**3. System Performance**

In this paper, two metrics are considered to evaluate the performance of FSO vertical links. First, the amount of harvested energy in the UAV is analyzed through the average EH parameter. Second, the received signal quality is studied in terms of the bit error rate (BER) parameter.

In particular, we must remark that the harvested energy analysis detailed below is valid for any modulation scheme; nevertheless, our BER analysis will be particularized for a variant of the OOK scheme previously optimized for energy harvesting under a fixed peak power constraint [31]. An OOK scheme was selected because of its simplicity and low power consumption, especially interesting due to limited on-board UAVs battery lifetime. In such a scheme, named OOK-EH, a power *P*1 = *Pm* is emitted for the transmission of a logical '1', where *Pm* is the peak power of the laser; whilst a fraction of *Pm* is carried for transmitting a logical '0', i.e.,

$$P\_0 = \zeta P\_{m\_{\prime}} \tag{13}$$

where 0 ≤ *ζ* < 1. In contrast to the classical OOK, where no light is emitted when a logical '0' is transmitted, our proposed scheme always carries a power level that will depend on the *ζ* parameter. This added DC component increases the average optical power *Pav* and can be used to improve energy collection by adjusting the *ζ* parameter appropriately. Furthermore, and in order to maximize the harvested energy, a power level *Pm* is also emitted when the transmitter is idle. However, any increase in the *P*0 level will reduce the dynamic range given by the peak average optical power ratio (PAOPR) and, consequently, its associated BER will be increased. The PAOPR is given by 2/(1 + *ζ*). Consequently, there is an important trade-off between increasing harvested energy and degrading BER. For any particular set of channel conditions, this trade-off is achieved for an optimal *ζ* that maximizes the harvested energy while keeping the BER performance below a predetermined target BER.

In order to evaluate the performance of any vertical FSO link with our proposed OOK-EH technique under realistic channel conditions, analytical closed-form expressions for the average EH and BER are derived in this section, considering the aforementioned main phenomena degrading the quality of the received signal, i.e., noise, turbulence, path loss, geometric spread and pointing errors.

## *3.1. EH Performance*

An EH module is added to the receiver to collect the energy from the received signal. This module extracts the DC component, *I*DC, of the electrical signal that is obtained by the PD, *i*. The DC current is then either stored or directly used to feed other modules such as the information detection module and the UAV's motor, which translates into extending the UAV's battery lifetime. Figure 2 illustrates a block diagram of the receiver where the EH and information detection modules are shown. As it can be observed, the EH module

is formed by a Schottky diode and a low-pass filter (LPF), which are passive devices, and thus, its associated power consumption is limited and could be included in the conversion efficiency, *ζ* [46]. Furthermore, this approach does not require either any control logic or any additional module to obtain the DC current.

**Figure 2.** Block diagram of the proposed receiver structure with EH. An optical circular converging lens would be placed in front of the PD.

As described in [29,31,47], the harvested energy per second in the optical receiver is given by

$$EE = fV\_t I\_{DC} \ln\left(1 + \frac{I\_{DC}}{I\_o}\right),\tag{14}$$

where *f* , *Vt* and *Io* stand for the photo-detector's fill factor, thermal voltage, and dark saturation current, respectively. The DC component of the output current, *I*DC, is written as [47]:

$$I\_{\rm DC} = R h P\_{\rm av}.\tag{15}$$

Here, *R* denotes the PD responsivity in A/W, with *h* being the composite channel attenuation coefficient, whereas *Pav* represents the average transmitted power.

As can be noticed, the harvested energy is random due to the influence of the channel attenuation coefficient *h*. Therefore, to obtain the collected energy over a long period of time, the average EH (AEH) is derived as:

$$\text{AEH} = \int\_{h>0} f\_h(h) f V\_t R h P\_{av} \ln\left(1 + \frac{R h P\_{av}}{I\_o}\right) dh,\tag{16}$$

where *fh*(*h*) is the pdf of the composite channel attenuation according to (12). Employing the expressions [48] (Eqs. (07.34.21.0011.01) and (07.34.21.0085.01)), the solution of this integral can be written in a closed-form as

$$\text{AEH} = \frac{\gamma^2 f V\_l R P\_{av} A\_o h\_l}{a \beta \Gamma(a) \Gamma(\beta)} G\_{5,3}^{1,5} \left( \begin{array}{c} A\_o h\_l R P\_{av} \\ I\_o a \beta \end{array} \Big| \begin{array}{c} 1, 1, -\gamma^2, -a, -\beta \\ 1, -1-\gamma^2, 0 \end{array} \right). \tag{17}$$

Moreover, when only the effect of turbulent fading is considered, and *fh*(*h*) is given by (5), the above equation reduces to

$$\text{AEH} = \frac{fV\_{\text{l}}RP\_{\text{av}}(A\_o h\_l)^2}{(a\beta)^2 \Gamma(a)\Gamma(\beta)} G\_{5,3}^{1,5} \left(\frac{A\_o h\_l RP\_{\text{av}}}{I\_o a\beta}\Big|\begin{array}{c} 1, 1, -1, -1-a, -1-\beta\\1, -2, 0 \end{array}\right) . \tag{18}$$

where the term *Pav*, which is present in Equations (16)–(18), is expressed as *Pav* = *Pm*(1 + *ζ*)/2 in case of OOK-EH. From Equations (17) and (18), the amount of harvested energy is not depending on the modulation scheme itself but on the average transmitted power. Of course, the particular channel conditions and the type of PD and rest of modules shown in Figure 2 will influence on the amount of energy any UAV can extract.

## *3.2. EH Optimization*

In view of the expressions for the average harvested energy derived above, it is straightforward to observe that they depend on: first, the channel and link conditions (i.e., attenuation and turbulence, fundamentally); second, on the transmitter and receiver systems with a given pointing misalignment loss, geometric spread, and maximum peak power, *Pm*; and third, on the parameter *ζ*. In this section, we consider all these parameters as uncontrollable variables that depend on the channel and the physical devices, except for the *ζ* variable, that can be adjusted to maximize the harvested energy.

Therefore, we notice that if *ζ* increases, then the average harvested energy also increases; however, this fact adversely affects on the BER performance of the system. In this respect, there exists an interesting trade-off between EH and reliability. We consider in this work that the communication system includes a simple forward error correction (FEC) decoder, which defines a maximum pre-FEC BER, normally referred to as BERtarget. Such a target is the maximum BER of the coded bits before the FEC decoder to guarantee an error-free transmission with high probability.

We assume two widely adopted target BER values of BERtarget =5 × 10−<sup>5</sup> and BERtarget = 10−<sup>8</sup> related to different FEC options as defined in [49]. Interestingly, the latter target BER is related to a BASE-R code, which is also know as Fire Code FEC (FC-FEC) as is defined in (clause 74 of [49]). This latter code requires a very small computational complexity for its decoding.

Thus, we pose the optimization problem for the design of the proposed FSO system with EH as follows

$$\begin{array}{c}\text{arg\,min}\_{\mathbb{Q}\in[0,1)}\text{AEH}\\\text{s.t.}\text{BER} < \text{BER}\_{\text{target}}\end{array}\tag{19}$$

The above problem is solved numerically in Section 4 to determine the optimal *ζ* value and assess the maximum AEH that can be obtained with the proposed framework under realistic and diverse channel conditions.
