Abstract

Calculations of the SNR obtainable with a Fourier transform spectrometer as well as that obtainable with a photometer or scanning device are presented. It is shown that the SNR obtained with a Fourier spectrometer is (N/8)1/2 greater than that obtained with a scanning device (where N is the desired number of spectral elements scanned). Reasons why this factor, known as the multiplex advantage, differs from other values found in the literature are discussed.

© 1977 Optical Society of America

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References

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  1. P. Fellgett, J. Phys. Radium 19, 187 (1958).
    [CrossRef]
  2. G. A. Vanasse, H. Sakai, in Progress in Optics, E. Wolf, Ed. (North Holland, Amsterdam, 1967), p. 261.
  3. M. H. Tai, M. Harwit, Appl. Opt. 15, 2664 (1976).
    [CrossRef] [PubMed]
  4. W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).
  5. P. W. Kruse, L. D. McGlauchlin, R. B. McQuistan, Elements of Infrared Technology: Generation, Transmission, and Detection (Wiley, New York, 1962).
  6. J. Connes, Rev. Opt. 40, 45, 116, 171, and 231 (1961).

1976 (1)

1961 (1)

J. Connes, Rev. Opt. 40, 45, 116, 171, and 231 (1961).

1958 (1)

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

Connes, J.

J. Connes, Rev. Opt. 40, 45, 116, 171, and 231 (1961).

Davenport, W. B.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

Fellgett, P.

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

Harwit, M.

Kruse, P. W.

P. W. Kruse, L. D. McGlauchlin, R. B. McQuistan, Elements of Infrared Technology: Generation, Transmission, and Detection (Wiley, New York, 1962).

McGlauchlin, L. D.

P. W. Kruse, L. D. McGlauchlin, R. B. McQuistan, Elements of Infrared Technology: Generation, Transmission, and Detection (Wiley, New York, 1962).

McQuistan, R. B.

P. W. Kruse, L. D. McGlauchlin, R. B. McQuistan, Elements of Infrared Technology: Generation, Transmission, and Detection (Wiley, New York, 1962).

Root, W. L.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

Sakai, H.

G. A. Vanasse, H. Sakai, in Progress in Optics, E. Wolf, Ed. (North Holland, Amsterdam, 1967), p. 261.

Tai, M. H.

Vanasse, G. A.

G. A. Vanasse, H. Sakai, in Progress in Optics, E. Wolf, Ed. (North Holland, Amsterdam, 1967), p. 261.

Appl. Opt. (1)

J. Phys. Radium (1)

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

Rev. Opt. (1)

J. Connes, Rev. Opt. 40, 45, 116, 171, and 231 (1961).

Other (3)

G. A. Vanasse, H. Sakai, in Progress in Optics, E. Wolf, Ed. (North Holland, Amsterdam, 1967), p. 261.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

P. W. Kruse, L. D. McGlauchlin, R. B. McQuistan, Elements of Infrared Technology: Generation, Transmission, and Detection (Wiley, New York, 1962).

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Figures (3)

Fig. 1
Fig. 1

Generalized model of the spectrometer.

Fig. 2
Fig. 2

Output of scanning spectrometer as a function of time.

Fig. 3
Fig. 3

The spectrum of frequencies contained in the scan shown in Fig. 2.

Equations (32)

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signal = P R - d t A ( t ) M ( t ) .
noise 2 = E [ - d t x ( t ) A ( t ) ] 2 = - d t d t A ( t ) A ( t ) E x ( t ) x ( t ) .
E x ( t ) x ( t ) = - d f S ( f ) exp [ 2 π i f ( t - t ) ] .
noise 2 = - d f S ( f ) A ˜ ( f ) 2 ,
A ˜ ( f ) = - d t A ( t ) exp ( - 2 π i f t ) .
noise 2 = S 0 - d f A ˜ ( f ) 2 = S 0 - d t A ( t ) 2 .
signal noise = P R ( S 0 ) 1 / 2 - d t A ( t ) M ( t ) [ - d t A ( t ) 2 ] 1 / 2 .
NEP = ( 2 S ) 1 / 2 R W / Hz 1 / 2 .
signal noise = P NEP 2 1 / 2 - d t A ( t ) M ( t ) [ - d t A ( t ) 2 ] 1 / 2 .
rect ( t / T ) = { 1 for 0 t T 0 elsewhere ,
A ( t ) = M ( t ) = rect ( t / T ) ,
signal noise = P NEP ( 2 T ) 1 / 2 .
M ( t ) = rect ( t / T ) and A ( t ) = rect ( t / T ) half the time , M ( t ) = 0 and A ( t ) = - rect ( t / T ) half the time .
signal noise = P NEP ( T / 2 ) 1 / 2 .
M ( t ) = { rect ( t / T ) when A ( t ) > 0 , 0 when A ( t ) < 0 , A ( t ) = cos ( 2 π f t ) rect ( t / T ) .
signal noise = P NEP ( T 4 / π 2 ) 1 / 2 .
M ( t ) = ½ ( 1 + cos 2 π f t ) rect ( t / T ) .
A ( t ) = cos ( 2 π f t ) rect ( t / T ) .
signal noise = P NEP ( T 4 ) 1 / 2 .
SNR FTS SNR SCAN = ( N 8 ) 1 / 2 .
signal noise = P NEP ( 2 T ) 1 / 2 .
signal noise = P R [ - 1 / T 1 / T d f S ( f ) ] 1 / 2 = P NEP T .
I ( t ) = R 4 - d f B ( f ) exp ( 2 π i f t ) ,
B ( f ) = 4 - d t A ( t ) I ( t ) exp ( - 2 π i f t ) .
B ( f ) = signal = R - d f A ˜ ( f ) B ( f - f ) ,
N ( f ) = 4 - d t A ( t ) x ( t ) exp ( - 2 π i f t ) .
noise 2 = E [ Real N ( f ) ] 2 = ( ½ ) E N * ( f ) N ( f ) .
E N * ( f ) N ( f ) = 16 - d t d t A ( t ) × A ( t ) E x ( t ) x ( t ) exp [ 2 π i f ( t - t ) ]
noise 2 = 8 - d f S ( f ) A ˜ ( f - f ) 2 .
signal noise = R - d f A ( f ) B ( f - f ) [ 8 - d f S ( f ) A ˜ ( f - f ) 2 ] 1 / 2 .
δ σ = 1 / ( 2 L max ) = 1 / ( T v ) .
signal noise = F ( σ ) δ σ NEP ( T 4 ) 1 / 2 .

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