Abstract

Any desired diffraction pattern can be produced in the Fourier plane by the specification of a corresponding input-plane transparency. Complex-valued transmittance is generally required, but in practice phase-only transmittance is used. Many design procedures use numerically intensive, constrained optimization. We instead introduce a noniterative procedure that directly translates the desired but unavailable complex transparency into an appropriate phase transparency. At each pixel the value of phase is pseudorandomly selected from a random distribution whose standard deviation is specified by the desired amplitude. We also derive statistical expressions and use them to evaluate the approximation errors between the desired and achieved diffraction patterns.

© 1994 Optical Society of America

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References

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  1. H. Dammann, K. Gortler, “High-efficiency multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
    [CrossRef]
  2. J. Jahns, M. M. Downs, M. E. Prise, N. Streibel, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
  3. F. B. McCormick, “Generation of large spot arrays from a single laser beam by multiple imaging with binary phase gratings,” Opt. Eng. 28, 299–304 (1989).
  4. M. P. Dames, R. J. Dowling, P. McKee, D. Wood, “Efficient optical elements to generate intensity weighted spot arrays: design and fabrication,” Appl. Opt. 30, 2685–2691 (1991).
    [CrossRef] [PubMed]
  5. N. C. Gallagher, B. Liu, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973).
    [CrossRef] [PubMed]
  6. R. W. Gerchberg, W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–250 (1972).
  7. If the complex Fourier transform pair is not known, then one FFT would have to be computed. Although such direct methods are much faster than the iterative methods, it is impractical to include electronic calculation of the FFT in real-time optoelectronic processors. Optical Fourier transform and interferometric detection with video cameras can be performed at real-time rates and might be practically employed in the latter case. See, for example, R. W. Cohn, “Adaptive real-time architectures for phase-only correlation,” Appl. Opt. 32, 718–725 (1993).
    [CrossRef] [PubMed]
  8. O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. 28, pp. 1–86, and references therein, including reviews by W. J. Dallas and W. -H. Lee.
    [CrossRef]
  9. C. B. Burckhardt, “Use of random phase mask for the recording of Fourier transform holograms of data masks,” Appl. Opt. 9, 695–700 (1970).
    [CrossRef] [PubMed]
  10. R. Brauer, U. Wojak, F. Wyrowski, O. Bryngdahl, “Digital diffusers for optical holography,” Opt. Lett. 16, 1427–1429 (1991).
    [CrossRef] [PubMed]
  11. R. W. Cohn, R. J. Nonnenkamp, “Statistical moments of the transmittance of phase-only spatial light modulators,” in Miniature and Micro-Optics: Fabrication, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1751, 289–297 (1992).
  12. R. M. Boysel, J. M. Florence, W. R. Wu, “Deformable mirror light modulators for image processing,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1151, 183–194 (1989).
  13. J. Amako, T. Sonehara, “Kinoform using an electrically controlled biréfringent liquid-crystal spatial light modulator,” Appl. Opt. 30, 4622–4628 (1991).
    [CrossRef] [PubMed]
  14. N. Konforti, E. Marom, S. -T. Wu, “Phase-only modulation with twisted nematic liquid-crystal spatial light modulators,” Opt. Lett. 13, 251–253 (1988).
    [CrossRef] [PubMed]
  15. R. W. Cohn, “Random phase errors and pseudorandom phase modulation of deformable mirror spatial light modulators,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1772, 360–368 (1992).
  16. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 1st ed. (McGraw-Hill, New York, 1965).
  17. The overbar in Eq. (6) is distinguished from the 〈 〉 operator as a way to indicate that the overbarred variable is a result of ensemble averaging. At times, when the meaning is clear, we will interchange the use of the overbar and the 〈 〉 operator in order to improve the readability.
  18. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), App. C.
  19. J. C. Dainty, ed., Laser Speckle and Related Phenomena, 2nd ed. (Springer, Berlin, 1984);Ref. 18, Chap. 8.
  20. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, England, 1986).
  21. Equation (24) is identical for both Gaussian and uniform distributions.
  22. B. V. K. V. Kumar, L. G. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
    [CrossRef] [PubMed]
  23. J. L. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992).
    [CrossRef] [PubMed]
  24. R. W. Cohn, J. L. Horner, “Effects of systematic phase errors on phase-only correlation,” Appl. Opt. (to be published).
  25. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-HallEnglewood Cliffs, N.J., 1975), Sec. 5.5.
  26. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
    [CrossRef]
  27. H. Kamoda, H. Goto, K. Imanaka, “Two dimensional optical position sensor using a dual axis miniature optical scanner,” in Miniature and Micro-Optics: Fabrication, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1751, 280–288 (1992).
  28. R. C. Gonzalez, P. Wintz, Digital Image Processing, (Addison-Wesley, Reading, Mass., 1977), Sec. 6.5.
  29. R. W. Cohn, “Adaptive real-time architectures for phase-only correlation,” Appl. Opt. 32, 718–725 (1993).
    [CrossRef] [PubMed]

1993

1992

1991

1990

1989

J. Jahns, M. M. Downs, M. E. Prise, N. Streibel, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

F. B. McCormick, “Generation of large spot arrays from a single laser beam by multiple imaging with binary phase gratings,” Opt. Eng. 28, 299–304 (1989).

1988

1978

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

1973

1972

R. W. Gerchberg, W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–250 (1972).

1971

H. Dammann, K. Gortler, “High-efficiency multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

1970

Amako, J.

Boysel, R. M.

R. M. Boysel, J. M. Florence, W. R. Wu, “Deformable mirror light modulators for image processing,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1151, 183–194 (1989).

Brauer, R.

Bryngdahl, O.

R. Brauer, U. Wojak, F. Wyrowski, O. Bryngdahl, “Digital diffusers for optical holography,” Opt. Lett. 16, 1427–1429 (1991).
[CrossRef] [PubMed]

O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. 28, pp. 1–86, and references therein, including reviews by W. J. Dallas and W. -H. Lee.
[CrossRef]

Burckhardt, C. B.

Cohn, R. W.

R. W. Cohn, “Adaptive real-time architectures for phase-only correlation,” Appl. Opt. 32, 718–725 (1993).
[CrossRef] [PubMed]

If the complex Fourier transform pair is not known, then one FFT would have to be computed. Although such direct methods are much faster than the iterative methods, it is impractical to include electronic calculation of the FFT in real-time optoelectronic processors. Optical Fourier transform and interferometric detection with video cameras can be performed at real-time rates and might be practically employed in the latter case. See, for example, R. W. Cohn, “Adaptive real-time architectures for phase-only correlation,” Appl. Opt. 32, 718–725 (1993).
[CrossRef] [PubMed]

R. W. Cohn, R. J. Nonnenkamp, “Statistical moments of the transmittance of phase-only spatial light modulators,” in Miniature and Micro-Optics: Fabrication, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1751, 289–297 (1992).

R. W. Cohn, “Random phase errors and pseudorandom phase modulation of deformable mirror spatial light modulators,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1772, 360–368 (1992).

R. W. Cohn, J. L. Horner, “Effects of systematic phase errors on phase-only correlation,” Appl. Opt. (to be published).

Dames, M. P.

Dammann, H.

H. Dammann, K. Gortler, “High-efficiency multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Dowling, R. J.

Downs, M. M.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibel, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, England, 1986).

Florence, J. M.

R. M. Boysel, J. M. Florence, W. R. Wu, “Deformable mirror light modulators for image processing,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1151, 183–194 (1989).

Gallagher, N. C.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–250 (1972).

Gonzalez, R. C.

R. C. Gonzalez, P. Wintz, Digital Image Processing, (Addison-Wesley, Reading, Mass., 1977), Sec. 6.5.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), App. C.

Gortler, K.

H. Dammann, K. Gortler, “High-efficiency multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Goto, H.

H. Kamoda, H. Goto, K. Imanaka, “Two dimensional optical position sensor using a dual axis miniature optical scanner,” in Miniature and Micro-Optics: Fabrication, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1751, 280–288 (1992).

Harris, F. J.

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Hassebrook, L. G.

Horner, J. L.

J. L. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992).
[CrossRef] [PubMed]

R. W. Cohn, J. L. Horner, “Effects of systematic phase errors on phase-only correlation,” Appl. Opt. (to be published).

Imanaka, K.

H. Kamoda, H. Goto, K. Imanaka, “Two dimensional optical position sensor using a dual axis miniature optical scanner,” in Miniature and Micro-Optics: Fabrication, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1751, 280–288 (1992).

Jahns, J.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibel, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Kamoda, H.

H. Kamoda, H. Goto, K. Imanaka, “Two dimensional optical position sensor using a dual axis miniature optical scanner,” in Miniature and Micro-Optics: Fabrication, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1751, 280–288 (1992).

Konforti, N.

Kumar, B. V. K. V.

Liu, B.

Marom, E.

McCormick, F. B.

F. B. McCormick, “Generation of large spot arrays from a single laser beam by multiple imaging with binary phase gratings,” Opt. Eng. 28, 299–304 (1989).

McKee, P.

Nonnenkamp, R. J.

R. W. Cohn, R. J. Nonnenkamp, “Statistical moments of the transmittance of phase-only spatial light modulators,” in Miniature and Micro-Optics: Fabrication, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1751, 289–297 (1992).

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-HallEnglewood Cliffs, N.J., 1975), Sec. 5.5.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 1st ed. (McGraw-Hill, New York, 1965).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, England, 1986).

Prise, M. E.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibel, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–250 (1972).

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-HallEnglewood Cliffs, N.J., 1975), Sec. 5.5.

Sonehara, T.

Streibel, N.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibel, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, England, 1986).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, England, 1986).

Walker, S. J.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibel, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Wintz, P.

R. C. Gonzalez, P. Wintz, Digital Image Processing, (Addison-Wesley, Reading, Mass., 1977), Sec. 6.5.

Wojak, U.

Wood, D.

Wu, S. -T.

Wu, W. R.

R. M. Boysel, J. M. Florence, W. R. Wu, “Deformable mirror light modulators for image processing,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1151, 183–194 (1989).

Wyrowski, F.

R. Brauer, U. Wojak, F. Wyrowski, O. Bryngdahl, “Digital diffusers for optical holography,” Opt. Lett. 16, 1427–1429 (1991).
[CrossRef] [PubMed]

O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. 28, pp. 1–86, and references therein, including reviews by W. J. Dallas and W. -H. Lee.
[CrossRef]

Appl. Opt.

M. P. Dames, R. J. Dowling, P. McKee, D. Wood, “Efficient optical elements to generate intensity weighted spot arrays: design and fabrication,” Appl. Opt. 30, 2685–2691 (1991).
[CrossRef] [PubMed]

N. C. Gallagher, B. Liu, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973).
[CrossRef] [PubMed]

C. B. Burckhardt, “Use of random phase mask for the recording of Fourier transform holograms of data masks,” Appl. Opt. 9, 695–700 (1970).
[CrossRef] [PubMed]

J. Amako, T. Sonehara, “Kinoform using an electrically controlled biréfringent liquid-crystal spatial light modulator,” Appl. Opt. 30, 4622–4628 (1991).
[CrossRef] [PubMed]

B. V. K. V. Kumar, L. G. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
[CrossRef] [PubMed]

J. L. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992).
[CrossRef] [PubMed]

If the complex Fourier transform pair is not known, then one FFT would have to be computed. Although such direct methods are much faster than the iterative methods, it is impractical to include electronic calculation of the FFT in real-time optoelectronic processors. Optical Fourier transform and interferometric detection with video cameras can be performed at real-time rates and might be practically employed in the latter case. See, for example, R. W. Cohn, “Adaptive real-time architectures for phase-only correlation,” Appl. Opt. 32, 718–725 (1993).
[CrossRef] [PubMed]

R. W. Cohn, “Adaptive real-time architectures for phase-only correlation,” Appl. Opt. 32, 718–725 (1993).
[CrossRef] [PubMed]

Opt. Commun.

H. Dammann, K. Gortler, “High-efficiency multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Opt. Eng.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibel, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

F. B. McCormick, “Generation of large spot arrays from a single laser beam by multiple imaging with binary phase gratings,” Opt. Eng. 28, 299–304 (1989).

Opt. Lett.

Optik (Stuttgart)

R. W. Gerchberg, W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–250 (1972).

Proc. IEEE

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Other

H. Kamoda, H. Goto, K. Imanaka, “Two dimensional optical position sensor using a dual axis miniature optical scanner,” in Miniature and Micro-Optics: Fabrication, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1751, 280–288 (1992).

R. C. Gonzalez, P. Wintz, Digital Image Processing, (Addison-Wesley, Reading, Mass., 1977), Sec. 6.5.

O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. 28, pp. 1–86, and references therein, including reviews by W. J. Dallas and W. -H. Lee.
[CrossRef]

R. W. Cohn, J. L. Horner, “Effects of systematic phase errors on phase-only correlation,” Appl. Opt. (to be published).

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-HallEnglewood Cliffs, N.J., 1975), Sec. 5.5.

R. W. Cohn, R. J. Nonnenkamp, “Statistical moments of the transmittance of phase-only spatial light modulators,” in Miniature and Micro-Optics: Fabrication, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1751, 289–297 (1992).

R. M. Boysel, J. M. Florence, W. R. Wu, “Deformable mirror light modulators for image processing,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1151, 183–194 (1989).

R. W. Cohn, “Random phase errors and pseudorandom phase modulation of deformable mirror spatial light modulators,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1772, 360–368 (1992).

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 1st ed. (McGraw-Hill, New York, 1965).

The overbar in Eq. (6) is distinguished from the 〈 〉 operator as a way to indicate that the overbarred variable is a result of ensemble averaging. At times, when the meaning is clear, we will interchange the use of the overbar and the 〈 〉 operator in order to improve the readability.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), App. C.

J. C. Dainty, ed., Laser Speckle and Related Phenomena, 2nd ed. (Springer, Berlin, 1984);Ref. 18, Chap. 8.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, England, 1986).

Equation (24) is identical for both Gaussian and uniform distributions.

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Figures (7)

Fig. 1
Fig. 1

Expected transmittance of random phase-only pixels (both piston-only and tilt-only pixels).

Fig. 2
Fig. 2

Pseudorandom phase-only design of elliptical aperture. (a) Desired amplitude modulation. (b) Phase-only approximation to (a). (c) Expected value of (d) the intensity of the far-field diffraction pattern. Only the central 128 × 64 pixels of the 128 × 128 modulation are shown in (a) and (b). Only the central 128 × 128 pixels of the 512 × 512 FFT are shown in (c) and (d). We have nonlinearly transformed the intensities in (c) and (d) by the exponent 1.3 (i.e., gamma) to increase the contrast of low-lying sidelobes of the Airy pattern.

Fig. 3
Fig. 3

Diffraction pattern of phase-only approximated elliptical aperture. The actual pattern (thick curve) is compared with the expected intensity plus and minus the error limits of one standard deviation. Horizontal dashed lines indicate the saturation (full-white) level of corresponding figures, Figs. 2(c) and 2(d). The spatial coordinates are normalized so that one unit corresponds to one of the 128 resolvable scan positions of the SLM or four samples of the FFT window.

Fig. 4
Fig. 4

SNR at diffraction peak as a function of number of nonrandom pixels for pseudorandom encoding of binary amplitudes.

Fig. 5
Fig. 5

Pseudorandom phase-only design of Dolph-windowed sinc apodization: (a) Desired amplitude modulation. (b) Phase-only approximation to (a). (c) Expected value of (d) the intensity of the far-field diffraction pattern. All units and settings are the same as in Fig. 2 except the recording gamma, which is unity for (c) and (d).

Fig. 6
Fig. 6

Diffraction pattern of a phase-only approximated Dolph-windowed sine apodization. (a) Cross section across vertical axis (0, fy). The actual pattern (thick curve) is compared with the expected intensity and the expected intensity plus and minus the errors limits of one standard deviation. Horizontal dashed lines indicate the saturation (full-white) level of corresponding figures, Figs. 5(c) and 5(d). Legend and units identical to those in Fig. 3. (b) Cross section across horizontal axis (fx, 0). The expected intensity (thick curve) is compared with the expected intensity sinc2(fx) if no Dolph windowing had been employed.

Fig. 7
Fig. 7

SNR as a function of effective number of nonrandom pixels for pseudorandom encoding of continuous amplitudes. SNR(0) is plotted for Gaussian and binary, and average SNR across passband is plotted for Dolph and sine designs. The sine is a rectangularly symmetric function, and the Gaussian is a circularly symmetric function.

Equations (32)

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

t ( x ) = i = 1 N a i ( x ) = r r ( x x i ) exp ( j ψ i ) ,
I ( f x ) = T ( f x ) T * ( f x ) = F { t ( x ) t ( x ) } ,
a ( x ) b ( x ) = a ( x + x ) b * ( x ) d x .
I 2 = T T * T * T = F { [ t ( x ) t ( x ) ] [ t ( x ) t ( x ) ] } .
M ( ω ) = exp ( j ωψ ) = exp ( j ωψ ) p ψ ( ψ ) d ψ = 2 π F 1 { p ψ ( ψ ) } ,
a ¯ i ( x ) = a i ( x ) = r ( x x i ) exp ( j ψ i ) , = r ( x x i ) M i ( 1 ) , = r ( x x i ) exp ( j ψ ¯ i ) M i ( 1 ) ,
M i ( 1 ) = exp ( 1 2 σ i 2 ) ,
υ i = 12 σ i ,
M i ( 1 ) = sinc [ υ i 2 π ] .
t ¯ ( x ) = i = 1 N a i ( x ) = i r ( x x i ) p i 1 / 2 exp ( j ψ ¯ i ) ,
p i = M i 2 ( 1 )
T ¯ ( f x ) = F { t ¯ ( x ) } .
I ¯ ( f x ) = i j A i ( f x ) A j * ( f x ) , = i j A i A j * + i | A i | 2 , = i j A i A j * i | A i | 2 + i | A i | 2 , = | T ¯ | 2 + i [ | A i | 2 | A ¯ i | 2 ] ,
I 2 = i j k l A i A j * A k * A l = 2 [ I ¯ 2 | T ¯ | 4 ] + | T ¯ 2 + i ( A i 2 A ¯ i 2 ) | 2 + 4 Re [ T ¯ * i ( | A i | 2 A i A i 2 A ¯ i * + 2 | A ¯ i | 2 A i ¯ 2 | A i | 2 A i ¯ ) ] + i [ | A i | 4 6 | A i ¯ | 4 + 8 | A i | 2 | A ¯ i | 2 | A i 2 | 2 2 | A i | 2 2 + 4 Re ( A i 2 A ¯ i * 2 | A i | 2 A i A ¯ i * ) ] .
σ I 2 = I 2 I ¯ 2 .
I ¯ ( f x ) = F { t ¯ ( x ) t ¯ ( x ) + [ r ( x ) r ( x ) ] i q i } , = | T ¯ ( f x ) | 2 + R 2 ( f x ) i q i ,
q i = 1 p i ,
σ 2 ( f x ) = I ¯ 2 2 | T ¯ | 4 + | T ¯ 2 T A | 2 4 Re [ T ¯ T B * ] G A .
g n ( x ) = [ r ( x ) r ( x ) ] n ,
t A ( x ) = i g 2 ( x 2 x i ) exp ( j 2 ψ ¯ i ) q i p i , t B ( x ) = i g 3 ( x x i ) exp ( j ψ ¯ i ) q i 2 p i 1 / 2 , g A ( x ) = g 4 ( x ) i q i 4 ,
t A ( x ) = i g 2 ( x 2 x i ) exp ( j 2 ψ ¯ i ) ( p i d i ) , t B ( x ) = i g 3 ( x x i ) exp ( j ψ ¯ i ) ( q i p i + d i ) p i 1 / 2 , g A ( x ) = g 4 ( x ) i ( q i 3 p i + 6 p i 2 4 p i d i + d i 2 ) ,
d i = sinc [ υ i π ] .
SNR ( f x ) = I ¯ ( f x ) σ I ( f x ) ,
SNR ( 0 ) = N N 2 + N R ( 2 N R N N 2 N R + N R 2 ) 1 / 2 N N 2 N R
PNR I ¯ ( 0 ) σ I ( f x ) N N 2 N R 2 SNR 2 ( 0 ) I ¯ ( 0 ) I ¯ ( f x ) ,
8 ψ ( x i ) = | ψ i ψ ¯ i | .
N ˆ i N q i .
I 2 = i j k l A i A j * A k * A l = i j k l A ¯ i A ¯ j * A ¯ k * A ¯ l + i | A i | 4 + 4 Re i j | A i | 2 A i * A ¯ j + 2 i j | A i | 2 | A j | 2 + i j A i 2 A i * 2 + 4 i j k | A i | 2 A ¯ j A ¯ k * + 2 Re i j k A i 2 A ¯ j * A ¯ k * .
T i j k l = i j k l a i b j c k d l , T i j k l i j k l = i j k l a i b j c k d l .
T i j i j = T i j T i i , T i j k i j k = T i j k T i i i T i i j i j T i j i i j T i j j i j , = T i j k + 2 T i i i T i i j T i j i T i j j .
T i j k l i j k l = T i j k l 6 T i i i i + T i i j j + T i j i j + T i j j i + 2 ( T i i i j + T i i j i + T i j i i + T i j j j ) ( T i i j k + T i j i k + T i j j k + T i j k i + T i j k j + T i j k k ) .
T i j k l i j k l = T i j k l 6 T i i i i + 2 T i i j j + T i j j i + 8 Re T i i i j 4 T i i j k 2 Re T i j k i .

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