## Abstract

A new upper bound on the capacity of power- and bandwidth-constrained optical wireless links using selection transmit diversity over exponential atmospheric turbulence channels with intensity modulation and direct detection is derived when non-uniform on-off keying (OOK) formats are used. In this strong turbulence free-space optical (FSO) scenario, average capacity is investigated subject to an average optical power constraint and not only to an average electrical power constraint when the transmit diversity technique assumed is based on the selection of the optical path with a greater value of irradiance. Simulation results for the mutual information are further demonstrated to confirm the analytical results for different diversity orders.

© 2010 Optical Society of America

## 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 solve the “*last mile*” *problem*, above all in densely populated urban areas. However, atmospheric turbulence produces fluctuations in the irradiance of the transmitted optical beam, which is known as *atmospheric scintillation*, severely degrading the link performance [1]. Spatial diversity can be used over FSO links to mitigate turbulence-induced fading [2]. In [3–5], selection transmit diversity is proposed for FSO links over strong turbulence channels, where the transmit diversity technique based on the selection of the optical path with a greater value of irradiance has shown to be able to extract full diversity as well as providing better performance compared to general FSO space-time codes (STCs) designs, such as conventional orthogonal space-time block codes (OSTBCs) and repetition codes (RCs). In this paper, a new upper bound on the capacity of power- and bandwidth-constrained optical wireless links using selection transmit diversity over exponential atmospheric turbulence channels with intensity modulation and direct detection is derived when non-uniform OOK formats are used. Unlike previous capacity bounds derived from the classical capacity formula corresponding to the electrical equivalent AWGN channel with uniform input distribution [6, 7], a new closed-form upper bound on the capacity is here derived from the expression obtained in [8] by bounding the mutual information subject to an average optical power constraint and not only to an average electrical power constraint. This 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 non-negativity constraint. This bound presents a tighter performance at lower optical signal-to-noise ratio (SNR) and shows the fact that the input distribution that maximizes the mutual information varies with the turbulence strength and the SNR. Simulation results for the mutual information are further demonstrated to confirm the analytical 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 be intensity-modulated only and all pointed towards a distant photodetector, and to be ideal noncoherent (direct-detection) receiver. The sources and the detector are physically situated so that all transmitters are simultaneously observed by the receiver. Additionaly, the fading experienced between source-detector pairs *I _{j}* is assumed to be statistically independent. Following the transmit laser selection (TLS) scheme based on the selection of the optical path with a greater value of fading gain (irradiance) [3], our MISO system model can be considered as an equivalent single-input-single-output (SISO) system model where the channel irradiance corresponding to the TLS scheme,

*I*, can be written as

_{m}In this way, having a SISO system as a reference, the instantaneous current in the receiving photodetector, *y*(*t*), can be written as

where the ⊗ symbol denotes convolution, *η* is the detector responsivity, assumed hereinafter to be the unity, *X* ≜ *x*(*t*) represents the optical power supplied by the source, *I _{m}* ≜

*i*(

_{m}*t*) the equivalent real-valued fading gain (irradiance), and

*h*(

*t*) the impulse response of an ideal low-pass filter, which cuts out all frequencies greater than

*W*hertz, modelling the fact that these systems are intrinsically bandwidth limited due to the use of large inexpensive optoelectronic components;

*Z*≜

*z*(

*t*) is assumed to include any front-end receiver thermal noise as well as shot noise caused by ambient light much stronger than the desired signal at detector. In this case, the noise can usually be modeled to high accuracy as AWGN with zero mean and variance

*σ*

^{2}=

*N*

_{0}/2, i.e.

*Z*~

*N*(0,

*N*

_{0}/2), independent of the on/off state of the received bit. Since the transmitted signal is an intensity,

*X*must satisfy ∀

*t*

*x*(

*t*) ≥ 0. Due to eye and skin safety regulations, the average optical power is limited and, hence, the average amplitude of

*X*is limited. The received electrical signal

*Y*≜

*y*(

*t*), however, can assume negative amplitude values. We use

*Y*,

*X*,

*I*and

_{m}*Z*to denote random variables and

*y*(

*t*),

*x*(

*t*),

*i*(

_{m}*t*) and

*z*(

*t*) their corresponding realizations.

Considering a limiting case of strong turbulence conditions [1, 4], the turbulence-induced fading is modelled as a multiplicative random process which follows the negative exponential distribution, whose probability density function (PDF) is given by

This PDF has also been adopted in different works [9–14] to describe turbulence-induced fading, leading to an easier mathematical treatment. Since the mean value of this turbulence model is *E*[*I _{j}*] = 1 and the second moment is given by

*E*[

*I*

^{2}

_{j}] = 2, the scintillation index (

*SI*), a parameter of interest used to describe the strength of atmospheric fading

_{j}*I*experienced between source-detector pairs [1], is defined as

_{j}*SI*=

_{j}*E*[

*I*

^{2}

_{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, ${f}_{{I}_{m}}\left({i}_{m}\right)$ , of the resulting channel irradiance corresponding to the transmit laser selection scheme,

*I*, is

_{m}where $\left(\begin{array}{c}a\\ b\end{array}\right)$ is the binomial coefficient.

We consider OOK formats with any pulse shape and reduced duty cycle, allowing the increase of the peak-to-average optical power ratio (PAOPR) parameter. A new basis function *ϕ*(*t*) is defined as
$\varphi \left(t\right)=\frac{g\left(t\right)}{\sqrt{{E}_{g}}}$
where *g*(*t*) represents any normalized pulse shape satisfying the non-negativity constraint, with 0 ≤ *g*(*t*) ≤ 1 in the bit period and 0 otherwise, and *E _{g}* =

*∫*

^{∞}

_{−∞}

*g*

^{2}(

*t*)

*dt*is the electrical energy. In this way, an expression for the optical intensity can be written as

where *G*(*f* = 0) represents the Fourier transform of *g*(*t*) evaluated at frequency *f* = 0, i.e. the area of the employed pulse shape, and *T _{b}* parameter is the bit period. The random variable

*a*follows a Bernoulli distribution with parameter

_{k}*p*, 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

*P*

_{opt}. The constellation here defined for the OOK format using any pulse shape consists of two points (

*x*

_{0}= 0 and

*x*

_{1}=

*d*) in a one-dimensional space with an Euclidean distance of

where *ξ* = *T _{b}E_{g}*/

*G*

^{2}(

*f*= 0) 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. This parameter

*d*represents the distance between the two possible transmitted signals and, hence, in this case, the square root of the energy of the baseband signal waveform corresponding to the bit “1” (on pulse) [15, Chapter 3]. Assuming

*h*(

*t*) as the impulse response of an ideal low-pass filter, which cuts out all frequencies greater than

*W*hertz, and the use of a matched filter at the receiver, as in [15, Chapter 6] or [16, Chapter 9], the electrical power of

*X*

_{rx}, random variable corresponding to the signal at the detector output, conditioned to the irradiance, can be written as

*P*

_{el}=

*pd*

^{2}

*i*

^{2}

*= (1/*

_{m}θ*p*)

*P*

^{2}

_{opt}i^{2}

*where*

_{m}T_{b}ξθ*θ*is obtained from

with 0 < *θ* < 1, representing the fact that the channel under study is constrained to *κ* = 2*WT _{b}* degrees of freedom. The channel is assumed to be memoryless, stationary and ergodic, with independent and identically distributed intensity fast fading statistics. In spite of 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 [5–7]. 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.

## 3. Upper bound on channel capacity

Considering the channel capacity as a random variable and perfect CSI available at both transmitter and receiver [17, 18], we can use the theory derived for discrete-time Gaussian channels [16], expressing the ergodic capacity in bits per channel use as

i.e., the maximum, over all distributions on the input that satisfy the average optical power constraint at a level *P*
_{opt}, of the conditional mutual information between the input and output, *I*(*X*;*Y*∣*i _{m}*), averaged over the PDF corresponding to the equivalent turbulence model. It must be noted that unlike the approach followed in [6, 7, 18–20], 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 non-uniform 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 [17], since the input distribution that maximizes mutual information is not the same regardless of the channel state [21–24]. The constraint in optical domain implies that

*E*[

*X*

^{2}

_{rx}], the second moment of

*X*

_{rx}, takes a value of up to

*P*

_{el}. Additionally, in our channel model, assuming an unidimensional space where the non-negativity constraint is satisfied and one of the two points of the constellation takes the value of 0, it is easy to deduce that an average electrical power constraint of

*P*

_{el}, and, hence,

*E*[

*X*

^{2}] ≤

*P*

_{el}/(

*i*

^{2}

*), implies an Euclidean distance as $d=\left(\frac{1}{p}\right){P}_{\mathrm{opt}}\sqrt{{T}_{b}\xi}$ and, hence, an average optical power constraint of*

_{m}θ*P*

_{opt}. Thus, an average electrical power constraint of

*P*

_{el}is necessary and sufficient condition for satisfying an average optical power constraint of

*P*

_{opt}. This is only valid for OOK signaling, representing the basis of our work in order to achieve a tighter performance if compared with previously reported bounds. In relation to the equivalent discrete-time channel, it must be emphasized that the transmitted optical signal is represented by the random variable

*X*, the atmospheric turbulence-induced signal is represented by the product

*XI*and the corresponding signal performed in electrical domain is represented by

_{m}*X*

_{rx}, being the latter the signal to be finally considered in our analysis. As presented in [8], applying the fact that the Gaussian distribution maximizes the entropy over all distributions with the same variance [16, Theorem 8.6.5], we obtain

where ${\sigma}_{{X}_{rx}}^{2}=E\left[{({X}_{rx}-E\left[{X}_{rx}\right])}^{2}\right]$ and represents the variance of the optical signal detected in electrical domain, resulting in

This expression bounds the conditional mutual information of the bandlimited optical intensity channel corrupted by white Gaussian noise with two-sided spectral density of *N _{o}*/2 watts/Hz and average optical power constraint of

*P*

_{opt}watts. In this way, assuming that the channel is constrained to

*κ*dimensions and even without maximizing over the input distribution, the channel capacity

*C*(

*γ*,

*p*), channel capacity depending on SNR and input distribution

*p*, can be obtained by averaging over the PDF in (4) as follows

where
$\gamma =\frac{{P}_{\mathrm{opt}}}{\sqrt{{N}_{o}W}}$
is the SNR definition, as in [25], different to the expression used in [6, 7, 18, 20, 26], and *H _{B}*(

*p*) = −

*p*log

_{2}

*p*− (1 −

*p*) log

_{2}(1 −

*p*) represents the entropy of the Bernoulli random variable

*a*in (5), presenting the maximum value achievable because of OOK is the signaling technique considered in this analysis. After a simple transformation of the random variable

_{k}*I*as

_{m}*I*=

_{n}*I*(

_{m}h*p,γ*), being $h(p,\gamma )=\gamma \sqrt{(\frac{1}{p}-1)\kappa \xi \theta}$ , the above integral can be evaluated by using [27, eqn. (4.338-1)], obtaining a closed-form solution for

*C*(

*γ, p*) as

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 *H _{B}*(

*p*), the ergodic capacity in bits per channel use is obtained by maximizing

*C*(

*γ, p*) over the parameter

*p*as

## 4. Numerical results and conclusions

In this section, we numerically evaluate mutual information for our channel model using OOK signaling 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 information varies with the SNR, the statistical channel model is normalized by replacing *Y* by *Y*/*σ* and now considering *X* ∈ {0,1}. In this way, our channel model can be rewritten as

where
$A=\left(\frac{1}{p}\right)\gamma \sqrt{\xi \theta \kappa}$
. The conditional mutual information *I*(*X*;*Y*∣*i _{m}*) for this channel is derived as in [8] as follows

where *P _{X}* (

*x*= 1) =

*p*,

*P*(

_{X}*x*= 0) = 1 −

*p*, ${f}_{Y}(y\mid x=1,{i}_{m})=\left(\frac{1}{\sqrt{2\pi}}\right)\mathrm{exp}\left(\frac{-{(y-A{i}_{m})}^{2}}{2}\right)$ , and ${f}_{Y}(y\mid x=0,{i}_{m})={f}_{Y}(y\mid x=0)=\left(\frac{1}{\sqrt{2\pi}}\right)\mathrm{exp}\left(\frac{-{y}^{2}}{2}\right)$ . In this way, even without maximizing over the input distribution, the mutual information

*I*(

*X*;

*Y*), function depending on SNR and input distribution

*p*, can be numerically obtained by averaging (15) over the PDF in (4) as follows

Then, the ergodic capacity in bits per channel use is numerically obtained by maximizing (16) over the parameter *p* as

This expression is computed using a symbolic mathematics package [28]. In Fig. 1, maximization of the capacity bound in (12), i.e. *C*
_{1}(*γ*), and mutual information, i.e. *C*
_{2}(*γ*), for the exponential atmospheric turbulent optical channel and non-turbulent optical channel are displayed for 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 considering the impact of the atmospheric turbulence. In Fig. 1, a value of *κ* = 20 has been considered and, hence, a value of *θ* = 0.9898 when using a rectangular pulse of duration *T _{b}* has been obtained from (7). For this rectangular pulse shape, the integral in (7) can be written as

Changing the variable *f* as *q* = *πT _{b}f* and using integration by parts, it is easy to deduce that

*θ*= (2

*πκ*si(

*πκ*) + 2cos(

*πκ*) +

*π*

^{2}

*κ*− 2)/(

*π*

^{2}

*κ*). As a result, a relevant improvement in average capacity for IM/DD exponential atmospheric turbulence FSO links is obtained when a transmit diversity technique based on the selection of the optical path with a greater value of irradiance is adopted, showing the fact that a non-uniform input signaling improves the channel capacity.

This is also corroborated in Fig. 2 where mutual information in (16) versus the input distribution *p* for different values of SNR and diversity orders is displayed. From this figure it can be deduced the fact that a non-uniform input signaling improves the channel capacity, especially at low SNR [23], depending the maximizing input distribution on the SNR and the diversity order corresponding to the transmit laser selection scheme here analyzed.

In this strong turbulence FSO scenario, it is worth commenting that a greater capacity can be achieved compared with the non-turbulent case when a diversity order *L* ≥ 4 is assumed. This can be justified from the greater robustness provided by the transmit laser selection scheme here studied against fluctuations in the irradiance of the transmitted optical beam proper to the atmospheric turbulence. A better comprehension of this fact can be achieved by calculating the equivalent scintillation index (*SI _{m}*), used to describe the strength of the equivalent fading gain

*I*and, hence, defined as

_{m}*SI*=

_{m}*E*[

*I*

^{2}

_{m}]/(

*E*[

*I*])

_{m}^{2}− 1, where

*E*[

*I*] and

_{m}*E*[

*I*

^{2}

_{m}] represent the mean value and the second moment of the equivalent turbulence model configured by the MISO system under study, respectively. The mean value of the equivalent irradiance,

*E*[

*I*], is obtained as follows

_{m}where [27, eqn. (0.155-4)] has been used for simplifying the sum. The second moment of the equivalent turbulence model, *E*[*I*
^{2}
_{m}], is also obtained as follows

In Fig. 3, the mean value (*E*[*I _{m}*]) and scintillation index (

*SI*) of the equivalent turbulence model configured by the MISO system under study versus the number of laser sources

_{m}*L*are displayed.

From this figure, it can be observed that the greater the number of laser sources, the greater the 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 of *L* ≥ 4 if compared with the non-turbulent case can be explained from the decreasing strength of the equivalent fading gain as *L* is increased together with the fact that the mean value of the 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 shown in Fig. 2. In this fashion, since a non-uniform input signaling improves the channel capacity especially at low SNR [8, 23], the increasing mean value of the equivalent fading gain implies a higher and higher equivalent SNR and, hence, the fact that the maximizing input distribution tends to be closer and closer to uniform signaling.

Finally, from the impact of the parameter *ξ* in (6), expression corresponding to the Euclidean distance in the constellation here defined for the OOK format, it is shown that a relevant 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 relevant improvement in terms of capacity here obtained when using non-uniform signaling OOK and the transmit diversity technique based on the selection of the optical path with a greater value of irradiance is adopted over exponential atmospheric turbulence channels, investigating the performance in alternative FSO scenarios covering a wider range of atmospheric turbulence conditions as well as incorporating pointing error effects are interesting topics for future research.

## Acknowledgments

The authors would like to thank the anonymous reviewers for their useful comments that helped to improve the presentation of the paper.

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