Abstract

Free-space optical (FSO) communication links provide high data rates; however, their reliability is heavily dependent on weather conditions. This paper presents our experimental urban 1.87 km FSO link based on a customized commercial system and develops a library of channel measurements in clear and light rain weather conditions. A channel model for the link is proposed and experimentally quantified. Channel measurements are obtained by modulating a 60 mW laser source. At the receiver, a 2 GSa/s data converter is used and 16 fast-Fourier transform cores are implemented in the hardware to improve noise immunity. The resulting signal-to-noise ratio of the channel samples is around 40 dB under clear weather conditions. Fittings with log-normal, gamma–gamma, and Erlang distributions are presented, and the scintillation index and coherence time are measured. A computationally efficient finite-state Markov chain is derived for the channel to model both the distribution and the autocorrelation of the fading and is verified by the measurements. The Markov models and channel measurements in a variety of atmospheric conditions are available for download to permit easy verification of communication algorithms on this urban FSO channel.

© 2012 OSA

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  25. V. Bhaskar, “Finite-state Markov model for lognormal, chi–square (central), chi–square (non-central), and K-distributions,” Int. J. Wireless Inf. Networks, vol. 14, no. 4, pp. 237–250, Oct.2007.
    [CrossRef]
  26. X. Zhu and J. M. Kahn, “Markov chain model in maximum-likelihood sequence detection for free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 51, no. 3, pp. 509–516, Mar.2003.
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2008 (1)

M. L. B. Riediger, R. Schober, and L. Lampe, “Multiple-symbol detection for photon-counting MIMO free-space optical communications,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 5369–5379, Dec.2008.
[CrossRef]

2007 (2)

F. S. Vetelino, C. Young, L. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence and spherical waves in the atmosphere,” J. Appl. Opt., vol. 46, no. 11, pp. 2099–2108, Apr.2007.
[CrossRef]

V. Bhaskar, “Finite-state Markov model for lognormal, chi–square (central), chi–square (non-central), and K-distributions,” Int. J. Wireless Inf. Networks, vol. 14, no. 4, pp. 237–250, Oct.2007.
[CrossRef]

2006 (1)

F. S. Vetelino, B. Clare, K. Corbett, C. Young, K. Grant, and L. Andrews, “Characterizing the propagation path in moderate to strong optical turbulence,” J. Appl. Opt., vol. 45, no. 15, pp. 3534–3543, May2006.
[CrossRef]

2003 (1)

X. Zhu and J. M. Kahn, “Markov chain model in maximum-likelihood sequence detection for free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 51, no. 3, pp. 509–516, Mar.2003.
[CrossRef]

2001 (1)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng., vol. 40, pp. 1554–1562, Aug.2001.
[CrossRef]

1999 (1)

Q. Zhang and S. Kassam, “Finite-state Markov model for Rayleigh fading channels,” IEEE Trans. Commun., vol. 47, no. 11, pp. 1688–1692, Nov.1999.
[CrossRef]

1998 (1)

I. Kim, J. Koontz, H. Hakakha, P. Adhikari, R. Stieger, C. Moursund, M. Barclay, A. Stanford, R. Ruigrok, J. Schuster, and E. Korevaar, “Measurement of scintillation and link margin for the terralink laser communication system,” Proc. SPIE, vol. 3232, pp. 100–118, Jan.1998.
[CrossRef]

1995 (1)

H. S. Wang and N. Moayeri, “Finite-state Markov channel - a useful model for radio communication channels,” IEEE Trans. Veh. Technol., vol. 44, no. 1, pp. 163–171, Feb.1995.
[CrossRef]

1965 (1)

M. E. Gracheva and A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the Earth,” Radiofiz., vol. 8, no. 4, pp. 717–724, 1965.

Adhikari, P.

I. Kim, J. Koontz, H. Hakakha, P. Adhikari, R. Stieger, C. Moursund, M. Barclay, A. Stanford, R. Ruigrok, J. Schuster, and E. Korevaar, “Measurement of scintillation and link margin for the terralink laser communication system,” Proc. SPIE, vol. 3232, pp. 100–118, Jan.1998.
[CrossRef]

Al-Habash, M. A.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng., vol. 40, pp. 1554–1562, Aug.2001.
[CrossRef]

Andrews, L.

F. S. Vetelino, C. Young, L. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence and spherical waves in the atmosphere,” J. Appl. Opt., vol. 46, no. 11, pp. 2099–2108, Apr.2007.
[CrossRef]

F. S. Vetelino, B. Clare, K. Corbett, C. Young, K. Grant, and L. Andrews, “Characterizing the propagation path in moderate to strong optical turbulence,” J. Appl. Opt., vol. 45, no. 15, pp. 3534–3543, May2006.
[CrossRef]

Andrews, L. C.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng., vol. 40, pp. 1554–1562, Aug.2001.
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed.SPIE, Bellingham, Washington, 2005.

Barclay, M.

I. Kim, J. Koontz, H. Hakakha, P. Adhikari, R. Stieger, C. Moursund, M. Barclay, A. Stanford, R. Ruigrok, J. Schuster, and E. Korevaar, “Measurement of scintillation and link margin for the terralink laser communication system,” Proc. SPIE, vol. 3232, pp. 100–118, Jan.1998.
[CrossRef]

Beleffi, G. M. T.

F. S. Marzano, S. Mori, F. Frezza, P. Nocito, G. M. T. Beleffi, G. Incerti, E. Restuccia, and F. Consalvi, “Free-space optical high-speed link in the urban area of southern Rome: preliminary experimental set up and channel modelling,” in Proc. of the 5th European Conf. on Antennas and Propagation (EUCAP), Apr. 2011, pp. 2737–2741.

Bhaskar, V.

V. Bhaskar, “Finite-state Markov model for lognormal, chi–square (central), chi–square (non-central), and K-distributions,” Int. J. Wireless Inf. Networks, vol. 14, no. 4, pp. 237–250, Oct.2007.
[CrossRef]

Bregenzer, J.

M. Gebhart, E. Leitgeb, and J. Bregenzer, “Atmospheric effects on optical wireless links,” in Proc. of the 7th Int. Conf. on Telecommunications (ConTEL), Zagreb, Croatia, June 2003, vol. 2, pp. 395–401.

Clare, B.

F. S. Vetelino, B. Clare, K. Corbett, C. Young, K. Grant, and L. Andrews, “Characterizing the propagation path in moderate to strong optical turbulence,” J. Appl. Opt., vol. 45, no. 15, pp. 3534–3543, May2006.
[CrossRef]

Consalvi, F.

F. S. Marzano, S. Mori, F. Frezza, P. Nocito, G. M. T. Beleffi, G. Incerti, E. Restuccia, and F. Consalvi, “Free-space optical high-speed link in the urban area of southern Rome: preliminary experimental set up and channel modelling,” in Proc. of the 5th European Conf. on Antennas and Propagation (EUCAP), Apr. 2011, pp. 2737–2741.

Corbett, K.

F. S. Vetelino, B. Clare, K. Corbett, C. Young, K. Grant, and L. Andrews, “Characterizing the propagation path in moderate to strong optical turbulence,” J. Appl. Opt., vol. 45, no. 15, pp. 3534–3543, May2006.
[CrossRef]

Cowley, W. G.

A. Khatoon, W. G. Cowley, and N. Letzepis, “Channel measurement and estimation for free space optical communications,” in AusCTW, Jan.–Feb. 2011.

Frezza, F.

F. S. Marzano, S. Mori, F. Frezza, P. Nocito, G. M. T. Beleffi, G. Incerti, E. Restuccia, and F. Consalvi, “Free-space optical high-speed link in the urban area of southern Rome: preliminary experimental set up and channel modelling,” in Proc. of the 5th European Conf. on Antennas and Propagation (EUCAP), Apr. 2011, pp. 2737–2741.

Gebhart, M.

M. Gebhart, E. Leitgeb, and J. Bregenzer, “Atmospheric effects on optical wireless links,” in Proc. of the 7th Int. Conf. on Telecommunications (ConTEL), Zagreb, Croatia, June 2003, vol. 2, pp. 395–401.

Ghuman, B. S.

H. Willebrand and B. S. Ghuman, Free Space Optics: Enabling Optical Connectivity in Today’s Networks. Sams Publishing, 2002.

Gracheva, M. E.

M. E. Gracheva and A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the Earth,” Radiofiz., vol. 8, no. 4, pp. 717–724, 1965.

Grant, K.

F. S. Vetelino, B. Clare, K. Corbett, C. Young, K. Grant, and L. Andrews, “Characterizing the propagation path in moderate to strong optical turbulence,” J. Appl. Opt., vol. 45, no. 15, pp. 3534–3543, May2006.
[CrossRef]

Gurvich, A. S.

M. E. Gracheva and A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the Earth,” Radiofiz., vol. 8, no. 4, pp. 717–724, 1965.

Hakakha, H.

I. Kim, J. Koontz, H. Hakakha, P. Adhikari, R. Stieger, C. Moursund, M. Barclay, A. Stanford, R. Ruigrok, J. Schuster, and E. Korevaar, “Measurement of scintillation and link margin for the terralink laser communication system,” Proc. SPIE, vol. 3232, pp. 100–118, Jan.1998.
[CrossRef]

Higashino, T.

K.-H. Kim, T. Higashino, K. Tsukamoto, and S. Komaki, “Optical fading analysis considering spectrum of optical scintillation in terrestrial free-space optical channel,” in Int. Conf. on Space Optical Systems and Applications, Santa Monica, CA, May 2011, pp. 58–66.

Incerti, G.

F. S. Marzano, S. Mori, F. Frezza, P. Nocito, G. M. T. Beleffi, G. Incerti, E. Restuccia, and F. Consalvi, “Free-space optical high-speed link in the urban area of southern Rome: preliminary experimental set up and channel modelling,” in Proc. of the 5th European Conf. on Antennas and Propagation (EUCAP), Apr. 2011, pp. 2737–2741.

Kahn, J. M.

X. Zhu and J. M. Kahn, “Markov chain model in maximum-likelihood sequence detection for free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 51, no. 3, pp. 509–516, Mar.2003.
[CrossRef]

Kassam, S.

Q. Zhang and S. Kassam, “Finite-state Markov model for Rayleigh fading channels,” IEEE Trans. Commun., vol. 47, no. 11, pp. 1688–1692, Nov.1999.
[CrossRef]

Kay, S. M.

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

Khatoon, A.

A. Khatoon, W. G. Cowley, and N. Letzepis, “Channel measurement and estimation for free space optical communications,” in AusCTW, Jan.–Feb. 2011.

Kim, I.

I. Kim, J. Koontz, H. Hakakha, P. Adhikari, R. Stieger, C. Moursund, M. Barclay, A. Stanford, R. Ruigrok, J. Schuster, and E. Korevaar, “Measurement of scintillation and link margin for the terralink laser communication system,” Proc. SPIE, vol. 3232, pp. 100–118, Jan.1998.
[CrossRef]

Kim, K.-H.

K.-H. Kim, T. Higashino, K. Tsukamoto, and S. Komaki, “Optical fading analysis considering spectrum of optical scintillation in terrestrial free-space optical channel,” in Int. Conf. on Space Optical Systems and Applications, Santa Monica, CA, May 2011, pp. 58–66.

Komaki, S.

K.-H. Kim, T. Higashino, K. Tsukamoto, and S. Komaki, “Optical fading analysis considering spectrum of optical scintillation in terrestrial free-space optical channel,” in Int. Conf. on Space Optical Systems and Applications, Santa Monica, CA, May 2011, pp. 58–66.

Koontz, J.

I. Kim, J. Koontz, H. Hakakha, P. Adhikari, R. Stieger, C. Moursund, M. Barclay, A. Stanford, R. Ruigrok, J. Schuster, and E. Korevaar, “Measurement of scintillation and link margin for the terralink laser communication system,” Proc. SPIE, vol. 3232, pp. 100–118, Jan.1998.
[CrossRef]

Korevaar, E.

I. Kim, J. Koontz, H. Hakakha, P. Adhikari, R. Stieger, C. Moursund, M. Barclay, A. Stanford, R. Ruigrok, J. Schuster, and E. Korevaar, “Measurement of scintillation and link margin for the terralink laser communication system,” Proc. SPIE, vol. 3232, pp. 100–118, Jan.1998.
[CrossRef]

Lampe, L.

M. L. B. Riediger, R. Schober, and L. Lampe, “Multiple-symbol detection for photon-counting MIMO free-space optical communications,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 5369–5379, Dec.2008.
[CrossRef]

Leitgeb, E.

M. Gebhart, E. Leitgeb, and J. Bregenzer, “Atmospheric effects on optical wireless links,” in Proc. of the 7th Int. Conf. on Telecommunications (ConTEL), Zagreb, Croatia, June 2003, vol. 2, pp. 395–401.

Letzepis, N.

A. Khatoon, W. G. Cowley, and N. Letzepis, “Channel measurement and estimation for free space optical communications,” in AusCTW, Jan.–Feb. 2011.

Marzano, F. S.

F. S. Marzano, S. Mori, F. Frezza, P. Nocito, G. M. T. Beleffi, G. Incerti, E. Restuccia, and F. Consalvi, “Free-space optical high-speed link in the urban area of southern Rome: preliminary experimental set up and channel modelling,” in Proc. of the 5th European Conf. on Antennas and Propagation (EUCAP), Apr. 2011, pp. 2737–2741.

Moayeri, N.

H. S. Wang and N. Moayeri, “Finite-state Markov channel - a useful model for radio communication channels,” IEEE Trans. Veh. Technol., vol. 44, no. 1, pp. 163–171, Feb.1995.
[CrossRef]

Mori, S.

F. S. Marzano, S. Mori, F. Frezza, P. Nocito, G. M. T. Beleffi, G. Incerti, E. Restuccia, and F. Consalvi, “Free-space optical high-speed link in the urban area of southern Rome: preliminary experimental set up and channel modelling,” in Proc. of the 5th European Conf. on Antennas and Propagation (EUCAP), Apr. 2011, pp. 2737–2741.

Moursund, C.

I. Kim, J. Koontz, H. Hakakha, P. Adhikari, R. Stieger, C. Moursund, M. Barclay, A. Stanford, R. Ruigrok, J. Schuster, and E. Korevaar, “Measurement of scintillation and link margin for the terralink laser communication system,” Proc. SPIE, vol. 3232, pp. 100–118, Jan.1998.
[CrossRef]

Nocito, P.

F. S. Marzano, S. Mori, F. Frezza, P. Nocito, G. M. T. Beleffi, G. Incerti, E. Restuccia, and F. Consalvi, “Free-space optical high-speed link in the urban area of southern Rome: preliminary experimental set up and channel modelling,” in Proc. of the 5th European Conf. on Antennas and Propagation (EUCAP), Apr. 2011, pp. 2737–2741.

Osche, G. R.

G. R. Osche, Optical Detection Theory for Laser Applications. Wiley-Interscience, 2002.

Phillips, R. L.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng., vol. 40, pp. 1554–1562, Aug.2001.
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed.SPIE, Bellingham, Washington, 2005.

Recolons, J.

F. S. Vetelino, C. Young, L. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence and spherical waves in the atmosphere,” J. Appl. Opt., vol. 46, no. 11, pp. 2099–2108, Apr.2007.
[CrossRef]

Restuccia, E.

F. S. Marzano, S. Mori, F. Frezza, P. Nocito, G. M. T. Beleffi, G. Incerti, E. Restuccia, and F. Consalvi, “Free-space optical high-speed link in the urban area of southern Rome: preliminary experimental set up and channel modelling,” in Proc. of the 5th European Conf. on Antennas and Propagation (EUCAP), Apr. 2011, pp. 2737–2741.

Riediger, M. L. B.

M. L. B. Riediger, R. Schober, and L. Lampe, “Multiple-symbol detection for photon-counting MIMO free-space optical communications,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 5369–5379, Dec.2008.
[CrossRef]

Ruigrok, R.

I. Kim, J. Koontz, H. Hakakha, P. Adhikari, R. Stieger, C. Moursund, M. Barclay, A. Stanford, R. Ruigrok, J. Schuster, and E. Korevaar, “Measurement of scintillation and link margin for the terralink laser communication system,” Proc. SPIE, vol. 3232, pp. 100–118, Jan.1998.
[CrossRef]

Schober, R.

M. L. B. Riediger, R. Schober, and L. Lampe, “Multiple-symbol detection for photon-counting MIMO free-space optical communications,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 5369–5379, Dec.2008.
[CrossRef]

Schuster, J.

I. Kim, J. Koontz, H. Hakakha, P. Adhikari, R. Stieger, C. Moursund, M. Barclay, A. Stanford, R. Ruigrok, J. Schuster, and E. Korevaar, “Measurement of scintillation and link margin for the terralink laser communication system,” Proc. SPIE, vol. 3232, pp. 100–118, Jan.1998.
[CrossRef]

Stanford, A.

I. Kim, J. Koontz, H. Hakakha, P. Adhikari, R. Stieger, C. Moursund, M. Barclay, A. Stanford, R. Ruigrok, J. Schuster, and E. Korevaar, “Measurement of scintillation and link margin for the terralink laser communication system,” Proc. SPIE, vol. 3232, pp. 100–118, Jan.1998.
[CrossRef]

Stieger, R.

I. Kim, J. Koontz, H. Hakakha, P. Adhikari, R. Stieger, C. Moursund, M. Barclay, A. Stanford, R. Ruigrok, J. Schuster, and E. Korevaar, “Measurement of scintillation and link margin for the terralink laser communication system,” Proc. SPIE, vol. 3232, pp. 100–118, Jan.1998.
[CrossRef]

Tsukamoto, K.

K.-H. Kim, T. Higashino, K. Tsukamoto, and S. Komaki, “Optical fading analysis considering spectrum of optical scintillation in terrestrial free-space optical channel,” in Int. Conf. on Space Optical Systems and Applications, Santa Monica, CA, May 2011, pp. 58–66.

Vetelino, F. S.

F. S. Vetelino, C. Young, L. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence and spherical waves in the atmosphere,” J. Appl. Opt., vol. 46, no. 11, pp. 2099–2108, Apr.2007.
[CrossRef]

F. S. Vetelino, B. Clare, K. Corbett, C. Young, K. Grant, and L. Andrews, “Characterizing the propagation path in moderate to strong optical turbulence,” J. Appl. Opt., vol. 45, no. 15, pp. 3534–3543, May2006.
[CrossRef]

Wang, H. S.

H. S. Wang and N. Moayeri, “Finite-state Markov channel - a useful model for radio communication channels,” IEEE Trans. Veh. Technol., vol. 44, no. 1, pp. 163–171, Feb.1995.
[CrossRef]

Willebrand, H.

H. Willebrand and B. S. Ghuman, Free Space Optics: Enabling Optical Connectivity in Today’s Networks. Sams Publishing, 2002.

Young, C.

F. S. Vetelino, C. Young, L. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence and spherical waves in the atmosphere,” J. Appl. Opt., vol. 46, no. 11, pp. 2099–2108, Apr.2007.
[CrossRef]

F. S. Vetelino, B. Clare, K. Corbett, C. Young, K. Grant, and L. Andrews, “Characterizing the propagation path in moderate to strong optical turbulence,” J. Appl. Opt., vol. 45, no. 15, pp. 3534–3543, May2006.
[CrossRef]

Zhang, Q.

Q. Zhang and S. Kassam, “Finite-state Markov model for Rayleigh fading channels,” IEEE Trans. Commun., vol. 47, no. 11, pp. 1688–1692, Nov.1999.
[CrossRef]

Zhu, X.

X. Zhu and J. M. Kahn, “Markov chain model in maximum-likelihood sequence detection for free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 51, no. 3, pp. 509–516, Mar.2003.
[CrossRef]

IEEE Trans. Commun. (2)

Q. Zhang and S. Kassam, “Finite-state Markov model for Rayleigh fading channels,” IEEE Trans. Commun., vol. 47, no. 11, pp. 1688–1692, Nov.1999.
[CrossRef]

X. Zhu and J. M. Kahn, “Markov chain model in maximum-likelihood sequence detection for free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 51, no. 3, pp. 509–516, Mar.2003.
[CrossRef]

IEEE Trans. Veh. Technol. (1)

H. S. Wang and N. Moayeri, “Finite-state Markov channel - a useful model for radio communication channels,” IEEE Trans. Veh. Technol., vol. 44, no. 1, pp. 163–171, Feb.1995.
[CrossRef]

IEEE Trans. Wireless Commun. (1)

M. L. B. Riediger, R. Schober, and L. Lampe, “Multiple-symbol detection for photon-counting MIMO free-space optical communications,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 5369–5379, Dec.2008.
[CrossRef]

Int. J. Wireless Inf. Networks (1)

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J. Appl. Opt. (2)

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Figures (14)

Fig. 1
Fig. 1

(Color online) Aerial view of the FSO link locations (MIP Lat.: 43 ° 1 5 23 . 7 5  N, Long.: 79 ° 5 4 2 . 0 6  W, BH Lat.: 43 ° 1 5 57 . 4 7  N, Long.: 79 ° 5 5 10 . 9 9  W) (satellite image © 2012 DigitalGlobe Inc. [11]).

Fig. 2
Fig. 2

(Color online) The experimental setup: (a) site-views, (b) block diagram.

Fig. 3
Fig. 3

Proposed channel model for the FSO link.

Fig. 4
Fig. 4

(Color online) Normalized magnitude frequency response of the FSO link.

Fig. 5
Fig. 5

(Color online) Thermal plus background noise samples at the receiver along with the Gaussian fit.

Fig. 6
Fig. 6

Block diagram of the measurement system used at the receiver.

Fig. 7
Fig. 7

(Color online) Measured channel fluctuations at different time scales: (a) 1 ms, (b) 10 ms, (c) 100 ms, (d) 1 s (October 30, 2011, 23:32 h, clear weather).

Fig. 8
Fig. 8

(Color online) Normalized autocovariance of the irradiance fluctuations. (October 30, 2011, 23:32 h, clear weather).

Fig. 9
Fig. 9

(Color online) (a) Average intensity and (b) scintillation index of the irradiance fluctuations during a 14 h duration (October 31, 2011, starting from 00:00 h until 14:00 h).

Fig. 10
Fig. 10

(Color online) Histogram of normalized channel gain h ˆ under a large scintillation index condition along with log-normal { σ ln } , gamma–gamma { α , β } , and Erlang { θ , λ } fits in clear weather conditions (samples measured on November 1, 2011, at 23:07 h).

Fig. 11
Fig. 11

(Color online) Histogram of normalized channel gain h ˆ under a small scintillation index condition along with log-normal { σ ln } , gamma–gamma { α , β } , and Erlang { θ , λ } fits in clear weather conditions (samples measured on November 2, 2011, at 09:18 h).

Fig. 12
Fig. 12

(Color online) Histogram of normalized channel gain h ˆ along with log-normal { σ ln } , gamma–gamma { α , β } , and Erlang { θ , λ } fits in light rain conditions (temperature: 1.9  ° C , wind speed: 24.46 km/h, rain level: 0.9 mm/h) during two measurement durations: (a) 1.29 s and (b) 165 s (samples measured on November 23, 2011, at 00:34 h).

Fig. 13
Fig. 13

(Color online) The histograms of the channel realizations obtained from the measurements and the realizations generated by the Markov model (samples measured on October 30, 2011, 23:32 h, clear weather).

Fig. 14
Fig. 14

(Color online) Normalized autocovariance of the channel realizations obtained from the measurements and generated by the Markov model.

Tables (2)

Tables Icon

Table I State-Transition Probability Matrix of the Markov Chain Model When K = 8

Tables Icon

Table II Comparison Between Measurements and the Markov Model

Equations (23)

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r ( t ) = R G h g h l h s c ( t ) [ f L D [ s ( t ) * h T x ( t ) ] * h R x ( t ) ] + w ( t ) ,
r ( t ) = R G h g h l h s c ( t ) [ s ( t ) * h T x ( t ) * h R x ( t ) ] + w ( t ) ,
r ( t ) = h ( t ) [ s ( t ) * h T x R x ( t ) ] + w ( t ) ,
h g D r x 2 ( D t x + θ t x L ) 2 D r x 2 ( θ t x L ) 2 , D t x θ t x L ,
37 dB  Power loss = 27 dB  Propagation loss h g + 1 dB  Atmospheric loss h l + 9 dB  Other losses .
h ̃ = h h = h 1 N i = 1 N h i ,
R h ̃ h ̃ ( m ) = { n = 1 N m h ̃ n + m h ̃ n   N 1 m 0 R h ̃ h ̃ ( m )   N + 1 m < 0 .
R ˆ h ̃ h ̃ ( m ) = R h ̃ h ̃ ( m ) R h ̃ h ̃ ( 0 ) .
R ˆ h ̃ h ̃ ( T c ) = 1 / e ,
SI = h 2 h 2 h 2 ,
h 2 h 2 = 1 N i = 1 N ( h i h ) 2 and h = 1 N i = 1 N h i ,
p h ˆ , LN ( h ˆ ) = 1 2 π σ ln h ˆ h ˆ exp ( ( ln h ˆ + 1 2 σ ln h ˆ 2 ) 2 2 σ ln h ˆ 2 ) , h ˆ > 0 , h ˆ = 1 ,
SI LN = exp ( σ ln h ˆ 2 ) 1 .
p h ˆ , GG ( h ˆ ) = 2 ( α β ) α + β 2 Γ ( α ) Γ ( β ) h ˆ ( α + β ) 2 1 K α β ( 2 α β h ˆ ) , h ˆ > 0 , h ˆ = 1 ,
SI GG = 1 α + 1 β + 1 α β .
p h ˆ , ER ( h ˆ ) = λ θ ( θ 1 ) ! h ˆ θ 1 e λ h ˆ , h ˆ > 0 , λ > 0 , θ { 1 , 2 , 3 , } .
i = 1 N ln { 1 2 π σ ln h ˆ 2 h i ˆ exp ( ( ln h i ˆ + 1 2 σ ln h ˆ 2 ) 2 2 σ ln h ˆ 2 ) } .
σ ̄ ln h ˆ 2 = 1 N i = 1 N ( ln h ˆ i 1 N i = 1 N ln h ˆ i ) 2 ,
i = 1 N ln { λ θ * ( θ * 1 ) ! h ˆ i θ * 1 e λ h ˆ i }
λ * = N θ * i = 1 N h ˆ i = θ * .
Δ = h max h min K ,
h min + k Δ h < h min + ( k + 1 ) Δ
t j , k = Pr ( S n + 1 = s k | S n = s j ) = N j , k l = 0 K 1 N l , j = N j , k N j , j , k { 0 , 1 , 2 , , K 1 } ,