Abstract

We derive the spectral signal-to-noise ratio (SNR) trade-offs associated with coarsely sampled Fourier transform spectroscopy using a step-and-integrate measurement scheme. We show that there is no SNR penalty in the shot noise limit and a slight SNR benefit in the detector noise limit for the case of coarse sampling to achieve the same spectral resolution as a baseline Nyquist sampling scenario, where the total detector integration time is the same for both sampling cases.

© 2007 Optical Society of America

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References

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  1. H. Nyquist, "Certain topics in telegraph transmission theory," Trans. Am. Inst. Electr. Eng. 47, 617-644 (1928).
    [CrossRef]
  2. E. T. Whittaker, "On the functions which are represented by the expansions of the interpolation theory," Proc. R. Soc. Edinburgh, Sect. A: Math. Phys. Sci. 35, 181-194 (1915).
  3. C. E. Shannon, "Communication in the presence of noise," Proc. Inst. Radio Eng. 37, 10-21 (1949).
  4. J. Connes, "Spectroscopic studies using Fourier transformations," Rev. Opt. Theor. Instrum. 40, 116 (1961).
  5. G. A. Vanasse and H. Sakai, "Fourier spectroscopy," in Progress in Optics, Vol. 6, E.Wolf, ed. (North-Holland, 1967), pp. 261-330.
  6. J. Kauppinen, "Correction of the linear phase errors of one-sided interferograms," Infrared Phys. 16, 359-366 (1976).
    [CrossRef]
  7. D. A. Naylor, B. G. Gom, M. K. Tahic, and G. R. Davis, "Astronomical spectroscopy using an aliased, step-and-integrate, Fourier transform spectrometer," Proc. SPIE 5498, 685-694 (2004).
    [CrossRef]
  8. D. A. Naylor, B. G. Gom, T. R. Fulton, M. K. Tahic, and G. R. Davis, "Increased efficiency through undersampling in Fourier transform spectroscopy," presented at the OSA Topical Meeting, "Fourier Transform Spectroscopy/Hyperspectral Imaging and Sounding of the Environment," Alexandria, Va., January 31-February 3, 2005.
  9. J. Kauppinen and J. Partanen, Fourier Transforms in Spectroscopy (Wiley-VCH, 2001).
    [CrossRef]
  10. ASTM International, "Standard tables for reference solar spectral irradiances: direct normal and hemispherical on 37° tilted surface," standard G173-03, http://www.astm.org (2003).

2004 (1)

D. A. Naylor, B. G. Gom, M. K. Tahic, and G. R. Davis, "Astronomical spectroscopy using an aliased, step-and-integrate, Fourier transform spectrometer," Proc. SPIE 5498, 685-694 (2004).
[CrossRef]

1976 (1)

J. Kauppinen, "Correction of the linear phase errors of one-sided interferograms," Infrared Phys. 16, 359-366 (1976).
[CrossRef]

1961 (1)

J. Connes, "Spectroscopic studies using Fourier transformations," Rev. Opt. Theor. Instrum. 40, 116 (1961).

1949 (1)

C. E. Shannon, "Communication in the presence of noise," Proc. Inst. Radio Eng. 37, 10-21 (1949).

1928 (1)

H. Nyquist, "Certain topics in telegraph transmission theory," Trans. Am. Inst. Electr. Eng. 47, 617-644 (1928).
[CrossRef]

1915 (1)

E. T. Whittaker, "On the functions which are represented by the expansions of the interpolation theory," Proc. R. Soc. Edinburgh, Sect. A: Math. Phys. Sci. 35, 181-194 (1915).

Connes, J.

J. Connes, "Spectroscopic studies using Fourier transformations," Rev. Opt. Theor. Instrum. 40, 116 (1961).

Davis, G. R.

D. A. Naylor, B. G. Gom, M. K. Tahic, and G. R. Davis, "Astronomical spectroscopy using an aliased, step-and-integrate, Fourier transform spectrometer," Proc. SPIE 5498, 685-694 (2004).
[CrossRef]

D. A. Naylor, B. G. Gom, T. R. Fulton, M. K. Tahic, and G. R. Davis, "Increased efficiency through undersampling in Fourier transform spectroscopy," presented at the OSA Topical Meeting, "Fourier Transform Spectroscopy/Hyperspectral Imaging and Sounding of the Environment," Alexandria, Va., January 31-February 3, 2005.

Fulton, T. R.

D. A. Naylor, B. G. Gom, T. R. Fulton, M. K. Tahic, and G. R. Davis, "Increased efficiency through undersampling in Fourier transform spectroscopy," presented at the OSA Topical Meeting, "Fourier Transform Spectroscopy/Hyperspectral Imaging and Sounding of the Environment," Alexandria, Va., January 31-February 3, 2005.

Gom, B. G.

D. A. Naylor, B. G. Gom, M. K. Tahic, and G. R. Davis, "Astronomical spectroscopy using an aliased, step-and-integrate, Fourier transform spectrometer," Proc. SPIE 5498, 685-694 (2004).
[CrossRef]

D. A. Naylor, B. G. Gom, T. R. Fulton, M. K. Tahic, and G. R. Davis, "Increased efficiency through undersampling in Fourier transform spectroscopy," presented at the OSA Topical Meeting, "Fourier Transform Spectroscopy/Hyperspectral Imaging and Sounding of the Environment," Alexandria, Va., January 31-February 3, 2005.

Kauppinen, J.

J. Kauppinen, "Correction of the linear phase errors of one-sided interferograms," Infrared Phys. 16, 359-366 (1976).
[CrossRef]

J. Kauppinen and J. Partanen, Fourier Transforms in Spectroscopy (Wiley-VCH, 2001).
[CrossRef]

Naylor, D. A.

D. A. Naylor, B. G. Gom, M. K. Tahic, and G. R. Davis, "Astronomical spectroscopy using an aliased, step-and-integrate, Fourier transform spectrometer," Proc. SPIE 5498, 685-694 (2004).
[CrossRef]

D. A. Naylor, B. G. Gom, T. R. Fulton, M. K. Tahic, and G. R. Davis, "Increased efficiency through undersampling in Fourier transform spectroscopy," presented at the OSA Topical Meeting, "Fourier Transform Spectroscopy/Hyperspectral Imaging and Sounding of the Environment," Alexandria, Va., January 31-February 3, 2005.

Nyquist, H.

H. Nyquist, "Certain topics in telegraph transmission theory," Trans. Am. Inst. Electr. Eng. 47, 617-644 (1928).
[CrossRef]

Partanen, J.

J. Kauppinen and J. Partanen, Fourier Transforms in Spectroscopy (Wiley-VCH, 2001).
[CrossRef]

Sakai, H.

G. A. Vanasse and H. Sakai, "Fourier spectroscopy," in Progress in Optics, Vol. 6, E.Wolf, ed. (North-Holland, 1967), pp. 261-330.

Shannon, C. E.

C. E. Shannon, "Communication in the presence of noise," Proc. Inst. Radio Eng. 37, 10-21 (1949).

Tahic, M. K.

D. A. Naylor, B. G. Gom, M. K. Tahic, and G. R. Davis, "Astronomical spectroscopy using an aliased, step-and-integrate, Fourier transform spectrometer," Proc. SPIE 5498, 685-694 (2004).
[CrossRef]

D. A. Naylor, B. G. Gom, T. R. Fulton, M. K. Tahic, and G. R. Davis, "Increased efficiency through undersampling in Fourier transform spectroscopy," presented at the OSA Topical Meeting, "Fourier Transform Spectroscopy/Hyperspectral Imaging and Sounding of the Environment," Alexandria, Va., January 31-February 3, 2005.

Vanasse, G. A.

G. A. Vanasse and H. Sakai, "Fourier spectroscopy," in Progress in Optics, Vol. 6, E.Wolf, ed. (North-Holland, 1967), pp. 261-330.

Whittaker, E. T.

E. T. Whittaker, "On the functions which are represented by the expansions of the interpolation theory," Proc. R. Soc. Edinburgh, Sect. A: Math. Phys. Sci. 35, 181-194 (1915).

Infrared Phys. (1)

J. Kauppinen, "Correction of the linear phase errors of one-sided interferograms," Infrared Phys. 16, 359-366 (1976).
[CrossRef]

Proc. Inst. Radio Eng. (1)

C. E. Shannon, "Communication in the presence of noise," Proc. Inst. Radio Eng. 37, 10-21 (1949).

Proc. R. Soc. Edinburgh, Sect. A: Math. Phys. Sci. (1)

E. T. Whittaker, "On the functions which are represented by the expansions of the interpolation theory," Proc. R. Soc. Edinburgh, Sect. A: Math. Phys. Sci. 35, 181-194 (1915).

Proc. SPIE (1)

D. A. Naylor, B. G. Gom, M. K. Tahic, and G. R. Davis, "Astronomical spectroscopy using an aliased, step-and-integrate, Fourier transform spectrometer," Proc. SPIE 5498, 685-694 (2004).
[CrossRef]

Rev. Opt. Theor. Instrum. (1)

J. Connes, "Spectroscopic studies using Fourier transformations," Rev. Opt. Theor. Instrum. 40, 116 (1961).

Trans. Am. Inst. Electr. Eng. (1)

H. Nyquist, "Certain topics in telegraph transmission theory," Trans. Am. Inst. Electr. Eng. 47, 617-644 (1928).
[CrossRef]

Other (4)

G. A. Vanasse and H. Sakai, "Fourier spectroscopy," in Progress in Optics, Vol. 6, E.Wolf, ed. (North-Holland, 1967), pp. 261-330.

D. A. Naylor, B. G. Gom, T. R. Fulton, M. K. Tahic, and G. R. Davis, "Increased efficiency through undersampling in Fourier transform spectroscopy," presented at the OSA Topical Meeting, "Fourier Transform Spectroscopy/Hyperspectral Imaging and Sounding of the Environment," Alexandria, Va., January 31-February 3, 2005.

J. Kauppinen and J. Partanen, Fourier Transforms in Spectroscopy (Wiley-VCH, 2001).
[CrossRef]

ASTM International, "Standard tables for reference solar spectral irradiances: direct normal and hemispherical on 37° tilted surface," standard G173-03, http://www.astm.org (2003).

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Figures (1)

Fig. 1
Fig. 1

Example spectra recovered from simulated FTS measurements (a) without noise, (b) with detector and shot noise, and (c)–(e) additionally with undersampling. For (a) and (b) α = 1 (Nyquist sampling), N = 384 ; for (c) α = 2 , N = 192 ; for (d) α = 3 , N = 128 ; and for (e) α = 4 , N = 96 . For the coarse sampling cases ( α > 1 ) , the unshaded regions indicate the extent of unaliased spectra, while the shaded regions contain aliased spectra. The vertical dotted lines in (c)–(e) indicate the extent of each length N DFT of the coarsely sampled data.

Tables (2)

Tables Icon

Table 1 Coarse versus Nyquist Sampling Trade-Offs in the Detector Noise Limit

Tables Icon

Table 2 Coarse versus Nyquist Sampling Trade-Offs in the Shot Noise Limit

Equations (28)

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h ( τ ) = t d 2 0 S ( ν ) [ 1 + cos ( 2 π ν τ ) ] d ν ,
f ( τ ) = h ( τ ) h ¯ = t d 2 0 S ( ν ) cos ( 2 π ν τ ) d ν ,
h ¯ = t d 2 0 S ( ν ) d ν .
F ( ν ) = f ( τ ) exp ( i 2 π ν τ ) d τ = t d 4 [ S ( ν ) + S ( ν ) ] .
g n = f ( n Δ τ ) ,
G p = 1 N n = N 2 N 2 1 g n exp ( i 2 π n p N ) ,
G p = 1 N Δ τ { F ( ν ) [ 1 Δ ν sinc ( ν Δ ν ) ] [ 1 N Δ ν comb ( ν N Δ ν ) ] } δ ( ν p Δ ν ) d ν ,
Δ ν = 1 ( N Δ τ )
G p = 1 N Δ τ F ( ν ) 1 N Δ ν comb ( ν p Δ ν N Δ ν ) d ν = t d 4 N Δ τ q = S [ ( p q N ) Δ ν ] + S [ ( p q N ) Δ ν ] ,
ε m ε n = σ τ , n 2 δ m , n ,
σ τ , n 2 = σ d 2 + h ( n Δ τ ) ,
σ τ , n 2 σ d 2 + h ¯ = σ d 2 + t d 2 0 S ( ν ) d ν .
σ ν , p 2 = [ 1 N n = N 2 N 2 ε n cos ( i 2 π p n N ) ] 2 = 1 N m = N 2 N 2 n = N 2 N 2 ε m ε n cos ( 2 π p m N ) cos ( 2 π p n N ) ,
= 1 N n = N 2 N 2 σ τ , n 2 cos 2 ( 2 π p n N ) ,
σ ν , p 2 = [ σ d 2 2 + t d 4 0 S ( ν ) d ν ] ( 1 + δ p , 0 + δ p , N 2 ) .
SNR p = G p σ ν , p .
σ d 2 t d 2 0 S ( ν ) d ν ,
SNR d , p = t d q = S [ ( p q N ) Δ ν ] + S [ ( p q N ) Δ ν ] 2 2 N σ d Δ τ ( 1 + δ p , 0 + δ p , N 2 ) ,
SNR d , p t d N Δ τ = N t d Δ ν .
t d 2 0 S ( ν ) d ν σ d 2 ,
SNR s , p = t d q = S [ ( p q N ) Δ ν ] + S [ ( p q N ) Δ ν ] 2 N Δ τ 0 S ( ν ) d ν ( 1 + δ p , 0 + δ p , N 2 ) ,
SNR s , p t d N Δ τ = N t d Δ ν .
t d = T N ,
t d = T N t s N .
t 0 2 0 S ( ν ) d ν = 1.3 × 10 4   photoelectrons
G p = 1 N n = N 2 N 2 1 f ( τ ) exp ( i 2 π ν τ ) δ ( τ n Δ τ ) δ ( ν p Δ ν ) d τ d ν .
G p = Lim ε 0 + 1 N f ( τ ) rect ( τ + ε N Δ τ ) [ 1 Δ τ comb ( τ Δ τ ) ] × exp ( i 2 π ν τ ) δ ( ν p Δ ν ) d τ d ν ,
G p = 1 N Δ τ { F ( ν ) [ 1 Δ ν sinc ( ν Δ ν ) ] [ 1 N Δ ν comb ( ν N Δ ν ) ] } δ ( ν p Δ ν ) d ν .

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