Abstract

An effective procedure for generating m equal-power log-normal scintillation sequences with any desired cross-correlation coefficient is presented. The method is focused on a multichannel generalization of the autoregressive variate generation method in order to satisfy Taylor’s hypothesis of frozen turbulence. Then the effect of the turbulent kinetic energy dissipation rate, ε, is included in the model, breaking the uniformity of the frozen-in hypothesis and obtaining a highly realistic behavior in the generated sequences when realistic scenarios are considered in which turbulence-induced fading may be correlated.

© 2009 Optical Society of America

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2007 (5)

M. C. R. Kalapureddy, K. K. Kumar, V. Sivakumar, A. K. Ghosh, A. R. Jain, K. K. Reddy, “Diurnal and seasonal variability of TKE dissipation rate in the ABL over a tropical station using UHF wind profiler,” J. Atmos. Sol.-Terr. Phys., vol. 69, nos. 4–5, pp. 419–430, April 2007.
[CrossRef]

A. Jurado Navas, A. García Zambrana, A. Puerta Notario, “Efficient lognormal channel model for turbulent FSO communications,” Electron. Lett., vol. 43, no. 3, pp. 178–179, Feb. 2007.
[CrossRef]

A. Christen, E. van Gorsel, R. Vogt, “Coherent structures in urban roughness sublayer turbulence,” Int. J. Climatol., vol. 27, no. 14, pp. 1955–1968, Oct. 2007.
[CrossRef]

J. F. Monserrat, R. Fraile, L. Rubio, “Application of alternating projection method to ensure feasibility of shadowing cross-correlation models,” Electron. Lett., vol. 43, no. 13, pp. 724–725, June 2007.
[CrossRef]

J. A. Anguita, M. A. Neifeld, B. V. Vasic, “Spatial correlation and irradiance statistics in a multiple-beam terrestrial free-space optical communication link,” Appl. Opt., vol. 46, no. 26, pp. 6561–6571, Sept. 2007.
[CrossRef] [PubMed]

2006 (1)

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag., vol. 44, no. 11, pp. 46–51, Nov. 2006.
[CrossRef]

2005 (3)

M. Razavi, J. H. Shapiro, “Wireless optical communications via diversity reception and optical preamplification,” IEEE Trans. Wireless Commun., vol. 4, no. 3, pp. 975–983, May 2005.
[CrossRef]

T. Burghelea, E. Segre, V. Steinberg, “Validity of the Taylor hypothesis in a random spatially smooth flow,” Phys. Fluids, vol. 17, no. 10, paper 103101, Oct. 2005.
[CrossRef]

C. I. Moore, H. R. Barris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE, vol. 5793, pp. 78–88, 2005.
[CrossRef]

2004 (2)

E. J. Lee, V. W. S. Chan, “Part I: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun., vol. 22, no. 9, pp. 1896–1906, Nov. 2004.
[CrossRef]

M. Al Naboulsi, H. Sizun, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng., vol. 43, no. 2, pp. 319–329, Feb. 2004.
[CrossRef]

2003 (1)

2002 (2)

X. Zhu, J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 50, no. 8, pp. 1293–1300, Aug. 2002.
[CrossRef]

S. M. Haas, J. H. Shapiro, V. Tarokh, “Space-time codes for wireless optical communications,” EURASIP J. Appl. Signal Process., vol. 2002, no. 3, pp. 211–220, March 2002.
[CrossRef]

2001 (1)

Y. I. Abramovich, N. K. Spencer, A. Y. Gorokhov, “Detection-estimation of more uncorrelated Gaussian sources than sensors in nonuniform linear antenna arrays—part I: Fully augmentable arrays,” IEEE Trans. Signal Process., vol. 49, no. 5, pp. 959–971, May 2001.
[CrossRef]

2000 (2)

A. Belmonte, B. Martin, W. I. Goldburg, “Experimental study of Taylor’s hypothesis in a turbulent soap film,” Phys. Fluids, vol. 12, no. 4, pp. 835–845, April 2000.
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, “Aperture averaging of optical scintillations power fluctuations and the temporal spectrum,” Waves Random Complex Media, vol. 10, no. 1, pp. 53–70, Jan. 2000.
[CrossRef]

1999 (2)

N. C. Beaulieu, “Generation of correlated Rayleigh fading envelopes,” IEEE Commun. Lett., vol. 3, no. 6, pp. 172–174, June 1999.
[CrossRef]

M. W. Rotach, “On the influence of the urban roughness sublayer on turbulence and dispersion,” Atmos. Environ., vol. 33, pp. 4001–4008, Oct. 1999.
[CrossRef]

1998 (1)

R. B. Ertel, J. H. Reed, “Generation of two equal power correlated Rayleigh fading envelopes,” IEEE Commun. Lett., vol. 2, pp. 276–278, Oct. 1998.
[CrossRef]

1997 (1)

W. K. Hocking, P. K. L. Mu, “Upper and middle tropospheric kinetic energy dissipation rates from measurements of Cn2—review of theories, in-situ investigations, and experimental studies using the Buckland park atmospheric radar in Australia,” J. Atmos. Sol.-Terr. Phys., vol. 59, no. 14, pp. 1779–1803, Sept. 1997.
[CrossRef]

1996 (1)

M. M. Ibrahim, A. M. Ibrahim, “Performance analysis of optical receivers with space diversity reception,” IEE Proc.-Commun., vol. 143, no. 6, pp. 369–372, Dec. 1996.
[CrossRef]

1993 (1)

G. Alefeld, G. Mayer, “The Cholesky method for interval data,” Linear Algebra Its Appl., vol. 194, pp. 161–182, Nov. 1993.
[CrossRef]

1976 (1)

G. E. Willis, J. W. Deardorff, “On the use of Taylor’s hypothesis for diffusion in the mixed layer,” Q. J. R. Meteorol. Soc., vol. 102, no. 434, pp. 817–822, Oct. 1976.
[CrossRef]

1972 (1)

P. R. Halmos, “Positive approximants of operators,” Indiana Univ. Math. J., vol. 21, no. 10, pp. 951–960, Oct. 1972.
[CrossRef]

1969 (1)

N. K. Vinnichenko, J. A. Dutton, “Empirical studies of atmospheric structure and spectra in the free atmosphere,” Radio Sci., vol. 4, no. 12, pp. 1115–1126, Dec. 1969.
[CrossRef]

1967 (1)

1965 (1)

J. L. Lumley, “Interpretation of time spectra measured in high-intensity shear flows,” Phys. Fluids, vol. 8, no. 6 pp. 1056–1062, June 1965.
[CrossRef]

Abramovich, Y. I.

Y. I. Abramovich, N. K. Spencer, A. Y. Gorokhov, “Detection-estimation of more uncorrelated Gaussian sources than sensors in nonuniform linear antenna arrays—part I: Fully augmentable arrays,” IEEE Trans. Signal Process., vol. 49, no. 5, pp. 959–971, May 2001.
[CrossRef]

Al Naboulsi, M.

M. Al Naboulsi, H. Sizun, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng., vol. 43, no. 2, pp. 319–329, Feb. 2004.
[CrossRef]

Alefeld, G.

G. Alefeld, G. Mayer, “The Cholesky method for interval data,” Linear Algebra Its Appl., vol. 194, pp. 161–182, Nov. 1993.
[CrossRef]

Andrews, L. C.

L. C. Andrews, R. L. Phillips, C. Y. Hopen, “Aperture averaging of optical scintillations power fluctuations and the temporal spectrum,” Waves Random Complex Media, vol. 10, no. 1, pp. 53–70, Jan. 2000.
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, Laser Beam, Scintillation With Applications. Bellingham, WA: SPIE, 2001.
[CrossRef]

L. C. Andrews, R. L. Phillips, Laser Beam Propagation Through Random Media. Bellingham, WA: SPIE, 1998.

Anguita, J. A.

Arnon, S.

Baddour, K. E.

K. E. Baddour, N. C. Beaulieu, “Accurate simulation of multiple cross-correlated fading channels,” in IEEE Int. Conf. on Communications, 2002, New York, NY, April 28–May 2, 2002, vol. 1, pp. 267–271.

Barris, H. R.

C. I. Moore, H. R. Barris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE, vol. 5793, pp. 78–88, 2005.
[CrossRef]

Beaulieu, N. C.

N. C. Beaulieu, “Generation of correlated Rayleigh fading envelopes,” IEEE Commun. Lett., vol. 3, no. 6, pp. 172–174, June 1999.
[CrossRef]

K. E. Baddour, N. C. Beaulieu, “Accurate simulation of multiple cross-correlated fading channels,” in IEEE Int. Conf. on Communications, 2002, New York, NY, April 28–May 2, 2002, vol. 1, pp. 267–271.

N. C. Beaulieu, M. L. Merani, “Efficient simulation of correlated diversity channels,” in IEEE Wireless Communications and Networking Conf., 2000. WCNC 2000, Chicago, IL, Sept. 23–28, 2000, vol. 1, pp. 207–210.

Belmonte, A.

A. Belmonte, B. Martin, W. I. Goldburg, “Experimental study of Taylor’s hypothesis in a turbulent soap film,” Phys. Fluids, vol. 12, no. 4, pp. 835–845, April 2000.
[CrossRef]

Burghelea, T.

T. Burghelea, E. Segre, V. Steinberg, “Validity of the Taylor hypothesis in a random spatially smooth flow,” Phys. Fluids, vol. 17, no. 10, paper 103101, Oct. 2005.
[CrossRef]

Chan, V. W. S.

E. J. Lee, V. W. S. Chan, “Part I: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun., vol. 22, no. 9, pp. 1896–1906, Nov. 2004.
[CrossRef]

Christen, A.

A. Christen, E. van Gorsel, R. Vogt, “Coherent structures in urban roughness sublayer turbulence,” Int. J. Climatol., vol. 27, no. 14, pp. 1955–1968, Oct. 2007.
[CrossRef]

A. Christen, M. W. Rotach, R. Vogt, “Experimental determination of the turbulent kinetic energy budget within and above an urban canopy,” in Fifth Conf. on Urban Environment, Vancouver, Canada, Aug. 23–27, 2004, paper 6.4.

Deardorff, J. W.

G. E. Willis, J. W. Deardorff, “On the use of Taylor’s hypothesis for diffusion in the mixed layer,” Q. J. R. Meteorol. Soc., vol. 102, no. 434, pp. 817–822, Oct. 1976.
[CrossRef]

Dutton, J. A.

N. K. Vinnichenko, J. A. Dutton, “Empirical studies of atmospheric structure and spectra in the free atmosphere,” Radio Sci., vol. 4, no. 12, pp. 1115–1126, Dec. 1969.
[CrossRef]

Dwivedi, A.

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag., vol. 44, no. 11, pp. 46–51, Nov. 2006.
[CrossRef]

Ertel, R. B.

R. B. Ertel, J. H. Reed, “Generation of two equal power correlated Rayleigh fading envelopes,” IEEE Commun. Lett., vol. 2, pp. 276–278, Oct. 1998.
[CrossRef]

Fraile, R.

J. F. Monserrat, R. Fraile, L. Rubio, “Application of alternating projection method to ensure feasibility of shadowing cross-correlation models,” Electron. Lett., vol. 43, no. 13, pp. 724–725, June 2007.
[CrossRef]

Frehlich, R.

R. Hill, R. Frehlich, W. Otto, The Probability Distribution of Irradiance Scintillation, NOAA Technical Memorandum ERL ETL-274, Jan. 1997.

Fried, D. L.

Frisch, U.

U. Frisch, Turbulence. The Legacy of A. N. Kolmogorov. New York, NY: Cambridge Univ. Press, 1995.

García Zambrana, A.

A. Jurado Navas, A. García Zambrana, A. Puerta Notario, “Efficient lognormal channel model for turbulent FSO communications,” Electron. Lett., vol. 43, no. 3, pp. 178–179, Feb. 2007.
[CrossRef]

Ghosh, A. K.

M. C. R. Kalapureddy, K. K. Kumar, V. Sivakumar, A. K. Ghosh, A. R. Jain, K. K. Reddy, “Diurnal and seasonal variability of TKE dissipation rate in the ABL over a tropical station using UHF wind profiler,” J. Atmos. Sol.-Terr. Phys., vol. 69, nos. 4–5, pp. 419–430, April 2007.
[CrossRef]

Gilbreath, G. C.

C. I. Moore, H. R. Barris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE, vol. 5793, pp. 78–88, 2005.
[CrossRef]

Goldburg, W. I.

A. Belmonte, B. Martin, W. I. Goldburg, “Experimental study of Taylor’s hypothesis in a turbulent soap film,” Phys. Fluids, vol. 12, no. 4, pp. 835–845, April 2000.
[CrossRef]

Golub, G. H.

G. H. Golub, C. F. van Loan, Matrix Computations. Baltimore, MD: John Hopkins Univ. Press, 1996.

Gorokhov, A. Y.

Y. I. Abramovich, N. K. Spencer, A. Y. Gorokhov, “Detection-estimation of more uncorrelated Gaussian sources than sensors in nonuniform linear antenna arrays—part I: Fully augmentable arrays,” IEEE Trans. Signal Process., vol. 49, no. 5, pp. 959–971, May 2001.
[CrossRef]

Haas, S. M.

S. M. Haas, J. H. Shapiro, V. Tarokh, “Space-time codes for wireless optical communications,” EURASIP J. Appl. Signal Process., vol. 2002, no. 3, pp. 211–220, March 2002.
[CrossRef]

Halmos, P. R.

P. R. Halmos, “Positive approximants of operators,” Indiana Univ. Math. J., vol. 21, no. 10, pp. 951–960, Oct. 1972.
[CrossRef]

Hammons, A. R.

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag., vol. 44, no. 11, pp. 46–51, Nov. 2006.
[CrossRef]

Hill, R.

R. Hill, R. Frehlich, W. Otto, The Probability Distribution of Irradiance Scintillation, NOAA Technical Memorandum ERL ETL-274, Jan. 1997.

Hocking, W. K.

W. K. Hocking, P. K. L. Mu, “Upper and middle tropospheric kinetic energy dissipation rates from measurements of Cn2—review of theories, in-situ investigations, and experimental studies using the Buckland park atmospheric radar in Australia,” J. Atmos. Sol.-Terr. Phys., vol. 59, no. 14, pp. 1779–1803, Sept. 1997.
[CrossRef]

Holton, G. A.

G. A. Holton, Value at Risk: Theory and Practice. San Diego, CA: Academic, 2004.

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, C. Y. Hopen, “Aperture averaging of optical scintillations power fluctuations and the temporal spectrum,” Waves Random Complex Media, vol. 10, no. 1, pp. 53–70, Jan. 2000.
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, Laser Beam, Scintillation With Applications. Bellingham, WA: SPIE, 2001.
[CrossRef]

Ibrahim, A. M.

M. M. Ibrahim, A. M. Ibrahim, “Performance analysis of optical receivers with space diversity reception,” IEE Proc.-Commun., vol. 143, no. 6, pp. 369–372, Dec. 1996.
[CrossRef]

Ibrahim, M. M.

M. M. Ibrahim, A. M. Ibrahim, “Performance analysis of optical receivers with space diversity reception,” IEE Proc.-Commun., vol. 143, no. 6, pp. 369–372, Dec. 1996.
[CrossRef]

Jain, A. R.

M. C. R. Kalapureddy, K. K. Kumar, V. Sivakumar, A. K. Ghosh, A. R. Jain, K. K. Reddy, “Diurnal and seasonal variability of TKE dissipation rate in the ABL over a tropical station using UHF wind profiler,” J. Atmos. Sol.-Terr. Phys., vol. 69, nos. 4–5, pp. 419–430, April 2007.
[CrossRef]

Jones, S. D.

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag., vol. 44, no. 11, pp. 46–51, Nov. 2006.
[CrossRef]

Juarez, J. C.

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag., vol. 44, no. 11, pp. 46–51, Nov. 2006.
[CrossRef]

Jurado Navas, A.

A. Jurado Navas, A. García Zambrana, A. Puerta Notario, “Efficient lognormal channel model for turbulent FSO communications,” Electron. Lett., vol. 43, no. 3, pp. 178–179, Feb. 2007.
[CrossRef]

Kahn, J. M.

X. Zhu, J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 50, no. 8, pp. 1293–1300, Aug. 2002.
[CrossRef]

Kalapureddy, M. C. R.

M. C. R. Kalapureddy, K. K. Kumar, V. Sivakumar, A. K. Ghosh, A. R. Jain, K. K. Reddy, “Diurnal and seasonal variability of TKE dissipation rate in the ABL over a tropical station using UHF wind profiler,” J. Atmos. Sol.-Terr. Phys., vol. 69, nos. 4–5, pp. 419–430, April 2007.
[CrossRef]

Kay, S. M.

S. M. Kay, Modern Spectral Estimation: Theory and Application. Englewood Cliffs, NJ: Prentice-Hall, 1988.

Kohldorfer, P.

S. S. Muhammad, P. Kohldorfer, E. Leitgeb, “Channel modeling for terrestrial free space optical links,” in Proc. 2005 7th Int. Conf. on Transparent Optical Networks, Barcelona, Spain, July 3–7, 2005, vol. 1, pp. 407–410.

Kumar, K. K.

M. C. R. Kalapureddy, K. K. Kumar, V. Sivakumar, A. K. Ghosh, A. R. Jain, K. K. Reddy, “Diurnal and seasonal variability of TKE dissipation rate in the ABL over a tropical station using UHF wind profiler,” J. Atmos. Sol.-Terr. Phys., vol. 69, nos. 4–5, pp. 419–430, April 2007.
[CrossRef]

Kyle, T.

T. Kyle, Atmospheric Transmission, Emission and Scattering. Oxford, UK: Pergamon, 1993.

Lee, E. J.

E. J. Lee, V. W. S. Chan, “Part I: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun., vol. 22, no. 9, pp. 1896–1906, Nov. 2004.
[CrossRef]

Leitgeb, E.

S. S. Muhammad, P. Kohldorfer, E. Leitgeb, “Channel modeling for terrestrial free space optical links,” in Proc. 2005 7th Int. Conf. on Transparent Optical Networks, Barcelona, Spain, July 3–7, 2005, vol. 1, pp. 407–410.

Lumley, J. L.

J. L. Lumley, “Interpretation of time spectra measured in high-intensity shear flows,” Phys. Fluids, vol. 8, no. 6 pp. 1056–1062, June 1965.
[CrossRef]

Mahon, R.

C. I. Moore, H. R. Barris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE, vol. 5793, pp. 78–88, 2005.
[CrossRef]

Martin, B.

A. Belmonte, B. Martin, W. I. Goldburg, “Experimental study of Taylor’s hypothesis in a turbulent soap film,” Phys. Fluids, vol. 12, no. 4, pp. 835–845, April 2000.
[CrossRef]

Mayer, G.

G. Alefeld, G. Mayer, “The Cholesky method for interval data,” Linear Algebra Its Appl., vol. 194, pp. 161–182, Nov. 1993.
[CrossRef]

Merani, M. L.

N. C. Beaulieu, M. L. Merani, “Efficient simulation of correlated diversity channels,” in IEEE Wireless Communications and Networking Conf., 2000. WCNC 2000, Chicago, IL, Sept. 23–28, 2000, vol. 1, pp. 207–210.

Monserrat, J. F.

J. F. Monserrat, R. Fraile, L. Rubio, “Application of alternating projection method to ensure feasibility of shadowing cross-correlation models,” Electron. Lett., vol. 43, no. 13, pp. 724–725, June 2007.
[CrossRef]

Moore, C. I.

C. I. Moore, H. R. Barris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE, vol. 5793, pp. 78–88, 2005.
[CrossRef]

Mu, P. K. L.

W. K. Hocking, P. K. L. Mu, “Upper and middle tropospheric kinetic energy dissipation rates from measurements of Cn2—review of theories, in-situ investigations, and experimental studies using the Buckland park atmospheric radar in Australia,” J. Atmos. Sol.-Terr. Phys., vol. 59, no. 14, pp. 1779–1803, Sept. 1997.
[CrossRef]

Muhammad, S. S.

S. S. Muhammad, P. Kohldorfer, E. Leitgeb, “Channel modeling for terrestrial free space optical links,” in Proc. 2005 7th Int. Conf. on Transparent Optical Networks, Barcelona, Spain, July 3–7, 2005, vol. 1, pp. 407–410.

Neifeld, M. A.

Nichols, R. A.

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag., vol. 44, no. 11, pp. 46–51, Nov. 2006.
[CrossRef]

Otto, W.

R. Hill, R. Frehlich, W. Otto, The Probability Distribution of Irradiance Scintillation, NOAA Technical Memorandum ERL ETL-274, Jan. 1997.

Phillips, R. L.

L. C. Andrews, R. L. Phillips, C. Y. Hopen, “Aperture averaging of optical scintillations power fluctuations and the temporal spectrum,” Waves Random Complex Media, vol. 10, no. 1, pp. 53–70, Jan. 2000.
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, Laser Beam, Scintillation With Applications. Bellingham, WA: SPIE, 2001.
[CrossRef]

L. C. Andrews, R. L. Phillips, Laser Beam Propagation Through Random Media. Bellingham, WA: SPIE, 1998.

Puerta Notario, A.

A. Jurado Navas, A. García Zambrana, A. Puerta Notario, “Efficient lognormal channel model for turbulent FSO communications,” Electron. Lett., vol. 43, no. 3, pp. 178–179, Feb. 2007.
[CrossRef]

Rabinovich, W. S.

C. I. Moore, H. R. Barris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE, vol. 5793, pp. 78–88, 2005.
[CrossRef]

Razavi, M.

M. Razavi, J. H. Shapiro, “Wireless optical communications via diversity reception and optical preamplification,” IEEE Trans. Wireless Commun., vol. 4, no. 3, pp. 975–983, May 2005.
[CrossRef]

Reddy, K. K.

M. C. R. Kalapureddy, K. K. Kumar, V. Sivakumar, A. K. Ghosh, A. R. Jain, K. K. Reddy, “Diurnal and seasonal variability of TKE dissipation rate in the ABL over a tropical station using UHF wind profiler,” J. Atmos. Sol.-Terr. Phys., vol. 69, nos. 4–5, pp. 419–430, April 2007.
[CrossRef]

Reed, J. H.

R. B. Ertel, J. H. Reed, “Generation of two equal power correlated Rayleigh fading envelopes,” IEEE Commun. Lett., vol. 2, pp. 276–278, Oct. 1998.
[CrossRef]

Rotach, M. W.

M. W. Rotach, “On the influence of the urban roughness sublayer on turbulence and dispersion,” Atmos. Environ., vol. 33, pp. 4001–4008, Oct. 1999.
[CrossRef]

A. Christen, M. W. Rotach, R. Vogt, “Experimental determination of the turbulent kinetic energy budget within and above an urban canopy,” in Fifth Conf. on Urban Environment, Vancouver, Canada, Aug. 23–27, 2004, paper 6.4.

Rubio, L.

J. F. Monserrat, R. Fraile, L. Rubio, “Application of alternating projection method to ensure feasibility of shadowing cross-correlation models,” Electron. Lett., vol. 43, no. 13, pp. 724–725, June 2007.
[CrossRef]

Scharpf, W. J.

C. I. Moore, H. R. Barris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE, vol. 5793, pp. 78–88, 2005.
[CrossRef]

Segre, E.

T. Burghelea, E. Segre, V. Steinberg, “Validity of the Taylor hypothesis in a random spatially smooth flow,” Phys. Fluids, vol. 17, no. 10, paper 103101, Oct. 2005.
[CrossRef]

Shapiro, J. H.

M. Razavi, J. H. Shapiro, “Wireless optical communications via diversity reception and optical preamplification,” IEEE Trans. Wireless Commun., vol. 4, no. 3, pp. 975–983, May 2005.
[CrossRef]

S. M. Haas, J. H. Shapiro, V. Tarokh, “Space-time codes for wireless optical communications,” EURASIP J. Appl. Signal Process., vol. 2002, no. 3, pp. 211–220, March 2002.
[CrossRef]

Sivakumar, V.

M. C. R. Kalapureddy, K. K. Kumar, V. Sivakumar, A. K. Ghosh, A. R. Jain, K. K. Reddy, “Diurnal and seasonal variability of TKE dissipation rate in the ABL over a tropical station using UHF wind profiler,” J. Atmos. Sol.-Terr. Phys., vol. 69, nos. 4–5, pp. 419–430, April 2007.
[CrossRef]

Sizun, H.

M. Al Naboulsi, H. Sizun, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng., vol. 43, no. 2, pp. 319–329, Feb. 2004.
[CrossRef]

Spencer, N. K.

Y. I. Abramovich, N. K. Spencer, A. Y. Gorokhov, “Detection-estimation of more uncorrelated Gaussian sources than sensors in nonuniform linear antenna arrays—part I: Fully augmentable arrays,” IEEE Trans. Signal Process., vol. 49, no. 5, pp. 959–971, May 2001.
[CrossRef]

Steinberg, V.

T. Burghelea, E. Segre, V. Steinberg, “Validity of the Taylor hypothesis in a random spatially smooth flow,” Phys. Fluids, vol. 17, no. 10, paper 103101, Oct. 2005.
[CrossRef]

Stell, M. F.

C. I. Moore, H. R. Barris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE, vol. 5793, pp. 78–88, 2005.
[CrossRef]

Strang, G.

G. Strang, Linear Algebra and Its Applications. 41st ed., Belmont, CA: Thomson, 2006.

Strohbehn, J. W.

J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere. New York, NY: Springer, 1978, chap. 3, pp. 45–106.
[CrossRef]

Suite, M. R.

C. I. Moore, H. R. Barris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE, vol. 5793, pp. 78–88, 2005.
[CrossRef]

Tarokh, V.

S. M. Haas, J. H. Shapiro, V. Tarokh, “Space-time codes for wireless optical communications,” EURASIP J. Appl. Signal Process., vol. 2002, no. 3, pp. 211–220, March 2002.
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation. Jerusalem, Israel: Program for Scientific Translations, 1971.

van Gorsel, E.

A. Christen, E. van Gorsel, R. Vogt, “Coherent structures in urban roughness sublayer turbulence,” Int. J. Climatol., vol. 27, no. 14, pp. 1955–1968, Oct. 2007.
[CrossRef]

van Loan, C. F.

G. H. Golub, C. F. van Loan, Matrix Computations. Baltimore, MD: John Hopkins Univ. Press, 1996.

Vasic, B. V.

Vinnichenko, N. K.

N. K. Vinnichenko, J. A. Dutton, “Empirical studies of atmospheric structure and spectra in the free atmosphere,” Radio Sci., vol. 4, no. 12, pp. 1115–1126, Dec. 1969.
[CrossRef]

Vogt, R.

A. Christen, E. van Gorsel, R. Vogt, “Coherent structures in urban roughness sublayer turbulence,” Int. J. Climatol., vol. 27, no. 14, pp. 1955–1968, Oct. 2007.
[CrossRef]

A. Christen, M. W. Rotach, R. Vogt, “Experimental determination of the turbulent kinetic energy budget within and above an urban canopy,” in Fifth Conf. on Urban Environment, Vancouver, Canada, Aug. 23–27, 2004, paper 6.4.

Wasiczko, L.

C. I. Moore, H. R. Barris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE, vol. 5793, pp. 78–88, 2005.
[CrossRef]

Weerackody, V.

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag., vol. 44, no. 11, pp. 46–51, Nov. 2006.
[CrossRef]

Wheelon, A. D.

A. D. Wheelon, Electromagnetic Scintillation: II. Weak Scattering. New York, NY: Cambridge Univ. Press, 2003.
[CrossRef]

Willis, G. E.

G. E. Willis, J. W. Deardorff, “On the use of Taylor’s hypothesis for diffusion in the mixed layer,” Q. J. R. Meteorol. Soc., vol. 102, no. 434, pp. 817–822, Oct. 1976.
[CrossRef]

Zhu, X.

X. Zhu, J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 50, no. 8, pp. 1293–1300, Aug. 2002.
[CrossRef]

Appl. Opt. (1)

Atmos. Environ. (1)

M. W. Rotach, “On the influence of the urban roughness sublayer on turbulence and dispersion,” Atmos. Environ., vol. 33, pp. 4001–4008, Oct. 1999.
[CrossRef]

Electron. Lett. (2)

A. Jurado Navas, A. García Zambrana, A. Puerta Notario, “Efficient lognormal channel model for turbulent FSO communications,” Electron. Lett., vol. 43, no. 3, pp. 178–179, Feb. 2007.
[CrossRef]

J. F. Monserrat, R. Fraile, L. Rubio, “Application of alternating projection method to ensure feasibility of shadowing cross-correlation models,” Electron. Lett., vol. 43, no. 13, pp. 724–725, June 2007.
[CrossRef]

EURASIP J. Appl. Signal Process. (1)

S. M. Haas, J. H. Shapiro, V. Tarokh, “Space-time codes for wireless optical communications,” EURASIP J. Appl. Signal Process., vol. 2002, no. 3, pp. 211–220, March 2002.
[CrossRef]

IEE Proc.-Commun. (1)

M. M. Ibrahim, A. M. Ibrahim, “Performance analysis of optical receivers with space diversity reception,” IEE Proc.-Commun., vol. 143, no. 6, pp. 369–372, Dec. 1996.
[CrossRef]

IEEE Commun. Lett. (2)

R. B. Ertel, J. H. Reed, “Generation of two equal power correlated Rayleigh fading envelopes,” IEEE Commun. Lett., vol. 2, pp. 276–278, Oct. 1998.
[CrossRef]

N. C. Beaulieu, “Generation of correlated Rayleigh fading envelopes,” IEEE Commun. Lett., vol. 3, no. 6, pp. 172–174, June 1999.
[CrossRef]

IEEE Commun. Mag. (1)

J. C. Juarez, A. Dwivedi, A. R. Hammons, S. D. Jones, V. Weerackody, R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag., vol. 44, no. 11, pp. 46–51, Nov. 2006.
[CrossRef]

IEEE J. Sel. Areas Commun. (1)

E. J. Lee, V. W. S. Chan, “Part I: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun., vol. 22, no. 9, pp. 1896–1906, Nov. 2004.
[CrossRef]

IEEE Trans. Commun. (1)

X. Zhu, J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 50, no. 8, pp. 1293–1300, Aug. 2002.
[CrossRef]

IEEE Trans. Signal Process. (1)

Y. I. Abramovich, N. K. Spencer, A. Y. Gorokhov, “Detection-estimation of more uncorrelated Gaussian sources than sensors in nonuniform linear antenna arrays—part I: Fully augmentable arrays,” IEEE Trans. Signal Process., vol. 49, no. 5, pp. 959–971, May 2001.
[CrossRef]

IEEE Trans. Wireless Commun. (1)

M. Razavi, J. H. Shapiro, “Wireless optical communications via diversity reception and optical preamplification,” IEEE Trans. Wireless Commun., vol. 4, no. 3, pp. 975–983, May 2005.
[CrossRef]

Indiana Univ. Math. J. (1)

P. R. Halmos, “Positive approximants of operators,” Indiana Univ. Math. J., vol. 21, no. 10, pp. 951–960, Oct. 1972.
[CrossRef]

Int. J. Climatol. (1)

A. Christen, E. van Gorsel, R. Vogt, “Coherent structures in urban roughness sublayer turbulence,” Int. J. Climatol., vol. 27, no. 14, pp. 1955–1968, Oct. 2007.
[CrossRef]

J. Atmos. Sol.-Terr. Phys. (2)

M. C. R. Kalapureddy, K. K. Kumar, V. Sivakumar, A. K. Ghosh, A. R. Jain, K. K. Reddy, “Diurnal and seasonal variability of TKE dissipation rate in the ABL over a tropical station using UHF wind profiler,” J. Atmos. Sol.-Terr. Phys., vol. 69, nos. 4–5, pp. 419–430, April 2007.
[CrossRef]

W. K. Hocking, P. K. L. Mu, “Upper and middle tropospheric kinetic energy dissipation rates from measurements of Cn2—review of theories, in-situ investigations, and experimental studies using the Buckland park atmospheric radar in Australia,” J. Atmos. Sol.-Terr. Phys., vol. 59, no. 14, pp. 1779–1803, Sept. 1997.
[CrossRef]

J. Opt. Soc. Am. (1)

Linear Algebra Its Appl. (1)

G. Alefeld, G. Mayer, “The Cholesky method for interval data,” Linear Algebra Its Appl., vol. 194, pp. 161–182, Nov. 1993.
[CrossRef]

Opt. Eng. (1)

M. Al Naboulsi, H. Sizun, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng., vol. 43, no. 2, pp. 319–329, Feb. 2004.
[CrossRef]

Opt. Lett. (1)

Phys. Fluids (3)

A. Belmonte, B. Martin, W. I. Goldburg, “Experimental study of Taylor’s hypothesis in a turbulent soap film,” Phys. Fluids, vol. 12, no. 4, pp. 835–845, April 2000.
[CrossRef]

T. Burghelea, E. Segre, V. Steinberg, “Validity of the Taylor hypothesis in a random spatially smooth flow,” Phys. Fluids, vol. 17, no. 10, paper 103101, Oct. 2005.
[CrossRef]

J. L. Lumley, “Interpretation of time spectra measured in high-intensity shear flows,” Phys. Fluids, vol. 8, no. 6 pp. 1056–1062, June 1965.
[CrossRef]

Proc. SPIE (1)

C. I. Moore, H. R. Barris, M. F. Stell, L. Wasiczko, M. R. Suite, R. Mahon, W. S. Rabinovich, G. C. Gilbreath, W. J. Scharpf, “Atmospheric turbulence studies of a 16 km maritime path,” Proc. SPIE, vol. 5793, pp. 78–88, 2005.
[CrossRef]

Q. J. R. Meteorol. Soc. (1)

G. E. Willis, J. W. Deardorff, “On the use of Taylor’s hypothesis for diffusion in the mixed layer,” Q. J. R. Meteorol. Soc., vol. 102, no. 434, pp. 817–822, Oct. 1976.
[CrossRef]

Radio Sci. (1)

N. K. Vinnichenko, J. A. Dutton, “Empirical studies of atmospheric structure and spectra in the free atmosphere,” Radio Sci., vol. 4, no. 12, pp. 1115–1126, Dec. 1969.
[CrossRef]

Waves Random Complex Media (1)

L. C. Andrews, R. L. Phillips, C. Y. Hopen, “Aperture averaging of optical scintillations power fluctuations and the temporal spectrum,” Waves Random Complex Media, vol. 10, no. 1, pp. 53–70, Jan. 2000.
[CrossRef]

Other (16)

L. C. Andrews, R. L. Phillips, C. Y. Hopen, Laser Beam, Scintillation With Applications. Bellingham, WA: SPIE, 2001.
[CrossRef]

A. D. Wheelon, Electromagnetic Scintillation: II. Weak Scattering. New York, NY: Cambridge Univ. Press, 2003.
[CrossRef]

S. M. Kay, Modern Spectral Estimation: Theory and Application. Englewood Cliffs, NJ: Prentice-Hall, 1988.

G. A. Holton, Value at Risk: Theory and Practice. San Diego, CA: Academic, 2004.

G. H. Golub, C. F. van Loan, Matrix Computations. Baltimore, MD: John Hopkins Univ. Press, 1996.

R. Hill, R. Frehlich, W. Otto, The Probability Distribution of Irradiance Scintillation, NOAA Technical Memorandum ERL ETL-274, Jan. 1997.

G. Strang, Linear Algebra and Its Applications. 41st ed., Belmont, CA: Thomson, 2006.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation. Jerusalem, Israel: Program for Scientific Translations, 1971.

U. Frisch, Turbulence. The Legacy of A. N. Kolmogorov. New York, NY: Cambridge Univ. Press, 1995.

T. Kyle, Atmospheric Transmission, Emission and Scattering. Oxford, UK: Pergamon, 1993.

A. Christen, M. W. Rotach, R. Vogt, “Experimental determination of the turbulent kinetic energy budget within and above an urban canopy,” in Fifth Conf. on Urban Environment, Vancouver, Canada, Aug. 23–27, 2004, paper 6.4.

J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere. New York, NY: Springer, 1978, chap. 3, pp. 45–106.
[CrossRef]

K. E. Baddour, N. C. Beaulieu, “Accurate simulation of multiple cross-correlated fading channels,” in IEEE Int. Conf. on Communications, 2002, New York, NY, April 28–May 2, 2002, vol. 1, pp. 267–271.

S. S. Muhammad, P. Kohldorfer, E. Leitgeb, “Channel modeling for terrestrial free space optical links,” in Proc. 2005 7th Int. Conf. on Transparent Optical Networks, Barcelona, Spain, July 3–7, 2005, vol. 1, pp. 407–410.

L. C. Andrews, R. L. Phillips, Laser Beam Propagation Through Random Media. Bellingham, WA: SPIE, 1998.

N. C. Beaulieu, M. L. Merani, “Efficient simulation of correlated diversity channels,” in IEEE Wireless Communications and Networking Conf., 2000. WCNC 2000, Chicago, IL, Sept. 23–28, 2000, vol. 1, pp. 207–210.

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Figures (6)

Fig. 1
Fig. 1

Block diagram representing the generation of m equal-power log-normal scintillation sequences using an AR model.

Fig. 2
Fig. 2

Scintillation coefficients, α sc ( t ) , generated from an AR model with σ χ 2 = 0.01 , u = 1 m s , ρ d 12 = 0.47 , and ρ d 13 = 5 × 10 6 .

Fig. 3
Fig. 3

Scintillation coefficients, α sc ( t ) , generated from an AR model with σ χ 2 = 0.01 , u = 1 m s , ρ d 12 = 0.47 , and ρ d 13 = 5 × 10 6 .

Fig. 4
Fig. 4

Normalized covariance function plotted as a function of τ scaled by the turbulence correlation time, τ 0 , for a plane wave with u = 1 m s and L = 2.5 km .

Fig. 5
Fig. 5

Block diagram representing the generation of m equal-power log-normal scintillation sequences using an AR model and including the dynamic evolution of atmospheric turbulence.

Fig. 6
Fig. 6

Two equal-power log-amplitude scintillation sequences, χ ( t ) , generated from Fig. 5, registered by two receivers displaced d 12 = 2 cm , where ρ l = 12 % . The laser wavelength is established at λ = 830 nm ; L = 2.5 km is the propagation path length, with σ χ 2 = 0.1 and u = 1 m s .

Equations (44)

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

u ( r ) = A ( r ) exp [ j φ ( r ) ] = u 0 ( r ) exp [ Φ 1 ] ,
u 0 ( r ) = A 0 ( r ) exp [ j φ 0 ( r ) ] ,
Φ 1 = log [ A ( r ) A 0 ( r ) ] + j [ φ ( r ) φ 0 ( r ) ] = χ + j S .
I ( t ) = I 0 exp [ 2 χ ( t ) ]
Y ( t ) = α sc ( t ) X ( t ) + N ( t ) ,
C χ = ( σ χ 2 σ χ 2 ρ d 12 σ χ 2 ρ d 1 m σ χ 2 ρ d 21 σ χ 2 σ χ 2 ρ d 2 m σ χ 2 ρ d m 1 σ χ 2 ρ d m 2 σ χ 2 ) m × m ,
A = σ I 2 ( D ) σ I 2 ( 0 ) = [ 1 + 1.062 ( k D 2 4 L ) ] 7 6 ,
C w = C χ [ 0 ] + k = 1 p C χ [ k ] A H [ k ] .
R z z = E { z [ n ] z T [ n ] } = I m ,
χ [ n ] = k = 1 p A [ k ] χ [ n k ] + w [ n ] .
Q T C w Q = Λ = diag ( λ 1 , , λ m ) ,
δ ( C w ) = min C ̃ w = C ̃ w T 0 C w C ̃ w ,
C w C ̃ w F = tr { ( C w C ̃ w ) T ( C w C ̃ w ) } ,
C w C ̃ w F = tr { ( Λ Y ) ( Λ Y ) T } = Λ Y F .
Λ Y F 2 = i j y 2 ( i , i ) + i = 1 m ( λ i y ( i , i ) ) 2 λ i < 0 ( λ i y ( i , i ) ) 2 λ i < 0 ( λ i ) 2 .
Y = Λ + Υ ,
υ i = { λ i , if λ i < 0 0 , if λ i 0 } i = 1 , 2 , , m .
C ̃ w = Q Y Q T
C w C ̃ w 2 = ρ { ( C w C ̃ w ) H ( C w C ̃ w ) } ,
ρ { ( C w C ̃ w ) T ( C w C ̃ w ) } = ρ { Υ T Υ } ,
C w = ( 2.173 × 10 12 3.031 × 10 13 1.790 × 10 13 3.804 × 10 13 4.921 × 10 12 3.031 × 10 13 5.076 × 10 14 3.673 × 10 14 9.848 × 10 14 9.395 × 10 13 1.790 × 10 13 3.673 × 10 14 2.830 × 10 14 8.608 × 10 14 8.205 × 10 13 3.804 × 10 13 9.848 × 10 14 8.608 × 10 14 2.466 × 10 13 3.002 × 10 12 4.921 × 10 12 9.395 × 10 13 8.205 × 10 13 3.002 × 10 12 9.964 × 10 11 ) 5 × 5 .
C ̃ w = ( 2.179 × 10 12 3.031 × 10 13 1.790 × 10 13 3.804 × 10 13 4.921 × 10 12 3.031 × 10 13 5.676 × 10 14 3.673 × 10 14 9.848 × 10 14 9.395 × 10 13 1.790 × 10 13 3.673 × 10 14 3.430 × 10 14 8.608 × 10 14 8.205 × 10 13 3.804 × 10 13 9.848 × 10 14 8.608 × 10 14 2.523 × 10 13 3.002 × 10 12 4.921 × 10 12 9.395 × 10 13 8.205 × 10 13 3.002 × 10 12 9.965 × 10 11 ) 5 × 5 .
C w = Q U Q T ,
K = Q ( Λ + Υ ) 1 2 ,
K K T = ( Q ( Λ + Υ ) 1 2 ) ( Q ( Λ + Υ ) 1 2 ) T = Q ( Λ + Υ ) 1 2 ( ( Λ + Υ ) 1 2 ) T Q T = Q ( Λ + Υ ) Q T = C ̃ ω ,
R Z Z = E { z [ n ] z T [ n ] } = I m ,
w [ n ] = [ w ( 1 ) [ n ] , w ( 2 ) [ n ] , , w ( m ) [ n ] ] T ,
E { w [ n ] × w T [ n ] } = E { K × z [ n ] × z [ n ] T × K T } = K K T = C ̃ ω ,
I ( r , t + τ ) = I ( r u τ , t ) ,
Δ τ 12 = | 1.01 0.97 | = 0.04 s ,
Δ τ 13 = | 1.01 0.85 | = 0.16 s ,
Δ τ 23 = | 0.97 0.85 | = 0.12 s .
Δ τ Taylor = d 12 u = 0.04 s ,
Δ τ Taylor = d 13 u = 0.16 s ,
Δ τ Taylor = d 23 u = 0.12 s .
τ e = ( λ L ) 2 3 ε 1 3 ,
u ( ε λ L ) 1 3 .
ρ l = τ 0 τ e = ( ε λ L ) 1 3 u < 1 ,
C ̃ χ = Q ( Λ + Υ ) Q T ,
E [ z corr [ n ] z corr T [ n ] ] = E [ K z [ n ] z T [ n ] K T ] = K K T = C ̃ χ .
χ [ n ] = ( 1 ρ l ) ( 1 ρ l ) 2 + ρ l 2 χ AR [ n ] + ρ l ( 1 ρ l ) 2 + ρ l 2 χ sep [ n ] ,
E [ χ AR [ n ] χ AR T [ n ] ] = E [ χ sep [ n ] χ sep T [ n ] ] = C χ .
E [ χ AR [ n ] χ sep T [ n ] ] = E [ χ sep [ n ] χ AR T [ n ] ] = 0 m ,
E [ χ [ n ] × χ T [ n ] ] = C χ .