## I Introduction

Visible light communications (VLC) is a promising complementary technology to radio-frequency (RF) based wireless systems, as it can be used to offload users from RF bands for releasing spectrum resources for other users while also simultaneously providing illumination [burchardt2014vlc]. Recently, hybrid VLC-RF systems are receiving attention in the literature, which can be designed to take advantage of both technologies, e.g., high-speed data transmission by VLC links while achieving seamless coverage through RF links [basnayaka2015hybrid]. In particular, a hybrid VLC-RF system is desirable for indoor applications such as the Internet of things (IoT) or wireless sensor networks[delgado2020hybrid]. Given that power constraint is a key performance bottleneck in such networks, a potential approach is to scavenge energy from the surrounding environment through energy harvesting (EH).

The current literature on hybrid VLC-RF systems that also use EH is mainly limited to finding the optimal value of DC bias to either maximize the data rate or to minimize the outage probability

[rakia2016optimal, yapici2020energy, peng2020performance]. In this extended abstract, we investigate the performance of EH for an indoor hybrid VLC-RF scenario. We formulate the optimization problem in the sense of maximizing the data rate. In particular, we consider both the DC bias and the assigned time duration to EH as the design parameters. We split the joint optimization problem into two sub-problems and cyclically solve them. First, we fix the assigned time duration to EH and solve the non-convex problem for DC bias by employing the majorization-minimization (MM) procedure. Second, we fix the DC bias obtained from the previous step and solve the optimization problem for the assigned time duration to EH.## Ii System Model

Fig. 1 illustrates the hybrid VLC-RF system under consideration. We assume a relay equipped with a single photo-detector (PD), energy-harvesting circuity, and a transmit antenna for RF communications. Let denote the block transmission time which is measured in seconds. Also, (unitless value) is the portion of time that is allocated to transmit information and energy to the relay node in the time block. Thus, the duration of this phase in seconds is . We assume that the block transmission time is constant. Hereafter, we drop the superscript in the sequel to simplify notation. Fig. 2 depicts the transmission block under consideration. Without loss of generality, we assume that second.

### Ii-a VLC Link

In the first hop, the LED transmits both energy and information to the relay node through the VLC link. The non-negativity of the transmitted optical signal can be assured by adding DC bias (i.e., ) to the modulated signal, i.e., where denotes the LED power per unit (in W/A) and is the modulated electrical signal. We assume that the information-bearing signal is zero-mean, and satisfies the peak-intensity constraint of the optical channel such that [1]

(1) |

where denotes the peak amplitude of the input electrical signal (i.e., ), and with and being the maximum and minimum input currents of the DC offset, respectively.

Let denote the double-sided signal bandwidth. The information rate associated with the optical link between the AP and relay node within a block with second, is given as [1]

(2) |

where is the photo-detector responsivity in A/W and is the optical DC channel gain. In (2), is the power of shot noise at the PD which is given as where is the charge of an electron and is the induced current due to the ambient light. The optical DC channel gain of VLC link can be written as

(3) |

where and are the vertical and horizontal distances, respectively, between AP and the relay node, and and are the respective angle of irradiance and incidence, respectively. The Lambertian order is where is the half-power beamwidth of each LED, and and are the detection area and half field-of-view (FoV) of the PD, respectively. The function is 1 whenever

, and is 0 otherwise. We also assume that the relay node distance follows a Uniform distribution with

, and that relay PD is looking directly upward. The harvested energy at this phase can be computed as(4) |

where is the thermal voltage, is the dark saturation current, and is the DC part of the output current given as . In the time period , the aim is to maximize the harvested energy while the relay transmits the information to the far user over the RF link. Thus, during second phase the LED eliminates the alternating current (AC) part and maximizes the DC bias, i.e., and . Mathematically speaking, the harvested energy during the second phase can be expressed as

(5) |

The total harvested energy at the relay can be calculated as where and .

### Ii-B RF Link

In the second hop, the relay re-transmits the information to the far user through the RF link by utilizing the harvested energy. The relaying operation is of decode-and-forward (DF) type. We assumed that the user is off the AP horizontally by a distance , which follows a Uniform distribution with . Let denote the bandwidth for the RF system and denote the noise power which can be defined as where is the thermal noise power, and is the noise figure. Further, assume that the relay re-transmits the electrical signal with normalized power. The respective information rate is given as

(6) |

where denotes the Rayleigh channel coefficients, is the transmit power and is the path loss model for RF link and can be expressed as

(7) |

where being the used RF carrier wavelength, m is the reference distance, and is the path loss exponent, which generally takes value between [1.6, 1.8].

The achievable information rate is limited by the smaller information rate between the VLC link and the RF link and can be expressed as [guo2021achievable]

(8) |

Our aim is to maximize and it can be written as

(9) | ||||

s.t. | ||||

where is a predefined threshold value, and are imposed to avoid any clipping and guarantee that the LED operates in its linear region and is added to satisfy the minimum required data rate. The above joint-optimization problem is non-smooth (due to the minimization operation) and non-convex (due to the objective function and ). To remove the non-smoothness in the objective function, we can rewrite (9) in the epigraph form as

(10) | ||||

s.t. | ||||

Although (10) solves the non-smoothness, it is still non-convex. Substituting (2) and (6) in and , we have

(11) | ||||

s.t. | ||||

where , and . In (11), is still non-smooth and can be replaced with [guo2021achievable]. The constraints are jointly non-convex. Consequently, the joint optimization is split into two subproblems, which are then cyclically solved.

If accepted, our full paper build on this extended abstract to present additional analysis and will provide numerical results demonstrating the effectiveness of our proposed optimization approach.