Abstract

The least-squares approximation of cosine polynomials is used to construct the spectrum from the simulated nonuniform samples of the interferogram given by a step-mirror-based static Fourier transform spectrometer. Numerical and experimental results show the stability of the algorithm and a spectrum-constructing error of 0.03%.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. R. Griffiths and J. A. de Haseth, “Signal-to-noise ratio,” in Fourier Transform Infrared Spectrometry (Wiley & Sons, 2007), pp. 167–168
  2. O. Manzardo, “Micro-sized Fourier spectrometers,” Ph.D. dissertation (University of Neuchatel, 2002).
  3. J. Sin, W. H. Lee, D. Popa, and H. E. Stephanou, “Assembled Fourier transform micro-spectrometer,” Proc. SPIE 6109, 1–8(2006).
    [CrossRef]
  4. U. Wallrabe, C. Solf, J. Mohr, and J. G. Korvink, “Miniaturized Fourier transform spectrometer for the near infrared wavelength regime incorporating an electromagnetic linear actuator,” Sens. Actuators A 123, 459–467 (2005).
    [CrossRef]
  5. G. Boer, P. Ruffieux, T. Scharf, P. Seitz, and R. Dandliker, “Compact liquid-crystal-polymer Fourier-transform spectrometer,” Appl. Opt. 43, 2201–2208 (2004).
    [CrossRef] [PubMed]
  6. B. Martin, A. Morand, P. Benech, G. Leblond, S. Blaize, G. Lerondel, P. Royer, P. Kern, and E. L. Coarer, “Design of a compact static Fourier transform spectrometer in integrated optics based on a leaky loop structure,” Opt. Lett. 34, 184–186(2009).
    [CrossRef] [PubMed]
  7. K. D. Moller, “Wave-front-dividing array interferometers without moving parts for real-time spectroscopy from the IR to the UV,” Appl. Opt. 34, 1493–1501 (1995).
    [CrossRef] [PubMed]
  8. A. Lacan, F.-M. Bréon, A. Rosak, F. Brachet, L. Roucayrol, P. Etcheto, C. Casteras, and Y. Salaün, “A static Fourier transform spectrometer for atmospheric sounding: concept and experimental implementation,” Opt. Express 18, 8311–8331(2010).
    [CrossRef] [PubMed]
  9. E. V. Ivanov, “Static Fourier transform spectroscopy with enhanced resolving power,” J. Opt. A 2, 519–528 (2000).
    [CrossRef]
  10. Y. M. Kong, J. Q. Liang, Z. Z. Liang, B. Wang, and J. Zhang, “Microassembled Fourier transform spectrometer,” Proc. SPIE 7283, 728304 (2009).
    [CrossRef]
  11. C. Feng, B. Wang, Z. Liang, and J. Liang, “Miniaturization of step mirrors in a static Fourier transform spectrometer theory and simulation,” J. Opt. Soc. Am. B 28, 128–133 (2010).
    [CrossRef]
  12. H. G. Feichtinger, K. Grochenig, and T. Strohmer, “Efficient numerical methods in non-uniform sampling theory,” Numer. Math. 69, 423–440 (1995).
    [CrossRef]
  13. F. Marvasti and M. Analoui, “Recovery of signals from nonuniform samples using iterative methods,” in Proceedings of IEEE International Symposium on Circuits and Systems (IEEE, 1989), pp. 1021–1024.
    [CrossRef]
  14. K. Grochenig, “A discrete theory of irregular sampling,” Linear Algebra Appl. 193, 129–150 (1993).
    [CrossRef]
  15. Y. Shen, C. Zhu, L. Liu, Q. Wang, J. Jin, and Y. Lin, “Explicit solution for nonuniform discrete fourier transform,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (I2MTC) (IEEE, 2010), pp. 1505–1509.
    [CrossRef]
  16. E. Margolis and Y. C. Eldar, “Nonuniform sampling of periodic bandlimited signals,” IEEE Trans. Signal Process. 56, 2728–2745 (2008).
    [CrossRef]
  17. Y. D. Siem, E. de Klerk, and D. den Hertog, “Discrete least-norm approximation by nonnegative (trigonometric) polynomials and rational functions,” Struct. Multidisc. Optim. 35, 327–339 (2008).
    [CrossRef]
  18. T. Strohmer, “Numerical analysis of the non-uniform sampling problem,” J. Comput. Appl. Math. 122, 297–316(2000).
    [CrossRef]
  19. K. Yao and J. B. Thomas, “On some stability and interpolatory properties of nonuniform sampling expansions,” IEEE Trans. Circuit Theory CT-14, 404–408 (1967).
    [CrossRef]
  20. R. F. Bass and K. Grochenig, “Random sampling of multivariate trigonometric polynomials,” SIAM J. Math. Anal. 36, 773–795 (2005).
    [CrossRef]
  21. Q. H. Liu and N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFTs),” IEEE Microwave Guided Wave Lett. 8, 18–20 (1998).
    [CrossRef]
  22. L. Palchetti and D. Lastrucci, “Spectral noise due to sampling errors in Fourier-transform spectroscopy,” Appl. Opt. 40, 3235–3243 (2001).
    [CrossRef]
  23. E. E. Bell and R. B. Sanderson, “Spectral errors resulting from random sampling-position errors in Fourier transform spectroscopy,” Appl. Opt. 11, 688–689 (1972).
    [CrossRef] [PubMed]
  24. D. L. Cohen, “Noise-equivalent change in radiance for sampling noise in a double-sided interferogram,” Appl. Opt. 42, 2289–2300 (2003).
    [CrossRef] [PubMed]
  25. A. Papoulis, “Error analysis in sampling theory,” Proc. IEEE 54, 947–955 (1966).
    [CrossRef]
  26. E. Sarkissian and K. W. Bowman, “Application of a nonuniform spectral resampling transform in Fourier-transform spectrometry,” Appl. Opt. 42, 1122–1131 (2003).
    [CrossRef] [PubMed]

2010

2009

2008

E. Margolis and Y. C. Eldar, “Nonuniform sampling of periodic bandlimited signals,” IEEE Trans. Signal Process. 56, 2728–2745 (2008).
[CrossRef]

Y. D. Siem, E. de Klerk, and D. den Hertog, “Discrete least-norm approximation by nonnegative (trigonometric) polynomials and rational functions,” Struct. Multidisc. Optim. 35, 327–339 (2008).
[CrossRef]

2006

J. Sin, W. H. Lee, D. Popa, and H. E. Stephanou, “Assembled Fourier transform micro-spectrometer,” Proc. SPIE 6109, 1–8(2006).
[CrossRef]

2005

U. Wallrabe, C. Solf, J. Mohr, and J. G. Korvink, “Miniaturized Fourier transform spectrometer for the near infrared wavelength regime incorporating an electromagnetic linear actuator,” Sens. Actuators A 123, 459–467 (2005).
[CrossRef]

R. F. Bass and K. Grochenig, “Random sampling of multivariate trigonometric polynomials,” SIAM J. Math. Anal. 36, 773–795 (2005).
[CrossRef]

2004

2003

2001

2000

E. V. Ivanov, “Static Fourier transform spectroscopy with enhanced resolving power,” J. Opt. A 2, 519–528 (2000).
[CrossRef]

T. Strohmer, “Numerical analysis of the non-uniform sampling problem,” J. Comput. Appl. Math. 122, 297–316(2000).
[CrossRef]

1998

Q. H. Liu and N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFTs),” IEEE Microwave Guided Wave Lett. 8, 18–20 (1998).
[CrossRef]

1995

K. D. Moller, “Wave-front-dividing array interferometers without moving parts for real-time spectroscopy from the IR to the UV,” Appl. Opt. 34, 1493–1501 (1995).
[CrossRef] [PubMed]

H. G. Feichtinger, K. Grochenig, and T. Strohmer, “Efficient numerical methods in non-uniform sampling theory,” Numer. Math. 69, 423–440 (1995).
[CrossRef]

1993

K. Grochenig, “A discrete theory of irregular sampling,” Linear Algebra Appl. 193, 129–150 (1993).
[CrossRef]

1972

1967

K. Yao and J. B. Thomas, “On some stability and interpolatory properties of nonuniform sampling expansions,” IEEE Trans. Circuit Theory CT-14, 404–408 (1967).
[CrossRef]

1966

A. Papoulis, “Error analysis in sampling theory,” Proc. IEEE 54, 947–955 (1966).
[CrossRef]

Analoui, M.

F. Marvasti and M. Analoui, “Recovery of signals from nonuniform samples using iterative methods,” in Proceedings of IEEE International Symposium on Circuits and Systems (IEEE, 1989), pp. 1021–1024.
[CrossRef]

Bass, R. F.

R. F. Bass and K. Grochenig, “Random sampling of multivariate trigonometric polynomials,” SIAM J. Math. Anal. 36, 773–795 (2005).
[CrossRef]

Bell, E. E.

Benech, P.

Blaize, S.

Boer, G.

Bowman, K. W.

Brachet, F.

Bréon, F.-M.

Casteras, C.

Coarer, E. L.

Cohen, D. L.

Dandliker, R.

de Haseth, J. A.

P. R. Griffiths and J. A. de Haseth, “Signal-to-noise ratio,” in Fourier Transform Infrared Spectrometry (Wiley & Sons, 2007), pp. 167–168

de Klerk, E.

Y. D. Siem, E. de Klerk, and D. den Hertog, “Discrete least-norm approximation by nonnegative (trigonometric) polynomials and rational functions,” Struct. Multidisc. Optim. 35, 327–339 (2008).
[CrossRef]

den Hertog, D.

Y. D. Siem, E. de Klerk, and D. den Hertog, “Discrete least-norm approximation by nonnegative (trigonometric) polynomials and rational functions,” Struct. Multidisc. Optim. 35, 327–339 (2008).
[CrossRef]

Eldar, Y. C.

E. Margolis and Y. C. Eldar, “Nonuniform sampling of periodic bandlimited signals,” IEEE Trans. Signal Process. 56, 2728–2745 (2008).
[CrossRef]

Etcheto, P.

Feichtinger, H. G.

H. G. Feichtinger, K. Grochenig, and T. Strohmer, “Efficient numerical methods in non-uniform sampling theory,” Numer. Math. 69, 423–440 (1995).
[CrossRef]

Feng, C.

Griffiths, P. R.

P. R. Griffiths and J. A. de Haseth, “Signal-to-noise ratio,” in Fourier Transform Infrared Spectrometry (Wiley & Sons, 2007), pp. 167–168

Grochenig, K.

R. F. Bass and K. Grochenig, “Random sampling of multivariate trigonometric polynomials,” SIAM J. Math. Anal. 36, 773–795 (2005).
[CrossRef]

H. G. Feichtinger, K. Grochenig, and T. Strohmer, “Efficient numerical methods in non-uniform sampling theory,” Numer. Math. 69, 423–440 (1995).
[CrossRef]

K. Grochenig, “A discrete theory of irregular sampling,” Linear Algebra Appl. 193, 129–150 (1993).
[CrossRef]

Ivanov, E. V.

E. V. Ivanov, “Static Fourier transform spectroscopy with enhanced resolving power,” J. Opt. A 2, 519–528 (2000).
[CrossRef]

Jin, J.

Y. Shen, C. Zhu, L. Liu, Q. Wang, J. Jin, and Y. Lin, “Explicit solution for nonuniform discrete fourier transform,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (I2MTC) (IEEE, 2010), pp. 1505–1509.
[CrossRef]

Kern, P.

Kong, Y. M.

Y. M. Kong, J. Q. Liang, Z. Z. Liang, B. Wang, and J. Zhang, “Microassembled Fourier transform spectrometer,” Proc. SPIE 7283, 728304 (2009).
[CrossRef]

Korvink, J. G.

U. Wallrabe, C. Solf, J. Mohr, and J. G. Korvink, “Miniaturized Fourier transform spectrometer for the near infrared wavelength regime incorporating an electromagnetic linear actuator,” Sens. Actuators A 123, 459–467 (2005).
[CrossRef]

Lacan, A.

Lastrucci, D.

Leblond, G.

Lee, W. H.

J. Sin, W. H. Lee, D. Popa, and H. E. Stephanou, “Assembled Fourier transform micro-spectrometer,” Proc. SPIE 6109, 1–8(2006).
[CrossRef]

Lerondel, G.

Liang, J.

Liang, J. Q.

Y. M. Kong, J. Q. Liang, Z. Z. Liang, B. Wang, and J. Zhang, “Microassembled Fourier transform spectrometer,” Proc. SPIE 7283, 728304 (2009).
[CrossRef]

Liang, Z.

Liang, Z. Z.

Y. M. Kong, J. Q. Liang, Z. Z. Liang, B. Wang, and J. Zhang, “Microassembled Fourier transform spectrometer,” Proc. SPIE 7283, 728304 (2009).
[CrossRef]

Lin, Y.

Y. Shen, C. Zhu, L. Liu, Q. Wang, J. Jin, and Y. Lin, “Explicit solution for nonuniform discrete fourier transform,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (I2MTC) (IEEE, 2010), pp. 1505–1509.
[CrossRef]

Liu, L.

Y. Shen, C. Zhu, L. Liu, Q. Wang, J. Jin, and Y. Lin, “Explicit solution for nonuniform discrete fourier transform,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (I2MTC) (IEEE, 2010), pp. 1505–1509.
[CrossRef]

Liu, Q. H.

Q. H. Liu and N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFTs),” IEEE Microwave Guided Wave Lett. 8, 18–20 (1998).
[CrossRef]

Manzardo, O.

O. Manzardo, “Micro-sized Fourier spectrometers,” Ph.D. dissertation (University of Neuchatel, 2002).

Margolis, E.

E. Margolis and Y. C. Eldar, “Nonuniform sampling of periodic bandlimited signals,” IEEE Trans. Signal Process. 56, 2728–2745 (2008).
[CrossRef]

Martin, B.

Marvasti, F.

F. Marvasti and M. Analoui, “Recovery of signals from nonuniform samples using iterative methods,” in Proceedings of IEEE International Symposium on Circuits and Systems (IEEE, 1989), pp. 1021–1024.
[CrossRef]

Mohr, J.

U. Wallrabe, C. Solf, J. Mohr, and J. G. Korvink, “Miniaturized Fourier transform spectrometer for the near infrared wavelength regime incorporating an electromagnetic linear actuator,” Sens. Actuators A 123, 459–467 (2005).
[CrossRef]

Moller, K. D.

Morand, A.

Nguyen, N.

Q. H. Liu and N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFTs),” IEEE Microwave Guided Wave Lett. 8, 18–20 (1998).
[CrossRef]

Palchetti, L.

Papoulis, A.

A. Papoulis, “Error analysis in sampling theory,” Proc. IEEE 54, 947–955 (1966).
[CrossRef]

Popa, D.

J. Sin, W. H. Lee, D. Popa, and H. E. Stephanou, “Assembled Fourier transform micro-spectrometer,” Proc. SPIE 6109, 1–8(2006).
[CrossRef]

Rosak, A.

Roucayrol, L.

Royer, P.

Ruffieux, P.

Salaün, Y.

Sanderson, R. B.

Sarkissian, E.

Scharf, T.

Seitz, P.

Shen, Y.

Y. Shen, C. Zhu, L. Liu, Q. Wang, J. Jin, and Y. Lin, “Explicit solution for nonuniform discrete fourier transform,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (I2MTC) (IEEE, 2010), pp. 1505–1509.
[CrossRef]

Siem, Y. D.

Y. D. Siem, E. de Klerk, and D. den Hertog, “Discrete least-norm approximation by nonnegative (trigonometric) polynomials and rational functions,” Struct. Multidisc. Optim. 35, 327–339 (2008).
[CrossRef]

Sin, J.

J. Sin, W. H. Lee, D. Popa, and H. E. Stephanou, “Assembled Fourier transform micro-spectrometer,” Proc. SPIE 6109, 1–8(2006).
[CrossRef]

Solf, C.

U. Wallrabe, C. Solf, J. Mohr, and J. G. Korvink, “Miniaturized Fourier transform spectrometer for the near infrared wavelength regime incorporating an electromagnetic linear actuator,” Sens. Actuators A 123, 459–467 (2005).
[CrossRef]

Stephanou, H. E.

J. Sin, W. H. Lee, D. Popa, and H. E. Stephanou, “Assembled Fourier transform micro-spectrometer,” Proc. SPIE 6109, 1–8(2006).
[CrossRef]

Strohmer, T.

T. Strohmer, “Numerical analysis of the non-uniform sampling problem,” J. Comput. Appl. Math. 122, 297–316(2000).
[CrossRef]

H. G. Feichtinger, K. Grochenig, and T. Strohmer, “Efficient numerical methods in non-uniform sampling theory,” Numer. Math. 69, 423–440 (1995).
[CrossRef]

Thomas, J. B.

K. Yao and J. B. Thomas, “On some stability and interpolatory properties of nonuniform sampling expansions,” IEEE Trans. Circuit Theory CT-14, 404–408 (1967).
[CrossRef]

Wallrabe, U.

U. Wallrabe, C. Solf, J. Mohr, and J. G. Korvink, “Miniaturized Fourier transform spectrometer for the near infrared wavelength regime incorporating an electromagnetic linear actuator,” Sens. Actuators A 123, 459–467 (2005).
[CrossRef]

Wang, B.

C. Feng, B. Wang, Z. Liang, and J. Liang, “Miniaturization of step mirrors in a static Fourier transform spectrometer theory and simulation,” J. Opt. Soc. Am. B 28, 128–133 (2010).
[CrossRef]

Y. M. Kong, J. Q. Liang, Z. Z. Liang, B. Wang, and J. Zhang, “Microassembled Fourier transform spectrometer,” Proc. SPIE 7283, 728304 (2009).
[CrossRef]

Wang, Q.

Y. Shen, C. Zhu, L. Liu, Q. Wang, J. Jin, and Y. Lin, “Explicit solution for nonuniform discrete fourier transform,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (I2MTC) (IEEE, 2010), pp. 1505–1509.
[CrossRef]

Yao, K.

K. Yao and J. B. Thomas, “On some stability and interpolatory properties of nonuniform sampling expansions,” IEEE Trans. Circuit Theory CT-14, 404–408 (1967).
[CrossRef]

Zhang, J.

Y. M. Kong, J. Q. Liang, Z. Z. Liang, B. Wang, and J. Zhang, “Microassembled Fourier transform spectrometer,” Proc. SPIE 7283, 728304 (2009).
[CrossRef]

Zhu, C.

Y. Shen, C. Zhu, L. Liu, Q. Wang, J. Jin, and Y. Lin, “Explicit solution for nonuniform discrete fourier transform,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (I2MTC) (IEEE, 2010), pp. 1505–1509.
[CrossRef]

Appl. Opt.

IEEE Microwave Guided Wave Lett.

Q. H. Liu and N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFTs),” IEEE Microwave Guided Wave Lett. 8, 18–20 (1998).
[CrossRef]

IEEE Trans. Circuit Theory

K. Yao and J. B. Thomas, “On some stability and interpolatory properties of nonuniform sampling expansions,” IEEE Trans. Circuit Theory CT-14, 404–408 (1967).
[CrossRef]

IEEE Trans. Signal Process.

E. Margolis and Y. C. Eldar, “Nonuniform sampling of periodic bandlimited signals,” IEEE Trans. Signal Process. 56, 2728–2745 (2008).
[CrossRef]

J. Comput. Appl. Math.

T. Strohmer, “Numerical analysis of the non-uniform sampling problem,” J. Comput. Appl. Math. 122, 297–316(2000).
[CrossRef]

J. Opt. A

E. V. Ivanov, “Static Fourier transform spectroscopy with enhanced resolving power,” J. Opt. A 2, 519–528 (2000).
[CrossRef]

J. Opt. Soc. Am. B

Linear Algebra Appl.

K. Grochenig, “A discrete theory of irregular sampling,” Linear Algebra Appl. 193, 129–150 (1993).
[CrossRef]

Numer. Math.

H. G. Feichtinger, K. Grochenig, and T. Strohmer, “Efficient numerical methods in non-uniform sampling theory,” Numer. Math. 69, 423–440 (1995).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. IEEE

A. Papoulis, “Error analysis in sampling theory,” Proc. IEEE 54, 947–955 (1966).
[CrossRef]

Proc. SPIE

Y. M. Kong, J. Q. Liang, Z. Z. Liang, B. Wang, and J. Zhang, “Microassembled Fourier transform spectrometer,” Proc. SPIE 7283, 728304 (2009).
[CrossRef]

J. Sin, W. H. Lee, D. Popa, and H. E. Stephanou, “Assembled Fourier transform micro-spectrometer,” Proc. SPIE 6109, 1–8(2006).
[CrossRef]

Sens. Actuators A

U. Wallrabe, C. Solf, J. Mohr, and J. G. Korvink, “Miniaturized Fourier transform spectrometer for the near infrared wavelength regime incorporating an electromagnetic linear actuator,” Sens. Actuators A 123, 459–467 (2005).
[CrossRef]

SIAM J. Math. Anal.

R. F. Bass and K. Grochenig, “Random sampling of multivariate trigonometric polynomials,” SIAM J. Math. Anal. 36, 773–795 (2005).
[CrossRef]

Struct. Multidisc. Optim.

Y. D. Siem, E. de Klerk, and D. den Hertog, “Discrete least-norm approximation by nonnegative (trigonometric) polynomials and rational functions,” Struct. Multidisc. Optim. 35, 327–339 (2008).
[CrossRef]

Other

Y. Shen, C. Zhu, L. Liu, Q. Wang, J. Jin, and Y. Lin, “Explicit solution for nonuniform discrete fourier transform,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (I2MTC) (IEEE, 2010), pp. 1505–1509.
[CrossRef]

P. R. Griffiths and J. A. de Haseth, “Signal-to-noise ratio,” in Fourier Transform Infrared Spectrometry (Wiley & Sons, 2007), pp. 167–168

O. Manzardo, “Micro-sized Fourier spectrometers,” Ph.D. dissertation (University of Neuchatel, 2002).

F. Marvasti and M. Analoui, “Recovery of signals from nonuniform samples using iterative methods,” in Proceedings of IEEE International Symposium on Circuits and Systems (IEEE, 1989), pp. 1021–1024.
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Simplified configuration of the static FTS. 1. lower step mirror; 2. collimator; 3. beam splitter (50% transmission, 50% reflection); 4. higher step mirror; 5. detecting system.

Fig. 2
Fig. 2

Distribution of the 2000 condition numbers.

Fig. 3
Fig. 3

Distribution of the 2000 SCEs.

Fig. 4
Fig. 4

Distribution of the 2000 condition numbers when cosine polynomials are used.

Fig. 5
Fig. 5

Distribution of the 2000 SCEs when cosine polynomials are used.

Fig. 6
Fig. 6

Steps of a higher step mirror.

Fig. 7
Fig. 7

Steps of a lower step mirror.

Fig. 8
Fig. 8

Constructed spectrum with different constructing methods (SC source).

Fig. 9
Fig. 9

Constructed spectrum with different constructing methods (SD source).

Tables (2)

Tables Icon

Table 1 Testing Results of the Step Heights of the Step Mirrors

Tables Icon

Table 2 Testing Results for SC and SD Sources with Different Constructing Methods

Equations (17)

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

SNR max = 4 Δ x v ˜ max ,
δ 2 2 = i = 0 n δ i 2 = i = 0 n [ S * ( x i ) y i ] 2 = min S ( x ) φ i = 0 n [ S ( x i ) y i ] 2 ,
S ( x ) = a 0 φ 0 ( x ) + a 1 φ 1 ( x ) + + a m φ m ( x ) ( m < n ) .
A = ( a 0 , a 1 , a m ) T ,
Y = ( y 0 , y 1 , y n ) T ,
Φ k = ( φ k ( x 0 ) , φ k ( x 2 ) , φ k ( x n ) ) T ,
G = ( Φ 0 , Φ 1 , Φ m ) .
G H G A = G H Y ,
I ( OPD ) = σ min σ max 2 I ( σ ) cos ( 2 π σ OPD ) d σ ,
x j = 2 π N j ( j = 0 , 1 , N 1 ) ,
Φ k = ( exp ( i k x 0 ) , exp ( i k x 1 ) , exp ( i k x N 1 ) ) T , ( k = 0 , 1 , N 1 ) .
T j k = { 0 , j k N , j = k .
δ x X c δ b B ,
Φ k = ( exp ( i k 2 π 2048 x 0 1.25 ) , exp ( i k 2 π 2048 x 1 1.25 ) , exp ( i k 2 π 2048 x 2047 1.25 ) ) T ( k = 500 , 501 , , 700 , 1348 , , 1547 , 1548 )
SCE = k = 0 n | I real ( k ) I ideal ( k ) | k = 0 n I ideal ( k ) ,
Φ k = ( cos ( k 2 π 2048 x 0 1.25 ) , cos ( k 2 π 2048 x 1 1.25 ) , cos ( i k 2 π 2048 x 2047 1.25 ) ) T ( k = 500 , 501 , , 700 )
Φ k = ( cos ( k 2 π 2048 x 0 1.25 + φ ( k ) ) , cos ( k 2 π 2048 x 1 1.25 + φ ( k ) ) , ) T ,

Metrics