# Signal Quality Assessment for Transdermal Optical Wireless Communications under Pointing Errors +
## 2 System Model  
As demonstrated in Figure 1, the main components of a transdermal MI are the out-of-body unit, the channel (skin) and the in-body unit. The out-of-body unit consists of a data capturing unit that converts external stimulations into electrical signals, followed by the the digital signal processing (DSP) unit that digitizes and compresses the data into modulated signals. In turn, these signals are transmitted by the TX to the RX of the in-body unit. The received signal is processed via a DSP stimulation (STM) unit that are responsible for generating the appropriate stimulations. We assume that the transmitted signal, \(x\), conveys over the optical wireless channel, \(h\), with additive noise \(n\). Hence, the baseband equivalent received signal can be obtained as  
\[y=Rlnx+n, \tag{1}\]  
where \(R\) stands for the responsivity of the RX's PD and can be expressed as  
\[R=\eta\frac{\left|e\right|}{p\,v_{p}}, \tag{2}\]  
with \(\eta\) being the quantum efficiency of the PD, \(e\) the electron charge, \(v_{p}\) the photons' frequency and \(p\) the Planck constant. Next, we provide the channel model under consideration. The baseband equivalent channel, \(h\), can be written as in [15; 19]  
\[h=h_{l}h_{p}, \tag{3}\]  
where \(h_{l}\) and \(h_{p}\), respectively, represent the deterministic channel coefficient, due to the path gain, and the stochastic process that models the geometric spread, because of the pointing errors. We assume a Gaussian spatial intensity profile of \(w_{\delta}\) on the RX plane at distance \(\delta\) from the TX and circular aperture of radius \(\beta\). Likewise, \(h_{p}\) represents the fraction of the collected power, due to geometric spread with radial displacement \(r\) from the origin of the PD. By assuming that the elevation and the horizontal displacement (sway) follow independent and identical Gaussian distributions, a usually employed assumption used in OWCs [20], we observe that the radial displacement at the RX follows a Rayleigh distribution. Thus, the probability density function (PDF) of the random variable \(h_{p}\) can be obtained as [21]  
\[f_{h_{p}}\left(x\right)=\frac{\xi}{A_{0}^{\xi}}x^{\xi-1},\qquad 0\leq x\leq A_{0}, \tag{4}\]  
where  
\[A_{0}=[\mathrm{erf}\left(v\right)]^{2}, \tag{5}\]  
Figure 1: System model under consideration. stands for the the fraction of the collected power in case of zero radial displacement [22] and  
\[v=\frac{\sqrt{A}}{\sqrt{2}w_{\delta}}, \tag{6}\]  
with \(A\) and \(w_{\delta}\), respectively, denoting the RX's PD effective area and the beam waste (radius calculated at \(e^{-2}\)) on the RX plane at distance \(\delta\) from the TX. Meanwhile, \(\xi\) is the square ratio of the equivalent beam radius, \(w_{e}\), and the pointing error displacement standard deviation at the RX, which can be obtained as  
\[\xi=\frac{w_{e}^{2}}{4\sigma^{2}}, \tag{7}\]  
where \(\sigma^{2}\) is the pointing error jitter variance at the RX. Finally, \(w_{e}^{2}\) can be evaluated as  
\[w_{e}^{2}=w_{\delta}^{2}\frac{\sqrt{\pi}\operatorname{erf}\left(v\right)}{2v \exp\left(-v^{2}\right)}. \tag{8}\]  
Based on Equation (4), the cumulative distribution function (CDF) of \(h_{p}\) can be obtained as  
\[F_{h_{p}}\left(x\right)=\int_{0}^{x}f_{h_{p}}\left(y\right)\,\mathrm{d}y, \tag{9}\]  
or, by substituting Equation (4), as  
\[F_{h_{p}}\left(x\right)=\left\{ \begin{array}{cc}\int_{0}^{x}\frac{\xi}{A_{0 }^{5}}y^{\xi-1}\,\mathrm{d}y,&0\leq y\leq A_{0}\\ 1,&x\geq A_{0}\end{array} \right.. \tag{10}\]  
By employing ([18], equation 2.01/1), Equation (10) can be rewritten as  
\[F_{h_{p}}\left(x\right)=\left\{ \begin{array}{cc}\frac{1}{A_{0}^{5}}x^{\xi} &0\leq x\leq A_{0}\\ 1,&x\geq A_{0}\end{array} \right.. \tag{11}\]  
The deterministic term of the channel coefficient can be expressed as ([8], Equation (10.1))  
\[h_{l}=\exp\left(-\frac{1}{2}\alpha(\lambda)\delta\right), \tag{12}\]  
where \(\alpha(\lambda)\) and \(\delta\) represent the skin attenuation coefficient in the wavelength \(\lambda\), and the total dermis thickness, respectively.