Abstract

It is shown that amplitude weighting can improve the accuracy of measurements of the frequency offset of a signal contaminated by multiplicative Gaussian noise. The more general non-Gaussian case is investigated through study of the statistics of a simple phase-screen scattering model. Formulas are derived for the low-order moments of the intensity-weighted phase derivative. Numerical simulation is tested against these results and is used to generate full probability densities that are analytically intractable and to determine the optimum weighting for the non-Gaussian regime of the model. The results are relevant to a variety of remote-sensing and signal-processing problems.

© 2001 Optical Society of America

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  1. J. F. Miller, K. Schatzel, B. Vincent, “The determination of very small electrophoretic mobilities in polar and non-polar colloidal dispersions using phase analysis light scattering,” J. Colloid. Interface Sci. 143, 532–554 (1991).
    [CrossRef]
  2. A. L. Kachelmyer, K. I. Schultz, “Laser vibration sensing,” Lincoln Lab. J. 8, 3–28 (1995).
  3. I. Renhorn, C. Karlsson, D. Letalick, M. Millnert, R. Rutgers, “Coherent laser radar for vibrometry: robust design and adaptive signal processing ,” in Applied Radar Technology II, Gary W. Kamerman, ed., Proc. SPIE2472, 23–30 (1995).
    [CrossRef]
  4. R. S. Eng, C. Freed, R. H. Kingston, K. I. Schultz, A. L. Katchelmyer, W. E. Keicher, “The effect of laser phase noise on laser radar performance,” Proceedings of the International Conference on Lasers—LASER ’92 (STS Press, McLean, Va., 1993), pp. 605–620.
  5. K. D. Ridley, E. Jakeman, “FM demodulation in the presence of multiplicative and additive noise,” Inverse Probl. 15, 214–219 (1999).
    [CrossRef]
  6. D. K. Asano, S. Pasupathy, “Improved post detection processing for limiter-discriminator detection of CMP in a Rayleigh, fast-fading channel,” IEEE Trans. Veh. Technol. 44, 729–734 (1995).
    [CrossRef]
  7. R. H. Delano, “A theory of target glint or angular scintillation in radar tracking,” Proc. IRE 41, 1778–1784 (1953).
    [CrossRef]
  8. B. Borden, “Requirement for optimal glint reduction by diversity methods,” IEEE Trans. Aerosp. Electron. Syst. 30, 1108–1114 (1994).
    [CrossRef]
  9. R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. A 58, 382–400 (1962).
    [CrossRef]
  10. H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. London Ser. A 242, 579–609 (1950).
    [CrossRef]
  11. E. E. Salpeter, “Interplanetary scintillations: theory,” Astrophys. J. 147, 433–448 (1967).
    [CrossRef]
  12. L. S. Taylor, C. J. Infosino, “Diffraction theory of optical scintillations due to turbulent layers,” J. Opt. Soc. Am. 65, 78–84 (1975).
    [CrossRef]
  13. A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
    [CrossRef]
  14. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963; reprinted by Artech House Inc., Norwood, Mass., 1987).
  15. J. A. Ogilvy, Theory of Scattering from Random Rough Surfaces (Hilger, Bristol, 1991).
  16. J. Parent, A. Bourdillon, “A method to correct HF Skywave back-scattered signals for ionospheric frequency modulation,” IEEE Trans. Antennas Propag. 36, 127–135 (1988).
    [CrossRef]
  17. D. Middleton, An Introduction to Statistical Communication Theory (Institute of Electrical and Electronics Engineers, New York, 1996).
  18. S. O. Rice, “Statistical properties of a sine wave plus random noise,” Bell Syst. Tech. J. 27, 109–157 (1948).
    [CrossRef]
  19. B. A. van Tiggelen, P. Sebbah, M. Stoytchev, A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E 59, 7166–7172 (1999).
    [CrossRef]
  20. E. Jakeman, R. J. A. Tough, “Non-Gaussian models for the statistics of scattered waves,” Adv. Phys. 37, 471–529 (1988).
    [CrossRef]
  21. E. Jakeman, J. G. McWhirter, “Correlation model dependence of the scintillation behind a deep random phase screen,” J. Phys. A. Math. Gen. 10, 1599–1643 (1977).
    [CrossRef]
  22. E. Jakeman, “Fresnel scattering by a corrugated random surface with fractal slope,” J. Opt. Soc. Am. 72, 1034–1041 (1982).
    [CrossRef]
  23. E. Jakeman, J. G. McWhirter, “Fluctuations in radiation scattered into the Fresnel Region by a random phase screen in uniform motion,” J. Phys. A Math. Gen. 9, 785–797 (1976).
    [CrossRef]
  24. E. Jakeman, J. G. McWhirter, P. N. Pusey, “Enhanced fluctuations in radiation scattered by a moving random phase screen,” J. Opt. Soc. Am. 66, 1175–1182 (1976).
    [CrossRef]
  25. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  26. J. M. Martin, S. M. Flatte, “Simulation of point-source scintillation through three-dimensional random media,” J. Opt. Soc. Am. A 7, 838–846 (1990).
    [CrossRef]
  27. K. D. Ridley, E. Jakeman, “Incomplete phase conjugation through a random phase screen. II. Numerical simulations,” J. Opt. Soc. Am. A 13, 2393–2402 (1996).
    [CrossRef]
  28. K. D. Ridley, E. Jakeman, “Signal to noise analysis of FM demodulation in the presence of multiplicative and additive noise,” Signal Process. 80, 1895–1970 (2000).
    [CrossRef]
  29. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw–Hill, New York, 1965).
  30. R. Buckley, “Diffraction by a random phase-changing screen: a numerical experiment,” J. Atmos. Terr. Phys. 37, 1431–1446 (1975).
    [CrossRef]

2000 (1)

K. D. Ridley, E. Jakeman, “Signal to noise analysis of FM demodulation in the presence of multiplicative and additive noise,” Signal Process. 80, 1895–1970 (2000).
[CrossRef]

1999 (2)

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E 59, 7166–7172 (1999).
[CrossRef]

K. D. Ridley, E. Jakeman, “FM demodulation in the presence of multiplicative and additive noise,” Inverse Probl. 15, 214–219 (1999).
[CrossRef]

1996 (1)

1995 (2)

D. K. Asano, S. Pasupathy, “Improved post detection processing for limiter-discriminator detection of CMP in a Rayleigh, fast-fading channel,” IEEE Trans. Veh. Technol. 44, 729–734 (1995).
[CrossRef]

A. L. Kachelmyer, K. I. Schultz, “Laser vibration sensing,” Lincoln Lab. J. 8, 3–28 (1995).

1994 (1)

B. Borden, “Requirement for optimal glint reduction by diversity methods,” IEEE Trans. Aerosp. Electron. Syst. 30, 1108–1114 (1994).
[CrossRef]

1991 (1)

J. F. Miller, K. Schatzel, B. Vincent, “The determination of very small electrophoretic mobilities in polar and non-polar colloidal dispersions using phase analysis light scattering,” J. Colloid. Interface Sci. 143, 532–554 (1991).
[CrossRef]

1990 (1)

1988 (2)

E. Jakeman, R. J. A. Tough, “Non-Gaussian models for the statistics of scattered waves,” Adv. Phys. 37, 471–529 (1988).
[CrossRef]

J. Parent, A. Bourdillon, “A method to correct HF Skywave back-scattered signals for ionospheric frequency modulation,” IEEE Trans. Antennas Propag. 36, 127–135 (1988).
[CrossRef]

1982 (1)

1977 (1)

E. Jakeman, J. G. McWhirter, “Correlation model dependence of the scintillation behind a deep random phase screen,” J. Phys. A. Math. Gen. 10, 1599–1643 (1977).
[CrossRef]

1976 (2)

E. Jakeman, J. G. McWhirter, “Fluctuations in radiation scattered into the Fresnel Region by a random phase screen in uniform motion,” J. Phys. A Math. Gen. 9, 785–797 (1976).
[CrossRef]

E. Jakeman, J. G. McWhirter, P. N. Pusey, “Enhanced fluctuations in radiation scattered by a moving random phase screen,” J. Opt. Soc. Am. 66, 1175–1182 (1976).
[CrossRef]

1975 (3)

R. Buckley, “Diffraction by a random phase-changing screen: a numerical experiment,” J. Atmos. Terr. Phys. 37, 1431–1446 (1975).
[CrossRef]

L. S. Taylor, C. J. Infosino, “Diffraction theory of optical scintillations due to turbulent layers,” J. Opt. Soc. Am. 65, 78–84 (1975).
[CrossRef]

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

1967 (1)

E. E. Salpeter, “Interplanetary scintillations: theory,” Astrophys. J. 147, 433–448 (1967).
[CrossRef]

1962 (1)

R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. A 58, 382–400 (1962).
[CrossRef]

1953 (1)

R. H. Delano, “A theory of target glint or angular scintillation in radar tracking,” Proc. IRE 41, 1778–1784 (1953).
[CrossRef]

1950 (1)

H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. London Ser. A 242, 579–609 (1950).
[CrossRef]

1948 (1)

S. O. Rice, “Statistical properties of a sine wave plus random noise,” Bell Syst. Tech. J. 27, 109–157 (1948).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Asano, D. K.

D. K. Asano, S. Pasupathy, “Improved post detection processing for limiter-discriminator detection of CMP in a Rayleigh, fast-fading channel,” IEEE Trans. Veh. Technol. 44, 729–734 (1995).
[CrossRef]

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963; reprinted by Artech House Inc., Norwood, Mass., 1987).

Booker, H. G.

H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. London Ser. A 242, 579–609 (1950).
[CrossRef]

Borden, B.

B. Borden, “Requirement for optimal glint reduction by diversity methods,” IEEE Trans. Aerosp. Electron. Syst. 30, 1108–1114 (1994).
[CrossRef]

Bourdillon, A.

J. Parent, A. Bourdillon, “A method to correct HF Skywave back-scattered signals for ionospheric frequency modulation,” IEEE Trans. Antennas Propag. 36, 127–135 (1988).
[CrossRef]

Buckley, R.

R. Buckley, “Diffraction by a random phase-changing screen: a numerical experiment,” J. Atmos. Terr. Phys. 37, 1431–1446 (1975).
[CrossRef]

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

Delano, R. H.

R. H. Delano, “A theory of target glint or angular scintillation in radar tracking,” Proc. IRE 41, 1778–1784 (1953).
[CrossRef]

Eng, R. S.

R. S. Eng, C. Freed, R. H. Kingston, K. I. Schultz, A. L. Katchelmyer, W. E. Keicher, “The effect of laser phase noise on laser radar performance,” Proceedings of the International Conference on Lasers—LASER ’92 (STS Press, McLean, Va., 1993), pp. 605–620.

Flatte, S. M.

Freed, C.

R. S. Eng, C. Freed, R. H. Kingston, K. I. Schultz, A. L. Katchelmyer, W. E. Keicher, “The effect of laser phase noise on laser radar performance,” Proceedings of the International Conference on Lasers—LASER ’92 (STS Press, McLean, Va., 1993), pp. 605–620.

Genack, A. Z.

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E 59, 7166–7172 (1999).
[CrossRef]

Gochelashvily, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

Infosino, C. J.

Jakeman, E.

K. D. Ridley, E. Jakeman, “Signal to noise analysis of FM demodulation in the presence of multiplicative and additive noise,” Signal Process. 80, 1895–1970 (2000).
[CrossRef]

K. D. Ridley, E. Jakeman, “FM demodulation in the presence of multiplicative and additive noise,” Inverse Probl. 15, 214–219 (1999).
[CrossRef]

K. D. Ridley, E. Jakeman, “Incomplete phase conjugation through a random phase screen. II. Numerical simulations,” J. Opt. Soc. Am. A 13, 2393–2402 (1996).
[CrossRef]

E. Jakeman, R. J. A. Tough, “Non-Gaussian models for the statistics of scattered waves,” Adv. Phys. 37, 471–529 (1988).
[CrossRef]

E. Jakeman, “Fresnel scattering by a corrugated random surface with fractal slope,” J. Opt. Soc. Am. 72, 1034–1041 (1982).
[CrossRef]

E. Jakeman, J. G. McWhirter, “Correlation model dependence of the scintillation behind a deep random phase screen,” J. Phys. A. Math. Gen. 10, 1599–1643 (1977).
[CrossRef]

E. Jakeman, J. G. McWhirter, “Fluctuations in radiation scattered into the Fresnel Region by a random phase screen in uniform motion,” J. Phys. A Math. Gen. 9, 785–797 (1976).
[CrossRef]

E. Jakeman, J. G. McWhirter, P. N. Pusey, “Enhanced fluctuations in radiation scattered by a moving random phase screen,” J. Opt. Soc. Am. 66, 1175–1182 (1976).
[CrossRef]

Kachelmyer, A. L.

A. L. Kachelmyer, K. I. Schultz, “Laser vibration sensing,” Lincoln Lab. J. 8, 3–28 (1995).

Karlsson, C.

I. Renhorn, C. Karlsson, D. Letalick, M. Millnert, R. Rutgers, “Coherent laser radar for vibrometry: robust design and adaptive signal processing ,” in Applied Radar Technology II, Gary W. Kamerman, ed., Proc. SPIE2472, 23–30 (1995).
[CrossRef]

Katchelmyer, A. L.

R. S. Eng, C. Freed, R. H. Kingston, K. I. Schultz, A. L. Katchelmyer, W. E. Keicher, “The effect of laser phase noise on laser radar performance,” Proceedings of the International Conference on Lasers—LASER ’92 (STS Press, McLean, Va., 1993), pp. 605–620.

Keicher, W. E.

R. S. Eng, C. Freed, R. H. Kingston, K. I. Schultz, A. L. Katchelmyer, W. E. Keicher, “The effect of laser phase noise on laser radar performance,” Proceedings of the International Conference on Lasers—LASER ’92 (STS Press, McLean, Va., 1993), pp. 605–620.

Kingston, R. H.

R. S. Eng, C. Freed, R. H. Kingston, K. I. Schultz, A. L. Katchelmyer, W. E. Keicher, “The effect of laser phase noise on laser radar performance,” Proceedings of the International Conference on Lasers—LASER ’92 (STS Press, McLean, Va., 1993), pp. 605–620.

Letalick, D.

I. Renhorn, C. Karlsson, D. Letalick, M. Millnert, R. Rutgers, “Coherent laser radar for vibrometry: robust design and adaptive signal processing ,” in Applied Radar Technology II, Gary W. Kamerman, ed., Proc. SPIE2472, 23–30 (1995).
[CrossRef]

Martin, J. M.

McWhirter, J. G.

E. Jakeman, J. G. McWhirter, “Correlation model dependence of the scintillation behind a deep random phase screen,” J. Phys. A. Math. Gen. 10, 1599–1643 (1977).
[CrossRef]

E. Jakeman, J. G. McWhirter, “Fluctuations in radiation scattered into the Fresnel Region by a random phase screen in uniform motion,” J. Phys. A Math. Gen. 9, 785–797 (1976).
[CrossRef]

E. Jakeman, J. G. McWhirter, P. N. Pusey, “Enhanced fluctuations in radiation scattered by a moving random phase screen,” J. Opt. Soc. Am. 66, 1175–1182 (1976).
[CrossRef]

Mercier, R. P.

R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. A 58, 382–400 (1962).
[CrossRef]

Middleton, D.

D. Middleton, An Introduction to Statistical Communication Theory (Institute of Electrical and Electronics Engineers, New York, 1996).

Miller, J. F.

J. F. Miller, K. Schatzel, B. Vincent, “The determination of very small electrophoretic mobilities in polar and non-polar colloidal dispersions using phase analysis light scattering,” J. Colloid. Interface Sci. 143, 532–554 (1991).
[CrossRef]

Millnert, M.

I. Renhorn, C. Karlsson, D. Letalick, M. Millnert, R. Rutgers, “Coherent laser radar for vibrometry: robust design and adaptive signal processing ,” in Applied Radar Technology II, Gary W. Kamerman, ed., Proc. SPIE2472, 23–30 (1995).
[CrossRef]

Ogilvy, J. A.

J. A. Ogilvy, Theory of Scattering from Random Rough Surfaces (Hilger, Bristol, 1991).

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw–Hill, New York, 1965).

Parent, J.

J. Parent, A. Bourdillon, “A method to correct HF Skywave back-scattered signals for ionospheric frequency modulation,” IEEE Trans. Antennas Propag. 36, 127–135 (1988).
[CrossRef]

Pasupathy, S.

D. K. Asano, S. Pasupathy, “Improved post detection processing for limiter-discriminator detection of CMP in a Rayleigh, fast-fading channel,” IEEE Trans. Veh. Technol. 44, 729–734 (1995).
[CrossRef]

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

Pusey, P. N.

Ratcliffe, J. A.

H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. London Ser. A 242, 579–609 (1950).
[CrossRef]

Renhorn, I.

I. Renhorn, C. Karlsson, D. Letalick, M. Millnert, R. Rutgers, “Coherent laser radar for vibrometry: robust design and adaptive signal processing ,” in Applied Radar Technology II, Gary W. Kamerman, ed., Proc. SPIE2472, 23–30 (1995).
[CrossRef]

Rice, S. O.

S. O. Rice, “Statistical properties of a sine wave plus random noise,” Bell Syst. Tech. J. 27, 109–157 (1948).
[CrossRef]

Ridley, K. D.

K. D. Ridley, E. Jakeman, “Signal to noise analysis of FM demodulation in the presence of multiplicative and additive noise,” Signal Process. 80, 1895–1970 (2000).
[CrossRef]

K. D. Ridley, E. Jakeman, “FM demodulation in the presence of multiplicative and additive noise,” Inverse Probl. 15, 214–219 (1999).
[CrossRef]

K. D. Ridley, E. Jakeman, “Incomplete phase conjugation through a random phase screen. II. Numerical simulations,” J. Opt. Soc. Am. A 13, 2393–2402 (1996).
[CrossRef]

Rutgers, R.

I. Renhorn, C. Karlsson, D. Letalick, M. Millnert, R. Rutgers, “Coherent laser radar for vibrometry: robust design and adaptive signal processing ,” in Applied Radar Technology II, Gary W. Kamerman, ed., Proc. SPIE2472, 23–30 (1995).
[CrossRef]

Salpeter, E. E.

E. E. Salpeter, “Interplanetary scintillations: theory,” Astrophys. J. 147, 433–448 (1967).
[CrossRef]

Schatzel, K.

J. F. Miller, K. Schatzel, B. Vincent, “The determination of very small electrophoretic mobilities in polar and non-polar colloidal dispersions using phase analysis light scattering,” J. Colloid. Interface Sci. 143, 532–554 (1991).
[CrossRef]

Schultz, K. I.

A. L. Kachelmyer, K. I. Schultz, “Laser vibration sensing,” Lincoln Lab. J. 8, 3–28 (1995).

R. S. Eng, C. Freed, R. H. Kingston, K. I. Schultz, A. L. Katchelmyer, W. E. Keicher, “The effect of laser phase noise on laser radar performance,” Proceedings of the International Conference on Lasers—LASER ’92 (STS Press, McLean, Va., 1993), pp. 605–620.

Sebbah, P.

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E 59, 7166–7172 (1999).
[CrossRef]

Shinn, D. H.

H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. London Ser. A 242, 579–609 (1950).
[CrossRef]

Shishov, V. I.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963; reprinted by Artech House Inc., Norwood, Mass., 1987).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Stoytchev, M.

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E 59, 7166–7172 (1999).
[CrossRef]

Taylor, L. S.

Tough, R. J. A.

E. Jakeman, R. J. A. Tough, “Non-Gaussian models for the statistics of scattered waves,” Adv. Phys. 37, 471–529 (1988).
[CrossRef]

van Tiggelen, B. A.

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E 59, 7166–7172 (1999).
[CrossRef]

Vincent, B.

J. F. Miller, K. Schatzel, B. Vincent, “The determination of very small electrophoretic mobilities in polar and non-polar colloidal dispersions using phase analysis light scattering,” J. Colloid. Interface Sci. 143, 532–554 (1991).
[CrossRef]

Adv. Phys. (1)

E. Jakeman, R. J. A. Tough, “Non-Gaussian models for the statistics of scattered waves,” Adv. Phys. 37, 471–529 (1988).
[CrossRef]

Astrophys. J. (1)

E. E. Salpeter, “Interplanetary scintillations: theory,” Astrophys. J. 147, 433–448 (1967).
[CrossRef]

Bell Syst. Tech. J. (1)

S. O. Rice, “Statistical properties of a sine wave plus random noise,” Bell Syst. Tech. J. 27, 109–157 (1948).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst. (1)

B. Borden, “Requirement for optimal glint reduction by diversity methods,” IEEE Trans. Aerosp. Electron. Syst. 30, 1108–1114 (1994).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. Parent, A. Bourdillon, “A method to correct HF Skywave back-scattered signals for ionospheric frequency modulation,” IEEE Trans. Antennas Propag. 36, 127–135 (1988).
[CrossRef]

IEEE Trans. Veh. Technol. (1)

D. K. Asano, S. Pasupathy, “Improved post detection processing for limiter-discriminator detection of CMP in a Rayleigh, fast-fading channel,” IEEE Trans. Veh. Technol. 44, 729–734 (1995).
[CrossRef]

Inverse Probl. (1)

K. D. Ridley, E. Jakeman, “FM demodulation in the presence of multiplicative and additive noise,” Inverse Probl. 15, 214–219 (1999).
[CrossRef]

J. Atmos. Terr. Phys. (1)

R. Buckley, “Diffraction by a random phase-changing screen: a numerical experiment,” J. Atmos. Terr. Phys. 37, 1431–1446 (1975).
[CrossRef]

J. Colloid. Interface Sci. (1)

J. F. Miller, K. Schatzel, B. Vincent, “The determination of very small electrophoretic mobilities in polar and non-polar colloidal dispersions using phase analysis light scattering,” J. Colloid. Interface Sci. 143, 532–554 (1991).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

J. Phys. A Math. Gen. (1)

E. Jakeman, J. G. McWhirter, “Fluctuations in radiation scattered into the Fresnel Region by a random phase screen in uniform motion,” J. Phys. A Math. Gen. 9, 785–797 (1976).
[CrossRef]

J. Phys. A. Math. Gen. (1)

E. Jakeman, J. G. McWhirter, “Correlation model dependence of the scintillation behind a deep random phase screen,” J. Phys. A. Math. Gen. 10, 1599–1643 (1977).
[CrossRef]

Lincoln Lab. J. (1)

A. L. Kachelmyer, K. I. Schultz, “Laser vibration sensing,” Lincoln Lab. J. 8, 3–28 (1995).

Philos. Trans. R. Soc. London Ser. A (1)

H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. London Ser. A 242, 579–609 (1950).
[CrossRef]

Phys. Rev. E (1)

B. A. van Tiggelen, P. Sebbah, M. Stoytchev, A. Z. Genack, “Delay-time statistics for diffuse waves,” Phys. Rev. E 59, 7166–7172 (1999).
[CrossRef]

Proc. Cambridge Philos. Soc. A (1)

R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. A 58, 382–400 (1962).
[CrossRef]

Proc. IEEE (1)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, V. I. Shishov, “Laser irradiance propagation in turbulent media,” Proc. IEEE 63, 790–811 (1975).
[CrossRef]

Proc. IRE (1)

R. H. Delano, “A theory of target glint or angular scintillation in radar tracking,” Proc. IRE 41, 1778–1784 (1953).
[CrossRef]

Signal Process. (1)

K. D. Ridley, E. Jakeman, “Signal to noise analysis of FM demodulation in the presence of multiplicative and additive noise,” Signal Process. 80, 1895–1970 (2000).
[CrossRef]

Other (7)

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw–Hill, New York, 1965).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

I. Renhorn, C. Karlsson, D. Letalick, M. Millnert, R. Rutgers, “Coherent laser radar for vibrometry: robust design and adaptive signal processing ,” in Applied Radar Technology II, Gary W. Kamerman, ed., Proc. SPIE2472, 23–30 (1995).
[CrossRef]

R. S. Eng, C. Freed, R. H. Kingston, K. I. Schultz, A. L. Katchelmyer, W. E. Keicher, “The effect of laser phase noise on laser radar performance,” Proceedings of the International Conference on Lasers—LASER ’92 (STS Press, McLean, Va., 1993), pp. 605–620.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963; reprinted by Artech House Inc., Norwood, Mass., 1987).

J. A. Ogilvy, Theory of Scattering from Random Rough Surfaces (Hilger, Bristol, 1991).

D. Middleton, An Introduction to Statistical Communication Theory (Institute of Electrical and Electronics Engineers, New York, 1996).

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Figures (7)

Fig. 1
Fig. 1

Normalized second moments of intensity against propagation distance from the phase screen. Symbols, results obtained from simulation; solid curves, display theoretical predictions.

Fig. 2
Fig. 2

Same as Fig. 1, but for the J statistic. The dashed curves were obtained by numerical integration of Eq. (24).

Fig. 3
Fig. 3

Probability density functions of intensity at various propagation distances. Inset: PDF of intensity at -ln(q)=+5 on semilog scale.

Fig. 4
Fig. 4

Probability density functions of ϕ˙ at various propagation distances.

Fig. 5
Fig. 5

Probability density functions of J at various propagation distances. Note that results at -ln(q)=-5 are not displayed, as they are identical to those in Fig. 4.

Fig. 6
Fig. 6

Variance of ϕ˙ as a function of amplitude. Symbols, simulation results; solid line, curve fit with gradient -2.

Fig. 7
Fig. 7

Correlation between A and Aϕ˙ as a function of propagation distance.

Equations (69)

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S(t)=A(t)exp{i[ϕ(t)+ft]},
E(t)=A(t)exp[iϕ(t)]
θ˙=ϕ˙+f.
C({λn})=exp inλnX(tn)=exp-12n,mλnλmX(tn)X(tm).
E(t)=X(t)+iY(t).
P(I)=exp(-I/I¯)/I¯.
P(ϕ˙)=τc/2(1+τc2ϕ˙2)3/2,
τc-2=X˙2X2=-d2dt2X(0)X(t)X2t=0.
C(s, u)=exp{i[sF(X, Y)+uAϕ˙]}=exp{i[sF(X, Y)+u(XY˙-YX˙)/(X2+Y2)1/2]},
C(s, u)=exp[isF(X, Y)]exp(-u2X˙2/2).
(Aϕ˙)2=A2/2τc2
fˆ=n=1NAnvθ˙nn=1NAnv=f+n=1NAnvϕ˙nn=1NAnv.
fˆ=f
varfˆ=n,m=1NAnvAmvϕ˙nϕ˙mn=1NAnv2
varfˆ=n=1NAn2(v-1)n=1NAnv2(Aϕ˙)2.
varfˆf2=12(N-1)f2τc2.
P(J)=τcIexp-2|J|τcI.
varfˆ=1NA2vϕ˙2Av2=1NA2(v-1)Av2 (Aϕ˙)2.
E(x, t)=ik2πz-dx×expik2z (x-x)2+iϕ(x, t),
ϕ(x, t)=ϕ(x+ut, 0)=ϕ(x+ut),
E(x, t)=ik2πz-dx×expik2z (x-x+ut)2+iϕ(x).
J=A2ϕ˙=Iϕ˙=12iE E*t-Et E*
J=-k2u2πz2-dxdxxcos[ϕ(x+ut)-ϕ(x+ut)+(k/2z)(x2-x2)].
J=0,I=1,
varfˆf2=1Nf2 J2.
J2=-πz2kk2u2πz22-dxdy4ϕ02z2k2d2ρdx2+x2×coskz xyexp{-ϕ02[2-2ρ(x)-2ρ(y)+ρ(x+y)+ρ(x-y)]},
ϕ(y)ϕ(y+x)=ϕ02ρ(x)=ϕ02exp(-x2/ξ2)
J2=ϕ02u2ξ2erfc(q)+2qπexp(-q2)ln8γϕ023+π erfi(q)-R(q)-q2-2.
R(q)=2π0qdx exp(x2)erfc(x),
q=kξ22zϕ06.
2Ex2-2ik Ez=0.
E(x, z0)=exp[iϕ(x)].
E(x, z1)=F-1exp-i κ2(z1-z0)2kF[E(x, z0)],
ϕ˙ΔϕΔt=u ΔϕΔx.
Jn=Inϕ˙n=In(ϕn+1-ϕn),
Δxdϕdxπ.
dϕdx2=-d2ρdx2x=0,
Δxmin=πξ2ϕ0.
θ=1kdϕdx.
D(z)=zkdϕdx=2ϕ0zkξ
ei=s+bi,
iNF(ai)eiiNF(ai)=s+iNF(ai)biiNF(ai),
P(a1, a2 ,, ai ,, aN, bi)
=P(bi|ai)P(a1)P(a2)  P(ai)  P(aN),
P(bi|ai)=P(-bi|ai).
NF(a1)2b12iNF(ai)2.
N-  -F(a1)2b12iNF(ai)2
×P(b1|a1)P(a1)  P(aN)db1da1  daN.
b12a1=-b12P(b1|a1)db1.
F(a)=f(a)+αη(a),
-  -η(a1)f(a1)b12a1iNf(ai)3
×i=2Nf(ai)P(a1)  P(aN)da1  daN
=-  -f(a1)2b12a1iNf(ai)3×i=2Nη(ai)P(a1)  P(aN)da1  daN.
-  -η(a1)f(a1)f(a2)b12a1iNf(ai)3
×P(a1)  P(aN)da1  daN
=-  -η(a2)f(a1)f(a1)b12a1iNf(ai)3×P(a1)  P(aN)da1  daN.
P(ϕ˙|A)=τsA/(2πA2)1/2exp[-(A2τs2/2A2)ϕ˙2].
ϕ˙2A=A2/τs2A2=X˙2/A2.
ϕ˙2A=-ϕ˙2P(ϕ˙|A)dϕ˙.
ϕ˙2A=1A2-(Aϕ˙)2P(Aϕ˙, A)P(A)d(Aϕ˙).
ϕ˙2A=1A2-x2P(x)dx,
R=A|Aϕ˙|-A|Aϕ˙|[(A2-A2)(|Aϕ˙|2-|Aϕ˙|2)]1/2.
J2=12k2u2πz22(I1+I2),
I1=-dx1  dx4x1x3cosϕ(x1+ut)-ϕ (x2+ut)+ϕ(x3+ut)-ϕ(x4+ut)+k2z (x12-x22-x32-x42),
I2=-dx1  dx4x1x3cosϕ(x1+ut)-ϕ (x2+ut)-ϕ(x3+ut)+ϕ(x4+ut)+k2z (x12-x22-x32+x42),
I1=Re-dx1  dx4x1x3×expik2z (x12-x22+x32-x42)×exp{-ϕ02[2-ρ(x1-x2)-ρ(x2-x3)-ρ(x3-x4)-ρ(x1-x4)+ρ(x1-x3)+ρ(x2-x4)]},
I1=14Re-dy1  dy4[(y1+y3)2-(y2+y4)2]×exp-ikz (y1y3+y2y4)×exp{-ϕ02[2-ρ(y1+y2)-ρ(y1-y2)-ρ(y4+y1)-ρ(y4-y1)+ρ(y4+y2)+ρ(y4-y2)]}.
I1=-πz2kRe-dy2dy42ϕ02z2k22ρ(y2)y22+2ρ(y4)y42-2izk+(y2+y4)2exp-ikz y2y4×exp{-ϕ02[2-2ρ(y2)-2ρ(y4)+ρ(y2+y4)+ρ(y2-y4)]}.
I1+I2=-πz2k-dxdy4ϕ02z2k2 [ρ(x)+ρ(y)]+x2+y2×coskz xyexp{-ϕ02[2-2ρ(x)-2ρ(y)+ρ(x+y)+ρ(x-y)]}.

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