## Impact of pointing errors on the performance of generalized atmospheric optical channels |

Optics Express, Vol. 20, Issue 11, pp. 12550-12562 (2012)

http://dx.doi.org/10.1364/OE.20.012550

Acrobat PDF (880 KB)

### Abstract

Recently, a new and generalized statistical model, called ℳ or Málaga distribution, was proposed to model the irradiance fluctuations of an unbounded optical wavefront (plane and spherical waves) propagating through a turbulent medium under all irradiance fluctuation conditions in homogeneous, isotropic turbulence. Málaga distribution was demonstrated to have the advantage of unifying most of the proposed statistical models derived until now in the bibliography in a closed-form expression providing, in addition, an excellent agreement with published plane wave and spherical wave simulation data over a wide range of turbulence conditions (weak to strong). Now, such a model is completed by including the adverse effect of pointing error losses due to misalignment. In this respect, the well-known effects of aperture size, beam width and jitter variance are taken into account. Accordingly, after presenting the analytical expressions for the combined distribution of scintillation and pointing errors, we derive its centered moments of the overall probability distribution. Finally, we obtain the analytical expressions for the average bit error rate performance for the ℳ distribution affected by pointing errors. Numerical results show the impact of misalignment on link performance.

© 2012 OSA

## 1. Introduction

2. X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. **50**, 1293–1300 (2002). [CrossRef]

3. 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. **40**, 1554–1562 (2001). [CrossRef]

4. J. H. Churnside and S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A **4**, 1923–1930 (1987). [CrossRef]

5. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: *Numerical Simulations of Physical and Engineering Processes* (Intech, 2011). [CrossRef]

3. 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. **40**, 1554–1562 (2001). [CrossRef]

6. A. Abdi, W. C. Lau, M. S. Alouini, and M. A. Kaveh, “A new simple model for land mobile satellite channels: first- and second-order statistics,” IEEE Trans. Wireless Commun. **2**, 519–528 (2003). [CrossRef]

3. 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. **40**, 1554–1562 (2001). [CrossRef]

7. E. Jakerman, “On the statistics of K-distributed noise,” J. Phys. A **13**, 31–48 (1980). [CrossRef]

7. E. Jakerman, “On the statistics of K-distributed noise,” J. Phys. A **13**, 31–48 (1980). [CrossRef]

**40**, 1554–1562 (2001). [CrossRef]

5. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: *Numerical Simulations of Physical and Engineering Processes* (Intech, 2011). [CrossRef]

8. S. Arnon and N. S. Kopeika, “Laser satellite communication network-vibration effect and possible solutions,” Proc. IEEE **85**, 1646–1661 (1997). [CrossRef]

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

10. K. Kiasaleh, “On the probability density function of signal intensity in free-space optical communications systems impaired by pointing jitter and turbulence,” Opt. Eng. **33**, 3748–3757 (1994). [CrossRef]

11. A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. **25**, 1702–1710 (2007). [CrossRef]

12. H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. **12**, 44–46 (2008). [CrossRef]

12. H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. **12**, 44–46 (2008). [CrossRef]

5. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: *Numerical Simulations of Physical and Engineering Processes* (Intech, 2011). [CrossRef]

13. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett. **36**, 4095–4097 (2011). [CrossRef] [PubMed]

*Numerical Simulations of Physical and Engineering Processes* (Intech, 2011). [CrossRef]

## 2. System model

11. A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. **25**, 1702–1710 (2007). [CrossRef]

## 3. Generalized atmospheric distribution model

*h*introduced in Section 2. The starting point is the new statistical model for the irradiance fluctuations of an unbounded optical wavefront propagating through a turbulent medium under all irradiance fluctuation conditions in homogeneous, isotropic turbulence that was presented in [5

_{a}*Numerical Simulations of Physical and Engineering Processes* (Intech, 2011). [CrossRef]

4. J. H. Churnside and S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A **4**, 1923–1930 (1987). [CrossRef]

*Numerical Simulations of Physical and Engineering Processes* (Intech, 2011). [CrossRef]

*U*, the second one is the component which is quasi-forward scattered by the eddies on the propagation axis,

_{L}*h*can be written as: where

*U*and

_{L}*G*is a real variable following a gamma distribution with

*E*[

*G*]=1. It represents the slow fluctuation of the LOS component. Following the notation of [6

6. A. Abdi, W. C. Lau, M. S. Alouini, and M. A. Kaveh, “A new simple model for land mobile satellite channels: first- and second-order statistics,” IEEE Trans. Wireless Commun. **2**, 519–528 (2003). [CrossRef]

*E*[|

*U*|

_{L}^{2}] represents the average power of the LOS term whereas the average power of the total scatter components is denoted by

*ϕ*and

_{A}*ϕ*are the deterministic phases of the LOS and the coupled-to-LOS scatter terms, respectively. On another note, 0≤

_{B}*ρ*≤1 is the factor expressing the amount of scattering power coupled to the LOS component. Finally,

*U*′

*is a circular Gaussian complex random variable, and*

_{S}*χ*and

*S*are real random variables representing the log-amplitude and phase perturbation of the field induced by the atmospheric turbulence, respectively. A plausible justification for the coupled-to-LOS scattering component,

16. R. S. Kennedy, “Communication through optical scattering channels: an introduction,” Proc. IEEE **58**, 1651–1665 (1970). [CrossRef]

*U*in our proposed model. Of course, when the turbulent medium becomes so thick, then the unscattered component of the field can be neglected.

_{L}*h*, as we have adopted the notation given in [11

_{a}11. A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. **25**, 1702–1710 (2007). [CrossRef]

6. A. Abdi, W. C. Lau, M. S. Alouini, and M. A. Kaveh, “A new simple model for land mobile satellite channels: first- and second-order statistics,” IEEE Trans. Wireless Commun. **2**, 519–528 (2003). [CrossRef]

*U*′

*) and a Nakagami distribution,*

_{S}**2**, 519–528 (2003). [CrossRef]

*Y*is given by: where

*β*≜ (

*E*[

*G*])

^{2}/Var[

*G*] is the amount of fading parameter with Var[·] being the variance operator, where | · | denotes the absolute value. For simplicity, we have denoted

*Numerical Simulations of Physical and Engineering Processes* (Intech, 2011). [CrossRef]

_{1}

*F*

_{1}(

*a; c; x*) is the Kummer confluent hypergeometric function of the first kind.

*h*, the mixture of

_{a}*f*(

_{X}*x*) and

*f*(

_{Y}*y*) must be accomplished. After the corresponding and non-obvious algebraic manipulation, extensively detailed in [5

*Numerical Simulations of Physical and Engineering Processes* (Intech, 2011). [CrossRef]

*β*being a natural number. In Eq. (7),

*K*(·) is the modified Bessel function of the second kind and order

_{ν}*ν*.

*Numerical Simulations of Physical and Engineering Processes* (Intech, 2011). [CrossRef]

*β*being a real number: where

## 4. Misalignment fading model

*h*(

*t*) is considered through this paper to be a product of two random components: the scintillation,

*h*, detailed in Section 3, and geometric spread and pointing errors, denoted by

_{a}*h*. This last factor is now analytically detailed through this section. Thus, in atmospheric optical links, pointing accuracy is an important issue in determining link performance. For this reason, Farid and Hranilovic [11

_{p}**25**, 1702–1710 (2007). [CrossRef]

*f*

_{hp}(

*h*), assuming a Gaussian spatial intensity profile of beam waist,

_{p}*w*, at distance

_{z}*z*from the transmitter: where

**is the radial vector from the beam center. In addition, a circular detection aperture of radius**

*ρ**a*was considered and a Gaussian beam profile at the receiver,

*I*

_{beam}. Then, the fraction of the collected power at a receiver of radius

*a*in the transverse plane of the incident wave when a pointing error,

**r**, is present can be approximated as the Gaussian form: where

*r*= ||

**r**|| is the radial distance. In particular, due to the symmetry of the beam shape and the detector area,

*h*depends only on this magnitude,

_{p}*r*. On another note,

*A*

_{0}= [erf(

*v*)]

^{2}is the fraction of the collected power at

*r*= 0, where

*r*, at the receiver follows a Rayleigh distribution. where

*h*is given by: where

_{p}*g*=

*w*/(2

_{zeq}*σ*) is the ratio between the equivalent beam radius at the receiver,

_{s}*w*, and the pointing error displacement standard deviation at the receiver,

_{zeq}*σ*. See [11

_{s}**25**, 1702–1710 (2007). [CrossRef]

## 5. Combined channel statistical model

*f*(

_{h}*h*), for the channel state,

*h*, is obtained by calculating the mixture of the two distributions presented above in Eqs. (7) or (9) and (15): where

*f*

_{h|ha}(

*h*|

*h*) is the conditional probability given a turbulence state,

_{a}*h*, and it is expressed as: and where

_{a}*f*(

_{ha}*h*) was written in Eqs. (7) and (9) depending on whether parameter

_{a}*β*is a natural number or a real one, respectively. For the first case, we substitute Eq. (7) into the integral expression given by Eq. (16) to finally obtain

*f*(

_{h}*h*): where the parameters

*A*and

*a*were already defined in Eq. (8). From [19

_{k}19. Wolfram, http://functions.wolfram.com/

*K*(·) can be expressed as a special case of the Meijer G function, given by the following relationship: where a definition of the Meijer G function can be found at [19

_{v}19. Wolfram, http://functions.wolfram.com/

*a*= −

*b*= (

*α*−

*k*)/2 in Eq. (19) and substituting such an equation into Eq. (18), then:

19. Wolfram, http://functions.wolfram.com/

*f*(

_{h}*h*), for the channel state,

*h*, can be written in a closed-form expression: just after having simplified the G-Meijer function using the relationship gathered in [19

19. Wolfram, http://functions.wolfram.com/

*β*being a real number, the obtention of

*f*(

_{h}*h*) is not so evident because it is firstly mandatory to verify if the infinite summation of Eq. (9) and the integrate operator that appears when building

*f*(

_{h}*h*) can be interchanged. To prove that, by normalization, it is possible to express Eq. (9) as a discrete mixture [20

20. C. A. Charalambides, *Combinatorial Methods in Discrete Distributions* (John Wiley & Sons, 2005) [CrossRef]

*f*(·) being a Gamma-Gamma pdf. Since the cumulative distribution function (CDF) can be written as

_{k}*F*(

_{k}*h*) is the CDF of each

_{a}*f*(·), then we can build a new CDF in the form:

_{k}*F*(

_{k}*h*) converges absolutely so

_{a}*ϕ*(

_{n}*h*) is said to converge weakly to

_{a}*F*(

*h*). This condition and the fact that Eq. (17) is bounded and continuous allow us to apply the Helly-Bray theorem [22]. By proceeding in an analogous manner as in [13

_{a}13. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett. **36**, 4095–4097 (2011). [CrossRef] [PubMed]

*f*(

_{h}*h*) is written as an infinite series of Meijer-G functions as

### 5.1. Moments of the combined ℳ probability distribution

*k*centered moments of

^{th}*h*, denoted by

_{a}*m*(

_{k}*h*), following an ℳ distribution was derived in [5

_{a}*Numerical Simulations of Physical and Engineering Processes* (Intech, 2011). [CrossRef]

*X*and

*Y*, then: whereas the operator

*E*[·] stands for ensemble average. From [23, Eq. (2.23)], the moment of a Nakagami-m pdf is given by:

*β*being a natural number was derived in [5

*Numerical Simulations of Physical and Engineering Processes* (Intech, 2011). [CrossRef]

*Numerical Simulations of Physical and Engineering Processes* (Intech, 2011). [CrossRef]

*β*being a real number, the moment of a shadowed-Rician distribution was given in [23, Eq. (2.69)]: and, thus, Eq. (27) is now generalized as:

**25**, 1702–1710 (2007). [CrossRef]

*h*, we can obtain the

_{a}*k*centered moments of the channel state,

^{th}*h*, as

## 6. Average BER

19. Wolfram, http://functions.wolfram.com/

## 7. Numerical results and discussions

12. H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. **12**, 44–46 (2008). [CrossRef]

13. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett. **36**, 4095–4097 (2011). [CrossRef] [PubMed]

24. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, *Laser Beam Scintillation with Applications* (SPIE, 2001). [CrossRef]

24. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, *Laser Beam Scintillation with Applications* (SPIE, 2001). [CrossRef]

**25**, 1702–1710 (2007). [CrossRef]

*k*= 2

*π*/

*λ*is the optical wave number,

*λ*(m) is the wavelength, and

*L*(m) is the propagation path length between transmitter and receiver. To obtain typical values of

^{−14}m

^{−2/3}, and the temperature oscillated from 22 °C to 26.4 °C, and the received power was recorded from −22.52 dBm to −20.48 dBm. At night (near 1 a.m.),

^{−15}m

^{−2/3}), for a temperature of 15.1 °C and a received power of −22.91 dBm. At sunrise (6.45 a.m.),

*L*= 1 km at a height of 25 m. The optical power transmitted is 11.5 dBm with a responsivity of 0.8 A/W. With such a set of parameters, if we consider

^{−15}m

^{−2/3}(night), then

*b*

_{0}= 1).

*w*/

_{z}*a*=10, 20, and 25. Parameters of ℳ turbulence model were fixed to:

*α*= 10 and

*β*= 5 in all cases. Moreover, for each

*w*/

_{z}*a*value, three different behaviors were obtained:

*ρ*= 1, which corresponds to a Gamma-Gamma distribution;

*ρ*= 0.75, and

*ρ*= 0.25. For such values, the associated Rytov variance is

**36**, 4095–4097 (2011). [CrossRef] [PubMed]

*w*/

_{z}*a*= 10, 20 and 25, respectively, for a bit error rate of 10

^{−6}.

25. L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple antenna channels,” IEEE Trans. Inf. Theory **49**, 1073–1096 (2003). [CrossRef]

*β*is restricted to a natural number. Nevertheless, due to the high degree of freedom inherent to the ℳ distribution, demonstrated in the set of results shown in [5

*Numerical Simulations of Physical and Engineering Processes* (Intech, 2011). [CrossRef]

## 8. Concluding remarks

*Numerical Simulations of Physical and Engineering Processes* (Intech, 2011). [CrossRef]

*w*/

_{z}*a*is fixed, and for a given value of bit error rate.

## Acknowledgment

## References and links

1. | L. C. Andrews and R. L. Phillips, |

2. | X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. |

3. | 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. |

4. | J. H. Churnside and S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A |

5. | A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: |

6. | A. Abdi, W. C. Lau, M. S. Alouini, and M. A. Kaveh, “A new simple model for land mobile satellite channels: first- and second-order statistics,” IEEE Trans. Wireless Commun. |

7. | E. Jakerman, “On the statistics of K-distributed noise,” J. Phys. A |

8. | S. Arnon and N. S. Kopeika, “Laser satellite communication network-vibration effect and possible solutions,” Proc. IEEE |

9. | S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun. |

10. | K. Kiasaleh, “On the probability density function of signal intensity in free-space optical communications systems impaired by pointing jitter and turbulence,” Opt. Eng. |

11. | A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. |

12. | H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett. |

13. | A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett. |

14. | M. Al Naboulsi and H. Sizun, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng. |

15. | ITU-R Report F.2106-1 “Fixed service applications using free-space optical links,” Nov.2010. |

16. | R. S. Kennedy, “Communication through optical scattering channels: an introduction,” Proc. IEEE |

17. | T. Rappaport, |

18. | J. R. Clark and S. Karp, “Approximations for lognormally fading optical signals,” Proc. IEEE |

19. | Wolfram, http://functions.wolfram.com/ |

20. | C. A. Charalambides, |

21. | L. C. Andrews, |

22. | P. Billingsley, |

23. | M. K. Simon and M. S. Alouini, |

24. | L. C. Andrews, R. L. Phillips, and C. Y. Hopen, |

25. | L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple antenna channels,” IEEE Trans. Inf. Theory |

**OCIS Codes**

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(290.5930) Scattering : Scintillation

(060.2605) Fiber optics and optical communications : Free-space optical communication

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: March 9, 2012

Revised Manuscript: May 10, 2012

Manuscript Accepted: May 11, 2012

Published: May 18, 2012

**Citation**

Antonio Jurado-Navas, José María Garrido-Balsells, José Francisco Paris, Miguel Castillo-Vázquez, and Antonio Puerta-Notario, "Impact of pointing errors on the performance of generalized atmospheric optical channels," Opt. Express **20**, 12550-12562 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-11-12550

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### References

- L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 1998).
- X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun.50, 1293–1300 (2002). [CrossRef]
- 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.40, 1554–1562 (2001). [CrossRef]
- J. H. Churnside and S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A4, 1923–1930 (1987). [CrossRef]
- A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in: Numerical Simulations of Physical and Engineering Processes (Intech, 2011). [CrossRef]
- A. Abdi, W. C. Lau, M. S. Alouini, and M. A. Kaveh, “A new simple model for land mobile satellite channels: first- and second-order statistics,” IEEE Trans. Wireless Commun.2, 519–528 (2003). [CrossRef]
- E. Jakerman, “On the statistics of K-distributed noise,” J. Phys. A13, 31–48 (1980). [CrossRef]
- S. Arnon and N. S. Kopeika, “Laser satellite communication network-vibration effect and possible solutions,” Proc. IEEE85, 1646–1661 (1997). [CrossRef]
- S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun.2, 626–629 (2003). [CrossRef]
- K. Kiasaleh, “On the probability density function of signal intensity in free-space optical communications systems impaired by pointing jitter and turbulence,” Opt. Eng.33, 3748–3757 (1994). [CrossRef]
- A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol.25, 1702–1710 (2007). [CrossRef]
- H. G. Sandalidis, T. A. Tsiftsis, G. K. Karagiannidis, and M. Uysal, “BER performance of FSO links over strong atmospheric turbulence channels with pointing errors,” IEEE Commun. Lett.12, 44–46 (2008). [CrossRef]
- A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett.36, 4095–4097 (2011). [CrossRef] [PubMed]
- M. Al Naboulsi and H. Sizun, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng.43, 319–329 (2004). [CrossRef]
- ITU-R Report F.2106-1 “Fixed service applications using free-space optical links,” Nov.2010.
- R. S. Kennedy, “Communication through optical scattering channels: an introduction,” Proc. IEEE58, 1651–1665 (1970). [CrossRef]
- T. Rappaport, Wireless Communications: Principles and Practice, 2nd ed. (Prentice Hall, 2001.).
- J. R. Clark and S. Karp, “Approximations for lognormally fading optical signals,” Proc. IEEE58, 1964–1965 (1970). [CrossRef]
- Wolfram, http://functions.wolfram.com/
- C. A. Charalambides, Combinatorial Methods in Discrete Distributions (John Wiley & Sons, 2005) [CrossRef]
- L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE, 1998).
- P. Billingsley, Convergence of Probability Measures, 2nd ed. (John Wiley & Sons, 2005)
- M. K. Simon and M. S. Alouini, Digital Communications over Fading Channels, 2nd ed. (Wiley-Interscience, 2005).
- L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001). [CrossRef]
- L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple antenna channels,” IEEE Trans. Inf. Theory49, 1073–1096 (2003). [CrossRef]

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