- Research Article
- Open Access

# On the Capacity of FSO Links over Gamma-Gamma Atmospheric Turbulence Channels Using OOK Signaling

- Antonio García-Zambrana
^{1}Email author, - Carmen Castillo-Vázquez
^{2}and - Beatriz Castillo-Vázquez
^{1}

**2010**:127657

https://doi.org/10.1155/2010/127657

© Antonio García-Zambrana et al. 2010

**Received:**27 November 2009**Accepted:**26 March 2010**Published:**9 May 2010

## Abstractg

A new upper bound on the capacity of power- and bandwidth-constrained optical wireless links over gamma-gamma atmospheric turbulence channels with intensity modulation and direct detection is derived when on-off keying (OOK) formats are used. In this free-space optical (FSO) scenario, unlike previous capacity bounds derived from the classic capacity of the well-known additive white Gaussian noise (AWGN) channel with uniform input distribution, a new closed-form upper bound on the capacity is found by bounding the mutual information subject to an average optical power constraint and not only to an average electrical power constraint, showing the fact that the input distribution that maximizes the mutual information varies with the turbulence strength and the signal-to-noise ratio (SNR). Additionally, it is shown that an increase of the peak-to-average optical power ratio (PAOPR) provides higher capacity values. Simulation results for the mutual information are further demonstrated to confirm the analytical results under several turbulence conditions.

## Keywords

- Mutual Information
- Channel State Information
- Channel Capacity
- Atmospheric Turbulence
- Ergodic Capacity

## 1. Introduction

Optical wireless communications using intensity modulation and direct detection (IM/DD) can provide high-speed links for a variety of applications [1], providing an unregulated spectral segment and high security. Here, the transmit power must be constrained by power consumption concerns and eye-safety considerations. Moreover, these systems are intrinsically bandwidth limited due to the use of large inexpensive optoelectronic components. Recently, the use of atmospheric free-space optical (FSO) transmission is being specially interesting to solve the *"last mile" problem*, above all in densely populated urban areas, as well as a supplement to radio-frequency (RF) links [2] and the recent development of radio on free-space optical links (RoFSOLs) [3, 4]. However, atmospheric turbulence produces fluctuations in the irradiance of the transmitted optical beam, which is known as *atmospheric scintillation*, severely degrading the link performance [5, 6].

An upper bound on the capacity of the indoor optical wireless channel was determined in [7] for the specific case of multicarrier systems where the average optical amplitude in each disjoint symbol interval is fixed. By contrast, Hranilovic and Kschischang determine in [8] an upper bound by not assuming a particular signaling set and allowing for the average optical amplitude of each symbol to vary. This upper bound is improved at low signal-to-noise ratio for IM/DD channels with pulse amplitude modulation in [9]. In [10], a new closed-form upper bound on the capacity is found through a sphere-packing argument for channels using equiprobable binary pulse amplitude modulation (PAM) and subject to an average optical power constraint, presenting a tighter performance at lower optical signal-to-noise ratio (SNR) if compared with [8]. Recently, using a dual expression for channel capacity introduced in [11], Lapidoth et al. have derived new upper bounds on the capacity of the indoor optical wireless channel when the input is constrained in both its average and its peak power [12]. In the analysis of the capacity of the atmospheric FSO channel, several works can be cited [13–22]. In [13], numerical results for the capacity of gamma-gamma atmospheric turbulence channels using on-off keying (OOK) formats are presented by maximizing the mutual information for this channel over a binomial input distribution. In [14, 15], the capacity of log-normal optical wireless channel with OOK formats is computed for known channel state information (CSI) in a similar way to the capacity of the well-known additive white Gaussian noise (AWGN) channel with binary phase shift keying (BPSK) signaling, assuming the fact that the input distribution that maximizes mutual information is the same regardless of the channel state. In [16–18], closed-form mathematical expressions for the evaluation of the average channel capacity are presented when log-normal and gamma-gamma models are adopted for the atmospheric turbulence, assuming the same considerations as in [14, 15]. In [19], the availability of CSI and the effects of channel memory on the capacities of FSO communications channels are investigated by adopting an approach as in [14–18], using a definition of SNR proper to RF fading channels where performance depends on the average power of the electrical current, obtained by the conversion from the optical signal. In [20], closed form expressions for the bit-error rate and the outage probability are presented when pointing errors effects are considered. In [21], ergodic capacity is numerically evaluated for turbulence channels with pointing errors using OOK formats. Recently, Farid and Hranilovic have considered in [22] the design of capacity-approaching, nonuniform optical intensity signaling in the presence of average and peak amplitude constraints, presenting a practical algorithm by using multilevel coding followed by a mapper and multistage decoding at the receiver. The analysis of the channel capacity for alternative FSO scenarios has been considered in [23–25].

In this paper, a new upper bound on the capacity of power- and bandwidth-constrained optical wireless links over gamma-gamma atmospheric turbulence channels with intensity modulation and direct detection is derived when OOK formats are used. Because FSO channel is envisioned as the solution to the convectivity bottleneck problem and as a supplement to RF links, the complexity of transmitter and receiver must be low. Therefore, the use of IM/DD links with OOK formats is proposed as a reasonable choice. In this FSO scenario, unlike previous capacity bounds derived from the classical capacity formula corresponding to the electrical equivalent AWGN channel with uniform input distribution, a new closed-form upper bound on the capacity is found by bounding the mutual information subject to an average optical power constraint and not only to an average electrical power constraint, being considered in our system model the impact of a nonuniform input distribution. This new approach is based on the fact that a necessary and sufficient condition between average optical power and average electrical power constraints is satisfied for OOK signaling where an unidimensional space is assumed with one of the two points of the constellation taking the value of 0, corroborating the nonnegativity constraint. This bound presents a tighter performance at lower optical SNR if compared with previously reported bounds and shows the fact that the input distribution that maximizes the mutual information varies with the turbulence strength and the SNR. Additionally, it is shown that an increase of the peak-to-average optical power ratio (PAOPR) provides higher capacity values. Simulation results for the mutual information are further demonstrated to confirm the analytical results under several turbulence conditions.

## 2. Atmospheric Turbulence Channel Model

*f*= 0 represents the Fourier transform of evaluated at frequency

*f*= 0, that is, the area of the employed pulse shape. The random variable (RV) follows a Bernoulli distribution with parameter , taking the values of 0 for the bit "0" (off pulse) and 1 for the bit "1" (on pulse). From this expression, it is easy to deduce that the average optical power transmitted is . The constellation here defined for the OOK format using any pulse shape consists of two points in a one-dimensional space with an Euclidean distance of where represents the square of the increment in Euclidean distance due to the use of a pulse shape of high PAOPR, alternative to the classical rectangular pulse. Assuming maximum-likelihood detection and as the impulse response of an ideal lowpass filter, which cuts out all frequencies greater than hertz, the electrical power of , signal corresponding to at the detector output, conditionated to the irradiance, can be written as where is obtained from

with , representing the fact that the channel under study is constrained to degrees of freedom. In this way, the bandwidth constraint in our analysis is subject to the channel and not to the signaling technique, as in [8]. In our opinion, this is closer to the real scenario. It must be noted that the intersymbol interference between successive code words is considered negligible, assuming that this channel is able to support the transmission of at most dimensions per symbol. With the aid of the converse to the coding theorem it is easy to show that the intersymbol interference cannot reduce error probability. There is no problem since we can transmit, in principle, only one code word of arbitrarily long duration, showing that arbitrarily small error probabilities can be achieved at any rate less than capacity [30, Section ]. The channel is assumed to be memoryless, stationary, and ergodic, with independent and identically distributed intensity fast fading statistics. Although scintillation is a slow time varying process relative to typical symbol rates of an FSO system, having a coherence time on the order of milliseconds, this approach is valid because temporal correlation can in practice be overcome by means of long interleavers, being usually assumed both in the analysis from the point of view of information theory and error rate performance analysis of coded FSO links [13, 29, 31]. This assumption has to be considered like an ideal scenario where the latency introduced by the interleaver is not an inconvenience for the required application, being interpreted the results so obtained as upper bounds on the system performance. We also consider that the channel state information is available at both transmitter and receiver. In this way, the channel capacity must be considered as a random variable following the gamma-gamma distribution corresponding to the atmospheric turbulence model and, hence, its average value, known as ergodic capacity, will indicate the average best rate for error-free transmission [16–19].

## 3. Upper Bound on Channel Capacity

that is, the maximum, over all distributions on the input that satisfy the average optical power constraint at a level , of the conditional mutual information between the input and output, , averaged over the PDF in (2). It must be noted that unlike the approach followed in [14–18], where the capacity is computed in a similar way to the capacity of the well-known AWGN channel with BPSK signaling, assuming the fact that the input distribution that maximizes mutual information is the same regardless of the channel state, we consider in our system model the impact of a nonuniform input distribution. In this way, the exchange of integration and maximization is not possible because the channel we consider does not satisfy a compatibility constraint [32], since the input distribution that maximizes mutual information is not the same regardless of the channel state, as also considered in [13, 34, 35].

For the sake of easy comparison, we present a closed-form expression in terms of the Meijer's G-function following a similar approach as in works in the same context [16–18]. Nonetheless, it must be commented that Meijer's G-function has to be numerically calculated and, hence, the use of Monte Carlo integration to solve (10) may represent an alternative with less computational load.

## 4. Numerical Results

as in [13, 19, 21], where , , , and . Then, substituting (14) in (7), the ergodic capacity is numerically obtained after maximizing over the expectation with respect to the PDF in (2) of the conditional mutual information. This expression is computed using a symbolic mathematics package [37].

### 4.1. No Atmospheric Turbulence

where and are obtained as explained in [9], depending on SNR values. As a result, the new bound derived in (15) yields superior tightness over the bound in (17) and (18). It can be corroborated that the superiority of the proposed upper bound is even more significant when the value of is lower. Recently, using a dual expression for channel capacity introduced in [11], Lapidoth et al. have derived new upper bounds on the capacity of the indoor optical wireless channel when the input is constrained in both its average and its peak power [12]. They also present results on the asymptotic capacity at low power, showing precise results when an average- and a peak-power constraint are imposed, presenting asymptotic upper and lower bounds whose ratio tends to 1 as the power tends to 0. Nonetheless, this ratio tends to as the power tends to 0 when only an average-power constraint is imposed, context in which the upper bound proposed in this paper is evaluated.

### 4.2. With Gamma-Gamma Atmospheric Turbulence

## 5. Conclusions

As a result, a new upper bound on the capacity of power- and bandwidth-constrained optical wireless links over gamma-gamma atmospheric turbulence channels with intensity modulation and direct detection is derived when OOK formats are used. In this FSO scenario, unlike previous capacity bounds derived from the classic capacity of the well-known AWGN channel with uniform input distribution, a new closed-form upper bound on the capacity is found by bounding the mutual information subject to an average optical power constraint and not only to an average electrical power constraint. This bound presents a tighter performance at lower optical SNR if compared with previously reported bounds and shows the fact that the input distribution that maximizes the mutual information varies with the turbulence strength and the SNR. Additionally, it is shown that an increase of the PAOPR provides higher capacity values. Simulation results for the mutual information are further demonstrated to confirm the analytical results under different turbulence conditions. From the results here obtained when only an average-power constraint is imposed, investigating the impact of an input constrained in both its average and its peak power as well as misalignment fading on the system model here proposed for representing OOK signaling is an interesting topic for future research.

## Declarations

### Acknowledgment

The authors are grateful for financial support from the Junta de Andalucía (research group "Communications Engineering (TIC-0102)").

## Authors’ Affiliations

## References

- Kahn JM, Barry JR: Wireless infrared communications.
*Proceedings of the IEEE*1997, 85(2):265-298. 10.1109/5.554222View ArticleGoogle Scholar - Stotts LB, Andrews LC, Cherry PC,
*et al*.: Hybrid optical RF airborne communications.*Proceedings of the IEEE*2009, 97(6):1109-1127.View ArticleGoogle Scholar - Lim W, Yun C, Kim K: BER performance analysis of radio over free-space optical systems considering laser phase noise under Gamma-Gamma turbulence channels.
*Optics Express*2009, 17(6):4479-4484. 10.1364/OE.17.004479View ArticleGoogle Scholar - Tsukamoto K, Hashimoto A, Aburakawa Y, Matsumoto M: The case for free space.
*IEEE Microwave Magazine*2009, 10(5):84-92.View ArticleGoogle Scholar - Andrews L, Phillips R, Hopen C:
*Laser Beam Scintillation with Applications*. SPIE Press, Bellingham, Wash, USA; 2001.View ArticleGoogle Scholar - Zhu X, Kahn JM: Free-space optical communication through atmospheric turbulence channels.
*IEEE Transactions on Communications*2002, 50(8):1293-1300. 10.1109/TCOMM.2002.800829View ArticleGoogle Scholar - You R, Kahn JM: Upper-bounding the capacity of optical IM/DD channels with multiple-subcarrier modulation and fixed bias using trigonometric moment space method.
*IEEE Transactions on Information Theory*2002, 48(2):514-523. 10.1109/18.979327MATHMathSciNetView ArticleGoogle Scholar - Hranilovic S, Kschischang FR: Capacity bounds for power- and band-limited optical intensity channels corrupted by Gaussian noise.
*IEEE Transactions on Information Theory*2004, 50(5):784-795. 10.1109/TIT.2004.826649MATHMathSciNetView ArticleGoogle Scholar - Farid AA, Hranilovic S: Upper and lower bounds on the capacity of wireless optical intensity channels.
*Proceedings of the IEEE International Symposium on Information Theory (ISIT '07), June 2007, Nice, France*2416-2420.Google Scholar - Garcia-Zambrana A, del Castillo-Vazquez B: Improved upper bound on capacity of optical IM/DD channels using binary pulse amplitude modulation.
*Electronics Letters*2008, 44(12):760-761. 10.1049/el:20080595View ArticleGoogle Scholar - Lapidoth A, Moser SM: Capacity bounds via duality with applications to multiple-antenna systems on flat-fading channels.
*IEEE Transactions on Information Theory*2003, 49(10):2426-2467. 10.1109/TIT.2003.817449MATHMathSciNetView ArticleGoogle Scholar - Lapidoth A, Moser SM, Wigger MA: On the capacity of free-space optical intensity channels.
*IEEE Transactions on Information Theory*2009, 55(10):4449-4461.MathSciNetView ArticleGoogle Scholar - Anguita JA, Djordjevic IB, Neifeld MA, Vasic BV: Shannon capacities and error-correction codes for optical atmospheric turbulent channels.
*Journal of Optical Networking*2005, 4(9):586-601. 10.1364/JON.4.000586View ArticleGoogle Scholar - Li J, Uysal M: Optical wireless communications: system model, capacity and coding.
*Proceedings of the 58th IEEE Vehicular Technology Conference (VTC '03), October 2003, Orlando, Fla, USA*1: 168-172.Google Scholar - Li J, Uysal M: Achievable information rate for outdoor free space optical communication with intensity modulation and direct detection.
*Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM #39;03), 2003, San Francisco, Calif, USA*5: 2654-2658.View ArticleGoogle Scholar - Sandalidis HG, Tsiftsis TA: Outage probability and ergodic capacity of free-space optical links over strong turbulence.
*Electronics Letters*2008, 44(1):46-47. 10.1049/el:20082495View ArticleGoogle Scholar - Nistazakis HE, Karagianni EA, Tsigopoulos AD, Fafalios ME, Tombras GS: Average capacity of optical wireless communication systems over atmospheric turbulence channels.
*Journal of Lightwave Technology*2009, 27(8):974-979.View ArticleGoogle Scholar - Nistazakis HE, Tsiftsis TA, Tombras GS: Performance analysis of free-space optical communication systems over atmospheric turbulence channels.
*IET Communications*2009, 3(8):1402-1409. 10.1049/iet-com.2008.0212View ArticleGoogle Scholar - Denic SZ, Djordjevic I, Anguita J, Vasic B, Neifeld MA: Information theoretic limits for free-space optical channels with and without memory.
*Journal of Lightwave Technology*2008, 26(19):3376-3384.View ArticleGoogle Scholar - Sandalidis HG: Optimization models for misalignment fading mitigation in optical wireless links.
*IEEE Communications Letters*2008, 12(5):395-397.View ArticleGoogle Scholar - Borah DK, Voelz DG: Pointing error effects on free-space optical communication links in the presence of atmospheric turbulence.
*Journal of Lightwave Technology*2009, 27(18):3965-3973.View ArticleGoogle Scholar - Farid A, Hranilovic S: Channel capacity and non-uniform signalling for free-space optical intensity channels.
*IEEE Journal on Selected Areas in Communications*2009, 27(9):1553-1563.View ArticleGoogle Scholar - Haas SM, Shapiro JH: Capacity of wireless optical communications.
*IEEE Journal on Selected Areas in Communications*2003, 21(8):1346-1357. 10.1109/JSAC.2003.816618View ArticleGoogle Scholar - Belmonte A, Kahn JM: Capacity of coherent free-space optical links using atmospheric compensation techniques.
*Optics Express*2009, 17(4):2763-2773. 10.1364/OE.17.002763View ArticleGoogle Scholar - Belmonte A, Kahn JM: Capacity of coherent free-space optical links using diversity-combining techniques.
*Optics Express*2009, 17(15):12601-12611. 10.1364/OE.17.012601View ArticleGoogle Scholar - Hranilovic S, Kschischang FR: Optical intensity-modulated direct detection channels: signal space and lattice codes.
*IEEE Transactions on Information Theory*2003, 49(6):1385-1399. 10.1109/TIT.2003.811928MATHMathSciNetView ArticleGoogle Scholar - Al-Habash MA, Andrews LC, Phillips RL: Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media.
*Optical Engineering*2001, 40(8):1554-1562. 10.1117/1.1386641View ArticleGoogle Scholar - Gradshteyn IS, Ryzhik IM:
*Table of Integrals, Series and Products*. 7th edition. Academic Press, New York, NY, USA; 2007.MATHGoogle Scholar - Uysal M, Li J, Yu M: Error rate performance analysis of coded free-space optical links over gamma-gamma atmospheric turbulence channels.
*IEEE Transactions on Wireless Communications*2006, 5(6):1229-1233.View ArticleGoogle Scholar - Gallager RG:
*Information Theory and Reliable Communications*. John Wiley & Sons, New York, NY, USA; 1968.Google Scholar - Djordjevic IB, Denic S, Anguita J, Vasic B, Neifeld MA: LDPC-coded MIMO optical communication over the atmospheric turbulence channel.
*Journal of Lightwave Technology*2008, 26(5):478-487.View ArticleGoogle Scholar - Goldsmith AJ, Varaiya PP: Capacity of fading channels with channel side information.
*IEEE Transactions on Information Theory*1997, 43(6):1986-1992. 10.1109/18.641562MATHMathSciNetView ArticleGoogle Scholar - Cover TM, Thomas JA:
*Elements of Information Theory*. 2nd edition. John Wiley & Sons, New York, NY, USA; 2006.MATHGoogle Scholar - Farid AA, Hranilovic S: Design of non-uniform capacity-approachingsignaling for optical wireless intensity channels.
*Proceedings of the IEEE International Symposium on Information Theory (ISIT '08), July 2008, Toronto, Canada*2327-2331.Google Scholar - Farid AA, Hranilovic S: Outage capacity with non-uniform signaling for free-space optical channels.
*Proceedings of the 24th Biennial Symposium on Communications (BSC '08), June 2008, Kingston, Canada*204-207.Google Scholar - Adamchik VS, Marichev OI: Algorithm for calculating integrals of hypergeometric type functions and its realization in reduce system.
*Proceedings of the International Symposium on Symbolic and Algebraic Computation, 1990, Tokyo, Japan*212-224.View ArticleGoogle Scholar - Wolfram Research :
*Mathematica, Version 7.0*. Wolfram Research, Champaign, Ill, USA; 2008.Google Scholar - Abramowitz M, Stegun IA:
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*. 9th edition. Dover, New York, NY, USA; 1970.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.