*3.3. BER Performance*

Now, we derive closed-form expressions of the BER for the OOK scheme used for SLIPT. Hence, the received BER can be written as

$$\text{BER} = p\_0 P\_c(1|0) + p\_1 P\_c(0|1),\tag{20}$$

where *Pe*(1|0) and *Pe*(0|1) represent the probability of false alarm and the probability of missed detection, respectively. In our analysis, we assume equally-likely symbols, *p*0 = *p*1 = 0.5 in many different realistic scenarios.

First, for an ideal scenario without neither turbulence nor pointing error (in (3), *h* is reduced to *h* = *Aohl*), the terms *Pe*(1|0) and *Pe*(0|1) can be expressed as [50]:

$$P\_{\varepsilon}(1|0) = \frac{1}{2} \text{erfc}\left(\frac{i\_t - i\_{s\_0}}{\sqrt{2\sigma\_n^2}}\right) = \frac{1}{2} \text{erfc}\left(\frac{i\_t - i\_{s\_1}\zeta}{\sqrt{2\sigma\_n^2}}\right) \tag{21}$$

$$P\_{\mathcal{E}}(0|1) = \frac{1}{2} \text{erfc}\left(\frac{i\_{s\_1} - i\_t}{\sqrt{2\sigma\_n^2}}\right);\tag{22}$$

with *is*0 and *is*1 denoting the received photocurrent corresponding to the transmission of a logical '0' and a logical '1', respectively; where *it* is the optimal detection threshold and *σ*2*n* is the noise variance given in Equation (2). On another note, and from (13), the photocurrents *is*0 and *is*1 can be expressed as *is*0 = *RhζPm* and *is*1 = *RhPm*, respectively. In addition, *it* can be written as a function of *ζ* as

$$i\_t = \frac{i\_{s\_1} + i\_{s\_0}}{2} = \frac{i\_{s\_1}(1 + \zeta)}{2}. \tag{23}$$

Recall that, for this ideal AWGN channel, *h* has a deterministic behavior since neither turbulence-induced nor misalignment fadings were considered yet. Accordingly, its associated BER is obtained by substituting (21) and (22) into (20), and it is given by

$$\text{BER} = \text{erfc}\left(\frac{RhP\_m(\zeta - 1)/2}{\sqrt{2\sigma\_n^2}}\right). \tag{24}$$

Second, we consider a more realistic scenario where atmospheric turbulence is now considered, although misalignment fading is still not included Therefore, the channel attenuation coefficient, *h*, becomes a RV since the turbulence-induced scintillation now incorporated into *h* follows the Gamma-Gamma pdf shown in (5). Thus (24) must be averaged with the corresponding pdf describing the behavior of *h*, and [48] (Eqs. (07.34.21.0013.01) and (07.34.21.0085.01)) are, again, used to solve the resulting integral:

$$\text{BER} = \frac{2^{(a+\beta)}}{8\pi\sqrt{\pi}\Gamma(a)\Gamma(\beta)}G\_{5,2}^{2,4}\left(\frac{8(RA\_oh\_lP\_{ll}(\zeta-1)/2)^2}{(\sigma\_na\beta)^2}\Big|\begin{array}{c} \frac{1-a}{2}, \frac{2-a}{2}, \frac{1-\beta}{2}, \frac{2-\beta}{2}, 1\\ 0, \frac{1}{2} \end{array}\Big|\begin{array}{c} \frac{1}{2} \end{array}\right) \tag{25}$$

Finally, when pointing errors are also taken into account, (21) and (22) must be now averaged with respect to (12). Hence,

$$\begin{split} \text{BER} &= \frac{2^{(\alpha+\beta)}\gamma^{2}}{16\pi\sqrt{\pi}\Gamma(\alpha)\Gamma(\beta)} \\ &\qquad G\_{7,4}^{2,6}\left(\frac{8(RA\_{\text{o}}\text{h}\_{\text{l}}\text{P}\_{\text{m}}(\zeta-1)/2)^{2}}{(\sigma\_{\text{m}}a\beta)^{2}}\Bigg|\begin{array}{c} \frac{1-\gamma^{2}}{2},\frac{2-\gamma^{2}}{2},\frac{1-\alpha}{2},\frac{2-\kappa}{2},\frac{1-\beta}{2},\frac{2-\beta}{2},1 \\ 0,\frac{1}{2},\frac{-\gamma^{2}}{2},\frac{1-\gamma^{2}}{2} \end{array};1\right), \end{split} \tag{26}$$

after having used [48] (Eqs. (07.34.21.0013.01) and (07.34.21.0085.01)). As a previous step, either the complementary error function, (erfc(·)), and the Bessel's K function were expressed in terms of Meijer's G functions from [48] (Eqs. (03.04.26.0006.01) y (06.27.26.0006.01)).

As can be seen from these latter equations, the exact expression for the error probability is given in terms of Meijer's G-function, which may be difficult to facilitate further analytical studies. Hence, it would be possible to obtain simpler expressions after some mathematical approximations following the approximation given in [51] involving an upper bound and a lower bound for the Gaussian Q-function that represents the behavior in terms of error probability of an ideal Gaussian channel, as shown in Equation (22), and based on series expansion. Additionally, and for a future work, we have planned to use a

more generic model for the turbulence, the Málaga model [52] and its formulation from Generalized-K functions [53]. Thus, as commented in [54], its pdf can be approximated by a Gauss–Laguerre quadrature. Since the Gamma-Gamma model employed in our paper is a particular case of the Málaga model [52], then this Gauss–Laguerre quadrature can also be applied. Thus the upper bound given in [54] and [51] (Equation (19)) can be employed.

#### **4. Results and Discussions**

In this section we analyze the impact of FSO link parameters on system performance in terms of energy harvesting and quality using the AEH and BER expressions derived in the previous section. This analysis allows to identify which are the key design parameters and the trade-offs to be considered in order to properly choose such parameters.

For our numerical analysis, we have assumed the values given in Table 1. In particular, we consider a vertical FSO link consisting of a ground-based optical transmitter located at a height *z*0 and a receiver mounted on a UAV hovering on the ground at a height *Z*. Therefore, the separation between both is *L* = *Z* − *z*0. The transmitter transmits a light beam with a divergence angle *θT*= 1 mrad [6] and a peak power *Pm* operating at a wavelength of 1550 nm. The receiver, as in [44], is composed of an optical circular converging lens whose effective light collection area is characterized by an aperture radius, *r*, of 10 cm with a responsivity *R* = 0.5 A/W. Some commercial implementations following these features can be found at [55,56] Therefore, for *L* = 300 m [57], the ratio of the beam waist to the aperture radius of the receiver is (*<sup>ω</sup>z*/*r*) = 3 (remember that a source with a 1 mrad divergence angle was considered) while, for *L* = 1000 m, (*<sup>ω</sup>z*/*r*) = 10. To cover a large area on the ground, a field of view FOV = 45◦ is assumed. Due to this large FOV, shot noise caused by ambient light is the dominant source of noise in the receiver. A spectral radiance *Sn* = 1 mW/(cm<sup>2</sup> nm srad), with an optical bandwidth *Bo* = 10 nm and a noise bandwidth *Bn* = 1 GHz are assumed to obtain *σ*<sup>2</sup> *n* from (2).

Regarding the turbulence, we calculate the magnitudes of *α* and *β*, with the expressions (6) and (7), considering the values of *C*<sup>2</sup> *n*(*z*) provided by the Hufnagel–Valley model [39] for different link heights, assuming *C*<sup>2</sup> *n*(*<sup>z</sup>*0) = 1.7 × 10−<sup>13</sup> m<sup>−</sup>2/3. Thus, *α* and *β* vary between 25 and 30 for heights between 300 m and 1000 m, leading to *σ*<sup>2</sup> *R* << 1, which corresponds to a very weak turbulence condition. In addition, we consider a jitter of *σs* = 10 cm [58] to model the pointing error.

Finally, we use the attenuation coefficients corresponding to very clear air, clear air, and haze shown in Table 1.


**Table 1.** System parameters.

Figure 3 shows the typical attenuation of vertical optical links (Figure 3a) together with the potential energies harvested (Figure 3b) for different separation distances between any UAV and the ground BS, ranging between 200 m and 1000 m.

**Figure 3.** (**a**) Average optical channel loss as a function of link length. (**b**) Average harvested energy as a function of the link length. Both figures consider different UAV jitters and weather conditions and a beam divergence of 1 mrad. To obtain the AEH, a single transmitter with *Pav* =500 mW is assumed.

Channel loss depicted in Figure 3a is calculated by −10 log(¯ *h*), with ¯ *h* = (*hl Aoγ*<sup>2</sup>)/(<sup>1</sup><sup>+</sup> *γ*<sup>2</sup>) representing the average attenuation coefficient calculated from Equation (12). It is straightforward to check how the coverage provided by any deployed UAV enhances

with increasing its altitude but, as shown in Figure 3a, at the cost of a significant increase in power losses at the receiver. Specifically, that increase is mainly caused by the beam broadening induced by the transmitter divergence. Namely, for *L* =300 m (i.e., (*<sup>ω</sup>z*/*r*) =3, as commented above), a power loss of 8.5 dB is reached for *σs* = 0.1 m. That value grows to 18 dB for *L* =1000 m as (*<sup>ω</sup>z*/*r*) =10. In both cases, a transmission divergence of 1 mrad and a 10 cm aperture radius were considered.

In addition, Figure 3b shows that increasing the channel loss dramatically reduces the harvested energy. Thus, for a single transmitter with a *Pav* = 500 mW, the AEH drops from 18 mJ/s for the best case at 200 m; to less than 2 mJ/s at 1000 m. Note that the AEH in the figure has been calculated using Equation (17). Finally, Figure 3 shows how atmospheric losses are not relevant for the amount of energy the UAV can harvest due to the short link lengths considered in the analysis. Furthermore, they can be neglected without loss of accuracy when the UAV is affected by UAV's jitter caused by its motor vibration.

Moreover, both figures also show the impact on the link performance of the pointing deviation caused by the jitter inherent to the UAV that acts as receiving side. For example, it can be observed that an increase in the UAV's jitter (higher *<sup>σ</sup>s*) is more detrimental when flying at lower altitudes than at higher altitudes. The reason for this is that the *<sup>ω</sup>z*/*<sup>r</sup>* ratio decreases for lower altitudes due to the beam divergence angle and, thus, the random variations inherent to the UAV location cause a more serious channel loss than in higher altitudes, even causing that the communication link may even be interrupted if the UAV moves out of the transmitted beam footprint. Consequently, energy harvesting is less affected by jitter as the UAV is operating at a higher altitude and, accordingly, the *<sup>ω</sup>z*/*<sup>r</sup>* ratio increases. Figure 4 summarizes this discussion.

**Figure 4.** Adverse effect of UAV's jitter caused by its motor vibration versus altitude in the flight state. It is supposed that the same UAV is flying at two different altitudes and affected by a same value of *σs*. For the case of the UAV operating at the lower altitude, the communication link may even be interrupted due to the presence of UAV's jitter. Of course, for a higher altitude, the beam broadening induces more serious power losses at the receiver.

Following the analysis of the channel behavior from Figure 3, now, Figure 5 shows the BER performance of the OOK and OOK-EH schemes under different channel impairments as a function of the received SNR. To illustrate the behavior of the OOK-EH scheme, a *ζ*= 0.8 has been assumed. In addition, two ratios *<sup>ω</sup>z*/*<sup>r</sup>* have been taken as representative values: (*<sup>ω</sup>z*/*r*) = 3 and (*<sup>ω</sup>z*/*r*) =10, corresponding to UAV heights of 300 m and 1000 m, assuming a transmission divergence *θT* = 1 mrad. BER curves plotted in black represent the case of (*<sup>ω</sup>z*/*r*) = 3; whilst the curves plotted in red depict the performance for (*<sup>ω</sup>z*/*r*) = 10. In addition, solid lines represent the BER of the ideal channel with *h* = *Ao*, i.e., assuming only the geometric loss; whereas the dashed lines include the adverse effect of the random medium, i.e., either solely turbulence or both turbulence and pointing error.

**Figure 5.** Average BER for the OOK and OOK-EH schemes versus signal-to-noise ratio for a vertical FSO link under different channel impairments and different values of the ratio (*wz*/*r*). A transmitter with a divergence *θT* =1 mrad and a *Pm* up to 20 dBm is assumed.

These results, which are calculated from the expression (25) for the case of only considering turbulence-induced fading; and from (26) for the combined scenario with turbulence and pointing error, have been validated using Monte Carlo simulations. Note that BER curves for the ideal channel and the one associated to the turbulent channel with no pointing errors are identical for both (*<sup>ω</sup>z*/*r*) ratios. In this figure, simulation (Monte Carlo) results are drawn with markers whereas theoretical results are drawn with either solid or dashed lines. In all cases, a perfect match is shown between the simulated results and those obtained from the derived expressions.

From the aforementioned Figure 5, it can be seen that turbulence and pointing error affect the BER with different severity depending on the *<sup>ω</sup>z*/*<sup>r</sup>* ratio. As far as turbulence is concerned, it affects BER equally regardless of the ratio *<sup>ω</sup>z*/*<sup>r</sup>*. However, as expected, pointing error causes a very severe degradation for lower *<sup>ω</sup>z*/*<sup>r</sup>* ratios. The figure shows that, for (*<sup>ω</sup>z*/*r*) = 3, the SNR degradation caused by pointing errors is huge, on the order of 15 dB for a target BER of 5 × <sup>10</sup>−5, while for (*<sup>ω</sup>z*/*r*) = 10, the degradation is nearly negligible. In fact, the BER curves considering turbulence and turbulence with pointing error almost overlap.

## *Optimization for EH*

As described in the previous section, the process of optimizing the modified OOK scheme for EH consists of choosing the optimal *ζ* value that maximizes the average EH while keeping the BER below the target value for each channel condition. Note that, since the increase in EH is achieved at the cost of degrading the link quality, the maximum *ζ* value will depend on the considered channel impairments, i.e., turbulent fading, pointing error, and atmospheric attenuation. Consequently, a lower channel degradation will lead

to higher *ζ* values and, thus, to higher harvested energies. The value of the optimal *ζ* has been obtained numerically from expression (26) by setting the link parameters and the target BER.

To form a complete picture of the dependence of optimum *ζ* on the FSO link parameters, Figure 6 shows the optimal value of *ζ* as a function of transmitter peak power considering different *<sup>ω</sup>z*/*<sup>r</sup>* ratios, jitters and target BERs. In all cases, the turbulence fading and the path loss corresponding to "very clear air" are included. As can be seen in the figure, as the peak power of the transmitter increases, the optimum *ζ* value also increases. In our analysis, realistic power values up to 35 dBm have been considered. Note that, since the UAV is intended to collect as much energy as possible, the power value chosen in the link design will be high. Therefore, the optimal *ζ* values will also be high. For the highest of the powers considered (*Pm* =35 dBm), the value of *ζ* is always higher than 0.8. In addition, as expected, the optimal *ζ* values obtained for the 10−<sup>8</sup> target rate (red lines) are lower than for 5 × 10−<sup>5</sup> (blue lines) and, similarly, higher jitter values lead to lower *ζ* values. From that Figure 6, it can be concluded that values of *ζ* higher than 0.8 one can be selected for realistic propagation scenarios.

**Figure 6.** Optimal *ζ* values for the proposed OOK-EH scheme as a function of the transmitted peak power considering different (*<sup>ω</sup>z*/*r*) ratios, jitters and target BER. These *ζ* values maximize the average harvested energy while maintaining the target BER. Turbulence fading (*α* = 30, *β* = 30) and "very clear air" conditions are assumed.

On a different matter, Figure 7 depicts the average harvested energy for the OOK and OOK-EH schemes as a function of the peak transmitted power. For the OOK-EH scheme, the optimal *ζ* values obtained in Figure 6 have been here employed.

Hence, from the AEH results depicted in Figure 7, the following comments can be drawn. First, it is clearly observed that the energy collected with the OOK-EH scheme (black lines) is higher than that of the OOK scheme (red lines). In fact, the energy harvested by the OOK-EH scheme tends to twice that in OOK scheme as the transmitted power increases. Figure 7 also shows that a minimum *Pm* is required to achieve the previously selected BER target (a power below *Pm* cannot satisfy the performance required by the BER target and, accordingly, no energy would be harvested). This *Pm* is higher for the ratio *<sup>ω</sup>z*/*<sup>r</sup>* =3 than for *<sup>ω</sup>z*/*<sup>r</sup>* =10 due to what was explained when introducing Figures 3–6. Therefore, the choice of the ratio *<sup>ω</sup>z*/*<sup>r</sup>* has a huge impact on the value of the collected energy. The figure shows that the energy obtained with *<sup>ω</sup>z*/*<sup>r</sup>* = 3 is much higher than that

obtained with *<sup>ω</sup>z*/*<sup>r</sup>* =10. In particular, for the highest power considered (Pm=35 dB), the AEH is about 70 mJ/s for *<sup>ω</sup>z*/*<sup>r</sup>* =3, while it hardly reaches 10 mJ/s for *<sup>ω</sup>z*/*<sup>r</sup>* = 10. Note that the AEH values shown in that figure are consistent with those ones published by other authors in [31] and [47] for peak transmitted powers of 100 and 200 mW, respectively.

**Figure 7.** Average harvested energy for the OOK and OOK-EH schemes versus peak transmitted power considering different (*<sup>ω</sup>z*/*r*) ratios and target BER. Turbulence fading (*α* =30, *β* =30), a jitter (*<sup>σ</sup>s* = 0.1 m) and "very clear air" conditions are assumed. A single transmitter has been considered.

As explained above, one of the main features affecting energy harvesting is the *<sup>ω</sup>z*/*<sup>r</sup>* ratio. In this respect, it is possible to design transmitters for EH with an accurate control of their beam width since significant gains can be achieved with the appropriate selection of *<sup>ω</sup>z*/*<sup>r</sup>*. Thus, when increasing this ratio, the pointing problem is relaxed and, as well, the negative effect inherent to the jitter is reduced, i.e., when increasing *<sup>ω</sup>z*/*<sup>r</sup>* then fading induced by jitter suffered by the receiver is less intense since the received beam waist becomes (much) bigger than the receiver aperture radius. In this respect, although the received spot can wander due to the jitter effect, however, the total amount of caught power in the receiver side is maintained without significant variations since the size of the received beam waist is large compared to the receiver aperture radius. Consequently, the amount of power that can be captured from the section of the beam illuminating the receiver is, more or less, of the same magnitude. On the contrary, for smaller values of *<sup>ω</sup>z*/*<sup>r</sup>*, the sway in the received beam footprint caused by misalignment is more critical in the sense that such a sway may make all the received beam spot drop out of the receiver photosensitive area. For that critical situation, the amount of captured power in the receiver would drop to zero. Of course, large *<sup>ω</sup>z*/*<sup>r</sup>* values lead to both severe geometrical losses (since most of the received footprint area is spread out of the physical photosensitive area implemented in the receiver side) and, consequently, low values of harvested energy. Therefore, the design of any FSO link should try to minimize this ratio by using adaptive pointing tracking systems [18].

It is worth noting that all the the results of average EH shown so far are for a single vertical FSO link between a given BS and UAV. Nevertheless, each UAV could receive simultaneous transmissions from a number of BSs as long as their locations fall within the region, R ⊂ R2, covered by the FOV of the UAV's receiver. The area of such a region, |R|, can be expressed in terms of the FOV angle and the height of the UAV, *Z*, as follows:


As a conclusion, considering the AEH results shown in Figure 7 and the number of BSs covered by the UAV's receiver described above, the total AEH that can be obtained for considered scenario in very clear air conditions with a target BER of 5 × 10−<sup>5</sup> and assuming realistic transmitted powers between 30 and 35 dBm is in the range between 134 and 450 mJ/s for a UAV height of 300 m and between 164 and 558 mJ/s for a UAV height of 1000 m. This free energy collected from the information-carrying FSO signals complements the energy harvested from the terrestrial optical and RF wireless power transfer (WPT) charging stations and contributes to extend the battery life of UAVs.

## **5. Concluding Remarks**

link loss with UAV height.

In this paper, we have investigated the application of FSO, UAVs, and EH as an adaptable and efficient solution to provide backhaul/fronthaul connectivity to 5G+ networks. There are many benefits supporting this approach. First, FSO technology provides high bandwidth for the 5G RAN. Second, FSO communication links do not interfere with the RF based 5G RAN. Third, the UAVs, which act as flying nodes of a backhaul/fronthaul network, can adapt to changes in weather and traffic conditions to provide reliable links. Nevertheless, the limited battery of the UAVs causes service interruptions when the UAVs need to recharge them. For this reason, we propose the use of EH to collect energy from the transmission of information signals to combine with the EH from the terrestrial optical and RF WPT charging stations. All these techniques are thought to enhance the on-board battery lifetime of UAVs. To assess the benefits of the proposed approach we have considered a realistic ye<sup>t</sup> tractable channel model that includes the effect of turbulence fading, pointing errors and atmospheric attenuation. Using this model, analytical closed-form expressions of the average harvested energy and the bit error rate of an OOK scheme optimized for information transmission and power transfer are derived. The derived expressions allow to evaluate the performance of vertical FSO links between ground-based BSs and UAVs and to properly select the link parameter to optimize the harvested energy while guaranteeing a reliable connection.

Results show that there exists an interesting trade-off between reliability and harvested energy.

**Author Contributions:** M.C.-V. and A.J.-N. defined the scope of the review manuscript. All the authors compiled the necessary information to elaborate this review manuscript. All the authors derived the expressions and obtained the results. C.Á.-R., M.Á.-R., F.J.M.-V., M.C.-V., T.R. and A.J.-N. wrote the article. All the authors reviewed and edited the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work has been funded by the University of Málaga.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable. **Conflicts of Interest:** The authors declare no conflict of interest.
