Abstract

When used in an outdoor environment to expedite networking access, the performance of wireless optical communication systems is affected by transmitter sway. In the design of such systems, much attention has been paid to developing power-efficient schemes. However, the bandwidth efficiency is also an important issue. One of the most natural approaches to promote bandwidth efficiency is to use multilevel modulation. This leads to multilevel pulse amplitude modulation in the context of intensity modulation and direct detection. We develop a model based on the four-level pulse amplitude modulation. We show that the model can be formulated as an optimization problem in terms of the transmitter power, bit error probability, transmitter gain, and receiver gain. The technical challenges raised by modeling and solving the problem include the analytical and numerical treatments for the improper integrals of the Gaussian functions coupled with the erfc function. The results demonstrate that, at the optimal points, the power penalty paid to the doubled bandwidth efficiency is around 3dB.

© 2008 Optical Society of America

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References

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  1. S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun. 2, 626-629 (2003).
    [CrossRef]
  2. X. Liu, “The free-space optics system using QCL: models and solutions,” in IEEE International Conference on Communications (IEEE, 2007), pp. 2457-2461.
    [CrossRef]
  3. X. Liu, “Performance of the wireless optical communication system with variable wavelength and Bessel pointing loss factor,” in IEEE Wireless Communications and Networking Conference (IEEE, to be published).
  4. J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE 85, 265-298 (1997).
    [CrossRef]
  5. J. R. Barry, Wireless Infrared Communications (Kluwer Academic, 1994).
    [CrossRef]
  6. J. J. Degnan and B. J. Klein, “Optical antenna gain. 2: Receiving antennas,” Appl. Opt. 13, 2397-2401 (1974).
    [CrossRef] [PubMed]
  7. B. J. Klein and J. J. Degnan, “Optical antenna gain. 1: Transmitting antennas,” Appl. Opt. 13, 2134-2141 (1974).
    [CrossRef] [PubMed]
  8. J. C. Palais, Fiber Optic Communications, 5th ed. (Prentice-Hall, 2005).
  9. D.-S. Shiu and J. M. Kahn, “Shaping and nonequiprobable signaling for intensity-modulated signals,” IEEE Trans. Inf. Theory 45, 2661-2668 (1999).
    [CrossRef]
  10. S. Haykin, Digital Communications (Wiley, 1988).

2003 (1)

S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun. 2, 626-629 (2003).
[CrossRef]

1999 (1)

D.-S. Shiu and J. M. Kahn, “Shaping and nonequiprobable signaling for intensity-modulated signals,” IEEE Trans. Inf. Theory 45, 2661-2668 (1999).
[CrossRef]

1997 (1)

J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE 85, 265-298 (1997).
[CrossRef]

1974 (2)

Arnon, S.

S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun. 2, 626-629 (2003).
[CrossRef]

Barry, J. R.

J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE 85, 265-298 (1997).
[CrossRef]

J. R. Barry, Wireless Infrared Communications (Kluwer Academic, 1994).
[CrossRef]

D.-S. Shiu,

D.-S. Shiu and J. M. Kahn, “Shaping and nonequiprobable signaling for intensity-modulated signals,” IEEE Trans. Inf. Theory 45, 2661-2668 (1999).
[CrossRef]

Degnan, J. J.

Haykin, S.

S. Haykin, Digital Communications (Wiley, 1988).

Kahn, J. M.

D.-S. Shiu and J. M. Kahn, “Shaping and nonequiprobable signaling for intensity-modulated signals,” IEEE Trans. Inf. Theory 45, 2661-2668 (1999).
[CrossRef]

J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE 85, 265-298 (1997).
[CrossRef]

Klein, B. J.

Liu, X.

X. Liu, “Performance of the wireless optical communication system with variable wavelength and Bessel pointing loss factor,” in IEEE Wireless Communications and Networking Conference (IEEE, to be published).

X. Liu, “The free-space optics system using QCL: models and solutions,” in IEEE International Conference on Communications (IEEE, 2007), pp. 2457-2461.
[CrossRef]

Palais, J. C.

J. C. Palais, Fiber Optic Communications, 5th ed. (Prentice-Hall, 2005).

Appl. Opt. (2)

IEEE Trans. Inf. Theory (1)

D.-S. Shiu and J. M. Kahn, “Shaping and nonequiprobable signaling for intensity-modulated signals,” IEEE Trans. Inf. Theory 45, 2661-2668 (1999).
[CrossRef]

IEEE Trans. Wireless Commun. (1)

S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun. 2, 626-629 (2003).
[CrossRef]

Proc. IEEE (1)

J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE 85, 265-298 (1997).
[CrossRef]

Other (5)

J. R. Barry, Wireless Infrared Communications (Kluwer Academic, 1994).
[CrossRef]

J. C. Palais, Fiber Optic Communications, 5th ed. (Prentice-Hall, 2005).

X. Liu, “The free-space optics system using QCL: models and solutions,” in IEEE International Conference on Communications (IEEE, 2007), pp. 2457-2461.
[CrossRef]

X. Liu, “Performance of the wireless optical communication system with variable wavelength and Bessel pointing loss factor,” in IEEE Wireless Communications and Networking Conference (IEEE, to be published).

S. Haykin, Digital Communications (Wiley, 1988).

Cited By

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Figures (2)

Fig. 1
Fig. 1

Gray mapping between bits and symbols.

Fig. 2
Fig. 2

Profile of optimal solutions.

Equations (39)

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L T ( G T , θ ) = exp ( - G T θ 2 ) .
P R = P T G T G R ( λ 4 π d ) 2 η T η R L T ( G T , θ ) L R ( G R , γ ) ,
E c = μ σ T 2 = R P R 6 σ T 2 = R P R 6 σ T b = ( R η T η R G R 6 σ T b ) ( λ 4 π d ) 2 P T G T exp ( G T θ 2 ) = α P T G T exp ( G T θ 2 ) ,
α = ( R η T η R G R T b 6 σ ) ( λ 4 π d ) 2 .
E av = 1 4 ( 2 μ T + 4 μ T + 6 μ T ) = 3 μ T .
p ( r | s 0 ) = 1 σ 2 π exp ( - r 2 2 σ 2 ) ,
p ( r | s 1 ) = 1 σ 2 π exp [ - ( r - 2 μ T ) 2 2 σ 2 ] ,
p ( r | s 2 ) = 1 σ 2 π exp [ - ( r - 4 μ T ) 2 2 σ 2 ] ,
p ( r | s 3 ) = 1 σ 2 π exp [ - ( r - 6 μ T ) 2 2 σ 2 ] ,
P ( e | s 0 ) = 3 μ p ( r | s 0 ) d r = Q ( 3 μ T σ ) ,
P ( e | s 1 ) = 3 μ p ( r | s 1 ) d r = Q ( μ T σ ) ,
P ( e | s 2 ) = - 3 μ p ( r | s 2 ) d r = Q ( μ T σ ) ,
P ( e | s 3 ) = - 3 μ p ( r | s 3 ) d r = Q ( 3 μ T σ ) ,
Q ( z ) = 1 2 π z exp ( - t 2 / 2 ) d t .
P b 1 ( μ ) = 1 2 [ Q ( μ T σ ) + Q ( 3 μ T σ ) ] = 1 4 [ erfc ( μ σ T 2 ) + erfc ( 3 μ σ T 2 ) ] = 1 4 [ erfc ( E c ) + erfc ( 3 E c ) ] ,
P b 2 ( μ ) = 1 2 [ 2 Q ( μ T σ ) + Q ( 3 μ T σ ) - Q ( 5 μ T σ ) ] = 1 4 [ 2 erfc ( μ σ T 2 ) + erfc ( 3 μ σ T 2 ) - erfc ( 5 μ σ T 2 ) ] = 1 4 [ 2 erfc ( E c ) + erfc ( 3 E c ) - erfc ( 5 E c ) ] .
P b ( μ ) = 1 2 [ P b 1 ( μ ) + P b 2 ( μ ) ] = 1 8 [ 3 erfc ( E c ) + 2 erfc ( 3 E c ) - erfc ( 5 E c ) ] = k = 1 3 a k erfc ( b k E c ) ,
f ( θ ) = θ σ r 2 exp ( - θ 2 2 σ r 2 ) ( θ 0 ) .
w = 0 P b ( μ ( θ ) ) f ( θ ) d θ = 0 k = 1 3 a k erfc ( b k E c ) ( θ σ r 2 ) exp ( - θ 2 2 σ r 2 ) d θ = k = 1 3 a k 0 erfc [ b k α P T G T exp ( - G T θ 2 ) ] ( θ σ r 2 ) exp ( - θ 2 2 σ r 2 ) d θ = k = 1 3 a k 0 erfc [ b k α P T G T exp ( - 2 G T σ r 2 x ) ] exp ( - x ) d x ,
w = f w ( v , z ) = k = 1 3 a k 0 erfc [ b k v z exp ( - 2 z x ) ] exp ( - x ) d x .
minimize v = f v ( w , z ) ,
subject to g ( v , z ) = 0 ;
d v d z = - g / z g / v = - w / z w / v .
d d y erfc ( y ) = - 2 π exp ( - y 2 ) .
w v = - 2 z π k = 1 3 a k b k 0 exp [ - b k 2 v 2 z 2 exp ( - 4 x z ) - x - 2 x z ] d x = - 3 z 4 π 0 exp [ - v 2 z 2 exp ( - 4 x z ) - x - 2 x z ] d x - 3 z 2 π 0 exp [ - 9 v 2 z 2 exp ( - 4 x z ) - x - 2 x z ] d x + 5 z 4 π 0 exp [ - 25 v 2 z 2 exp ( - 4 x z ) - x - 2 x z ] d x .
exp [ - 9 v 2 z 2 exp ( - 4 x z ) - x - 2 x z ] > exp [ - 25 v 2 z 2 exp ( - 4 x z ) - x - 2 x z ] ,
3 2 exp [ - 9 v 2 z 2 exp ( - 4 x z ) - x - 2 x z ] > 5 4 exp [ - 25 v 2 z 2 exp ( - 4 x z ) - x - 2 x z ] .
w z = 2 v π k = 1 3 a k b k 0 exp [ - ( b k v z ) 2 exp ( - 4 z x ) - 2 z x - x ] ( 2 z x - 1 ) d x .
minimize w = f w ( v , z ) ,
subject to v = A ;
d w d z = 2 A π k = 1 3 a k b k 0 exp [ - ( A b k z ) 2 exp ( - 4 z x ) - 2 z x - x ] ( 2 z x - 1 ) d x ,
d 2 w d z 2 = 4 A π k = 1 3 a k b k 0 h ( x ; A , b k , z ) ( 1 - 2 z x ) [ 2 x + ( 1 + 2 z x ) A 2 b k 2 z exp ( - 4 z x ) ] d x ,
h ( x ; A , b k , z ) = exp [ - ( A b k z ) 2 exp ( - 4 z x ) - 2 z x - x ] .
w = f w ( v , z ) = 1 2 z ( 1 v z ) 1 2 z k = 1 3 a k b k - 1 / ( 2 z ) 0 b k v z erfc ( r ) r 1 / ( 2 z ) - 1 d r .
[ 0 , 2 μ 2 / R b , 4 μ 2 / R b , 6 μ 2 / R b ] .
p ( r | s 0 ) = 1 σ 2 π exp ( - r 2 2 σ 2 ) ,
p ( r | s 1 ) = 1 σ 2 π exp [ - ( r - 6 μ T ) 2 2 σ 2 ] .
P b ( μ ) = Q ( 3 μ T σ ) = 1 2 erfc ( 3 μ σ T 2 ) = 1 2 erfc ( 3 E c ) .
w = 0 P b ( μ ( θ ) ) f ( θ ) d θ = 1 2 0 erfc ( 3 E c ) ( θ σ r 2 ) exp ( - θ 2 2 σ r 2 ) d θ = 1 2 0 erfc [ ( 3 / 2 ) α P T G T exp ( - 2 G T σ r 2 x ) ] exp ( - x ) d x .

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