# Characterization of Light-To-Frequency Converter for Visible Light Communication Systems
## 2 VLC and LTF System Model  
### VLC System Model  
VLC systems are based on intensity modulation and direct detection (IM/DD), which is the most used method to implement optical wireless communications [1; 2; 7]. A typical VLC system is depicted in Figure 1.  
Once the photodiode (PD) detects an incident optical power or irradiance \(E(\lambda)\) on its photosensitive surface, it will generate a photocurrent \(i_{r}(t)\) proportional to device responsivity \(R(\lambda)\) and \(E(\lambda)\), which is corrupted by noise \(n(t)\). Such relation is illustrated in Equation (1):  
\[i_{r}(t)=R(\lambda)E(\lambda)+n(t) \tag{1}\]  
If emitted optical signal degradation effects due to communication channel are considered, the model in the work of [1] can be described through the expressions in Equation (2):  
\[\begin{array}{l}i_{r}(t)=R(\lambda)P_{t}(t)\otimes h(t)+n(t),\\ P_{t}(t)=i_{t}(t)\otimes h_{eo}(t),\end{array} \tag{2}\]  
where \(i_{t}(t)\) is the bias current of light-emitting-diode (LED); \(h_{eo}(t)\) is the impulse response of LED; \(P_{t}(t)\) is the emitted instantaneous optical power by the LED; \(h(t)\) represents the channel impulse response; \(i_{r}(t)\) is the sensor generated photocurrent; \(R(\lambda)\) is the photodiode responsivity; \(\otimes\) denotes the convolution operator; and \(n(t)\) is the system noise, which is modeled as additive white gaussian  
Figure 1: Block diagram of a visible light communication (VLC) system. LED—light-emitting-diode; PD—photodiode.  
noise (AWGN). The radiated optical power is always positive \(P_{t}(t)\geq 0\); moreover, it is important take into account that the required illumination in a space in which people are dwelling needs to be below a certain limit of the average total emitted optical power, in order to mitigate the possible harmful effects on the eyes [2]. The average optical power of the source can be estimated with Equation (3):  
\[P_{avg}=\lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^{T}P_{t}(t)dt \tag{3}\]  
VLC systems usually have two main threats: thermal noise and shot noise, both of which distort the signal of interest. The source of shot noise is the randomness in the photon absorption process and the electron-hole pair recombination within PD, whereas thermal noise depends on the environment temperature that perturbs enough the electrons in the receiver discrete devices [8]. The noise is a random process, thus it is characterized by a total variance. In the model described in the literature [8], the overall variance \(\sigma^{2}\) is equal to the sum of the shot noise variance \(\sigma^{2}_{shot}\) and the thermal noise variance \(\sigma^{2}_{thermal}\) as shown in Equation (4):  
\[\begin{gathered}\sigma^{2}=\sigma^{2}_{shot}+\sigma^{2}_{ thermal},\\ \sigma^{2}=2 qR(\lambda)(P_{r}+P_{n})B_{w}+i_{amp}{}^{2}B_{amp},\\ B_{n}=\beta B_{r},\end{gathered} \tag{4}\]  
where:  
\(q\), is the electron charge \(\left(1602\times 10^{-19}\text{ coulomb}\right)\),  
\(R(\lambda)\), is the PD responsivity,  
\(P_{r}\) is the signal power received,  
\(P_{n}\) is the noise power generated by external light sources,  
\(B_{w}\) is the channel equivalent noise bandwidth,  
\(i_{amp}\) is the parasitic current of the amplifier,  
\(B_{amp}\) is the bandwidth of the amplifier,  
\(\beta\) is the Bandwidth factor, and  
\(B_{r}\) is the signal bit rate.  
A commonly used figure of merit in telecommunications is the SNR, which is a ratio between the signal power and the power contributed by the noise described by \(\sigma^{2}\)[17; 18; 19; 20; 21; 22; 23]. In the case of a VLC system, the electrical SNR can be estimated with (5):  
\[\begin{gathered} P_{in}(t)=E(\lambda)Ar,\\ SNR=\frac{(R(\lambda),P_{n}(t))^{2}}{\sigma^{2}}\end{gathered} \tag{5}\]  
where:  
\(P_{in}(t)\) is the incident optical power,  
\(E(\lambda)\) is the irradiance, and  
\(Ar\), is the PD area.