Average capacity of FSO links withtransmit laser selection using
non-uniform OOK signaling overexponential atmospheric turbulence
channels
Antonio Garcıa-Zambrana,1,∗ Beatriz Castillo-Vazquez,1, andCarmen Castillo-Vazquez2
1Department of Communications Engineering, University of Malaga, E-29071 Malaga, Spain2Department of Statistics and Operations Research, University of Malaga,
Abstract: A new upper bound on the capacity of power- and bandwidth-constrained optical wireless links using selection transmit diversity overexponential atmospheric turbulence channels with intensity modulationand direct detection is derived when non-uniform on-off keying (OOK)formats are used. In this strong turbulence free-space optical (FSO) sce-nario, average capacity is investigated subject to an average optical powerconstraint and not only to an average electrical power constraint when thetransmit diversity technique assumed is based on the selection of the opticalpath with a greater value of irradiance. Simulation results for the mutualinformation are further demonstrated to confirm the analytical results fordifferent diversity orders.
References and links1. L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (Bellingham, WA: SPIE
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1. Introduction
Free-space optical (FSO) transmission using intensity modulation and direct detection (IM/DD)can provide high-speed links for a variety of applications and are specially interesting to solvethe “last mile” problem, above all in densely populated urban areas. However, atmospheric tur-bulence produces fluctuations in the irradiance of the transmitted optical beam, which is knownas atmospheric scintillation, severely degrading the link performance [1]. Spatial diversity canbe used over FSO links to mitigate turbulence-induced fading [2]. In [3–5], selection transmitdiversity is proposed for FSO links over strong turbulence channels, where the transmit diver-sity technique based on the selection of the optical path with a greater value of irradiance hasshown to be able to extract full diversity as well as providing better performance comparedto general FSO space-time codes (STCs) designs, such as conventional orthogonal space-timeblock codes (OSTBCs) and repetition codes (RCs). In this paper, a new upper bound on thecapacity of power- and bandwidth-constrained optical wireless links using selection transmitdiversity over exponential atmospheric turbulence channels with intensity modulation and di-rect detection is derived when non-uniform OOK formats are used. Unlike previous capacity
#131127 - $15.00 USD Received 6 Jul 2010; revised 26 Aug 2010; accepted 3 Sep 2010; published 10 Sep 2010(C) 2010 OSA 13 September 2010 / Vol. 18, No. 19 / OPTICS EXPRESS 20446
bounds derived from the classical capacity formula corresponding to the electrical equivalentAWGN channel with uniform input distribution [6, 7], a new closed-form upper bound on thecapacity is here derived from the expression obtained in [8] by bounding the mutual informa-tion subject to an average optical power constraint and not only to an average electrical powerconstraint. This approach is based on the fact that a necessary and sufficient condition betweenaverage optical power and average electrical power constraints is satisfied for OOK signalingwhere an unidimensional space is assumed with one of the two points of the constellation tak-ing the value of 0, corroborating the non-negativity constraint. This bound presents a tighterperformance at lower optical signal-to-noise ratio (SNR) and shows the fact that the input dis-tribution that maximizes the mutual information varies with the turbulence strength and theSNR. Simulation results for the mutual information are further demonstrated to confirm theanalytical results for different diversity orders.
2. System and channel model
We adopt a multiple-input-single-output (MISO) array based on L laser sources, assumed to beintensity-modulated only and all pointed towards a distant photodetector, and to be ideal nonco-herent (direct-detection) receiver. The sources and the detector are physically situated so that alltransmitters are simultaneously observed by the receiver. Additionaly, the fading experiencedbetween source-detector pairs I j is assumed to be statistically independent. Following the trans-mit laser selection (TLS) scheme based on the selection of the optical path with a greater valueof fading gain (irradiance) [3], our MISO system model can be considered as an equivalentsingle-input-single-output (SISO) system model where the channel irradiance corresponding tothe TLS scheme, Im, can be written as
Im = max j=1,2,···L Ij (1)
In this way, having a SISO system as a reference, the instantaneous current in the receivingphotodetector, y(t), can be written as
y(t) = η im(t)x(t)⊗h(t)+ z(t) (2)
where the ⊗ symbol denotes convolution, η is the detector responsivity, assumed hereinafter tobe the unity, X � x(t) represents the optical power supplied by the source, Im � im(t) the equiv-alent real-valued fading gain (irradiance), and h(t) the impulse response of an ideal low-passfilter, which cuts out all frequencies greater than W hertz, modelling the fact that these systemsare intrinsically bandwidth limited due to the use of large inexpensive optoelectronic compo-nents; Z � z(t) is assumed to include any front-end receiver thermal noise as well as shot noisecaused by ambient light much stronger than the desired signal at detector. In this case, the noisecan usually be modeled to high accuracy as AWGN with zero mean and variance σ2 = N0/2,i.e. Z ∼ N(0,N0/2), independent of the on/off state of the received bit. Since the transmittedsignal is an intensity, X must satisfy ∀t x(t) ≥ 0. Due to eye and skin safety regulations, the av-erage optical power is limited and, hence, the average amplitude of X is limited. The receivedelectrical signal Y � y(t), however, can assume negative amplitude values. We use Y , X , Im andZ to denote random variables and y(t), x(t), im(t) and z(t) their corresponding realizations.
Considering a limiting case of strong turbulence conditions [1, 4], the turbulence-inducedfading is modelled as a multiplicative random process which follows the negative exponentialdistribution, whose probability density function (PDF) is given by
fI j(i j) = e−i j , i j ≥ 0 (3)
This PDF has also been adopted in different works [9–14] to describe turbulence-induced fad-ing, leading to an easier mathematical treatment. Since the mean value of this turbulence model
#131127 - $15.00 USD Received 6 Jul 2010; revised 26 Aug 2010; accepted 3 Sep 2010; published 10 Sep 2010(C) 2010 OSA 13 September 2010 / Vol. 18, No. 19 / OPTICS EXPRESS 20447
is E[I j] = 1 and the second moment is given by E[I2j ] = 2, the scintillation index (SIj), a pa-
rameter of interest used to describe the strength of atmospheric fading Ij experienced betweensource-detector pairs [1], is defined as SIj = E[I2
j ]/(E[I j])2 −1 = 1. According to [4], for i.i.d.random variables of {I j} j=1,2,···L, the PDF, fIm(im), of the resulting channel irradiance corre-sponding to the transmit laser selection scheme, Im, is
fIm(im) = LL
∑n=1
(L−1n−1
)(−1)n−1e−nim (4)
where(a
b
)is the binomial coefficient.
We consider OOK formats with any pulse shape and reduced duty cycle, allowing the in-crease of the peak-to-average optical power ratio (PAOPR) parameter. A new basis functionφ(t) is defined as φ(t) = g(t)
/√Eg where g(t) represents any normalized pulse shape satis-
fying the non-negativity constraint, with 0 ≤ g(t) ≤ 1 in the bit period and 0 otherwise, andEg =
∫ ∞−∞ g2(t)dt is the electrical energy. In this way, an expression for the optical intensity can
be written as
x(t) =∞
∑k=−∞
ak(1/p)TbPopt
G( f = 0)g(t − kTb) =
∞
∑k=−∞
ak(1/p)PoptTb
√Eg
G( f = 0)φ (t − kTb) (5)
where G( f = 0) represents the Fourier transform of g(t) evaluated at frequency f = 0, i.e. thearea of the employed pulse shape, and Tb parameter is the bit period. The random variableak follows a Bernoulli distribution with parameter p, taking the values of 0 for the bit “0” (offpulse) and 1 for the bit “1” (on pulse). From this expression, it is easy to deduce that the averageoptical power transmitted is Popt. The constellation here defined for the OOK format using anypulse shape consists of two points (x0 = 0 and x1 = d) in a one-dimensional space with anEuclidean distance of
d = (1/p)Popt
√Tbξ (6)
where ξ = TbEg/G2( f = 0) represents the square of the increment in Euclidean distance dueto the use of a pulse shape of high PAOPR, alternative to the classical rectangular pulse. Thisparameter d represents the distance between the two possible transmitted signals and, hence, inthis case, the square root of the energy of the baseband signal waveform corresponding to thebit “1” (on pulse) [15, Chapter 3]. Assuming h(t) as the impulse response of an ideal low-passfilter, which cuts out all frequencies greater than W hertz, and the use of a matched filter at thereceiver, as in [15, Chapter 6] or [16, Chapter 9], the electrical power of Xrx, random variablecorresponding to the signal at the detector output, conditioned to the irradiance, can be writtenas Pel = pd2i2mθ = (1/p)P2
opt i2mTbξ θ where θ is obtained from
θ =∫ W
−W
∣∣∣∣∣1√Eg
G( f )
∣∣∣∣∣2
d f (7)
with 0 < θ < 1, representing the fact that the channel under study is constrained to κ = 2WTb
degrees of freedom. The channel is assumed to be memoryless, stationary and ergodic, withindependent and identically distributed intensity fast fading statistics. In spite of scintillationis a slow time varying process relative to typical symbol rates of an FSO system, having acoherence time on the order of milliseconds, this approach is valid because temporal correlationcan in practice be overcome by means of long interleavers, being usually assumed both in theanalysis from the point of view of information theory and error rate performance analysis ofcoded FSO links [5–7]. This assumption has to be considered like an ideal scenario where thelatency introduced by the interleaver is not an inconvenience for the required application, beinginterpreted the results so obtained as upper bounds on the system performance.
#131127 - $15.00 USD Received 6 Jul 2010; revised 26 Aug 2010; accepted 3 Sep 2010; published 10 Sep 2010(C) 2010 OSA 13 September 2010 / Vol. 18, No. 19 / OPTICS EXPRESS 20448
3. Upper bound on channel capacity
Considering the channel capacity as a random variable and perfect CSI available at both trans-mitter and receiver [17, 18], we can use the theory derived for discrete-time Gaussian chan-nels [16], expressing the ergodic capacity in bits per channel use as
C = maxp
∫ ∞
0I(X ;Y |im) fIm(im)dim (8)
i.e., the maximum, over all distributions on the input that satisfy the average optical powerconstraint at a level Popt, of the conditional mutual information between the input and output,I(X ;Y |im), averaged over the PDF corresponding to the equivalent turbulence model. It mustbe noted that unlike the approach followed in [6,7,18–20], where the capacity is computed in asimilar way to the capacity of the well-known AWGN channel with BPSK signaling, assumingthe fact that the input distribution that maximizes mutual information is the same regardlessof the channel state, we consider in our system model the impact of a non-uniform input dis-tribution. In this way, the exchange of integration and maximization is not possible because thechannel we consider does not satisfy a compatibility constraint [17], since the input distribu-tion that maximizes mutual information is not the same regardless of the channel state [21–24].The constraint in optical domain implies that E[X2
rx], the second moment of Xrx, takes a valueof up to Pel. Additionally, in our channel model, assuming an unidimensional space where thenon-negativity constraint is satisfied and one of the two points of the constellation takes thevalue of 0, it is easy to deduce that an average electrical power constraint of Pel, and, hence,E[X2] ≤ Pel/(i2mθ), implies an Euclidean distance as d = (1/p)Popt
√Tbξ and, hence, an av-
erage optical power constraint of Popt. Thus, an average electrical power constraint of Pel isnecessary and sufficient condition for satisfying an average optical power constraint of Popt.This is only valid for OOK signaling, representing the basis of our work in order to achieve atighter performance if compared with previously reported bounds. In relation to the equivalentdiscrete-time channel, it must be emphasized that the transmitted optical signal is representedby the random variable X , the atmospheric turbulence-induced signal is represented by theproduct XIm and the corresponding signal performed in electrical domain is represented by Xrx,being the latter the signal to be finally considered in our analysis. As presented in [8], apply-ing the fact that the Gaussian distribution maximizes the entropy over all distributions with thesame variance [16, Theorem 8.6.5], we obtain
I(X ;Y |im) ≤ 12
log2
(1+
σ2Xrx
No/2
)(9)
where σ2Xrx
= E[(Xrx −E[Xrx])2] and represents the variance of the optical signal detected inelectrical domain, resulting in
I(X ;Y |im) ≤ 12
log2
(1+
((1/p)−1)P2opti
2mTbξ θ
No/2
)(10)
This expression bounds the conditional mutual information of the bandlimited optical intensitychannel corrupted by white Gaussian noise with two-sided spectral density of No/2 watts/Hzand average optical power constraint of Popt watts. In this way, assuming that the channel is con-strained to κ dimensions and even without maximizing over the input distribution, the channelcapacity C(γ, p), channel capacity depending on SNR and input distribution p, can be obtainedby averaging over the PDF in (4) as follows
C(γ, p) ≤∫ ∞
0
12
log2
(1+(
1p−1)κξ θγ2i2m
)fIm(im)dim ≤ HB(p) (11)
#131127 - $15.00 USD Received 6 Jul 2010; revised 26 Aug 2010; accepted 3 Sep 2010; published 10 Sep 2010(C) 2010 OSA 13 September 2010 / Vol. 18, No. 19 / OPTICS EXPRESS 20449
where γ = Popt/√
NoW is the SNR definition, as in [25], different to the expression usedin [6, 7, 18, 20, 26], and HB(p) = −p log2 p− (1− p) log2(1− p) represents the entropy of theBernoulli random variable ak in (5), presenting the maximum value achievable because of OOKis the signaling technique considered in this analysis. After a simple transformation of the ran-
dom variable Im as In = Imh(p,γ), being h(p,γ) = γ√
( 1p −1)κξ θ , the above integral can be
evaluated by using [27, eqn. (4.338-1)], obtaining a closed-form solution for C(γ, p) as
C(γ, p) ≤−2LL
∑n=1
(−1)n−1(L−1
n−1
)(si
(n
h(p,γ)
)sin
(n
h(p,γ)
)+ ci
(n
h(p,γ)
)cos
(n
h(p,γ)
))n ln(4)
(12)
where si(·) and ci(·) represent the sine integral and cosine integral functions, respectively [27,eqn. (8.230)]. Knowing that C(γ, p) is also upper bounded by the binary entropy HB(p), theergodic capacity in bits per channel use is obtained by maximizing C(γ, p) over the parameterp as
C1(γ) = maxp
C(γ, p) (13)
4. Numerical results and conclusions
In this section, we numerically evaluate mutual information for our channel model using OOKsignaling to corroborate the tightness of the previous results. For the sake of simplicity [15,Chapter 6], showing the fact that the input distribution that maximizes the mutual informationvaries with the SNR, the statistical channel model is normalized by replacing Y by Y/σ andnow considering X ∈ {0,1}. In this way, our channel model can be rewritten as
Y = AXIm +Z, X ∈ {0,1}, Z ∼ N(0,1) (14)
where A = (1/p)γ√
ξ θκ . The conditional mutual information I(X ;Y |im) for this channel isderived as in [8] as follows
√2π)exp(−y2/2). In this way, even without maximizing
over the input distribution, the mutual information I(X ;Y ), function depending on SNR andinput distribution p, can be numerically obtained by averaging (15) over the PDF in (4) asfollows
I(X ;Y ) =∫ ∞
0I(X ;Y |im) fIm(im)dim (16)
Then, the ergodic capacity in bits per channel use is numerically obtained by maximizing (16)over the parameter p as
C2(γ) = maxp
I(X ;Y ) (17)
This expression is computed using a symbolic mathematics package [28]. In Fig. 1, maximiza-tion of the capacity bound in (12), i.e. C1(γ), and mutual information, i.e. C2(γ), for the expo-nential atmospheric turbulent optical channel and non-turbulent optical channel are displayedfor different diversity orders. It must be commented that the mutual information for the non-turbulent optical channel is numerically solved in a similar way as in (15) but not yet consider-ing the impact of the atmospheric turbulence. In Fig. 1, a value of κ = 20 has been considered
#131127 - $15.00 USD Received 6 Jul 2010; revised 26 Aug 2010; accepted 3 Sep 2010; published 10 Sep 2010(C) 2010 OSA 13 September 2010 / Vol. 18, No. 19 / OPTICS EXPRESS 20450
and, hence, a value of θ = 0.9898 when using a rectangular pulse of duration Tb has beenobtained from (7). For this rectangular pulse shape, the integral in (7) can be written as
θ =∫ κ/2Tb
−κ/2Tb
Tb sin2(πTb f )/(πTb f )2d f (18)
Changing the variable f as q = πTb f and using integration by parts, it is easy to deduce thatθ = (2πκsi(πκ)+2cos(πκ)+π2κ −2)/(π2κ). As a result, a relevant improvement in aver-age capacity for IM/DD exponential atmospheric turbulence FSO links is obtained when atransmit diversity technique based on the selection of the optical path with a greater value ofirradiance is adopted, showing the fact that a non-uniform input signaling improves the channelcapacity.
Fig. 1. Maximization over the input distribution p of the capacity bound, i.e. C1(γ) in (13),and mutual information, i.e. C2(γ) in (17), for the atmospheric turbulent optical channeland the non-turbulent case when κ = 20, a rectangular pulse shape with ξ = 1 and differentdiversity orders are adopted.
This is also corroborated in Fig. 2 where mutual information in (16) versus the input distribu-tion p for different values of SNR and diversity orders is displayed. From this figure it can bededuced the fact that a non-uniform input signaling improves the channel capacity, especiallyat low SNR [23], depending the maximizing input distribution on the SNR and the diversityorder corresponding to the transmit laser selection scheme here analyzed.
In this strong turbulence FSO scenario, it is worth commenting that a greater capacity can beachieved compared with the non-turbulent case when a diversity order L ≥ 4 is assumed. Thiscan be justified from the greater robustness provided by the transmit laser selection schemehere studied against fluctuations in the irradiance of the transmitted optical beam proper to theatmospheric turbulence. A better comprehension of this fact can be achieved by calculatingthe equivalent scintillation index (SIm), used to describe the strength of the equivalent fadinggain Im and, hence, defined as SIm = E[I2
m]/(E[Im])2 − 1, where E[Im] and E[I2m] represent the
mean value and the second moment of the equivalent turbulence model configured by the MISOsystem under study, respectively. The mean value of the equivalent irradiance, E[Im], is obtained
#131127 - $15.00 USD Received 6 Jul 2010; revised 26 Aug 2010; accepted 3 Sep 2010; published 10 Sep 2010(C) 2010 OSA 13 September 2010 / Vol. 18, No. 19 / OPTICS EXPRESS 20451
Fig. 2. Mutual information in (16) versus the input distribution p for values of SNR ofγ = −5 dB 2(a) and γ = −10 dB 2(b) and different diversity orders together with the non-turbulent case.
as follows
E[Im] = LL
∑n=1
(−1)n−1(
L−1n−1
)∫ ∞
0ime−nim dim =
L
∑n=1
(−1)n−1
n
(Ln
)=
L
∑n=1
1n
(19)
where [27, eqn. (0.155-4)] has been used for simplifying the sum. The second moment of the
#131127 - $15.00 USD Received 6 Jul 2010; revised 26 Aug 2010; accepted 3 Sep 2010; published 10 Sep 2010(C) 2010 OSA 13 September 2010 / Vol. 18, No. 19 / OPTICS EXPRESS 20452
equivalent turbulence model, E[I2m], is also obtained as follows
E[I2m] = L
L
∑n=1
(−1)n−1(
L−1n−1
)∫ ∞
0i2me−nim dim = 2
L
∑n=1
(−1)n−1
n2
(Ln
)(20)
In Fig. 3, the mean value (E[Im]) and scintillation index (SIm) of the equivalent turbulence modelconfigured by the MISO system under study versus the number of laser sources L are displayed.
Fig. 3. Mean value (E[Im]) 3(a) and equivalent scintillation index (SIm) 3(b) of the equiv-alent turbulence model configured by the MISO system under study versus the number oflaser sources L.
From this figure, it can be observed that the greater the number of laser sources, the greaterthe mean value and the lower the equivalent scintillation index are obtained. In this fashion,the better performance in terms of capacity corresponding to the TLS scheme for values ofL ≥ 4 if compared with the non-turbulent case can be explained from the decreasing strengthof the equivalent fading gain as L is increased together with the fact that the mean value ofthe equivalent fading gain and, hence, the resulting SNR has been improved more than double.This can also explain that the parameter p tends the value of 0.5 as L is increased, as shownin Fig. 2. In this fashion, since a non-uniform input signaling improves the channel capacityespecially at low SNR [8, 23], the increasing mean value of the equivalent fading gain impliesa higher and higher equivalent SNR and, hence, the fact that the maximizing input distributiontends to be closer and closer to uniform signaling.
#131127 - $15.00 USD Received 6 Jul 2010; revised 26 Aug 2010; accepted 3 Sep 2010; published 10 Sep 2010(C) 2010 OSA 13 September 2010 / Vol. 18, No. 19 / OPTICS EXPRESS 20453
Finally, from the impact of the parameter ξ in (6), expression corresponding to the Eu-clidean distance in the constellation here defined for the OOK format, it is shown that a rele-vant improvement in performance must be noted as a consequence of the pulse shape used [8],concluding that an increase of the PAOPR provides higher capacity values. From the relevantimprovement in terms of capacity here obtained when using non-uniform signaling OOK andthe transmit diversity technique based on the selection of the optical path with a greater valueof irradiance is adopted over exponential atmospheric turbulence channels, investigating theperformance in alternative FSO scenarios covering a wider range of atmospheric turbulenceconditions as well as incorporating pointing error effects are interesting topics for future re-search.
Acknowledgments
The authors would like to thank the anonymous reviewers for their useful comments that helpedto improve the presentation of the paper.
#131127 - $15.00 USD Received 6 Jul 2010; revised 26 Aug 2010; accepted 3 Sep 2010; published 10 Sep 2010(C) 2010 OSA 13 September 2010 / Vol. 18, No. 19 / OPTICS EXPRESS 20454
