proc. IEEE 57, 58 (1969).
[CrossRef]

Santini, R. E.

R. E. Santini, M. J. Milano, H. L. Pardue, D. W. Margerum, Anal. Chem. 44, 826 (1972).
[CrossRef] [PubMed]

Tourin, R. N.

Winefordner, J. D.

F. W. Plankey, J. D. Winefordner, Paper No. 200, Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy, 4–8 March (1974).

Anal. Chem. (2)

R. E. Santini, M. J. Milano, H. L. Pardue, D. W. Margerum, Anal. Chem. 44, 826 (1972).
[CrossRef] [PubMed]

G. Horlick, E. G. Coddington, Anal. Chem. 45, 1490 (1973).
[CrossRef]

Appl. Opt. (4)

Astrophys. J. (1)

F. D. Kahn, Astrophys. J. 129, 518 (1959).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Phys. Radium (1)

P. Fellgett, J. Phys. Radium 19, 187 (1958).
[CrossRef]

Phillips Tech. Rev. (1)

J. Adams, B. W. Manley, Phillips Tech. Rev. 28, 156 (1967).

Proc. IEEE (1)

W. K. Pratt, J. Kane, H. C. Andrews, Proc. IEEE 57, 58 (1969).
[CrossRef]

Other (7)

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.

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.

J. E. Carnes, Optical Spectra, page 27, March (1974).

F. W. Plankey, J. D. Winefordner, Paper No. 200, Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy, 4–8 March (1974).

Martin O. Harwit, John A. Decker, in Progress in Optics (North Holland Publishing Co., Amsterdam, 1974), Vol. 12.
[CrossRef]

δji is the Kronecker delta; δjk = 1 if i = j, δji = 0 if i ≠ j.

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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Figures (3)

Fig. 1
Fig. 1

(a) A noise-free spectrum typical of those obtained at low spectral energy (infrared absorption), (b) Computer simulation of the measurement using an SS instrument, of the spectrum shown in (a), where the noise is as described in Sec. III with q = 250 and B = 1. (c) Same as (b) but using an HTS instrument with N = 127.

Fig. 2
Fig. 2

(a) A noise-free sparse line-emission spectrum, typical of those obtained in the uv-visible spectral region, (b) Computer simulation of the measurement, using an SS instrument, of the spectrum shown in (a), where the noise is as described in Sec. IV with α = 0, β = 1, and B = 1. (c) Same as (b) but using an HTS instrument with N = 127.

Fig. 3
Fig. 3

(a) A noise-free molecular absorption spectrum typical of those found in the uv-visible region. (b) Computer simulation of the measurement, using an SS instrument, of the spectrum shown in (a), where the noise is as described in Sec. IV with α = 0, β = 1, and B = 1. (c) Same as (b) but using an HTS instrument with N = 127.

Equations (62)

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k = 1 N b j k a k i = δ j i . 16
x i + f o p ( λ i , x i , t k ) .
f i k = f o ( λ i , x i , t k ) .
x i + f i i .
D i C = x i + f i i + B + Q i C ,
Q i C = Q o p ( x i + f i i + B , t i ) .
E i C = D i C x i = f i i + B + Q i C .
S i C / N i C = D i C / E i C E i C 1 / 2 .
S i C / N i C = x i / E i C E i C 1 / 2 ,
i = 1 N a k i ( x i + f i k ) ,
y k = i = 1 N a k i ( x i + f i k ) + B + Q k H ,
Q k H = Q o p [ i = 1 N a k i ( x i + f i k ) + B , t k ] .
D j H = k = 1 N b j k y k .
D j H = x j + k = 1 N b j k i = 1 N a k i f i k + 2 B N + 1 + k = 1 N b j k Q k H .
E j H = D j H x j ,
E j H = k = 1 N b j k i = 1 N a k i f i k + 2 B N + 1 + k = 1 N b j k Q k H .
S j H / N j H = D j H / E j H E j H 1 / 2 .
Q k = 0 and Q k Q k = q 2 δ k k ;
f i k = f o p ( λ i , x i , t k ) 0.
E j C = B + Q j C .
E j C E j C = B 2 + B Q j C + Q j C B + Q j C Q j C = B 2 + q 2 δ j j .
N j C = ( B 2 + q 2 ) 1 / 2 .
S j C = D j C = x j + B + Q j C = x j + B .
S j C / N j C = ( x j + B ) / ( B 2 + q 2 ) 1 / 2 .
E j H = 2 B N + 1 + k = 1 N b j k Q k H .
E j H E j H = ( 2 B N + 1 ) 2 + ( 2 B N + 1 ) k = 1 N ( b j k + b j k ) Q k H + k = 1 N k = 1 N b j k b j k Q k H Q k H .
E j H E j H = ( 2 B N + 1 ) 2 + q 2 k = 1 N b j k b j k = ( 2 N + 1 ) 2 { B 2 + [ ( N + 1 ) δ j j 1 ] q 2 } .
N j H = [ 2 / ( N + 1 ) ] N [ ( B 2 / N ) + q 2 ] 1 / 2 .
S j H = x j + [ 2 B / ( N + 1 ) ] .
S j H / N j H = [ ( N + 1 ) / 2 N ] [ x j + 2 B / ( N + 1 ) ] / [ ( B 2 / N ) + q 2 ] 1 / 2 .
F j = ( S j H / N j H ) / ( S j C / N j C ) = [ ( N + 1 ) / 2 N ] { ( B 2 + q 2 ) / [ ( B 2 / N ) + q 2 ] } 1 / 2 × { [ x j + 2 B / ( N + 1 ) ] / ( x j + B ) } .
f j k = f o p ( λ i , x i , t k ) ,
f i k f i k = δ i i δ k k α x i ,
Q k = Q o p ( W , t k ) ,
Q k Q k = W β δ k k ,
E j C = B + f j j + Q j C ,
Q j C = Q o p ( x i + f i i + B , t j ) .
E j C E j C = B 2 + f j j f j j + Q j C Q j C = B 2 + ( α x j + β ( x j + f i i + B ) ) δ j j = B 2 + ( β B + ( α + β ) x j ) δ j j .
S j C / N j C = ( x j + B ) / [ B 2 + β B + ( α + β ) x j ] 1 / 2 .
E j H = 2 B N + 1 + k = 1 N i = 1 N b j k a k i f i k + k = 1 N b j k Q k H ,
Q k H = Q o p ( i = 1 N a k i x i + i = 1 N a k i f i k + B , t k ) .
E j H E j H = ( 2 B N + 1 ) 2 + k = 1 N i = 1 N k = 1 N i = 1 N b j k a k i b j k a k i × f i k f i k + k = 1 N k = 1 N b j k b j k Q k H Q k H ,
E j H E j H = ( 2 B N + 1 ) 2 + k = 1 N i = 1 N b j k b j k a k i 2 α x i + k = 1 N b j k b j k ( β i = 1 N a k i x i + β i = 1 N a k i f i k + β B ) .
E j H E j H = ( 2 B N + 1 ) 2 + ( α + β ) k = 1 N b j k b j k a k i x i + β B ( 2 N + 1 ) 2 [ ( N + 1 ) δ j j 1 ] .
k X k Y k k | X k | | Y k | .
k = 1 N b j k b j k a k i ( 2 N + 1 ) 2 k = 1 N a k i .
E j H E j H ( 2 N + 1 ) 2 { B 2 + β B [ ( N + 1 ) δ j j 1 ] } + ( α + β ) 2 N + 1 i = 1 N x i ,
E j H E j H ( 2 N + 1 ) 2 ( B 2 + β B N ) + ( α + β ) 2 N + 1 i = 1 N x i ,
S j H / N j H = { X j + 2 B / ( N + 1 ) } / { [ 2 / ( N + 1 ) ] 2 N B [ ( B / N ) + β ] + 2 ( α + β ) x ¯ } .
x ¯ = 1 N + 1 j = 1 N x j ;
F j = { [ x j + 2 B / ( N + 1 ) ] / ( x j + B ) } × ( [ B 2 + β B + ( α + β ) x j ] / { [ 2 / ( N + 1 ) ] 2 N B [ ( B / N ) + β ] + 2 ( α + β ) x ¯ } ) 1 / 2 .
F j = ( x j / 2 x ¯ ) 1 / 2 .
b j k = [ 2 / ( N + 1 ) ] ( 2 a j k 1 ) .
b ¯ j k = [ ( N + 1 ) / 2 ] b j k 1 j N , 1 k N ,
b ¯ N + 1 , k = b ¯ j , N + 1 = 1 for all j and k .
j = 1 N + 1 b ¯ j k b ¯ j k = ( N + 1 ) δ k k .
j = 1 N ( N + 1 2 b j k ) ( N + 1 2 b j k ) + b ¯ N + 1 , k b ¯ N + 1 , k = ( N + 1 ) δ k k ,
j = 1 N b j k b j k = [ ( N + 1 ) δ k k 1 ] ( 2 N + 1 ) 2 .
j = 1 N ( N + 1 2 b j k ) ( 1 ) + ( 1 ) ( 1 ) = 0 ,
j = 1 N b j k = 2 N + 1 .
j = 1 N 2 N + 1 ( 2 a j k 1 ) = 2 N + 1 ,
j = 1 N a j k = N + 1 2 .

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