## Capacity of coherent free-space optical links using diversity-combining techniques

Optics Express, Vol. 17, Issue 15, pp. 12601-12611 (2009)

http://dx.doi.org/10.1364/OE.17.012601

Acrobat PDF (223 KB)

### Abstract

We study the performance of diversity combining techniques applied to synchronous laser communication through the turbulent atmosphere. We assume that a single information-bearing signal is transmitted over two or more statistically independent fading channels, and that the multiple replicas are combined at the receiver to improve detection efficiency. We consider the effects of log-normal amplitude fluctuations and Gaussian phase fluctuations, in addition to local oscillator shot noise. We study the effect of various parameters, including the ratio of receiver aperture diameter to wavefront coherence diameter, the scintillation index, and the number of independent diversity branches combined at the receiver. We consider both maximal-ratio combining (MRC) and selective combining (SC) diversity schemes. We derive expressions for the outage Shannon capacity, thus placing upper bounds on the spectral efficiency achievable using these techniques.

© 2009 OSA

## 1. Introduction

1. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE **55**(1), 57–77 (1967). [CrossRef]

3. A. Belmonte and J. Khan, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express **16**(18), 14151–14162 (2008). [CrossRef] [PubMed]

4. A. Belmonte and J. M. Kahn, “Capacity of coherent free-space optical links using atmospheric compensation techniques,” Opt. Express **17**(4), 2763–2773 (2009). [CrossRef] [PubMed]

4. A. Belmonte and J. M. Kahn, “Capacity of coherent free-space optical links using atmospheric compensation techniques,” Opt. Express **17**(4), 2763–2773 (2009). [CrossRef] [PubMed]

5. S. Haas and J. H. Shapiro, “Capacity of wireless optical communications,” IEEE J. Sel. Areas Comm. **21**(8), 1346–1357 (2003). [CrossRef]

12. N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, ““Performance bounds for free-space optical MIMO systems with APD receivers in atmospheric turbulence,” IEEE,” J. Select. Areas. Commun. **26**(3), 3–12 (2008). [CrossRef]

## 2. Outage probability of coherent atmospheric links using diversity combining

*C*of a communication channel is the maximal rate at which the information may be transferred through the channel without error. For an additive white Gaussian noise (AWGN) channel, in the classical capacity formula with average power constraint

*P*and noise power spectral density

*N*

_{0}/2, given by

*C = B*log

_{2}(1

*+ γ*

_{0}), the spectral bandwidth

*B*, which has units of Hz, multiplies the maximal spectral efficiency log

_{2}(1

*+ γ*

_{0}), which has units of bits/s/Hz. Here,

*γ*

_{0}

*= P/N*

_{0}

*B*is the SNR per unit bandwidth

*B*. The SNR

*γ*

_{0}for a quantum or shot-noise limited signal can be interpreted as the detected number of photons (photocounts) per symbol when 1/

*B*is the symbol period [14]. Coherently detected signals are modeled as narrowband RF signals with additive white Gaussian noise (AWGN). In free-space optical communication through the turbulent atmosphere, we must consider fading channels, which are a class of channels with multiplicative noise. In the fading AWGN channel with average power constraint

*P*and noise power spectral density

*N*

_{0}/2, we let

*α*

^{2}denote the atmospheric channel power fading and (

*P/N*

_{0}

*B*)

*α*

^{2}=

*γ*

_{0}

*α*

^{2}denote the instantaneous received SNR per symbol. For a shot-noise-limited coherent optical receiver, the SNR of the envelope detector can be taken as the number of signal photons detected on the receiver aperture

*γ*

_{0}multiplied by a heterodyne mixing efficiency

*α*

^{2}: In addition to the effective delivery of the signal to the detector, the performance of the optical link also depends on the receiver sensitivity measured in terms of received photons. For systems with perfect spatial mode matching the heterodyne mixing efficiency is equal to 1. When the spatial modes are not properly matched, the contribution to the current signal from different parts of the receiver aperture can interfere destructively and result in the reduced instantaneous heterodyne mixing and consequent fading.

*α*, this is an AWGN channel with instantaneous received SNR

*γ = γ*

_{0}

*α*

^{2}. This quantity is a function of the random channel power gain

*α*

^{2}, and is therefore random. The statistical properties of the atmospheric random channel fade

*α*

^{2}, with probability density function (PDF) p

_{α}_{2}(

*α*

^{2}), provide a statistical characterization of the SNR

*γ = γ*

_{0}

*α*

^{2}and, consequently, of the maximal spectral efficiency achievable for the free-space optical link. We have already modeled the impact of atmospheric turbulence-induced phase and amplitude fluctuations on free-space optical links using synchronous detection and found that the SNR γ for a single monolithic-aperture coherent receiver is described by a noncentral chi-square probability distribution function (PDF) with two degrees of freedom [3

3. A. Belmonte and J. Khan, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express **16**(18), 14151–14162 (2008). [CrossRef] [PubMed]

*/r*consider turbulence effects. The model leading to the PDF in Eq. (1) is based on the observation that the downconverted signal current can be characterized as the sum of many contributions from

*N*different coherent regions within the aperture [3

3. A. Belmonte and J. Khan, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express **16**(18), 14151–14162 (2008). [CrossRef] [PubMed]

*/r*is a measure of the strength of the residual halo relative to the coherent component. The parameter

*r*ranges between 0 and ∞. It can be shown that when the constant term is very weak (

*r*→0), turbulence fading makes the SNR to become negative-exponential-distributed, just as in a speckle pattern. Likewise, when the dominant term is very strong (

*r*→∞), the density function becomes highly peaked around the mean value

*χ*(

**) and**

*r**f*(

**) represent the log-amplitude fluctuations (scintillation) and phase variations (aberrations), respectively, introduced by atmospheric turbulence. Consequently, both**

*r**/r*are described in terms of log-normal amplitude fluctuations and Gaussian phase fluctuations as characterized by their respective statistical variances,

*σ*

_{χ}^{2}and

*σ*

_{f}^{2},

*σ*

_{β}^{2}is often referred to as the scintillation index [15

15. J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. **10**(1), 59–70 (1975). [CrossRef]

*σ*

_{f}^{2}assumes that no terms are corrected by a receiver employing active modal compensation [16

16. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. **66**(3), 207–211 (1976). [CrossRef]

*D*is normalized by the wavefront coherence diameter

*r*

_{0}, which describes the spatial correlation of phase fluctuations in the receiver plane [1

1. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE **55**(1), 57–77 (1967). [CrossRef]

*γ*for a summing coherent MRC combiner is the power ratio of the phase-coherent addition of the signal amplitudes from each element of the combiner to the incoherent addition of the noise. If an optimum voltage gain proportional to the amplitude of the signal itself is assumed for each receiver in the combiner, and if equal noise powers are assumed, the resultant composite SNR for an

_{MRC}*L-*element MRC combiner is [17]

*γ*is the sum of the component array element SNR’s. For independent branch signals and equal average branch SNR

_{MRC}*γ*at the output of a perfect

_{MRC}*L*-branch MRC coherent combiner in the atmosphere would be described a sum of

*L*independent and identically distributed non-central chi-squared random variables with two degrees of freedom. This random variable has a noncentral chi-square distribution with 2

*L*degrees of freedom:

*L*times the single-element mean SNR

*L*

*γ*, which is defined as the probability that the output SNR

_{MRC}*γ*falls below a certain specified threshold

_{MRC}*γ*, or outage probability. After some algebra, we obtain the CDFwhere

_{t}*Q*(

_{L}*a,b*) is the generalized

*L*-order Marcum Q function.

*L*−1 diversity branches released. Assuming independent branch signals and equal average branch SNR

*γ*at the output of an

*L*-branch SC coherent combiner is found to be given by the

*L*th power of the CDF corresponding to a single-branch receiver [18]: The probability that the SNR of the

*l*-th branch is less than or equal to any specified value

*γ*is given by its CDF, so that the probability that the SNR of a perfect

_{t}*L*-branch SC coherent combiner is simultaneously less than

*γ*must be the product of the corresponding CDFs. Using Eq. (6) with

_{t}*L*= 1, it results the following expression for the SC CDF:where

*Q*now refers to the first-order Marcum Q function.

## 3. Outage capacity numerical results and comparisons

*R*, i.e.,

*p*(

_{out}*R*) = P{log

_{2}(1

*+ γ*)

*<R*}. Here, the operator P{A} indicates the probability of an event A. If

*γ*denotes the SNR that is required to support a rate

_{R}*R*, the probability of outage can be expressed in terms of SNR as

*p*(

_{out}*R*)

*=*P{

*γ<γ*}. This result can be expressed in terms of the cumulative distribution function CDF of the SNR

_{R}*γ*as

*p*(

_{out}*R*)

*= F*(

*γ*). From here, we can solve

_{R}*p*(

_{out}*R*)

*= ε*to obtain the SNR

*γ*producing a

*ε*-outage probability

*F*(

*γ*)

_{R}*= ε*, i.e.,

*γ*

_{R}= F^{−1}(

*ε*). Then, by definition, the

*ε*-outage spectral efficiency becomesIt is clear that the atmospheric outage spectral efficiency depends on the statistical distribution of SNR

*γ*through its CDF

*F*(

*γ*). Note that Eq. (8) requires the inversion of the CDFs given by Eqs. (6) and (7) modeled the impact of atmospheric turbulence-induced phase and amplitude fluctuations on free-space optical links using synchronous detection and MRC and SC diversity combining of the received signal. However, there is no known elementary inverse of the Marcum Q functions of any order

_{R}*L*. For

*L*>1, even the use of alternative representations of the generalized Marcum Q-functions does not alleviate this difficulty. Instead, we approach the problem numerically by using Newton’s method to converge to the solution of

*F*

^{−1}(

*ε*) for both MRC and SC diversity combining.

*L*-aperture heterodyne receiver system where the receivers are separated by more than one coherence length. Each receiver has a pupil area 1/

*L*times the pupil area of the single receiver system so that the received signal power in both the single and multiple receiver systems is the same. Equivalently, as it was noted before, diversity can also be obtained with just one of these reduced apertures by using

*L*wavelength channels separated by at least the coherence bandwidth of the channel or in time, by using

*L*time slots separated by the coherence time of the atmospheric channel.

*D*/

*r*

_{0}for a constant phase coherence length

*r*

_{0}. In (a), MRC combining is employed. In (b), a SC combiner is considered. In all cases, the outage probability is small and fixed at ε = 0.001, and the channel capacity per unit bandwidth is shown for different values of the number

*L*of combiner branches. The case

*L*= 1 corresponds to no receive diversity. The area

*πD*

^{2}describes the combined, multi-aperture system equivalent aperture. When no receive diversity is considered,

*D*equals the receiver aperture diameter. If a

*L*-aperture system is analyzed, each one of the aperture diameters equals

*D/√L*. For the smallest aperture considered, we assume γ

_{0}equal to 10 photons-per-symbol. For any other aperture diameter, the value of

*γ*

_{0}is proportional to

*D*

^{2}. Comparing Figs. 1(a) and 1(b) we see that, as expected, the SC scheme provides less diversity gain and a lower rate of improvement than the MRC scheme. The dependence on scintillation index

*σ*

_{β}^{2}is strong, as it can be seen in the plots, where the outage capacity for

*σ*

_{β}^{2}= 0 (solid line) and

*σ*

_{β}^{2}= 1 (dashed line) are compared. For relatively small apertures, amplitude scintillation is dominant. A receiver with MRC will coherently combine the diversity branches by weighting them by the complex conjugate of their respective fading gains and adding them. Consequently, it will provide some degree of protection against scintillation. On the other hand, SC combining does not allow for this kind of scintillation compensation and performance is less improved by the degree of diversity considered in the analysis.

*L*the outage spectral efficiency improves appreciably, the greatest improvement is still obtained in going from single- to two-branch combining. In any case, even for such small outage probability ε = 0.001, a multiple receiver system with

*L =*8 apertures reach an outage spectral efficiency of over 10 bits/s/Hz, very close to the optimal spectral rate of a single branch AWGN channel as defined by Shannon.

*L*= 1) or dual (

*L*= 2) receiver system is considered, the existence of an optimal aperture diameter in coherent free-space links is apparent. This optimal aperture diameter maximizes the ε-outage capacity. When the aperture is larger than the optimal value, phase distortion cannot be overcome by the increase in collected power, and we observe a decrease in capacity. For

*L*>2 receiver systems, or under strong scintillation conditions, no optimal value can be identified.

*D*/

*r*

_{0}are considered, the capacity tends toward an asymptotic value that is independent of normalized aperture diameter

*D*/

*r*

_{0}. In this regime, dominated by wavefront distortions, and where amplitude fluctuations and the level of scintillation

*σ*

_{β}^{2}are of little influence, the signal coherence term is weak and

*r*→0. Now, the fading PDF Eq. (5), when L independent signals are combined in a MRC combiner, becomes a gamma distribution, i.e.,

*a,x*) is the lower incomplete gamma function and Γ(

*a*) = Γ(

*a,0*) is the ordinary gamma function. It is easy to shown that, for reasonable small outage probabilities

*r*→0, the PDF Eq. (1) for a single receiver system becomes a negative-exponential distribution and, consequently, the fading CDF in a L-branch SC combiner is given by

*a*, Γ(1 +

*a*) =

*a*!. This fade margin increases with the number of branches in the combiner and the outage probability

*ε*, but is the same regardless of the mean SNR

_{0}. The normalized aperture

*D*/

*r*

_{0}= 2 remains constant for all the cases considered in the figure which corresponds, in most situations, to the exponential regime

*r*→0. From Eq. (13) and the study of Fig. 2, it becomes clear the superior power efficiency of the MRC combiner. As a further example, if the combining system is assumed to collect γ

_{0}200 photons per symbol in the absence of turbulence, then the maximum spectral rate that can be achieved using a 4-branch SC combining is

*C*= 3.5 bits/s/Hz, while when utilizing a 4-branch MRC combining is a 34% higher at

_{ε}/B*C*= 4.7 bits/s/Hz. Still, it is a spectral efficiency nearly a 40% smaller than it could be expected without turbulence, with an estimated Shannon’s rate of almost 8 bits/s/Hz, and extra gains in achievable rates are within reach by increasing the combiner complexity

_{ε}/B*L*.

*ε*and the maximum achievable rate C

_{ε}/B is analyzed for MRC and SC combining. The

*ε*versus

*C*tradeoff is governed by Eq. (8) and parameterized by the turbulence-free photons per symbol γ

_{ε}/B_{0}, the normalized aperture diameter

*D*/

*r*

_{0}, and the number of branches

*L*at the combiner. These information-theoretic limits quantify the optimum tradeoff between the pair

*ε*and

*C*, and show those set of pairs for which is impossible to find a reliable error-correcting codes which can realize any arbitrarily small probability of error. It is clear from Fig. 3 that, for a given probability of outage, there is a significant gain in the achievable rate when utilizing the MRC combiner over the SC combiner. For example, if the system is designed to meet

_{ε}/B*ε*= 10

^{−4}, then the maximum code rate that can be reliably transmitted over this channel using a 4-branch SC combining is

*C*= 2.1 bits/s/Hz, while when utilizing a 4-branch MRC combining,

_{ε}/B*C*= 3.1 bits/s/Hz, which is an increase of close to 50% in the achievable rate. Once again, sizeable additional gains in achievable rate are on hand by increasing the combiner branch number

_{ε}/B*L*, and the optimal tradeoff between

*ε*and

*C*shown in Fig. 3 can be used to select code rates in feasible coherent FSO channels.

_{ε}/B## 4. Conclusion

## Acknowledgments

## References

1. | D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE |

2. | J. Proakis, and M. Salehi, |

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

4. | A. Belmonte and J. M. Kahn, “Capacity of coherent free-space optical links using atmospheric compensation techniques,” Opt. Express |

5. | S. Haas and J. H. Shapiro, “Capacity of wireless optical communications,” IEEE J. Sel. Areas Comm. |

6. | E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” J. Select. Areas Commun. |

7. | S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. |

8. | J. A. Anguita, I. B. Djordjevic, M. Neifeld, and B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Netw. |

9. | I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Technol. Lett. |

10. | N. Letzepis, I. Holland, and W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wirel. Comm. |

11. | K. Chakraborty, S. Dey, and M. Franceschetti, “Outage capacity of MIMO Poisson fading channels,” IEEE Trans. Inf. Theory |

12. | N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, ““Performance bounds for free-space optical MIMO systems with APD receivers in atmospheric turbulence,” IEEE,” J. Select. Areas. Commun. |

13. | C. E. Shannon, “A mathematical theory of communications,” Bell Syst. Tech. J. |

14. | R. M. Gagliardi, and S. Karp, |

15. | J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. |

16. | R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. |

17. | J. D. Parsons, “Diversity techniques in communications receivers,” in |

18. | N. Kong, T. Eng, and L. B. Milstein, “A selection combining scheme for rake receivers,” in Proc. ICUPC, Japan, 1995, pp. 426–429. |

**OCIS Codes**

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(060.4510) Fiber optics and optical communications : Optical communications

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: May 13, 2009

Revised Manuscript: July 3, 2009

Manuscript Accepted: July 3, 2009

Published: July 20, 2009

**Citation**

Aniceto Belmonte and Joseph M. Kahn, "Capacity of coherent free-space optical links using diversity-combining techniques," Opt. Express **17**, 12601-12611 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12601

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### References

- D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55(1), 57–77 (1967). [CrossRef]
- J. Proakis, and M. Salehi, Digital Communications, (Mc Graw-Hill, 2007).
- A. Belmonte and J. Khan, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express 16(18), 14151–14162 (2008). [CrossRef] [PubMed]
- A. Belmonte and J. M. Kahn, “Capacity of coherent free-space optical links using atmospheric compensation techniques,” Opt. Express 17(4), 2763–2773 (2009). [CrossRef] [PubMed]
- S. Haas and J. H. Shapiro, “Capacity of wireless optical communications,” IEEE J. Sel. Areas Comm. 21(8), 1346–1357 (2003). [CrossRef]
- E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” J. Select. Areas Commun. 22(9), 1896–1906 (2004). [CrossRef]
- S. G. Wilson, M. Brandt-Pearce, Q. Cao, and J. H. Leveque, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun. 53(8), 1402–1412 (2005). [CrossRef]
- J. A. Anguita, I. B. Djordjevic, M. Neifeld, and B. V. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Netw. 4(9), 586–601 (2005). [CrossRef]
- I. B. Djordjevic, B. Vasic, and M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Technol. Lett. 18(14), 1491–1493 (2006). [CrossRef]
- N. Letzepis, I. Holland, and W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wirel. Comm. 7(5), 1744–1753 (2008). [CrossRef]
- K. Chakraborty, S. Dey, and M. Franceschetti, “Outage capacity of MIMO Poisson fading channels,” IEEE Trans. Inf. Theory 54(11), 4887–4907 (2008). [CrossRef]
- N. Cvijetic, S. G. Wilson, and M. Brandt-Pearce, ““Performance bounds for free-space optical MIMO systems with APD receivers in atmospheric turbulence,” IEEE,” J. Select. Areas. Commun. 26(3), 3–12 (2008). [CrossRef]
- C. E. Shannon, “A mathematical theory of communications,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
- R. M. Gagliardi, and S. Karp, Optical Communications (John Wiley & Sons, 1995).
- J. W. Strohbehn, T. Wang, and J. P. Speck, “On the probability distribution of line-of-sight fluctuations of optical signals,” Radio Sci. 10(1), 59–70 (1975). [CrossRef]
- R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976). [CrossRef]
- J. D. Parsons, “Diversity techniques in communications receivers,” in Advanced Signal Processing, D. A. Creasey, ed. (Peregrinus, 1985), Chap. 6.
- N. Kong, T. Eng, and L. B. Milstein, “A selection combining scheme for rake receivers,” in Proc. ICUPC, Japan, 1995, pp. 426–429.

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