A new upper bound on the capacity of power and bandwidthconstrained optical wireless links over gammagamma atmospheric turbulence channels with intensity modulation and direct detection is derived when onoff keying (OOK) formats are used. In this freespace optical (FSO) scenario, unlike previous capacity bounds derived from the classic capacity of the wellknown additive white Gaussian noise (AWGN) channel with uniform input distribution, a new closedform 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 signaltonoise ratio (SNR). Additionally, it is shown that an increase of the peaktoaverage 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.
1. Introduction
Optical wireless communications using intensity modulation and direct detection (IM/DD) can provide highspeed 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 eyesafety considerations. Moreover, these systems are intrinsically bandwidth limited due to the use of large inexpensive optoelectronic components. Recently, the use of atmospheric freespace 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 radiofrequency (RF) links [2] and the recent development of radio on freespace 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 signaltonoise ratio for IM/DD channels with pulse amplitude modulation in [9]. In [10], a new closedform upper bound on the capacity is found through a spherepacking 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 signaltonoise 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 gammagamma atmospheric turbulence channels using onoff keying (OOK) formats are presented by maximizing the mutual information for this channel over a binomial input distribution. In [14, 15], the capacity of lognormal optical wireless channel with OOK formats is computed for known channel state information (CSI) in a similar way to the capacity of the wellknown 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], closedform mathematical expressions for the evaluation of the average channel capacity are presented when lognormal and gammagamma 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 biterror 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 capacityapproaching, 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 bandwidthconstrained optical wireless links over gammagamma 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 closedform 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 peaktoaverage 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
The use of infrared technologies based on IM/DD links is considered, where the instantaneous current in the receiving photodetector, , can be written as
where the symbol denotes convolution, is the detector responsivity, assumed hereinafter to be the unity, represents the optical power supplied by the source, the impulse response of an ideal lowpass filter, which cuts out all frequencies greater than hertz, modelling the fact that these systems are intrinsically bandwidth limited due to the use of large inexpensive optoelectronic components, and the scintillation at the optical path; is assumed to include any frontend 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 , that is, , independent of the on/off state of the received bit [1]. Since the transmitted signal is an intensity, must satisfy for all . Due to eye and skin safety regulations, the average optical power is limited and, hence, the average amplitude of is limited. Although limits are placed on both the average and peak optical power transmitted, in the case of most practical modulated optical sources, it is the average optical power constraint that dominates [26]. The received electrical signal , however, can assume negative amplitude values. In this fashion, the atmospheric turbulence channel model consists of a multiplicative noise model, where the optical signal is multiplied by the channel irradiance. Here, we consider the gammagamma turbulence model proposed in [5, 27], where the normalized irradiance is defined as the product of two independent random variables, that is, , and representing largescale and smallscale turbulent eddies and each of them following a gamma distribution. This leads to the socalled gammagamma distribution, whose probability density function (PDF) is given by
where is the wellknown Gamma function, and is the thorder modified Bessel function of the second kind [28]. Assuming spherical wave propagation, the parameters and are related to the atmospheric conditions through the following expressions [27, 29]:
where and . Here, is the optical wave number, is the wavelength, is the diameter of the receiver collecting lens aperture, and is the link distance in meters. stands for the altitudedependent index of the refractive structure parameter and varies from for strong turbulence to for weak turbulence. Since the mean value of this turbulence model is and the second moment is given by , the scintillation index (SI), a parameter of interest used to describe the strength of atmospheric fading, is defined as
We consider OOK formats with any pulse shape and reduced duty cycle, allowing the increase of the PAOPR parameter. A new basis function is defined as where represents any normalized pulse shape satisfying the nonnegativity constraint, with in the bit period and otherwise, and is the electrical energy. In this way, an expression for the optical intensity can be written as
where 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 onedimensional 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 maximumlikelihood 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 gammagamma distribution corresponding to the atmospheric turbulence model and, hence, its average value, known as ergodic capacity, will indicate the average best rate for errorfree transmission [16–19].
3. Upper Bound on Channel Capacity
Considering the channel capacity as a random variable and perfect CSI available at both transmitter and receiver [14, 32], we can use the theory derived for discretetime Gaussian channels [33], expressing the ergodic capacity in bits per channel use as
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 wellknown 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].
The constraint in optical domain implies that , the second moment of , takes a value of up to . Additionally, in our channel model, assuming a unidimensional space where the nonnegativity 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 , and, hence, , implies an Euclidean distance as and, hence, an average optical power constraint of . Thus, an average electrical power constraint of is necessary and sufficient condition for satisfying an average optical power constraint of . 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 discretetime channel, it must be emphasized that the transmitted optical signal is represented by the random variable , the atmospheric turbulenceinduced signal is represented by the product and the corresponding signal performed in electrical domain is represented by , being the latter the signal to be finally considered in our analysis. Applying the fact that the Gaussian distribution maximizes the entropy over all distributions with the same variance [33, Theorem ], we obtain
where 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 twosided spectral density of watts/Hz and average optical power constraint of watts. Next, assuming that the channel is constrained to dimensions and even without maximizing over the input distribution, the channel capacity can be obtained by averaging over the PDF in (2) as follows:
where is the SNR definition, as in [8, 10], different to the expression used in [14, 16–19], and represents the entropy of the Bernoulli RV in (5), presenting the maximum value achievable because OOK is the signaling technique considered in this analysis. After substituting (2) in (10), we can use Meijer's Gfunction [28, equation ()], available in standard scientific software packages such as Mathematica and Maple, in order to transform the integral expresion to the form in [36, equation ()], expressing in (10) the modified Bessel function of the second kind [36, equation ()] and the logarithm function [36, equation ()] in terms of Meijer's Gfunction. Finally, after a simple power transformation of the RV in order to achieve a linear argument for Meijer's Gfunction related to the logarithm function and using [36, equation ()], a closedform solution for is derived as can be seen in
Knowing that is also upper bounded by the binary entropy , the ergodic capacity in bits per channel use is obtained by maximizing over the parameter as
For the sake of easy comparison, we present a closedform expression in terms of the Meijer's Gfunction following a similar approach as in works in the same context [16–18]. Nonetheless, it must be commented that Meijer's Gfunction 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
We now numerically evaluate mutual information for our channel model using OOK signaling to corroborate the tightness of the previous results. For the sake of simplicity, showing the fact that the input distribution that maximizes the mutual information varies with the turbulence strength and the SNR, the statistical channel model can be rewritten as
where . The conditional mutual information for this channel is, therefore, derived as can be seen in
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
Firstly, no atmospheric turbulence is considered to show the fact that the input distribution that maximizes the mutual information varies with the SNR. It is easy to deduce from the upper bound in (9) that the channel capacity in the absence of atmospheric turbulence is obtained by maximizing over , that is, , where is
At this point, the greater tightness of this upper bound can be corroborated if compared to the approach followed in [14–18], where the capacity is computed in a similar way to the capacity of the wellknown AWGN channel with BPSK signaling, assuming the fact that the input distribution that maximizes mutual information is the same regardless of the channel state and with a value of . With our notation, this capacity can be expressed as
Obtained results for the capacity in (15), with a value of , and in (16) are illustrated in Figure 1 when a rectangular pulse shape with duty cycle of is adopted, that is, . Here, and have been considered and, hence, values of and , respectively, are computed in (6) by direct integration in frequency domain using a symbolic mathematics package [37]. For this rectangular pulse shape, it is easy to deduce that , where is the sine integral function [38, equation ()]. In this figure, mutual information is also displayed, being numerically solved in a similar way as in (14) but not yet considering the impact of the atmospheric turbulence. It can be corroborated that the proposed upper bound in the absence of turbulence shows a tighter performance, regardless of the value of . Here, there must be commented the fact that the analysis in this paper is particularized for the OOK signaling and, hence, the improvement in performance for the capacity in (15) is sufficiently contrasted if compared to the mutual information, numerically solved for the OOK signaling. However, when no signaling schemes are particularized in the capacity analysis, upper bounds are usually corroborated by evaluating the asymptotic behavior with the corresponding lower bounds.
Figure 1. Capacity bounds and mutual information numerically solved for the nonturbulent optical channel with uniform input distribution when a rectangular pulse shape with duty cycle of , that is, , and values of (a) and (b) are adopted.
In Figure 1, we also include the upper bound on channel capacity determined in [8, expression (21)] by Hranilovic and Kschischang, based on a signal space representing the convex hull of a generalized Ncone with vertex at the origin. As in [8, Section V.A], this is adopted in the unidimensional case but using the new basis function proposed in this paper to consider the favorable impact of the increase of the PAOPR and, this way, to compare results in similar conditions. It must be noted that the mathematical treatment in [8] is more general since a particular signaling is not assumed when the spherepacking procedure is carried out. This modified upper bound can be written in bits/channel use as
Recently, a better representation at lower SNR for the channel capacity (in bits/channel use) has been derived by Farid and Hranilovic in [9, expression ()], compared to previous work in [8] with
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 peakpower 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 averagepower constraint is imposed, context in which the upper bound proposed in this paper is evaluated.
Since the input distribution that maximizes the mutual information varies with the SNR, numerical maximization of the capacity bound in (15) and mutual information over the input distribution for the nonturbulent channel are shown in Figure 2(a) when a rectangular pulse shape with duty cycle of , that is, , and are adopted. Figure 2(b) shows the fact that a nonuniform input distribution improves the channel capacity, especially at low SNR [34, 35]. Unlike other channels in which the gap between mutual information with uniform and nonuniform source distributions is small, this figure demonstrates that for optical wireless systems the use of nonuniform distributions provides a relevant improvement in performance.
Figure 2. (a) Maximization of the capacity bound in (15) and mutual information over the input distribution for the nonturbulent optical channel when and a rectangular pulse shape with are adopted. (b) Mutual information versus the input distribution for values of SNR of dB, dB, and dB.
4.2. With GammaGamma Atmospheric Turbulence
In this subsection, atmospheric turbulence is considered, showing the fact that the input distribution that maximizes the mutual information varies with the turbulence strength and the SNR, and corroborating the better performance for the upper bound in (11) if compared to previous capacity bounds derived from the classic capacity of the wellknown AWGN channel with uniform input distribution. In a similar way as derived in (11) but starting from the expression in (16), this capacity , corresponding to the approach followed in [14–18], can be written with our notation as can be seen in
Obtained results for the capacity in (11), with a value of , and in (19) are illustrated in Figure 3 when and a rectangular pulse shape with are adopted. In this figure, mutual information is also displayed, being numerically solved as in (14). Here, the greater tightness of the proposed upper bound in (11) can be corroborated when a uniform input distribution and different levels of turbulence strength are assumed, corresponding to values of scintillation index of and .
Figure 3. Capacity bounds and mutual information numerically solved for the atmospheric turbulent optical channel with uniform input distribution when , a rectangular pulse shape with and different levels of turbulence strength (a) and (b) are assumed.
As in nonturbulent case, since the input distribution that maximizes the mutual information is nonuniform, numerical maximization of the capacity bound in (11) and mutual information over the input distribution for the gammagamma atmospheric turbulent channel are shown in Figure 4(a) when and a rectangular pulse shape with are used. Figure 4(b) shows the fact that a nonuniform input signaling improves the channel capacity, especially at low SNR [35], depending on the maximizing input distribution on the SNR and the turbulence strength.
Figure 4. (a) Maximization of the capacity bound in (11) and mutual information over the input distribution for the atmospheric turbulent optical channel when , a rectangular pulse shape with and a scintillation index of are adopted. (b) Mutual information versus the input distribution for a value of SNR of dB and different levels of turbulence strength.
Additionally, from the result in (11) for the capacity proposed in this letter, a relevant improvement in performance must be noted as a consequence of the pulse shape used. To fully exploit this improvement, a pulse shape with a high PAOPR must be employed. So, for instance, when a rectangular pulse shape of duration , with , is adopted, a value of can be easily shown. Nonetheless, a significantly higher value of is obtained when a Gaussian pulse of duration as for all is adopted, where and the reduction of duty cycle is also here controlled by the parameter . In this fashion, of the average optical power of a Gaussian pulse shape is being considered. In Figure 5, maximization of the capacity bound in (11) and mutual information for the atmospheric turbulent optical channel are displayed when a scintillation index of and rectangular and Gaussian pulse shapes are adopted. Here, a value of has been considered and, hence, values of when using a rectangular pulse with and when using a Gaussian pulse shape with have been obtained from (6). For this Gaussian pulse shape, it is easy to deduce that , where is the error function [38, equation ()]. It is shown that OOK format using the classical rectangular pulse with duty cycle of 100% requires about 5 dB more optical SNR to yield similar values of capacity compared with OOK format with Gaussian pulses having a duty cycle of 25%.
Figure 5. Maximization of the capacity bound in (11) and mutual information over the input distribution for the atmospheric turbulent optical channel when and a scintillation index of are adopted with rectangular and Gaussian pulse shapes.
5. Conclusions
As a result, a new upper bound on the capacity of power and bandwidthconstrained optical wireless links over gammagamma 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 wellknown AWGN channel with uniform input distribution, a new closedform 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 averagepower 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.
Acknowledgment
The authors are grateful for financial support from the Junta de Andalucía (research group "Communications Engineering (TIC0102)").
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