## Performance analysis of coherent wireless optical communications with atmospheric turbulence |

Optics Express, Vol. 20, Issue 6, pp. 6515-6520 (2012)

http://dx.doi.org/10.1364/OE.20.006515

Acrobat PDF (709 KB)

### Abstract

Coherent wireless optical communication systems with heterodyne detection are analyzed for binary phase-shift keying (BPSK), differential PSK (DPSK), and *M*-ary PSK over Gamma-Gamma turbulence channels. Closed-form error rate expressions are derived using a series expansion approach. It is shown that, in the special case of *K*-distributed turbulence channel, the DPSK incurs a 3 dB signal-to-noise ratio (SNR) penalty compared to BPSK in the large SNR regime. The outage probability is also obtained, and a detailed outage truncation error analysis is presented and used to assess the accuracy in system performance estimation. It is shown that our series error rate expressions are simple to use and highly accurate for practical system performance estimation.

© 2012 OSA

## 1. Introduction

1. V. W. S. Chan, “Free-space optical communications,” J. Lightwave Technol. **24**, 4750–4762 (2006). [CrossRef]

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

3. E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in Gamma-Gamma fading,” IEEE Trans. Commun. **57**, 3415–3424 (2009). [CrossRef]

4. J. Li, J. Q. Liu, and D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun. **55**, 1598–1606 (2007). [CrossRef]

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

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

*M*-ary PSK (MPSK) and differential PSK (DPSK). Additional insights into coherent WOC systems are provided through asymptotic error performance analyses, and a detailed study of the truncation error is presented for the outage series solution.

## 2. Coherent WOC system

### 2.1. Receiver model

7. M. Niu, J. Cheng, and J. F. Holzman, “Error rate analysis of *M*-ary coherent free-space optical communication systems with *K*-distributed turbulence,” IEEE Trans. Commun. **59**, 664–668 (2011). [CrossRef]

*i*=

_{dc}*R*(

*P*+

_{s}*P*) and

_{LO}*n*(

*t*) is a zero-mean additive white Gaussian noise (AWGN) process due to shot noise. Since the local oscillator power,

*P*, is made to be sufficiently large, the shot noise can be modeled as AWGN with high accuracy [2

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

9. G. P. Agrawal, *Fiber-Optical Communication Systems* (Wiley, 2002). [CrossRef]

*R*is the photodetector responsivity,

*P*denotes the received optical signal power incident on the beamsplitter, the pre-modulated or differentially coded phase information is represented by

_{s}*ϕ*, and

*ω*=

_{IF}*ω*

_{0}–

*ω*is the intermediate frequency, with

_{LO}*ω*

_{0}and

*ω*denoting the carrier frequency and local oscillator frequency, respectively. In the following analysis, we assume the phase noise can be fully compensated at the receiver, as the phase noise is sufficiently slow compared to the extremely high data rates of WOC systems [6

_{LO}6. K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in *K*-distributed turbulence,” IEEE Trans. Commun. **54**, 604–607 (2006). [CrossRef]

9. G. P. Agrawal, *Fiber-Optical Communication Systems* (Wiley, 2002). [CrossRef]

*A*and irradiance, the SNR for coherent WOC systems at the input of the demodulator is

*γ*=

*RAI*/(

_{s}*q*Δ

*f*) where

*I*is the turbulence-dependent optical irradiance, Δ

_{s}*f*is the noise equivalent bandwidth of the photodetector, and

*q*is the electronic charge.

### 2.2. Turbulence model

*K*-distributed model (for strong turbulence regimes), and Gamma-Gamma model (from weak to strong turbulence regimes). In this work, we model the optical irradiance

*I*as a Gamma-Gamma random variable with a probability density function (PDF) [10

_{s}10. 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]

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

_{α−β}*α*

*−*

*β*. The positive parameters

*α*and

*β*are the effective number of large- and small-scale cells of atmospheric scattering processes, respectively. The parameters

*α*and

*β*are set by the Rytov variance [11

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

*α*>

*β*holds in WOC applications under an assumption of plane wave and negligible inner-scale [12

12. N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the Gamma-Gamma atmospheric turbulence model,” Opt. Express **18**, 12824–12831 (2010). [CrossRef] [PubMed]

## 3. Performance analysis of coherent WOC

### 3.1. Error rate analysis

*P*(

_{e}*I*) is given by

_{s}*γ̄*denotes the average SNR and

*Q*(·) is the Gaussian

*Q*-function defined as

3. E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in Gamma-Gamma fading,” IEEE Trans. Commun. **57**, 3415–3424 (2009). [CrossRef]

13. Wolfram Mathworld. [Online]: http://functions.wolfram.com/03.04.06.0002.01.

*I*can be expressed as [3

_{s}3. E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in Gamma-Gamma fading,” IEEE Trans. Commun. **57**, 3415–3424 (2009). [CrossRef]

*Q*-function, we can rewrite Eq. (3) as Substituting Eq. (4) into Eq. (6) and using integral identities [14, Eq. 3.478(1), Eq 3.621(1), Eq. 8.384(4)], one obtains the average bit-error rate (BER) for BPSK coherent WOC systems as

*B*(

*x,y*) = Γ(

*x*)Γ(

*y*)/Γ(

*x*+

*y*) as the Beta function [14, Eq. 8.384(1)] and the integral property [14, Eq. 3.621(1), Eq. 8.384(4)] With Eq. (7), one can examine error rate behavior in large SNR regimes with insight into coherent WOC systems. Since

*α*>

*β*, the term (

*γ̄*/2)

^{−(p+α)}decreases faster than the term (

*γ̄*/2)

^{−(p+β)}in Eq. (7) for the same

*p*values as

*γ̄*increases. Consequently, when

*γ̄*approaches ∞, the leading term of the series in Eq. (7) becomes dominant, and the error rate in large SNR regimes can be obtained as

7. M. Niu, J. Cheng, and J. F. Holzman, “Error rate analysis of *M*-ary coherent free-space optical communication systems with *K*-distributed turbulence,” IEEE Trans. Commun. **59**, 664–668 (2011). [CrossRef]

*g*(

_{p}*x*) denotes an integral identity from Mathematica

^{®}defined as

_{2}

*F*

_{1}(·,·;·;·) is the Gaussian hypergeometric function [14, Eq. 9.100].

15. M. Niu, J. Schlenker, J. Cheng, J. F. Holzman, and R. Schober, “Coherent wireless optical communications with predetection and postdetection EGC over Gamma-Gamma atmospheric turbulence channels,” J. Opt. Commun. Netw. **3**, 860–869 (2011). [CrossRef]

*γ*/2). Integrating over

*γ*gives the average BER of DPSK as The BER for high SNR regimes can be similarly obtained as Using Eq. (9) and Eq. (13), we obtain the SNR penalty factor for DPSK with respect to BPSK as

*β*, and the SNR penalty factor for DPSK is exactly 10log

_{10}(2) ≈ 3 dB over the

*K*-distributed turbulence channels when

*β*= 1.

### 3.2. Outage probability analysis

*I*in terms of a series as With

_{s}*γ*=

*γ̄I*, one obtains the outage probability as where

_{s}*γ*denotes the specific outage probability threshold.

_{th}### 3.3. Truncation error analysis

*J*+ 1 terms in Eq. (15). We first define the outage truncation error as The truncated error

*ε*can then be rewritten as where we have defined The summation term in Eq. (17) can be simplified through a Taylor series expansion of the exponential function. The result gives an upper bound of the truncation error in Eq. (16) through where

_{J}*c*(

_{p}*α,β*) ≜ |

*b*(

_{p}*α,β*)

*−b*(

_{p}*β,α*)|. It is straightforward to show that

*b*(

_{p}*α,β*) or

*c*(

_{p}*α,β*) decreases to zero when

*p*approaches ∞, and the truncation error diminishes with increasing index

*p*. From Eqs. (18) and (19), it is also clear that the truncation error diminishes rapidly with increasing average SNR. It can be shown that

*ε*is upper bounded by a term on the order of

_{J}*γ̄*

^{−β}when

*p*and

*γ̄*are large. Although only the outage truncation error analysis is presented here, similar truncation error analyses can be performed for differential and coherent PSK.

## 4. Numerical examples

*α*= 3.553,

*β*= 3.339), moderate (

*α*= 2.711,

*β*= 2.319) and strong (

*α*= 1.967,

*β*= 1.236) Gamma-Gamma turbulence channels are plotted for coherent WOC with BPSK and DPSK. Exact error rate is found by numerical integration based on Eq. (3), which has been confirmed by Monte Carlo simulations. Excellent agreement is seen between our series solutions and the exact BERs. As predicted in Section 3.1, the SNR penalty factor depends on channel turbulence levels (via the parameter

*β*). For weak turbulence with

*α*= 3.553 and

*β*= 3.339 at an average BER of 10

^{−10}, the SNR penalty factor is 1.6 dB, which agrees with the 1.58 dB theoretical prediction. For moderate turbulence with

*α*= 2.711 and

*β*= 2.319 at an average BER of 10

^{−8}, the SNR penalty factor is 2 dB, which agrees with the 1.96 dB theoretical prediction. The SNR penalty factor increases as

*β*decreases.

*α*= 2.711,

*β*= 2.319) and strong (

*α*= 1.967,

*β*= 1.236) turbulence. The SER degrades as

*M*increases for MPSK, as expected, due to average symbol power constraints. In Fig. 2(a), outage probability is presented for the same weak to strong turbulence conditions as Fig. 1(a) with

*γ*= 3 dB. Again, our series solutions match the exact error rate and outage curves.

_{th}^{−18}at an average SNR of 30 dB. Also shown in Fig. 2(b), the outage truncation error diminishes with increasing average SNR in a linear fashion on a log-log plot when

*p*and average SNR is large (as discussed in Section 3.3).

## 5. Conclusion

## References and links

1. | V. W. S. Chan, “Free-space optical communications,” J. Lightwave Technol. |

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

3. | E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in Gamma-Gamma fading,” IEEE Trans. Commun. |

4. | J. Li, J. Q. Liu, and D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun. |

5. | W. Huang, J. Takayanagi, T. Sakanaka, and M. Nakagawa, “Atmospheric optical communication system using subcarrier PSK modulation,” IEICE Trans. Commun . |

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

7. | M. Niu, J. Cheng, and J. F. Holzman, “Error rate analysis of |

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

9. | G. P. Agrawal, |

10. | 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. |

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

12. | N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the Gamma-Gamma atmospheric turbulence model,” Opt. Express |

13. | Wolfram Mathworld. [Online]: http://functions.wolfram.com/03.04.06.0002.01. |

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

15. | M. Niu, J. Schlenker, J. Cheng, J. F. Holzman, and R. Schober, “Coherent wireless optical communications with predetection and postdetection EGC over Gamma-Gamma atmospheric turbulence channels,” J. Opt. Commun. Netw. |

**OCIS Codes**

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(060.4510) Fiber optics and optical communications : Optical communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: November 14, 2011

Revised Manuscript: February 29, 2012

Manuscript Accepted: February 29, 2012

Published: March 6, 2012

**Citation**

Mingbo Niu, Xuegui Song, Julian Cheng, and Jonathan F. Holzman, "Performance analysis of coherent wireless optical communications with atmospheric turbulence," Opt. Express **20**, 6515-6520 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-6-6515

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### References

- V. W. S. Chan, “Free-space optical communications,” J. Lightwave Technol.24, 4750–4762 (2006). [CrossRef]
- X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun.50, 1293–1300 (2002). [CrossRef]
- E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in Gamma-Gamma fading,” IEEE Trans. Commun.57, 3415–3424 (2009). [CrossRef]
- J. Li, J. Q. Liu, and D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun.55, 1598–1606 (2007). [CrossRef]
- W. Huang, J. Takayanagi, T. Sakanaka, and M. Nakagawa, “Atmospheric optical communication system using subcarrier PSK modulation,” IEICE Trans. Commun. E76-B, 1169–1177 (1993).
- K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun.54, 604–607 (2006). [CrossRef]
- M. Niu, J. Cheng, and J. F. Holzman, “Error rate analysis of M-ary coherent free-space optical communication systems with K-distributed turbulence,” IEEE Trans. Commun.59, 664–668 (2011). [CrossRef]
- A. Belmonte and J. M. Kahn, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express16, 14151–14162 (2008). [CrossRef] [PubMed]
- G. P. Agrawal, Fiber-Optical Communication Systems (Wiley, 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]
- L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001). [CrossRef]
- N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the Gamma-Gamma atmospheric turbulence model,” Opt. Express18, 12824–12831 (2010). [CrossRef] [PubMed]
- Wolfram Mathworld. [Online]: http://functions.wolfram.com/03.04.06.0002.01 .
- I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 2000).
- M. Niu, J. Schlenker, J. Cheng, J. F. Holzman, and R. Schober, “Coherent wireless optical communications with predetection and postdetection EGC over Gamma-Gamma atmospheric turbulence channels,” J. Opt. Commun. Netw.3, 860–869 (2011). [CrossRef]

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