# Polarization Differential Visible Light Communication: Theory and Experimental Evaluation +
## 2 System Model
### Signal Processing and Signal-to-Noise Ratio (\(Snr\))  
In order to evaluate the impact of receiver rotation on the \(SNR\), we assume a single receiver pair using a pair of orthogonal polarizers as previously elaborated. The linearly polarized optical  
Figure 1: Setup consisting of two LEDs and two photodiodes, each equipped with mutually orthogonal linear polarizers (LP) [17]. signals transmitted by \(\mathrm{T_{X1}}\) and \(\mathrm{T_{X2}}\), thus after transmittance through the polarizers, are denoted as \(s_{1}(t)\) and \(s_{2}(t)\) respectively. As mentioned before, \(\mathrm{T_{X1}}\) and \(\mathrm{T_{X2}}\) are complementarily toggled resulting in polarization-based modulation. Ideally, the optical power in transmittance \(A\) is identical for these complementary signals as illustrated in the upper part of Figure 2.  
Taking the optical channel model and receiver polarizers into consideration, it can be shown that the received signals at the photodetectors are given by [19]:  
\[ \begin{bmatrix}r_{1}(t)\\ r_{2}(t)\end{bmatrix} =R_{p} \begin{bmatrix}\alpha_{11}\cos^{2}(\theta)&\alpha_{2 1}\cos^{2}(\theta+90^{\circ})\\ \alpha_{12}\cos^{2}(\theta+90^{\circ})&\alpha_{22}\cos^{2}(\theta)\end{bmatrix} \begin{bmatrix}s_{1}(t)\\ s_{2}(t)\end{bmatrix} + \begin{bmatrix}w_{1}(t)\\ w_{2}(t)\end{bmatrix} . \tag{2}\]  
Here \(R_{p}\) is the photodiode responsivity, assumed identical for both photodiodes, \(\alpha_{ij}\) is the optical channel gain from \(\mathrm{T_{Xi}}\) to \(\mathrm{R_{Xi}}\). \(\theta\) is the rotation angle of the receiver polarizer axes with regard to the transmitter polarizers axes. \(w_{1}(t)\) and \(w_{2}(t)\) are the noise contributions at each photodetector that can be modeled as Additive White Gaussian Noise (AWGN) [20]. These contributions are a sum of thermal and shot noise. While the transmitted signals \(s_{1}(t)\) and \(s_{2}(t)\) typically contain noise contributions as well, these noise contributions can be neglected compared to the thermal and shot noise contributions. Taking into account the differential signalling operation of \(s_{1}(t)\) and \(s_{2}(t)\), \(s(t)\) is defined as:  
\[s(t)=s_{1}(t)-s_{2}(t). \tag{3}\]  
Figure 2: Illustration of ideal complementary transmitted signals \(s_{1}(t)\) and \(s_{2}(t)\) (top) and DC-filtered differential signal \(s(t)\) (bottom) as defined by Equation (3) [17]. As shown in the bottom part of Figure 2, \(s(t)\) can also be expressed as a signal proportional to the AC-component of \(s_{1}(t)\) and \(s_{2}(t)\):  
\[ \begin{split} s(t)&=2\left[s_{1}(t)-\left\langle s_{ 1}(t)\right\rangle\right]\\ &=-2\left[s_{2}(t)-\left\langle s_{2}(t)\right\rangle\right], \end{split}  \tag{4}\]  
where \(\left\langle\ \right\rangle\) denotes the mean operator, thus corresponding to the DC-components of \(s_{1}(t)\) and \(s_{2}(t)\). Ideally, the mutual distances between both transmitters and receivers are very small, meaning the difference in path length and the difference in both irradiance and incidence angle are negligible. Under these circumstances, it can be assumed that \(\alpha=\alpha_{11}=\alpha_{12}=\alpha_{21}=\alpha_{22}\). Combining Equations (2) and (3) and applying \(\cos(\theta\pm 90^{\circ})=\mp\sin(\theta)\) and \(\cos(2\theta)=\cos^{2}(\theta)-\sin^{2}(\theta)\), the differential signal \(r(t)\) between the photodetectors is thus given by:  
\[r(t)=r_{1}(t)-r_{2}(t)=\alpha R_{p}s(t)\cos(2\theta)+w(t), \tag{5}\]  
where \(w(t)=w_{1}(t)-w_{2}(t)\). As both noise components \(w_{1}(t)\) and \(w_{2}(t)\) consist of differential and common-mode noise, the differential setup effectively suppresses the common-mode noise component by differentiating between the two photodetectors. The differential operation also ensures that interference from intensity-based VLC is eliminated as the contributions of unpolarized, modulated light has an equal impact on both \(r_{1}(t)\) and \(r_{2}(t)\). In practice, \(r(t)\) is also DC-filtered to account for any possible asymmetry between the detectors.