## 3 System Description
### System Model
#### 3.2.3 Channel Model  
As explained earlier, OBANs employ IM/DD for transmission and reception of data, respectively. For IM/DD, the received optical signal is given by:  
\[y(t)=H(0)\gamma x(t)+n(t) \tag{1}\]  
where \(y(t)\) is the received signal, \(H(0)\) is the static channel gain, \(\gamma\) is the receiver responsivity, \(x(t)\) is the transmitted signal, and \(n(t)\) denotes the additive white gaussian noise (AWGN) with double-sided noise power spectral density (PSD), \(N_{0}\). The DC channel gain, \(H(0)\), of an optical link is a key factor in determining \(P_{r}\), which can be calculated for LoS and NLoS links using (2) and (3), respectively:  
\[H_{los}(0)=\frac{A_{r}(m+1)}{2\pi d^{2}}\cos^{m}{(\phi)T_{s}(\psi)g(\psi)}\cos {(\psi)} \tag{2}\]  
\[H(0)_{nlos}^{1}=\frac{(m+1)}{2\pi}\sum_{j=1}^{\kappa}\ \rho_{j}\cos^{m}{ (\phi_{S_{nj}})}\frac{\cos{(\psi_{S_{nj}})}\ cos{(\psi_{Rj})}\triangle\text{AA} _{r}}{d_{S_{nj}}^{2}d_{Rj}^{2}} \tag{3}\]  
where \(A_{r}\) is the physical area of the detector, \(m\) is the Lambert's order, \(T_{s}(\psi)\) is the gain of the optical filter, \(g(\psi)\) represents the concentrator gain, \(\triangle A\) is the area of the differential element, \(\kappa\) is the total number of reflecting elements in the room, \(\rho_{j}\) is the reflectivity of each differential surface, \(d_{S_{nj}}\) is the distance from n\({}^{th}\) source \(S_{n}\) to reflector element j, and \(d_{Rj}\) is the distance between reflector j and receiver R. The total received power (\(P_{r-total}\)) can then be computed as:  
\[P_{r-total}=\sum_{n=1}^{N_{LED}}\left(H_{los}(0)+\sum_{nlos}H(0)_{nlos}^{n} \right)P_{t} \tag{4}\]  
The received signal quality in a VLC system can be evaluated using \(SNR\). It is a quantity used to compare the levels of the desired signal to the level of background noise and is measured in decibels. Without incorporating the BL, the \(SNR\) is expressed as [5;33]:  
\[SNR_{woBL}=\frac{{I_{sig}}^{2}(t)}{R_{b}N_{o}}=\frac{{\gamma^{2}H(0)^{2}{P_{t} }^{2}}}{\left({\sigma_{th}}^{2}+{\sigma_{shot}}^{2}+{\sigma_{d}}^{2}+{\sigma_{ amp}}^{2}\right)B} \tag{5}\]  
where \(I_{sig}\) is the received photoelectric current, \(R_{b}=B\), \(B\) is the bandwidth and \(N_{o}\) is the summation of \(\sigma_{th},\sigma_{shot},\sigma_{d}\) and \(\sigma_{amp}\), which are the thermal, shot, dark, and amplifier noise PSDs, calculated using (6), (7), (8), and (9), respectively:  
\[\sigma_{th}^{2}=\frac{{8\pi kT\xi A_{r}I_{2}B^{2}}}{G}+\frac{{16\pi^{2}kTTA_{r }^{2}\xi^{2}B^{3}I_{3}}}{{g_{m}}} \tag{6}\]  
\[{\sigma_{shot}}^{2}=2 qI_{sig} \tag{7}\]  
\[{\sigma_{d}}^{2}=2 qI_{d} \tag{8}\]  
\[{\sigma_{amp}}^{2}=2 qI_{amp} \tag{9}\]  
where \(k\) is the Boltzmann constant, \(T\) is the temperature in Kelvin, \(I_{2}\) is the noise bandwidth factor for rectangular pulse shape, \(I_{3}\) is the noise bandwidth factor for a full raised cosine and equalized pulse shape, \(\xi\) is the fixed capacitance per unit junction area, \(G\) is the open-loop gain, \(\Gamma\) is the FET channel noise factor, \(g_{m}\) is the FET transconductance, \(q\) is the electronic charge in Coulombs, \(I_{d}\) is the dark current, and \(I_{amp}\) is the input-referred RMS noise. All of these parameters have been summarized in Table 1 and are derived from [5;11]. The \(SNR\) obtained from (5) is for standard VLC communication without considering the BL. Nevertheless, for patient monitoring, the \(SNR\) received at the on-body PD must also incorporate the impact of body on the communication link. Therefore, we calculate \(SNR_{OBAN}\), as :  
\[SNR_{OBAN}=SNR_{woBL}-BL \tag{10}\]  
Although human body does not have significant effects on the communication performance when high body reflectivity and data rates lower than 250 Mbps are considered [30], an average BL value of 5 dB is assumed in this study to account for the reflective losses that occur due to body absorption and other NLoS reflections that fall below the receiver height.