Abstract

To replace the film recording aspect of performing optical correlation, conventional real-time joint-transform correlation (JTC) optical systems make use of a spatial light modulator (SLM) located in the Fourier plane to record the joint-transform power spectrum (JPS) to achieve real-time processing. The use of an SLM in the Fourier plane, however, is a major drawback in these systems because SLMs are limited in resolution, phase uniformity, and contrast ratio, which are, therefore, not desirable for robust applications. We propose a hybrid (optical/electronic) processing technique to achieve real-time joint-transform correlation. The technique employs acousto-optic heterodyning scanning. The proposed real-time JTC system does not require an SLM at the Fourier plane as in other real-time JTC systems. This departure from the conventional scheme is extremely important as the proposed approach does not depend on SLM issues. We develop the theory of the technique and substantiate it with experimental results.

© 2003 Optical Society of America

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References

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  1. A. VanderLugt, “Signal detection by complex spatial filter,” IEEE Trans. Inf. Theory IT-10, 139–146 (1964).
  2. C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  3. G. Indebetouw, “Scanning optical correlator,” Opt. Lett. 6, 10–12 (1981).
    [CrossRef] [PubMed]
  4. T.-C. Poon, R. Juday, T. Hara, eds., Feature Issue on Spatial Light Modulators, Appl. Opt. 37, 7471–7552 (1998).
  5. J. E. Rau, “Real-time complex spatial modulation,” J. Opt. Soc. Am. 57, 798–802 (1967).
    [CrossRef]
  6. P. Nisenson, R. A. Sprague, “Real-time optical correlation,” Appl. Opt. 14, 2602–2606 (1975).
    [CrossRef] [PubMed]
  7. F. T. S. Yu., X. Lu, “A real-time programmable joint transform correlator,” Opt. Commun 52, 10–16 (1984).
    [CrossRef]
  8. F. T. S. Yu, S. Jutamulia, T. W. Lin, D. Gregory, “Adaptive real-time pattern recognition using a liquid crystal TV based joint transform correlator,” Appl. Opt. 26, 1370–1372 (1987).
    [CrossRef] [PubMed]
  9. J. M. Florence, “Joint-transform correlator systems using deformable-mirror spatial light modulator,” Opt. Lett. 14, 341–316 (1989).
    [CrossRef] [PubMed]
  10. T. D. Hudson, D. W. Trivett, D. Gregory, J. C. Kirsch, “Real time optical correlator architectures using a deformable mirror spatial light modulator,” Appl. Opt. 28, 4853–4860 (1989).
    [CrossRef] [PubMed]
  11. G. Lu, Z. Zhang, S. Wu, F. T. S. Yu, “Implementation of a non-zero-order joint-transform correlator by use of phase-shifting techniques,” Appl. Opt. 38, 470–483 (1995).
  12. A. Gregory, “Time multiplexed miniature optical correlator,” Lett. rep. RD-RE-88-02, U.S. Army Missile Command, Alabama (1988).
  13. T. H. Bames, K. Matsuda, T. Eiju, K. Matsumoto, F. Johnson, “Joint transform correlator using a phase only spatial light modulator,” Jap. J. Appl. Phys. 29, L1293–L1296 (1990).
    [CrossRef]
  14. S. Jutamulia, G. M. Storti, D. A. Gregory, J. C. Kirsch, “Illumination-independent high-efficiency joint transform correlator,” Appl. Opt. 30, 4173–4175 (1991).
    [CrossRef] [PubMed]
  15. D. A. Gregory, J. C. Kirsch, E. C. Tam, “Full complex modulation using liquid-crystal televisions,” Appl. Opt. 31, 163–165 (1992).
    [CrossRef] [PubMed]
  16. Q. Tang, B. Javidi, “Technique for reducing the redundant and self-correlation terms in joint transform correlators,” Appl. Opt. 32, 1911–1918 (1993).
    [CrossRef] [PubMed]
  17. M. S. Alam, M. A. Karim, “Fringe-adjusted joint transform correlation,” Appl. Opt. 32, 4344–4350 (1993).
    [CrossRef] [PubMed]
  18. C. J. Kuo, “Joint transform correlator improved by means of the frequency-selective technique,” Opt. Eng. 33, 522–527 (1994).
    [CrossRef]
  19. T. J. Grycewicz, “Applying time-modulation to the joint transform correlator,” Opt. Eng. 33, 1813–1820 (1994).
    [CrossRef]
  20. M. S. Alam, M. A. Karim, “Multiple target detection using a modified fringe-adjusted joint transform correlator.” Opt. Eng. 33, 1610–1617 (1994).
    [CrossRef]
  21. M. S. Alam, “Fractional power fringe-adjusted joint transform correlation,” 34, 3208–3216 (1995).
  22. T. Nomura, Y. Yoshimura, K. Itoh, Y. Ichioka, “Incoherent-only joint-transform correlator,” Appl. Opt. 34, 1420–1425 (1995).
    [CrossRef] [PubMed]
  23. T. Nomura, “Phase-encoded joint transform correlator to reduce the influence of extraneous signals,” Appl. Opt. 37, 3651–3655 (1998).
    [CrossRef]
  24. I. Labastida, A. Carnicer, E. Martin-Badosa, I. Juvells, S. Vallmitjana, “On-axis joint transform correlation based on a four-level power spectrum,” Appl. Opt. 38, 6111–6116 (1999).
    [CrossRef]
  25. H.-J. Su, M. A. Karim, “Phase-shifting joint transform correlation with phase-iterative algorithm: effect of the dynamic range limit,” Appl. Opt. 39, 5556–5559 (1999).
    [CrossRef]
  26. A. Cherri, M. S. Alam, “Reference phase-encoded fringe-adjusted joint transform correlation,” Appl. Opt. 40, 1216–1225 (2001).
    [CrossRef]
  27. T.-C. Poon, A. Korpel, “Optical transfer function of an acousto-optic heterodyning image processor,” Opt. Lett. 4, 317–319 (1979).
    [CrossRef] [PubMed]
  28. A. W. Lohmann, W. T. Rhodes, “Two-pupil synthesis of optical transfer functions,” Appl. Opt. 17, 1145–1151 (1978).
    [CrossRef]
  29. T.-C. Poon, “Scanning holography and two-dimensional image processing by acousto-optic two-pupil synthesis,” J. Opt. Soc. Am. A 2, 621–627 (1985).
    [CrossRef]
  30. J. Mait, “Pupil-function design for complex incoherent spatial filtering,” J. Opt. Soc. Am. A 4, 1185–1193 (1987).
    [CrossRef]
  31. A. Korpel, “Acousto-Optics,” in Applied Solid State Science, R. Wolfe, ed., Vol. 3 (Academic, New York, 1972).
  32. T.-C. Poon, P. P. Banerjee, Contemporary Optical Image Processing With Matlab (Elsevier Science, Oxford, U.K., 2001).
  33. B. Y. Soon, M. S. Alam, M. A. Karim, “Improved feature extraction by use of a joint wavelet transform correlator,” Appl. Opt. 37, 821–827 (1998).
    [CrossRef]

2001

1999

I. Labastida, A. Carnicer, E. Martin-Badosa, I. Juvells, S. Vallmitjana, “On-axis joint transform correlation based on a four-level power spectrum,” Appl. Opt. 38, 6111–6116 (1999).
[CrossRef]

H.-J. Su, M. A. Karim, “Phase-shifting joint transform correlation with phase-iterative algorithm: effect of the dynamic range limit,” Appl. Opt. 39, 5556–5559 (1999).
[CrossRef]

1998

1995

G. Lu, Z. Zhang, S. Wu, F. T. S. Yu, “Implementation of a non-zero-order joint-transform correlator by use of phase-shifting techniques,” Appl. Opt. 38, 470–483 (1995).

T. Nomura, Y. Yoshimura, K. Itoh, Y. Ichioka, “Incoherent-only joint-transform correlator,” Appl. Opt. 34, 1420–1425 (1995).
[CrossRef] [PubMed]

1994

C. J. Kuo, “Joint transform correlator improved by means of the frequency-selective technique,” Opt. Eng. 33, 522–527 (1994).
[CrossRef]

T. J. Grycewicz, “Applying time-modulation to the joint transform correlator,” Opt. Eng. 33, 1813–1820 (1994).
[CrossRef]

M. S. Alam, M. A. Karim, “Multiple target detection using a modified fringe-adjusted joint transform correlator.” Opt. Eng. 33, 1610–1617 (1994).
[CrossRef]

1993

1992

1991

1990

T. H. Bames, K. Matsuda, T. Eiju, K. Matsumoto, F. Johnson, “Joint transform correlator using a phase only spatial light modulator,” Jap. J. Appl. Phys. 29, L1293–L1296 (1990).
[CrossRef]

1989

1987

1985

T.-C. Poon, “Scanning holography and two-dimensional image processing by acousto-optic two-pupil synthesis,” J. Opt. Soc. Am. A 2, 621–627 (1985).
[CrossRef]

1984

F. T. S. Yu., X. Lu, “A real-time programmable joint transform correlator,” Opt. Commun 52, 10–16 (1984).
[CrossRef]

1981

1979

1978

A. W. Lohmann, W. T. Rhodes, “Two-pupil synthesis of optical transfer functions,” Appl. Opt. 17, 1145–1151 (1978).
[CrossRef]

1975

1967

1966

1964

A. VanderLugt, “Signal detection by complex spatial filter,” IEEE Trans. Inf. Theory IT-10, 139–146 (1964).

Alam, M. S.

Bames, T. H.

T. H. Bames, K. Matsuda, T. Eiju, K. Matsumoto, F. Johnson, “Joint transform correlator using a phase only spatial light modulator,” Jap. J. Appl. Phys. 29, L1293–L1296 (1990).
[CrossRef]

Banerjee, P. P.

T.-C. Poon, P. P. Banerjee, Contemporary Optical Image Processing With Matlab (Elsevier Science, Oxford, U.K., 2001).

Carnicer, A.

Cherri, A.

Eiju, T.

T. H. Bames, K. Matsuda, T. Eiju, K. Matsumoto, F. Johnson, “Joint transform correlator using a phase only spatial light modulator,” Jap. J. Appl. Phys. 29, L1293–L1296 (1990).
[CrossRef]

Florence, J. M.

Goodman, J. W.

Gregory, A.

A. Gregory, “Time multiplexed miniature optical correlator,” Lett. rep. RD-RE-88-02, U.S. Army Missile Command, Alabama (1988).

Gregory, D.

Gregory, D. A.

Grycewicz, T. J.

T. J. Grycewicz, “Applying time-modulation to the joint transform correlator,” Opt. Eng. 33, 1813–1820 (1994).
[CrossRef]

Hudson, T. D.

Ichioka, Y.

Indebetouw, G.

Itoh, K.

Javidi, B.

Johnson, F.

T. H. Bames, K. Matsuda, T. Eiju, K. Matsumoto, F. Johnson, “Joint transform correlator using a phase only spatial light modulator,” Jap. J. Appl. Phys. 29, L1293–L1296 (1990).
[CrossRef]

Jutamulia, S.

Juvells, I.

Karim, M. A.

H.-J. Su, M. A. Karim, “Phase-shifting joint transform correlation with phase-iterative algorithm: effect of the dynamic range limit,” Appl. Opt. 39, 5556–5559 (1999).
[CrossRef]

B. Y. Soon, M. S. Alam, M. A. Karim, “Improved feature extraction by use of a joint wavelet transform correlator,” Appl. Opt. 37, 821–827 (1998).
[CrossRef]

M. S. Alam, M. A. Karim, “Multiple target detection using a modified fringe-adjusted joint transform correlator.” Opt. Eng. 33, 1610–1617 (1994).
[CrossRef]

M. S. Alam, M. A. Karim, “Fringe-adjusted joint transform correlation,” Appl. Opt. 32, 4344–4350 (1993).
[CrossRef] [PubMed]

Kirsch, J. C.

Korpel, A.

T.-C. Poon, A. Korpel, “Optical transfer function of an acousto-optic heterodyning image processor,” Opt. Lett. 4, 317–319 (1979).
[CrossRef] [PubMed]

A. Korpel, “Acousto-Optics,” in Applied Solid State Science, R. Wolfe, ed., Vol. 3 (Academic, New York, 1972).

Kuo, C. J.

C. J. Kuo, “Joint transform correlator improved by means of the frequency-selective technique,” Opt. Eng. 33, 522–527 (1994).
[CrossRef]

Labastida, I.

Lin, T. W.

Lohmann, A. W.

A. W. Lohmann, W. T. Rhodes, “Two-pupil synthesis of optical transfer functions,” Appl. Opt. 17, 1145–1151 (1978).
[CrossRef]

Lu, G.

G. Lu, Z. Zhang, S. Wu, F. T. S. Yu, “Implementation of a non-zero-order joint-transform correlator by use of phase-shifting techniques,” Appl. Opt. 38, 470–483 (1995).

Lu, X.

F. T. S. Yu., X. Lu, “A real-time programmable joint transform correlator,” Opt. Commun 52, 10–16 (1984).
[CrossRef]

Mait, J.

Martin-Badosa, E.

Matsuda, K.

T. H. Bames, K. Matsuda, T. Eiju, K. Matsumoto, F. Johnson, “Joint transform correlator using a phase only spatial light modulator,” Jap. J. Appl. Phys. 29, L1293–L1296 (1990).
[CrossRef]

Matsumoto, K.

T. H. Bames, K. Matsuda, T. Eiju, K. Matsumoto, F. Johnson, “Joint transform correlator using a phase only spatial light modulator,” Jap. J. Appl. Phys. 29, L1293–L1296 (1990).
[CrossRef]

Nisenson, P.

Nomura, T.

Poon, T.-C.

T.-C. Poon, “Scanning holography and two-dimensional image processing by acousto-optic two-pupil synthesis,” J. Opt. Soc. Am. A 2, 621–627 (1985).
[CrossRef]

T.-C. Poon, A. Korpel, “Optical transfer function of an acousto-optic heterodyning image processor,” Opt. Lett. 4, 317–319 (1979).
[CrossRef] [PubMed]

T.-C. Poon, P. P. Banerjee, Contemporary Optical Image Processing With Matlab (Elsevier Science, Oxford, U.K., 2001).

Rau, J. E.

Rhodes, W. T.

A. W. Lohmann, W. T. Rhodes, “Two-pupil synthesis of optical transfer functions,” Appl. Opt. 17, 1145–1151 (1978).
[CrossRef]

Soon, B. Y.

Sprague, R. A.

Storti, G. M.

Su, H.-J.

H.-J. Su, M. A. Karim, “Phase-shifting joint transform correlation with phase-iterative algorithm: effect of the dynamic range limit,” Appl. Opt. 39, 5556–5559 (1999).
[CrossRef]

Tam, E. C.

Tang, Q.

Trivett, D. W.

Vallmitjana, S.

VanderLugt, A.

A. VanderLugt, “Signal detection by complex spatial filter,” IEEE Trans. Inf. Theory IT-10, 139–146 (1964).

Weaver, C. S.

Wu, S.

G. Lu, Z. Zhang, S. Wu, F. T. S. Yu, “Implementation of a non-zero-order joint-transform correlator by use of phase-shifting techniques,” Appl. Opt. 38, 470–483 (1995).

Yoshimura, Y.

Yu, F. T. S.

G. Lu, Z. Zhang, S. Wu, F. T. S. Yu, “Implementation of a non-zero-order joint-transform correlator by use of phase-shifting techniques,” Appl. Opt. 38, 470–483 (1995).

F. T. S. Yu, S. Jutamulia, T. W. Lin, D. Gregory, “Adaptive real-time pattern recognition using a liquid crystal TV based joint transform correlator,” Appl. Opt. 26, 1370–1372 (1987).
[CrossRef] [PubMed]

Yu., F. T. S.

F. T. S. Yu., X. Lu, “A real-time programmable joint transform correlator,” Opt. Commun 52, 10–16 (1984).
[CrossRef]

Zhang, Z.

G. Lu, Z. Zhang, S. Wu, F. T. S. Yu, “Implementation of a non-zero-order joint-transform correlator by use of phase-shifting techniques,” Appl. Opt. 38, 470–483 (1995).

Appl. Opt.

C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
[CrossRef] [PubMed]

T.-C. Poon, R. Juday, T. Hara, eds., Feature Issue on Spatial Light Modulators, Appl. Opt. 37, 7471–7552 (1998).

P. Nisenson, R. A. Sprague, “Real-time optical correlation,” Appl. Opt. 14, 2602–2606 (1975).
[CrossRef] [PubMed]

F. T. S. Yu, S. Jutamulia, T. W. Lin, D. Gregory, “Adaptive real-time pattern recognition using a liquid crystal TV based joint transform correlator,” Appl. Opt. 26, 1370–1372 (1987).
[CrossRef] [PubMed]

S. Jutamulia, G. M. Storti, D. A. Gregory, J. C. Kirsch, “Illumination-independent high-efficiency joint transform correlator,” Appl. Opt. 30, 4173–4175 (1991).
[CrossRef] [PubMed]

D. A. Gregory, J. C. Kirsch, E. C. Tam, “Full complex modulation using liquid-crystal televisions,” Appl. Opt. 31, 163–165 (1992).
[CrossRef] [PubMed]

Q. Tang, B. Javidi, “Technique for reducing the redundant and self-correlation terms in joint transform correlators,” Appl. Opt. 32, 1911–1918 (1993).
[CrossRef] [PubMed]

M. S. Alam, M. A. Karim, “Fringe-adjusted joint transform correlation,” Appl. Opt. 32, 4344–4350 (1993).
[CrossRef] [PubMed]

T. D. Hudson, D. W. Trivett, D. Gregory, J. C. Kirsch, “Real time optical correlator architectures using a deformable mirror spatial light modulator,” Appl. Opt. 28, 4853–4860 (1989).
[CrossRef] [PubMed]

G. Lu, Z. Zhang, S. Wu, F. T. S. Yu, “Implementation of a non-zero-order joint-transform correlator by use of phase-shifting techniques,” Appl. Opt. 38, 470–483 (1995).

T. Nomura, Y. Yoshimura, K. Itoh, Y. Ichioka, “Incoherent-only joint-transform correlator,” Appl. Opt. 34, 1420–1425 (1995).
[CrossRef] [PubMed]

T. Nomura, “Phase-encoded joint transform correlator to reduce the influence of extraneous signals,” Appl. Opt. 37, 3651–3655 (1998).
[CrossRef]

I. Labastida, A. Carnicer, E. Martin-Badosa, I. Juvells, S. Vallmitjana, “On-axis joint transform correlation based on a four-level power spectrum,” Appl. Opt. 38, 6111–6116 (1999).
[CrossRef]

H.-J. Su, M. A. Karim, “Phase-shifting joint transform correlation with phase-iterative algorithm: effect of the dynamic range limit,” Appl. Opt. 39, 5556–5559 (1999).
[CrossRef]

A. Cherri, M. S. Alam, “Reference phase-encoded fringe-adjusted joint transform correlation,” Appl. Opt. 40, 1216–1225 (2001).
[CrossRef]

A. W. Lohmann, W. T. Rhodes, “Two-pupil synthesis of optical transfer functions,” Appl. Opt. 17, 1145–1151 (1978).
[CrossRef]

B. Y. Soon, M. S. Alam, M. A. Karim, “Improved feature extraction by use of a joint wavelet transform correlator,” Appl. Opt. 37, 821–827 (1998).
[CrossRef]

IEEE Trans. Inf. Theory

A. VanderLugt, “Signal detection by complex spatial filter,” IEEE Trans. Inf. Theory IT-10, 139–146 (1964).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

T.-C. Poon, “Scanning holography and two-dimensional image processing by acousto-optic two-pupil synthesis,” J. Opt. Soc. Am. A 2, 621–627 (1985).
[CrossRef]

J. Mait, “Pupil-function design for complex incoherent spatial filtering,” J. Opt. Soc. Am. A 4, 1185–1193 (1987).
[CrossRef]

Jap. J. Appl. Phys.

T. H. Bames, K. Matsuda, T. Eiju, K. Matsumoto, F. Johnson, “Joint transform correlator using a phase only spatial light modulator,” Jap. J. Appl. Phys. 29, L1293–L1296 (1990).
[CrossRef]

Opt. Commun

F. T. S. Yu., X. Lu, “A real-time programmable joint transform correlator,” Opt. Commun 52, 10–16 (1984).
[CrossRef]

Opt. Eng.

C. J. Kuo, “Joint transform correlator improved by means of the frequency-selective technique,” Opt. Eng. 33, 522–527 (1994).
[CrossRef]

T. J. Grycewicz, “Applying time-modulation to the joint transform correlator,” Opt. Eng. 33, 1813–1820 (1994).
[CrossRef]

M. S. Alam, M. A. Karim, “Multiple target detection using a modified fringe-adjusted joint transform correlator.” Opt. Eng. 33, 1610–1617 (1994).
[CrossRef]

Opt. Lett.

Other

A. Gregory, “Time multiplexed miniature optical correlator,” Lett. rep. RD-RE-88-02, U.S. Army Missile Command, Alabama (1988).

A. Korpel, “Acousto-Optics,” in Applied Solid State Science, R. Wolfe, ed., Vol. 3 (Academic, New York, 1972).

T.-C. Poon, P. P. Banerjee, Contemporary Optical Image Processing With Matlab (Elsevier Science, Oxford, U.K., 2001).

M. S. Alam, “Fractional power fringe-adjusted joint transform correlation,” 34, 3208–3216 (1995).

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Figures (8)

Fig. 1
Fig. 1

Conventional real-time JTC system.

Fig. 2
Fig. 2

Basic optical heterodyne scanning system.

Fig. 3
Fig. 3

Practical implementation of Fig. 2.

Fig. 4
Fig. 4

Autocorrelation result of two identical animals (animal 1).

Fig. 5
Fig. 5

Chirp grating used in the experiment.

Fig. 6
Fig. 6

Correlation result of two different animals.

Fig. 7
Fig. 7

Autocorrelation result of two identical random patterns.

Fig. 8
Fig. 8

Cross-correlation result of a random pattern and its contrast reversal pattern.

Equations (23)

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

Jkx, ky=|Fg1x-x0, y+Fg2x+x0, y|2=|G1kx, ky|2+| G2kx, ky|2+G1*kx, kyG2kx, kyexpj2kxx0+G1kx, kyG2*kx, kyexp-j2kxx0,
Gikx, ky=gix, yexpjkxx+kyydxdy=Fgix, ykx, ky,
FJk0xf, k0yfkx, ky=C11-x, -y+C22-x, -y+C12-x-2x0, -y+C21-x+2x0, -y,
Cijx, y=gix, ygjx, y= gi*x, y×gjx+x, y+ydxdy,
G1*kx, ky G2kx, kyexpj2kxx0+G1kx, ky G2*kx, kyexp-j2kxx0=2|G1kx, ky G2kx, ky|cos2kxx0+θ,
P1k0xf, k0yfexpjω0t+P2k0xf, k0yf×expjω0+Ωt,
ix, y=AP1k0xf, k0yfexpjω0t+P2k0xf, k0yfexpjω0+Ωt×Tox+x, y+y2dxdy.
iΩx, y=ReA P1*k0xf, k0yfP2k0xf, k0yf×|Tox+x, y+y|2dxdy expjΩt,
iΩx, y=ReiΩpx, yexpjΩt,
iΩpx, y=AP1*k0xf, k0yf P2k0xf, x0yf×|Tox+xy+y|2dxdy
iΩpx=AP1*k0xf, k0yf P2k0xf, x0yf×1+12expjax+x2+12exp-jax+x2dxdy =I0+12t1x+12t2x,
I0=A P1*k0xf, k0yf P2k0xf, k0yf×dxdy,
t1x=AP1*k0xf, k0yf P2k0xf, k0yf×exp [jax+x2dxdy,
t2x=A P1*k0xf, k0yf P2k0xf, k0yf×exp-jax+x2dxdy,
t1x=expjax2A P1*k0xf, k0yf P2k0xf, k0yf×expjax2expj2axxdxdy=expjax2FP1*k0xf, k0yf P2k0xf, k0yf×expjax2kx=2ax, ky=0.
t1x=expjax2 FP1*k0xf, k0yf×P2k0xf, k0yfkx=2ax, ky=0.
t1x=expjax2P1- fkxk0, - fkyk0P2- fkxk0, - fkyk0kx=2ax, ky=0.
t2x=exp-jax2P1fkxk0, fkyk0P2fkxk0, fkyk0kx=2ax, ky=0.
iΩx, y=ReiΩpx, yexpjΩt,
iΩpx=I0+12expjax2P1- fkxk0, - fkyk0P2- fkxk0, - fkyk0kx=2ax, ky=0+12exp-jax2p1fkxk0, fkyk0p2fkxk0, fkyk0kx=2ax, ky=0.
iΩpx=I0+12expjax2C12-2axfk0-2x0, 0+12exp-jax2C122axfk0-2x0, 0,
iΩx=|I0|cosΩt+θ0+12C12-2axfk0-2x0, 0 ×cos(Ωt+ax2+θ1)+12C122axfk0-2x0, 0×cos(Ωt-ax2+θ2),
ilock-inx|I0|cosθ0+12C12-2axfk0-2x0, 0×cosax2+θ1+12C122axfk0-2x0, 0×cosax2-θ2,

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