## Exact error rate analysis of equal gain and selection diversity for coherent free-space optical systems on strong turbulence channels

Optics Express, Vol. 18, Issue 13, pp. 13915-13926 (2010)

http://dx.doi.org/10.1364/OE.18.013915

Acrobat PDF (742 KB)

### Abstract

Exact error rate performances are studied for coherent free-space optical communication systems under strong turbulence with diversity reception. Equal gain and selection diversity are considered as practical schemes to mitigate turbulence. The exact bit-error rate for binary phase-shift keying and outage probability are developed for equal gain diversity. Analytical expressions are obtained for the bit-error rate of differential phase-shift keying and asynchronous frequency-shift keying, as well as for outage probability using selection diversity. Furthermore, we provide the closed-form expressions of diversity order and coding gain with both diversity receptions. The analytical results are verified by computer simulations and are suitable for rapid error rates calculation.

© 2010 Optical Society of America

## 1. Introduction

1. M. M. Ibrahim and A. M. Ibrahim, “Performance analysis of optical receivers with space diversity reception,” IEE Proc. Commun. **143**, 369–372 (1996). [CrossRef]

1. M. M. Ibrahim and A. M. Ibrahim, “Performance analysis of optical receivers with space diversity reception,” IEE Proc. Commun. **143**, 369–372 (1996). [CrossRef]

2. T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wirel. Commun. **8**, 951–957 (2009). [CrossRef]

*K*-distributed [4

4. K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in *K*-distributed turbulence,” IEEE Trans. Commun. **54**, 604–607 (2006). [CrossRef]

5. T. A. Tsiftsis, “Performance of heterodyne wireless optical communication systems over Gamma-Gamma atmospheric turbulence channels,” Electron. Lett. **44**, 373–375 (2008). [CrossRef]

*et al.*[6

6. H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol. **27**, 4440–4445 (2009). [CrossRef]

7. A. Belmonte and J. M. Kahn, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express **16**, 14151–14162 (2008). [CrossRef] [PubMed]

8. A. Belmonte and J. M. Kahn, “Capacity of coherent free-space optical links using diversity combining techniques,” Opt. Express **17**, 12601–12611 (2009). [CrossRef] [PubMed]

*K*distribution [4

4. K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in *K*-distributed turbulence,” IEEE Trans. Commun. **54**, 604–607 (2006). [CrossRef]

10. R. L. Phillips and L. C. Andrews, “Measured statistics for laser light scattering in atmospheric turbulence,” J. Opt. Soc. Am. **71**, 1440–1445 (1981). [CrossRef]

## 2. The Coherent FSO System Model and Statistics

### 2.1. Channel Model

*l*th branch of a coherent FSO system using BPSK, with received and local oscillator beams mixed in perfect spatial coherence over a sufficiently small photodetector area. Practical issues such as beam combining and polarization are discussed in [11

11. M. Jafar, D. C. O’Brien, C. J. Stevens, and D. J. Edwards, “Evaluation of coverage area for a wide line-of-sight indoor optical free-space communication system empolying coherent detection,” IET Commun. **2**, 18–26 (2008). [CrossRef]

*i*=

_{dc,l}*R*(

*P*+

_{s,l}*P*) and

_{LO}*R*is the detector responsivity, and

*n*(

_{l}*t*) is a shot-noise-limited additive white Gaussian noise (AWGN) process. The power PLO is sufficiently large so that shot noise is dominant and can be modeled as AWGN [12

12. G. P. Agrawal, *Fiber-Optical Communication Systems*, (New York: Wiley, 2002). [CrossRef]

*P*is the received

_{s,l}*l*th optical signal power incident on the beamsplitter,

*P*is the local oscillator power assumed to be the same for all branches, and

_{LO}*ω*=

_{IF}*ω*

_{0}−

*ω*is the intermediate frequency, with

_{LO}*ω*

_{0}and

*ω*denoting the carrier frequency and local oscillator frequency, respectively. The phase information is

_{LO}*ϕ*∈ {0,

*π*}, and

*ϕ*is the

_{s,l}*l*th branch phase noise. It is assumed that each diversity branch turbulence is independent. For EGC cases, we assume the signal phase is well tracked at the coherent FSO detector and the phase noise can be fully compensated [3], as the optical phase varies slowly over 1 or 2 bits at a high data rate.

### 2.2. SNR Analysis

*i*

^{2}

_{ac}(

*t*)〉 to the total noise variance [12

12. G. P. Agrawal, *Fiber-Optical Communication Systems*, (New York: Wiley, 2002). [CrossRef]

*L*diversity branches, the received signals at different branches are co-phased and added with equal weight. In such a case, as shown in the Appendix, the instantaneous SNR at the output of the combiner becomes

*q*denotes the electronic charge, Δ

*f*is the noise equivalent bandwidth of the photodetector,

*I*is the normalized

_{N,l}*K*turbulent coefficient (

*E*[

*I*] = 1) of the

_{N,l}*l*th branch,

*N*

_{0}/2 =

*qRP*is the noise power spectral density, and

_{LO}*E*=

_{b}*η*

^{2}

*AT*denotes the energy per bit. Here,

_{b}*E*[·] denotes the expectation operation,

*η*

^{2}is the mean irradiance,

*A*denotes the detector area, and

*T*is the bit duration. With the relationship

_{b}*P*=

_{s,l}*AI*where

_{s,l}*I*is the instantaneous received optical irradiance at the

_{s,l}*l*th branch, we can rewrite the instantaneous SNR as

*l*th branch instantaneous SNR as

*γ*=

_{l}*RP*/(

_{s,l}*q*Δ

*f*) [9], the instantaneous SNR for SD is

*P*. This is an important property of coherent FSO communication (in contrast to coherent radio frequency literature), as the local oscillator power does not affect coherent FSO system performance.

_{LO}### 2.3. Analysis for the Square Root of the Irradiance

*I*. In this work, we assume a signal irradiance

_{s}*I*with a

_{s}*K*-distributed model having a probability density function (PDF) given by [4

4. K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in *K*-distributed turbulence,” IEEE Trans. Commun. **54**, 604–607 (2006). [CrossRef]

*σ*

^{2}

_{si}in the

*K*-distributed model is (2,3) [4

*K*-distributed turbulence,” IEEE Trans. Commun. **54**, 604–607 (2006). [CrossRef]

*α*is in (1,2). In the strong turbulence regime, the scintillation index can be calculated from the Rytov variance [13

13. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. **16**, 1417–1429 (1999). [CrossRef]

13. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. **16**, 1417–1429 (1999). [CrossRef]

*C*

^{2}

_{n}is the index-of-refraction structure parameter,

*k*is the optical wave number, and

*d*is the propagation distance. For this coherent FSO system in the strong turbulence regime, one would expect

*C*

^{2}

_{n}= 10

^{−13}(

*m*

^{−2/3}) for a link length beyond 1

*km*and aperture sizes below 1

*mm*

^{2}. This avoids aperture averaging (and a loss of coherence) and provides sufficient distance between individual photodetectors [10

10. R. L. Phillips and L. C. Andrews, “Measured statistics for laser light scattering in atmospheric turbulence,” J. Opt. Soc. Am. **71**, 1440–1445 (1981). [CrossRef]

*z*can be shown as [14

14. E. Jakeman and R. J. A. Tough, “Generalized *K* distribution: a statistical model for weak scattering,” J. Opt. Soc. Am. A **4**, 1764–1772 (1987). [CrossRef]

*z*, we let

*z*=

*xy*, where

*α*≥ 1/2) PDFs, respectively, as

*z*conditioned on

*z*as

_{2}

*F*

_{1}(·, ·, ·; ·) is the Gaussian hypergeometric function, ℜ{·} and ℑ{·} denote the real and imaginary operators, respectively. The CF of

*z*can also be obtained from the moment generating function (MGF) of a product of two independent Nakagami RVs [15

15. G. K. Karagiannidis, N. C. Sagias, and P. T. Mathiopoulos, “N*Nakagami: A Novel Stochastic Model for Cascaded Fading Channels,” IEEE Trans. Commun. **55**, 1453–1458 (2007). [CrossRef]

## 3. Performance Analysis of EGC FSO Systems

### 3.1. Error Rate Analysis

*z*is the sum of

_{s}*L*square roots of the irradiance, i.e.,

*f*(

*z*) is the PDF of

_{s}*z*and

_{s}*P*(

_{e}*z*) denotes the conditional bit-error probability given by

_{s}*erfc*(·) is the complementary error function. In general, it is difficult to obtain a closed-form expression for the PDF of

*z*. Hence, we use the Fourier inversion theorem to obtain the PDF of

_{s}*z*by

_{s}*j*

^{2}= −1,

*z*and

_{s}*z*(with

_{l}16. A. Annamalai, C. Tellambura, and V. K. Bhargava, “Exact evaluation of maximal-ratio and equal-gain diversity receivers for M-ary QAM on Nakagami fading channels,” IEEE Trans. Commun. **47**, 1335–1344 (1999). [CrossRef]

*ω*= tan

*θ*, it is possible to rewrite Eq. (20) in terms of a definite integration for easier numerical evaluation

*M*-ary quadrature amplitude modulation.

### 3.2. Outage Probability Analysis

*z*as [18

_{s}18. Q. T. Zhang, “Outage probability in cellular mobile ratio due to nakagami signal and interferers with arbitrary parameters,” IEEE Trans. Veh. Technol. **45**, 364–372 (1996). [CrossRef]

*z*in the polar form as

_{l}*z*are obtained from Eq. (13) and Eq. (14). The numerator of the integrand in Eq. (22) can now be written as

_{l}*γ*

^{*}is the specified outage threshold. Equation (25) can provide accurate outage calculation with a simple numerical integration algorithm.

## 4. Performance Analysis of SD FSO Systems

### 4.1. Outage Probability Analysis

*I*, at the

_{s,l}*l*th branch from Eq. (5) as

### 4.2. Error Rate Analysis for Differential and Asynchronous Detection

*I*≜ max{

_{s,SD}*I*,

_{s,l}*l*= 1, …,

*L*}. For identically distributed turbulence, we can show the PDF of

*I*as

_{s,SD}12. G. P. Agrawal, *Fiber-Optical Communication Systems*, (New York: Wiley, 2002). [CrossRef]

*is known to be*γ ¯

*β*= 1/2 for DPSK (assuming moderate frequency offset) and

*β*= 1/4 for asynchronous FSK [12

12. G. P. Agrawal, *Fiber-Optical Communication Systems*, (New York: Wiley, 2002). [CrossRef]

## 5. Analysis of Diversity Order and Coding Gain

*G*and

_{d}*G*respectively denote diversity order and coding gain [9]. Similarly we denote

_{c}*G*and

_{dl}*G*as the diversity order and coding gain of the

_{cl}*l*th branch.

*K*-distributed RV and following [9], one can show that the diversity order and coding gain offered by the

*l*th branch are

*G*= 1 and

_{dl}*G*= 2

_{cl}*βη*

^{2}(

*α*−1)/

*α*respectively. Consequently, the diversity order of SD becomes

*G*= ∑

_{d,SD}

^{L}_{l=1}

*G*=

_{dl}*L*and the coding gain of SD can be derived as

*l*th branch can be found to be

*G*= 1 and

_{dl}*G*= 2

_{cl}*η*

^{2}(

*α*− 1)/

*α*respectively. The overall EGC diversity order is

*G*=

_{d,EGC}*L*, and the expression for the overall EGC coding gain,

*G*can be found in [19

_{c,EGC}19. Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. **51**, 1389–1398 (2003). [CrossRef]

*m*fading model).

## 6. Numerical Results

*=*γ ¯

*RAη*

^{2}/(

*q*Δ

*f*). BER curves for BPSK-modulated signals with EGC diversity reception (

*α*= 1.8) are presented in Fig. 1 for average SNRs from 0 to 30 dB. Asymptotic BERs are also shown and agree with our analytical results when the SNR is large. When the number of diversity branches increases, the BER performance improves. At an average SNR=30 dB, a three-branch EGC reception achieves an error rate of 2.5 × 10

^{−8}. Though it is not shown in Fig. 1, all analytical results have been verified with computer simulations and they have excellent agreement.

^{−7}, 1 dB SNR (or less) is required for EGC to achieve the MRC performance. Hence, EGC offers a good balance between system performance and complexity.

*σ*

^{2}

_{si}ranging from 2.1 to 2.9 for both MRC and EGC. As expected, BER performance improves for coherent FSO communication as the value of

*σ*

^{2}

_{si}decreases.

*σ*

^{2}

_{si}= 2.1 outperforms MRC with

*σ*

^{2}

_{si}= 2.5 for the same number of diversity branches. For practical applications, a modest parametrization error in estimating the scintillation indices or channel parameters will overshadow the benefits provided by MRC. In such a case, EGC may be a better choice due to its reduced complexity.

## 7. Conclusion

## Appendix

*l*th photodetector, the received photocurrent is given in Eq. (1). Let

*σ*

^{2}

_{n,l}= 2

*qRP*Δ

_{LO}*f,l*= 1, …,

*L*, is the noise variance in the

*l*th diversity branch at the receiver side, and it is assumed to be the same for all diversity branches.

## Acknowledgments

## References and links

1. | M. M. Ibrahim and A. M. Ibrahim, “Performance analysis of optical receivers with space diversity reception,” IEE Proc. Commun. |

2. | T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wirel. Commun. |

3. | E. Lee and V. Chan, “Diversity coherent receivers for optical communication over the clear turbulent atmosphere,” in Proceedings of IEEE International Conference on Communications (Institute of Electrical and Electronics Engineers, New York, 2007). |

4. | K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in |

5. | T. A. Tsiftsis, “Performance of heterodyne wireless optical communication systems over Gamma-Gamma atmospheric turbulence channels,” Electron. Lett. |

6. | H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol. |

7. | A. Belmonte and J. M. Kahn, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express |

8. | A. Belmonte and J. M. Kahn, “Capacity of coherent free-space optical links using diversity combining techniques,” Opt. Express |

9. | M. Niu, J. Cheng, J. F. Holzman, and L. McPhail, “Performance analysis of coherent free-space optical communication systems with |

10. | R. L. Phillips and L. C. Andrews, “Measured statistics for laser light scattering in atmospheric turbulence,” J. Opt. Soc. Am. |

11. | M. Jafar, D. C. O’Brien, C. J. Stevens, and D. J. Edwards, “Evaluation of coverage area for a wide line-of-sight indoor optical free-space communication system empolying coherent detection,” IET Commun. |

12. | G. P. Agrawal, |

13. | L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. |

14. | E. Jakeman and R. J. A. Tough, “Generalized |

15. | G. K. Karagiannidis, N. C. Sagias, and P. T. Mathiopoulos, “N*Nakagami: A Novel Stochastic Model for Cascaded Fading Channels,” IEEE Trans. Commun. |

16. | A. Annamalai, C. Tellambura, and V. K. Bhargava, “Exact evaluation of maximal-ratio and equal-gain diversity receivers for M-ary QAM on Nakagami fading channels,” IEEE Trans. Commun. |

17. | I. S. Gradshteyn and I. M. Ryzhik, |

18. | Q. T. Zhang, “Outage probability in cellular mobile ratio due to nakagami signal and interferers with arbitrary parameters,” IEEE Trans. Veh. Technol. |

19. | Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. |

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

(060.4510) Fiber optics and optical communications : Optical communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: March 30, 2010

Revised Manuscript: June 1, 2010

Manuscript Accepted: June 5, 2010

Published: June 14, 2010

**Citation**

Mingbo Niu, Julian Cheng, and Jonathan F. Holzman, "Exact error rate analysis of equal gain and selection diversity for coherent
free-space optical systems on strong turbulence channels," Opt. Express **18**, 13915-13926 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13915

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### References

- . M. M. Ibrahim and A. M. Ibrahim, “Performance analysis of optical receivers with space diversity reception,” IEE Proc. Commun. 143,369–372 (1996). [CrossRef]
- . T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wirel. Commun. 8,951–957 (2009). [CrossRef]
- . E. Lee and V. Chan, “Diversity coherent receivers for optical communication over the clear turbulent atmosphere,” in Proceedings of IEEE International Conference on Communications (Institute of Electrical and Electronics Engineers, New York, 2007).
- . K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun. 54,604–607 (2006). [CrossRef]
- . T. A. Tsiftsis, “Performance of heterodyne wireless optical communication systems over Gamma-Gamma atmospheric turbulence channels,” Electron. Lett. 44,373–375 (2008). [CrossRef]
- . H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol. 27,4440–4445 (2009). [CrossRef]
- . A. Belmonte and J. M. Kahn, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express 16,14151–14162 (2008). [CrossRef] [PubMed]
- . A. Belmonte and J. M. Kahn, “Capacity of coherent free-space optical links using diversity combining techniques,” Opt. Express 17,12601–12611 (2009). [CrossRef] [PubMed]
- . M. Niu, J. Cheng, J. F. Holzman, and L. McPhail, “Performance analysis of coherent free-space optical communication systems with K-distributed turbulence,” in Proceedings of IEEE International Conference on Communications (Institute of Electrical and Electronics Engineers, New York, 2009).
- . R. L. Phillips and L. C. Andrews, “Measured statistics for laser light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71,1440–1445 (1981). [CrossRef]
- . M. Jafar, D. C. O’Brien, C. J. Stevens, and D. J. Edwards, “Evaluation of coverage area for a wide line-of-sight indoor optical free-space communication system empolying coherent detection,” IET Commun. 2,18–26 (2008). [CrossRef]
- . G. P. Agrawal, Fiber-Optical Communication Systems, (New York: Wiley, 2002). [CrossRef]
- . L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. 16,1417–1429 (1999). [CrossRef]
- . E. Jakeman and R. J. A. Tough, “Generalized K distribution: a statistical model for weak scattering,” J. Opt. Soc. Am. A 4,1764–1772 (1987). [CrossRef]
- . G. K. Karagiannidis, N. C. Sagias, and P. T. Mathiopoulos, “N*Nakagami: A Novel Stochastic Model for Cascaded Fading Channels,” IEEE Trans. Commun. 55,1453–1458 (2007). [CrossRef]
- . A. Annamalai, C. Tellambura, and V. K. Bhargava, “Exact evaluation of maximal-ratio and equal-gain diversity receivers for M-ary QAM on Nakagami fading channels,” IEEE Trans. Commun. 47,1335–1344 (1999). [CrossRef]
- . I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, (San Diego: Academic Press, 2000).
- . Q. T. Zhang, “Outage probability in cellular mobile ratio due to nakagami signal and interferers with arbitrary parameters,” IEEE Trans. Veh. Technol. 45,364–372 (1996). [CrossRef]
- . Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51,1389–1398 (2003). [CrossRef]

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