Abstract

Fog is a highly dispersive medium at optical wavelengths, and the received pulse waveform may suffer significant distortion. Thus it is desirable to have the impulse response of the propagation channel to recover data transmitted through fog. The fog particle density and the particle size distribution both strongly influence the channel impulse response, yet it is difficult to estimate these parameters. We present a method using a dual-wavelength free-space optical system for estimating the average particle diameter and the particle number density and for approximating the particle distribution function. These parameters serve as inputs to estimate the atmospheric channel impulse response using simulation based on the modified vector radiative transfer theory. The estimated channel response is used to design a minimum mean-square-error equalization filter to improve the bit error rate by correcting distortion in the received signal waveform due to intersymbol interference and additive white Gaussian noise.

© 2008 Optical Society of America

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References

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  1. O. Bouchet, H. Sizun, C. Boisrobert, F. de Fornel, and P.-N. Favennec, Free-Space Optics: Propagation and Communication (ISTE, 2006).
    [CrossRef]
  2. S. Jaruwatanadilok, U. Ketprom, Y. Kuga, and A. Ishimaru, “Modeling the point-to-point wireless communication channel under adverse weather conditions,” IEICE Trans. Electron. E87-C, 1463-1466 (2004).
  3. U. Ketprom, Y. Kuga, S. Jaruwatanadilok, and A. Ishimaru, “Numerical studies on time-domain responses of on-off-keyed modulated optical signals through a dense fog,” Appl. Opt. 43, 496-505 (2004).
    [CrossRef] [PubMed]
  4. S. Chandrasekar, Radiative Transfer (Dover, 1960).
  5. L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
    [CrossRef]
  6. A. Ishimaru, Wave Propagation and Scattering in Random Media (Elsevier, 1978).
  7. G. Bal and M. Moscoso, “Theoretical and numerical analysis of polarization for time-dependent radiative transfer equations,” J. Quant. Spectrosc. Radiat. Transf. 70, 75-98(2001).
    [CrossRef]
  8. A. D. Kim, A. Ishimaru, and Y. Kuga, “Polarimetric pulse propagation through discrete random media,” Proc. SPIE , 3609, 101-110 (1999).
    [CrossRef]
  9. R. T. Cheung and A. Ishimaru, “Transmission, backscattering, and depolarization of waves in randomly distributed spherical particles,” Appl. Opt. 21, 3792-3798 (1982).
    [CrossRef] [PubMed]
  10. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  11. U. Ketprom, Y. Kuga, S. Jaruwatanadilok, A. Ishimaru, and J. A. Ritcey, “Channel modeling for optical wireless communication through dense fog,” J. Opt. Netw. 4, 291-299 (2005).
    [CrossRef]
  12. A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, “Polarized pulse waves in random discrete scatterers,” Appl. Opt. 40, 5495-5502 (2001).
    [CrossRef]
  13. L. Ma and R. K. Hanson, “Measurement of aerosol size distribution functions by wavelength-multiplexed laser extinction,” Appl. Phys. B 81, 567-576 (2005).
    [CrossRef]
  14. National Institute for Resources and Environment, “Measurement of size distribution of aerosol by bistatic system,” http://www.nire.go.jp/annual/1999/21.htm.
  15. B. Sklar, Digital Communications: Fundamentals and Applications, 2nd ed. (Prentice-Hall, 2002).
  16. J. R. Barry, E. A. Lee, and D. G. Messerschmitt, Digital Communication, 3rd ed. (Kluwer Academic, 2004).
  17. D. G. Manolakis, V. K. Ingle, and S. M. Kogon, Statistical and Adaptive Signal Processing: Spectral Estimation, Signal Modeling, Adaptive Filtering, and Array Processing (McGraw-Hill, 2000).
    [PubMed]
  18. J. A. Ritcey, “Electrical Engineering 507: Communications Theory II Course Lecture Notes” (University of Washington, 2004).

2006 (1)

O. Bouchet, H. Sizun, C. Boisrobert, F. de Fornel, and P.-N. Favennec, Free-Space Optics: Propagation and Communication (ISTE, 2006).
[CrossRef]

2005 (2)

L. Ma and R. K. Hanson, “Measurement of aerosol size distribution functions by wavelength-multiplexed laser extinction,” Appl. Phys. B 81, 567-576 (2005).
[CrossRef]

U. Ketprom, Y. Kuga, S. Jaruwatanadilok, A. Ishimaru, and J. A. Ritcey, “Channel modeling for optical wireless communication through dense fog,” J. Opt. Netw. 4, 291-299 (2005).
[CrossRef]

2004 (4)

U. Ketprom, Y. Kuga, S. Jaruwatanadilok, and A. Ishimaru, “Numerical studies on time-domain responses of on-off-keyed modulated optical signals through a dense fog,” Appl. Opt. 43, 496-505 (2004).
[CrossRef] [PubMed]

J. R. Barry, E. A. Lee, and D. G. Messerschmitt, Digital Communication, 3rd ed. (Kluwer Academic, 2004).

J. A. Ritcey, “Electrical Engineering 507: Communications Theory II Course Lecture Notes” (University of Washington, 2004).

S. Jaruwatanadilok, U. Ketprom, Y. Kuga, and A. Ishimaru, “Modeling the point-to-point wireless communication channel under adverse weather conditions,” IEICE Trans. Electron. E87-C, 1463-1466 (2004).

2002 (1)

B. Sklar, Digital Communications: Fundamentals and Applications, 2nd ed. (Prentice-Hall, 2002).

2001 (2)

G. Bal and M. Moscoso, “Theoretical and numerical analysis of polarization for time-dependent radiative transfer equations,” J. Quant. Spectrosc. Radiat. Transf. 70, 75-98(2001).
[CrossRef]

A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, “Polarized pulse waves in random discrete scatterers,” Appl. Opt. 40, 5495-5502 (2001).
[CrossRef]

2000 (2)

D. G. Manolakis, V. K. Ingle, and S. M. Kogon, Statistical and Adaptive Signal Processing: Spectral Estimation, Signal Modeling, Adaptive Filtering, and Array Processing (McGraw-Hill, 2000).
[PubMed]

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

1999 (1)

A. D. Kim, A. Ishimaru, and Y. Kuga, “Polarimetric pulse propagation through discrete random media,” Proc. SPIE , 3609, 101-110 (1999).
[CrossRef]

1982 (1)

1981 (1)

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

1978 (1)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Elsevier, 1978).

1960 (1)

S. Chandrasekar, Radiative Transfer (Dover, 1960).

Favennec, P.-N.

O. Bouchet, H. Sizun, C. Boisrobert, F. de Fornel, and P.-N. Favennec, Free-Space Optics: Propagation and Communication (ISTE, 2006).
[CrossRef]

Bal, G.

G. Bal and M. Moscoso, “Theoretical and numerical analysis of polarization for time-dependent radiative transfer equations,” J. Quant. Spectrosc. Radiat. Transf. 70, 75-98(2001).
[CrossRef]

Barry, J. R.

J. R. Barry, E. A. Lee, and D. G. Messerschmitt, Digital Communication, 3rd ed. (Kluwer Academic, 2004).

Boisrobert, C.

O. Bouchet, H. Sizun, C. Boisrobert, F. de Fornel, and P.-N. Favennec, Free-Space Optics: Propagation and Communication (ISTE, 2006).
[CrossRef]

Bouchet, O.

O. Bouchet, H. Sizun, C. Boisrobert, F. de Fornel, and P.-N. Favennec, Free-Space Optics: Propagation and Communication (ISTE, 2006).
[CrossRef]

Chandrasekar, S.

S. Chandrasekar, Radiative Transfer (Dover, 1960).

Cheung, R. T.

de Fornel, F.

O. Bouchet, H. Sizun, C. Boisrobert, F. de Fornel, and P.-N. Favennec, Free-Space Optics: Propagation and Communication (ISTE, 2006).
[CrossRef]

Ding, K.-H.

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

Hanson, R. K.

L. Ma and R. K. Hanson, “Measurement of aerosol size distribution functions by wavelength-multiplexed laser extinction,” Appl. Phys. B 81, 567-576 (2005).
[CrossRef]

Ingle, V. K.

D. G. Manolakis, V. K. Ingle, and S. M. Kogon, Statistical and Adaptive Signal Processing: Spectral Estimation, Signal Modeling, Adaptive Filtering, and Array Processing (McGraw-Hill, 2000).
[PubMed]

Ishimaru, A.

Jaruwatanadilok, S.

Ketprom, U.

Kim, A. D.

A. D. Kim, A. Ishimaru, and Y. Kuga, “Polarimetric pulse propagation through discrete random media,” Proc. SPIE , 3609, 101-110 (1999).
[CrossRef]

Kogon, S. M.

D. G. Manolakis, V. K. Ingle, and S. M. Kogon, Statistical and Adaptive Signal Processing: Spectral Estimation, Signal Modeling, Adaptive Filtering, and Array Processing (McGraw-Hill, 2000).
[PubMed]

Kong, J. A.

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

Kuga, Y.

Lee,

J. R. Barry, E. A. Lee, and D. G. Messerschmitt, Digital Communication, 3rd ed. (Kluwer Academic, 2004).

Ma, L.

L. Ma and R. K. Hanson, “Measurement of aerosol size distribution functions by wavelength-multiplexed laser extinction,” Appl. Phys. B 81, 567-576 (2005).
[CrossRef]

Manolakis, D. G.

D. G. Manolakis, V. K. Ingle, and S. M. Kogon, Statistical and Adaptive Signal Processing: Spectral Estimation, Signal Modeling, Adaptive Filtering, and Array Processing (McGraw-Hill, 2000).
[PubMed]

Messerschmitt, D. G.

J. R. Barry, E. A. Lee, and D. G. Messerschmitt, Digital Communication, 3rd ed. (Kluwer Academic, 2004).

Moscoso, M.

G. Bal and M. Moscoso, “Theoretical and numerical analysis of polarization for time-dependent radiative transfer equations,” J. Quant. Spectrosc. Radiat. Transf. 70, 75-98(2001).
[CrossRef]

Ritcey, J. A.

U. Ketprom, Y. Kuga, S. Jaruwatanadilok, A. Ishimaru, and J. A. Ritcey, “Channel modeling for optical wireless communication through dense fog,” J. Opt. Netw. 4, 291-299 (2005).
[CrossRef]

J. A. Ritcey, “Electrical Engineering 507: Communications Theory II Course Lecture Notes” (University of Washington, 2004).

Sizun, H.

O. Bouchet, H. Sizun, C. Boisrobert, F. de Fornel, and P.-N. Favennec, Free-Space Optics: Propagation and Communication (ISTE, 2006).
[CrossRef]

Sklar, B.

B. Sklar, Digital Communications: Fundamentals and Applications, 2nd ed. (Prentice-Hall, 2002).

Tsang, L.

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

Appl. Opt. (3)

Appl. Phys. B (1)

L. Ma and R. K. Hanson, “Measurement of aerosol size distribution functions by wavelength-multiplexed laser extinction,” Appl. Phys. B 81, 567-576 (2005).
[CrossRef]

IEICE Trans. Electron. (1)

S. Jaruwatanadilok, U. Ketprom, Y. Kuga, and A. Ishimaru, “Modeling the point-to-point wireless communication channel under adverse weather conditions,” IEICE Trans. Electron. E87-C, 1463-1466 (2004).

J. Opt. Netw. (1)

J. Quant. Spectrosc. Radiat. Transf. (1)

G. Bal and M. Moscoso, “Theoretical and numerical analysis of polarization for time-dependent radiative transfer equations,” J. Quant. Spectrosc. Radiat. Transf. 70, 75-98(2001).
[CrossRef]

Proc. SPIE (1)

A. D. Kim, A. Ishimaru, and Y. Kuga, “Polarimetric pulse propagation through discrete random media,” Proc. SPIE , 3609, 101-110 (1999).
[CrossRef]

Other (10)

O. Bouchet, H. Sizun, C. Boisrobert, F. de Fornel, and P.-N. Favennec, Free-Space Optics: Propagation and Communication (ISTE, 2006).
[CrossRef]

S. Chandrasekar, Radiative Transfer (Dover, 1960).

L. Tsang, J. A. Kong, and K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Elsevier, 1978).

National Institute for Resources and Environment, “Measurement of size distribution of aerosol by bistatic system,” http://www.nire.go.jp/annual/1999/21.htm.

B. Sklar, Digital Communications: Fundamentals and Applications, 2nd ed. (Prentice-Hall, 2002).

J. R. Barry, E. A. Lee, and D. G. Messerschmitt, Digital Communication, 3rd ed. (Kluwer Academic, 2004).

D. G. Manolakis, V. K. Ingle, and S. M. Kogon, Statistical and Adaptive Signal Processing: Spectral Estimation, Signal Modeling, Adaptive Filtering, and Array Processing (McGraw-Hill, 2000).
[PubMed]

J. A. Ritcey, “Electrical Engineering 507: Communications Theory II Course Lecture Notes” (University of Washington, 2004).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

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Figures (10)

Fig. 1
Fig. 1

Atmospheric channel impulse response for FOV = { 20 , 40 , 60 } mrad and (a)  τ 0 = 10 and (b)  τ 0 = 15 .

Fig. 2
Fig. 2

Simulated (a) transmitted and received signal waveforms for FOV = 40 mrad and (b)  τ 0 = 5 and (c)  τ 0 = 10 .

Fig. 3
Fig. 3

Relative significance of noise terms at optical depths (a)  τ 0 = 1 and (b)  τ 0 = 10 .

Fig. 4
Fig. 4

σ t ( 1.5 μm ) σ t ( 0.8 μm ) for (a) Gaussian and (b) lognormal particle size distributions.

Fig. 5
Fig. 5

Estimated channel responses based on Gaussian distributions at wavelengths of (a)  λ = 0.8 μm and (b)  λ = 1.5 μm .

Fig. 6
Fig. 6

Estimated channel responses based on lognormal distribution at wavelengths (a)  λ = 0.8 μm and (b)  λ = 1.5 μm .

Fig. 7
Fig. 7

Block diagram of basic FSO communications system.

Fig. 8
Fig. 8

Simulation of four-level transmitted, received, and MMSE-LE equalized signals.

Fig. 9
Fig. 9

Distributions of received signal amplitudes (a) without and (b) with equalization.

Fig. 10
Fig. 10

Bit-error-rate versus E b / N 0 for (a) OOK, (b) four-level, and (c) eight-level MMSE-LE (LN, lognormal distribution).

Tables (3)

Tables Icon

Table 1 Empirical Fog Particle Size Distribution n ( D )

Tables Icon

Table 2 Mean-Squared Error Performance for Gaussian Distribution Candidates

Tables Icon

Table 3 Mean-Squared Error Performance for Lognormal Distribution Candidates

Equations (26)

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( μ τ + 1 i ω τ 0 ) I ( ω , τ , μ , φ ) = 0 2 π 1 1 S ( μ , φ , μ , φ ) I ( ω , τ , μ , φ ) d μ d φ + F 0 ( τ , μ , φ ) exp ( τ ) for     0 τ τ 0 ,
I = [ I 1 = | E 1 | 2 I 2 = | E 2 | 2 U = 2 R e { E 1 E 2 * } V = 2 I m { E 1 E 2 * } ] T .
F 0 = S ( μ , ϕ , 1 , 0 ) I 0 ,
I 0 = [ 1 / 2 1 / 2 0 1 ] T .
I ri = I 0 f ( ω , τ ) exp ( τ ) δ ( ϕ ) δ ( μ 1 ) ,
I d ( τ = 0 ) = 0 for     0 μ 1 , I d ( τ = τ 0 ) = 0 for     1 μ 0.
L ( μ , μ ) = 0 2 π S ( μ , μ , ϕ ϕ ) d ( ϕ ϕ ) .
( μ τ + 1 + ( μ 1 ) i ω τ 0 ) I d ( ω , τ , μ ) = 1 1 L ( μ , μ ) I d ( ω , τ , μ ) d μ + F 0 ( μ ) exp ( τ ) for     0 τ τ 0 .
I d ( t , τ , μ ) = 1 2 π I d ( ω , τ , μ ) exp ( i ω τ / τ o i ω t ) d ω .
r ( t ) = y ( t ) + n ( t ) = x ( t ) * h ( t ) + n ( t ) ( continuous-time system ) , r ^ = x ^ T h ^ + n ^ ( equivalent discrete-time vector representation ) ,
I rcv I 0 exp [ τ total ] ,
τ total = τ abs + τ scat ,
I rcv = I 0 exp [ ( τ abs + τ scat ) ] = I 0 exp [ τ abs ] exp [ ρ σ scat L ] ,
I rcv _ 1.5 μm = I 0 exp [ τ 1.5 μm ] , I rcv _ 0.8 μm = I 0 exp [ τ 0.8 μm ] , τ 1.5 μm = ln [ I rcv _ 1.5 μm / I 0 ] , τ 0.8 μm = ln [ I rcv _ 0.8 μm / I 0 ] .
τ 1.5 μm τ 0.8 μm = ρ σ t _ 1.5 μm L ρ σ t _ 0.8 μm L = σ t _ 1.5 μm σ t _ 0.8 μm = ln [ I rcv _ 1.5 μm / I 0 ] ln [ I rcv _ 0.8 μm / I 0 ] .
ρ ( 1.5 μm ) ρ ( 0.8 μm ) = τ 1.5 μm / σ t _ 1.5 μm L τ 0.8 μm / σ t _ 0.8 μm L = τ 1.5 μm σ t _ 0.8 μm τ 0.8 μm σ t _ 1.5 μm .
ρ = τ 0 σ t ( D ^ avg ) L .
y ( k ) = h ( k ) * x ( k ) = n = 0 N 1 h ( n ) x ( k n ) , r ( k ) = y ( k ) + n ( k ) = h ( k ) * x ( k ) + n ( k ) , g ( k ) = w ( k ) * r ( k ) = w ( k ) * ( y ( k ) + n ( k ) ) , d ( k ) = b ( k ) * x ( k ) , e ( k ) = d ( k ) g ( k ) ,
y k = h x k , r k = y k + n k = h x k + n k , g k = w r k = w H ^ x k + w n k , d k = b x k ,
H ^ = [ h 0 h 1 h N 1 0 0 0 h 0 h 1 h N 1 0 0 0 0 0 h 0 h 1 h N 1 ] .
e ( k ) e k = d k g k = ( b w H ) x k w n k ,
e k 2 = e k e k = [ ( b w H ) x k w n k ] [ ( b w H ) x k w n k ] .
R x y ( τ ) = E [ x k y k τ ] .
MSE = E [ | e k | 2 ] = E [ e k e k ] = E [ ( d k g k ) ( d k g k ) ] = ( b w H ) R x x ( b H w ) + w R n n w .
( MSE ) w = ( b w H ) R x x H + w R n n = 0     w = b R x x H ( H R x x H + R n n ) 1 .
w = b Γ H ( I + Γ H H ) 1 .

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