Abstract

We describe a static aperture-coded, dispersive longwave infrared (LWIR) spectrometer that uses a microbolometer array at the detector plane. The two-dimensional aperture code is based on a row-doubled Hadamard mask with transmissive and opaque openings. The independent column code nature of the matrix makes for a mathematically well-defined pattern that spatially and spectrally maps the source information to the detector plane. Post-processing techniques on the data provide spectral estimates of the source. Comparative experimental results between a slit and coded aperture for emission spectroscopy from a CO2 laser are demonstrated.

© 2007 Optical Society of America

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References

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  1. S. C. Bates, P. W. M., and P. R. Solomon, “Infrared Monitoring of Combustion,” in SPIE, Vol. 1434 (1991).
    [Crossref]
  2. P. Jacquinot, “New developments in interference spectroscopy,” Rep. Prog. Phys. 23, 267–312 (1960).
    [Crossref]
  3. P. Fellgett, “The Multiplex Advantage,” Ph.D. thesis, Cambridge, UK (1951).
  4. E. C. Cull, M. E. Gehm, and D. Brady, “Dispersion multiplexing with broadband filtering for miniature spectrometers,” Appl. Opt. 46, 365–374 (2007)
    [Crossref] [PubMed]
  5. S. T. McCain, M. E. Gehm, Y. Wang, N. P. Pitsianis, and D. J. Brady, “Coded aperture Raman spectroscopy for quantitative measurements of ethanol in a tissue phantom solution,” Appl. Spectrosc. 60, 663–671 (2006).
    [Crossref] [PubMed]
  6. M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan, “Static two-dimensional aperture coding for multimodal, multiplex spectroscopy,” Appl. Opt. 45, 2965–2974 (2006).
    [Crossref] [PubMed]
  7. T. Schimert, D. Ratcliff, J. Brady, S. Ropson, R. Gooch, B. Ritchey, P. McCardel, K. Rachels, M. Wand, M. Wein-stein, and J. Wynn, “Low Cost, Low Power Uncooled a-Si-based Micro Infrared Camera for Unattended Ground Sensor Applications,” in SPIE Proceedings on Unattended Ground Sensor Technologies and Applications,  vol. 3713, pp. 101–111 (1999).
  8. P. Richards, “Bolometers for infrared and millimeter waves,” J. Appl. Phys. 76, 1–24 (1994).
    [Crossref]
  9. T. Schimert, N. Cunningham, G. Francisco, R. Gooch, J. Gooden, P. McCardel, B. Neal, B. Ritchey, J. Rife, A. J. Syllaios, J. Tregilgas, J. Brady, J. Gilstrap, and S. Ropson, “Low Cost, Low Power Uncooled 120×160 a-Si-based Micro Infrared Camera for Law Enforcement Applications,” in Proceedings of SPIE - The International Society for Optical Engineering,  vol. 4232, pp. 187–194 (2001).
  10. R. G. Driggers, P. Cox, and T. Edwards, Introduction to Infrared and Electro-Optical Systems (Artech House, London, 1999).
  11. G. Nitzsche and R. Riesenberg, “Noise, Fluctuation and HADAMARD-Transform-Spectrometry,” in Proceedings of SPIE Fluctuations and Noise in Photonics and Quantum Optics,  vol. 5111, pp. 273–282 (2003).
  12. N. Menn, Practical Optics (Elsevier Academic Press, New York, 2004).
  13. J. R. Wilson, Handbook of Optics, 2nd ed. (McGraw-Hill, Inc., 1995) Vol. 1.
  14. C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Inc, 1974).

2007 (1)

2006 (2)

2003 (1)

G. Nitzsche and R. Riesenberg, “Noise, Fluctuation and HADAMARD-Transform-Spectrometry,” in Proceedings of SPIE Fluctuations and Noise in Photonics and Quantum Optics,  vol. 5111, pp. 273–282 (2003).

2001 (1)

T. Schimert, N. Cunningham, G. Francisco, R. Gooch, J. Gooden, P. McCardel, B. Neal, B. Ritchey, J. Rife, A. J. Syllaios, J. Tregilgas, J. Brady, J. Gilstrap, and S. Ropson, “Low Cost, Low Power Uncooled 120×160 a-Si-based Micro Infrared Camera for Law Enforcement Applications,” in Proceedings of SPIE - The International Society for Optical Engineering,  vol. 4232, pp. 187–194 (2001).

1999 (1)

T. Schimert, D. Ratcliff, J. Brady, S. Ropson, R. Gooch, B. Ritchey, P. McCardel, K. Rachels, M. Wand, M. Wein-stein, and J. Wynn, “Low Cost, Low Power Uncooled a-Si-based Micro Infrared Camera for Unattended Ground Sensor Applications,” in SPIE Proceedings on Unattended Ground Sensor Technologies and Applications,  vol. 3713, pp. 101–111 (1999).

1994 (1)

P. Richards, “Bolometers for infrared and millimeter waves,” J. Appl. Phys. 76, 1–24 (1994).
[Crossref]

1991 (1)

S. C. Bates, P. W. M., and P. R. Solomon, “Infrared Monitoring of Combustion,” in SPIE, Vol. 1434 (1991).
[Crossref]

1960 (1)

P. Jacquinot, “New developments in interference spectroscopy,” Rep. Prog. Phys. 23, 267–312 (1960).
[Crossref]

Bates, S. C.

S. C. Bates, P. W. M., and P. R. Solomon, “Infrared Monitoring of Combustion,” in SPIE, Vol. 1434 (1991).
[Crossref]

Brady, D.

Brady, D. J.

Brady, J.

T. Schimert, N. Cunningham, G. Francisco, R. Gooch, J. Gooden, P. McCardel, B. Neal, B. Ritchey, J. Rife, A. J. Syllaios, J. Tregilgas, J. Brady, J. Gilstrap, and S. Ropson, “Low Cost, Low Power Uncooled 120×160 a-Si-based Micro Infrared Camera for Law Enforcement Applications,” in Proceedings of SPIE - The International Society for Optical Engineering,  vol. 4232, pp. 187–194 (2001).

T. Schimert, D. Ratcliff, J. Brady, S. Ropson, R. Gooch, B. Ritchey, P. McCardel, K. Rachels, M. Wand, M. Wein-stein, and J. Wynn, “Low Cost, Low Power Uncooled a-Si-based Micro Infrared Camera for Unattended Ground Sensor Applications,” in SPIE Proceedings on Unattended Ground Sensor Technologies and Applications,  vol. 3713, pp. 101–111 (1999).

Cox, P.

R. G. Driggers, P. Cox, and T. Edwards, Introduction to Infrared and Electro-Optical Systems (Artech House, London, 1999).

Cull, E. C.

Cunningham, N.

T. Schimert, N. Cunningham, G. Francisco, R. Gooch, J. Gooden, P. McCardel, B. Neal, B. Ritchey, J. Rife, A. J. Syllaios, J. Tregilgas, J. Brady, J. Gilstrap, and S. Ropson, “Low Cost, Low Power Uncooled 120×160 a-Si-based Micro Infrared Camera for Law Enforcement Applications,” in Proceedings of SPIE - The International Society for Optical Engineering,  vol. 4232, pp. 187–194 (2001).

Driggers, R. G.

R. G. Driggers, P. Cox, and T. Edwards, Introduction to Infrared and Electro-Optical Systems (Artech House, London, 1999).

Edwards, T.

R. G. Driggers, P. Cox, and T. Edwards, Introduction to Infrared and Electro-Optical Systems (Artech House, London, 1999).

Fellgett, P.

P. Fellgett, “The Multiplex Advantage,” Ph.D. thesis, Cambridge, UK (1951).

Francisco, G.

T. Schimert, N. Cunningham, G. Francisco, R. Gooch, J. Gooden, P. McCardel, B. Neal, B. Ritchey, J. Rife, A. J. Syllaios, J. Tregilgas, J. Brady, J. Gilstrap, and S. Ropson, “Low Cost, Low Power Uncooled 120×160 a-Si-based Micro Infrared Camera for Law Enforcement Applications,” in Proceedings of SPIE - The International Society for Optical Engineering,  vol. 4232, pp. 187–194 (2001).

Gehm, M. E.

Gilstrap, J.

T. Schimert, N. Cunningham, G. Francisco, R. Gooch, J. Gooden, P. McCardel, B. Neal, B. Ritchey, J. Rife, A. J. Syllaios, J. Tregilgas, J. Brady, J. Gilstrap, and S. Ropson, “Low Cost, Low Power Uncooled 120×160 a-Si-based Micro Infrared Camera for Law Enforcement Applications,” in Proceedings of SPIE - The International Society for Optical Engineering,  vol. 4232, pp. 187–194 (2001).

Gooch, R.

T. Schimert, N. Cunningham, G. Francisco, R. Gooch, J. Gooden, P. McCardel, B. Neal, B. Ritchey, J. Rife, A. J. Syllaios, J. Tregilgas, J. Brady, J. Gilstrap, and S. Ropson, “Low Cost, Low Power Uncooled 120×160 a-Si-based Micro Infrared Camera for Law Enforcement Applications,” in Proceedings of SPIE - The International Society for Optical Engineering,  vol. 4232, pp. 187–194 (2001).

T. Schimert, D. Ratcliff, J. Brady, S. Ropson, R. Gooch, B. Ritchey, P. McCardel, K. Rachels, M. Wand, M. Wein-stein, and J. Wynn, “Low Cost, Low Power Uncooled a-Si-based Micro Infrared Camera for Unattended Ground Sensor Applications,” in SPIE Proceedings on Unattended Ground Sensor Technologies and Applications,  vol. 3713, pp. 101–111 (1999).

Gooden, J.

T. Schimert, N. Cunningham, G. Francisco, R. Gooch, J. Gooden, P. McCardel, B. Neal, B. Ritchey, J. Rife, A. J. Syllaios, J. Tregilgas, J. Brady, J. Gilstrap, and S. Ropson, “Low Cost, Low Power Uncooled 120×160 a-Si-based Micro Infrared Camera for Law Enforcement Applications,” in Proceedings of SPIE - The International Society for Optical Engineering,  vol. 4232, pp. 187–194 (2001).

Hanson, R. J.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Inc, 1974).

Jacquinot, P.

P. Jacquinot, “New developments in interference spectroscopy,” Rep. Prog. Phys. 23, 267–312 (1960).
[Crossref]

Lawson, C. L.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Inc, 1974).

M., P. W.

S. C. Bates, P. W. M., and P. R. Solomon, “Infrared Monitoring of Combustion,” in SPIE, Vol. 1434 (1991).
[Crossref]

McCain, S. T.

McCardel, P.

T. Schimert, N. Cunningham, G. Francisco, R. Gooch, J. Gooden, P. McCardel, B. Neal, B. Ritchey, J. Rife, A. J. Syllaios, J. Tregilgas, J. Brady, J. Gilstrap, and S. Ropson, “Low Cost, Low Power Uncooled 120×160 a-Si-based Micro Infrared Camera for Law Enforcement Applications,” in Proceedings of SPIE - The International Society for Optical Engineering,  vol. 4232, pp. 187–194 (2001).

T. Schimert, D. Ratcliff, J. Brady, S. Ropson, R. Gooch, B. Ritchey, P. McCardel, K. Rachels, M. Wand, M. Wein-stein, and J. Wynn, “Low Cost, Low Power Uncooled a-Si-based Micro Infrared Camera for Unattended Ground Sensor Applications,” in SPIE Proceedings on Unattended Ground Sensor Technologies and Applications,  vol. 3713, pp. 101–111 (1999).

Menn, N.

N. Menn, Practical Optics (Elsevier Academic Press, New York, 2004).

Neal, B.

T. Schimert, N. Cunningham, G. Francisco, R. Gooch, J. Gooden, P. McCardel, B. Neal, B. Ritchey, J. Rife, A. J. Syllaios, J. Tregilgas, J. Brady, J. Gilstrap, and S. Ropson, “Low Cost, Low Power Uncooled 120×160 a-Si-based Micro Infrared Camera for Law Enforcement Applications,” in Proceedings of SPIE - The International Society for Optical Engineering,  vol. 4232, pp. 187–194 (2001).

Nitzsche, G.

G. Nitzsche and R. Riesenberg, “Noise, Fluctuation and HADAMARD-Transform-Spectrometry,” in Proceedings of SPIE Fluctuations and Noise in Photonics and Quantum Optics,  vol. 5111, pp. 273–282 (2003).

Pitsianis, N. P.

Potuluri, P.

Rachels, K.

T. Schimert, D. Ratcliff, J. Brady, S. Ropson, R. Gooch, B. Ritchey, P. McCardel, K. Rachels, M. Wand, M. Wein-stein, and J. Wynn, “Low Cost, Low Power Uncooled a-Si-based Micro Infrared Camera for Unattended Ground Sensor Applications,” in SPIE Proceedings on Unattended Ground Sensor Technologies and Applications,  vol. 3713, pp. 101–111 (1999).

Ratcliff, D.

T. Schimert, D. Ratcliff, J. Brady, S. Ropson, R. Gooch, B. Ritchey, P. McCardel, K. Rachels, M. Wand, M. Wein-stein, and J. Wynn, “Low Cost, Low Power Uncooled a-Si-based Micro Infrared Camera for Unattended Ground Sensor Applications,” in SPIE Proceedings on Unattended Ground Sensor Technologies and Applications,  vol. 3713, pp. 101–111 (1999).

Richards, P.

P. Richards, “Bolometers for infrared and millimeter waves,” J. Appl. Phys. 76, 1–24 (1994).
[Crossref]

Riesenberg, R.

G. Nitzsche and R. Riesenberg, “Noise, Fluctuation and HADAMARD-Transform-Spectrometry,” in Proceedings of SPIE Fluctuations and Noise in Photonics and Quantum Optics,  vol. 5111, pp. 273–282 (2003).

Rife, J.

T. Schimert, N. Cunningham, G. Francisco, R. Gooch, J. Gooden, P. McCardel, B. Neal, B. Ritchey, J. Rife, A. J. Syllaios, J. Tregilgas, J. Brady, J. Gilstrap, and S. Ropson, “Low Cost, Low Power Uncooled 120×160 a-Si-based Micro Infrared Camera for Law Enforcement Applications,” in Proceedings of SPIE - The International Society for Optical Engineering,  vol. 4232, pp. 187–194 (2001).

Ritchey, B.

T. Schimert, N. Cunningham, G. Francisco, R. Gooch, J. Gooden, P. McCardel, B. Neal, B. Ritchey, J. Rife, A. J. Syllaios, J. Tregilgas, J. Brady, J. Gilstrap, and S. Ropson, “Low Cost, Low Power Uncooled 120×160 a-Si-based Micro Infrared Camera for Law Enforcement Applications,” in Proceedings of SPIE - The International Society for Optical Engineering,  vol. 4232, pp. 187–194 (2001).

T. Schimert, D. Ratcliff, J. Brady, S. Ropson, R. Gooch, B. Ritchey, P. McCardel, K. Rachels, M. Wand, M. Wein-stein, and J. Wynn, “Low Cost, Low Power Uncooled a-Si-based Micro Infrared Camera for Unattended Ground Sensor Applications,” in SPIE Proceedings on Unattended Ground Sensor Technologies and Applications,  vol. 3713, pp. 101–111 (1999).

Ropson, S.

T. Schimert, N. Cunningham, G. Francisco, R. Gooch, J. Gooden, P. McCardel, B. Neal, B. Ritchey, J. Rife, A. J. Syllaios, J. Tregilgas, J. Brady, J. Gilstrap, and S. Ropson, “Low Cost, Low Power Uncooled 120×160 a-Si-based Micro Infrared Camera for Law Enforcement Applications,” in Proceedings of SPIE - The International Society for Optical Engineering,  vol. 4232, pp. 187–194 (2001).

T. Schimert, D. Ratcliff, J. Brady, S. Ropson, R. Gooch, B. Ritchey, P. McCardel, K. Rachels, M. Wand, M. Wein-stein, and J. Wynn, “Low Cost, Low Power Uncooled a-Si-based Micro Infrared Camera for Unattended Ground Sensor Applications,” in SPIE Proceedings on Unattended Ground Sensor Technologies and Applications,  vol. 3713, pp. 101–111 (1999).

Schimert, T.

T. Schimert, N. Cunningham, G. Francisco, R. Gooch, J. Gooden, P. McCardel, B. Neal, B. Ritchey, J. Rife, A. J. Syllaios, J. Tregilgas, J. Brady, J. Gilstrap, and S. Ropson, “Low Cost, Low Power Uncooled 120×160 a-Si-based Micro Infrared Camera for Law Enforcement Applications,” in Proceedings of SPIE - The International Society for Optical Engineering,  vol. 4232, pp. 187–194 (2001).

T. Schimert, D. Ratcliff, J. Brady, S. Ropson, R. Gooch, B. Ritchey, P. McCardel, K. Rachels, M. Wand, M. Wein-stein, and J. Wynn, “Low Cost, Low Power Uncooled a-Si-based Micro Infrared Camera for Unattended Ground Sensor Applications,” in SPIE Proceedings on Unattended Ground Sensor Technologies and Applications,  vol. 3713, pp. 101–111 (1999).

Solomon, P. R.

S. C. Bates, P. W. M., and P. R. Solomon, “Infrared Monitoring of Combustion,” in SPIE, Vol. 1434 (1991).
[Crossref]

Sullivan, M. E.

Syllaios, A. J.

T. Schimert, N. Cunningham, G. Francisco, R. Gooch, J. Gooden, P. McCardel, B. Neal, B. Ritchey, J. Rife, A. J. Syllaios, J. Tregilgas, J. Brady, J. Gilstrap, and S. Ropson, “Low Cost, Low Power Uncooled 120×160 a-Si-based Micro Infrared Camera for Law Enforcement Applications,” in Proceedings of SPIE - The International Society for Optical Engineering,  vol. 4232, pp. 187–194 (2001).

Tregilgas, J.

T. Schimert, N. Cunningham, G. Francisco, R. Gooch, J. Gooden, P. McCardel, B. Neal, B. Ritchey, J. Rife, A. J. Syllaios, J. Tregilgas, J. Brady, J. Gilstrap, and S. Ropson, “Low Cost, Low Power Uncooled 120×160 a-Si-based Micro Infrared Camera for Law Enforcement Applications,” in Proceedings of SPIE - The International Society for Optical Engineering,  vol. 4232, pp. 187–194 (2001).

Wand, M.

T. Schimert, D. Ratcliff, J. Brady, S. Ropson, R. Gooch, B. Ritchey, P. McCardel, K. Rachels, M. Wand, M. Wein-stein, and J. Wynn, “Low Cost, Low Power Uncooled a-Si-based Micro Infrared Camera for Unattended Ground Sensor Applications,” in SPIE Proceedings on Unattended Ground Sensor Technologies and Applications,  vol. 3713, pp. 101–111 (1999).

Wang, Y.

Wein-stein, M.

T. Schimert, D. Ratcliff, J. Brady, S. Ropson, R. Gooch, B. Ritchey, P. McCardel, K. Rachels, M. Wand, M. Wein-stein, and J. Wynn, “Low Cost, Low Power Uncooled a-Si-based Micro Infrared Camera for Unattended Ground Sensor Applications,” in SPIE Proceedings on Unattended Ground Sensor Technologies and Applications,  vol. 3713, pp. 101–111 (1999).

Wilson, J. R.

J. R. Wilson, Handbook of Optics, 2nd ed. (McGraw-Hill, Inc., 1995) Vol. 1.

Wynn, J.

T. Schimert, D. Ratcliff, J. Brady, S. Ropson, R. Gooch, B. Ritchey, P. McCardel, K. Rachels, M. Wand, M. Wein-stein, and J. Wynn, “Low Cost, Low Power Uncooled a-Si-based Micro Infrared Camera for Unattended Ground Sensor Applications,” in SPIE Proceedings on Unattended Ground Sensor Technologies and Applications,  vol. 3713, pp. 101–111 (1999).

Appl. Opt. (2)

Appl. Spectrosc. (1)

in SPIE, (1)

S. C. Bates, P. W. M., and P. R. Solomon, “Infrared Monitoring of Combustion,” in SPIE, Vol. 1434 (1991).
[Crossref]

J. Appl. Phys. (1)

P. Richards, “Bolometers for infrared and millimeter waves,” J. Appl. Phys. 76, 1–24 (1994).
[Crossref]

Rep. Prog. Phys. (1)

P. Jacquinot, “New developments in interference spectroscopy,” Rep. Prog. Phys. 23, 267–312 (1960).
[Crossref]

Other (8)

P. Fellgett, “The Multiplex Advantage,” Ph.D. thesis, Cambridge, UK (1951).

T. Schimert, N. Cunningham, G. Francisco, R. Gooch, J. Gooden, P. McCardel, B. Neal, B. Ritchey, J. Rife, A. J. Syllaios, J. Tregilgas, J. Brady, J. Gilstrap, and S. Ropson, “Low Cost, Low Power Uncooled 120×160 a-Si-based Micro Infrared Camera for Law Enforcement Applications,” in Proceedings of SPIE - The International Society for Optical Engineering,  vol. 4232, pp. 187–194 (2001).

R. G. Driggers, P. Cox, and T. Edwards, Introduction to Infrared and Electro-Optical Systems (Artech House, London, 1999).

G. Nitzsche and R. Riesenberg, “Noise, Fluctuation and HADAMARD-Transform-Spectrometry,” in Proceedings of SPIE Fluctuations and Noise in Photonics and Quantum Optics,  vol. 5111, pp. 273–282 (2003).

N. Menn, Practical Optics (Elsevier Academic Press, New York, 2004).

J. R. Wilson, Handbook of Optics, 2nd ed. (McGraw-Hill, Inc., 1995) Vol. 1.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Inc, 1974).

T. Schimert, D. Ratcliff, J. Brady, S. Ropson, R. Gooch, B. Ritchey, P. McCardel, K. Rachels, M. Wand, M. Wein-stein, and J. Wynn, “Low Cost, Low Power Uncooled a-Si-based Micro Infrared Camera for Unattended Ground Sensor Applications,” in SPIE Proceedings on Unattended Ground Sensor Technologies and Applications,  vol. 3713, pp. 101–111 (1999).

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Figures (7)

Fig. 1.
Fig. 1.

3D-Mechanical layout for the LWIR spectrometer.(M)Mask at input. (L1-L4)Lenses in the optical system. (A) Gold mirror. (G) Grating. (FP) Microbolometer focal plane array.

Fig. 2.
Fig. 2.

Optical design layout for the LWIR spectrometer.

Fig. 3.
Fig. 3.

Hadamard mask N=12.

Fig. 4.
Fig. 4.

Reconstruction procedure for a slit system.(a) Microbolometer raw image of a CO2 laser emission spectrum at the focal plane. (b) Microbolometer image after binning the rows. (c) Reconstructed emission spectrum of a CO2 laser.

Fig. 5.
Fig. 5.

Reconstruction procedure for an aperture-coded system. (a) Microbolometer raw image of a CO2 laser emission spectrum. (b) Microbolometer data binned by rows. (b1)Image of a row-doubled Hadamard matrix. (c) Spectral estimates shown after a column inversion using the NNLS algorithm is applied to the binned data. (d) Spectral estimates are aligned for row summation. (e) Spectral reconstruction of the CO2 source after row summation.

Fig. 6.
Fig. 6.

Slit (a) and Mask (b) reconstructions for calibration of the x-axis.

Fig. 7.
Fig. 7.

Spectral reconstruction of a slit (left column) and mask (right column) for SNR comparison. Reconstructed plot are shown at wavelengths (1) 9.673 μm, (2) 10.247 μm (3) 10.349 μm, (4) 10.696 μm and (5) 10.764 μm.

Equations (16)

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

M H = 2 nfp + ( 2 n 1 ) dp
M W = npf ,
M OA = ( 2 npf ) ( npf ) 2 ,
Γ λ = [ Δ f 2 + Δ s 2 + Δ r 2 ] 1 2 .
σ s = ( N = 1 M σ N 2 ) 1 2 .
σ p = ( σ t 2 + σ b 2 ) 1 2
σ c = ( σ j 2 + σ f 2 ) 1 2
σ s = ( σ p 2 + σ c 2 + σ k 2 ) 1 2 .
I x y = H ( x , x , y , y ; λ ) T x y S ( x , y ; λ ) dλdxdy
α = dx = mf Λ cos ( θ R . )
I x y = T x y S ( x , y ; x x α + λ c ) dx
I x y = S ( 0 , y ; λ c x α )
Minimize Ex f x 0
Ψ K = λ P λ ( E hc ) .
Ψ K D M OA .
SNR = I M ( λ ) N ̅ σ N .

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