# Securing Smart Grid Enabled Home Area Networks with Retro-Reflective Visible Light Communication
## 2 Methodology
### BER Analysis for Multi-pixel Design  
As shown in Figure 3, the multi-pixel tag reflects the downlink light from the VLC access point (AP) back to the photodiode (PD) embedded on the VLC AP, and modulates the uplink optical signal through separate control of each individual pixel. A multi-level signal generated by the multi-pixel design can be utilized to emulate a sinusoidal wave and thus enables more spectral-efficient modulation schemes than OOK in the single-pixel mode. The distortion of the sinusoidal wave is determined by the number of pixels. In this subsection, we quantify the impact of the distortion on the amplitude and phase offsets of a general M-QAM signal and use 4-QAM as a paradigm to discuss the effect on BER performance. With \(n\) equally sized pixels, \(n+1\) amplitude values can be attained. A quantized arbitrary waveform can be constructed using the \(n+1\) signal levels. Consider an arbitrary signal \(s(t)\) within the range \([-a,a]\); the quantized \(s(t)\) can be computed as follows:  
\[Q[s(t)]=\lfloor\frac{n(s(t)+a)}{2a}\rceil. \tag{1}\]  
The right-hand side of equation first changes the signal range from \([-a,a]\) to \([0,n]\) and then rounds each value to the nearest integer. Each integer level corresponds to a certain number of pixels during the process of signal generation. In general, a QAM signal can be written as [12]  
\[s(t)=\Re\{u(t)e^{2\pi if_{c}t}\}, \tag{2}\]  
where \(u(t)=x_{I}(t)+jx_{Q}(t)\) is a complex waveform with the real part \(x_{I}(t)\) representing the inphase amplitude and imaginary part \(x_{Q}(t)\) representing the quadrature amplitude; \(f_{c}\) is the carrier frequency. A noiseless channel is assumed to achieve isolation of the I-Q offset caused by additive white Gaussian noise (AWGN). The main noise factors at the optical receiver are shot noise and thermal noise [13; 14; 15]. In most of the optical wireless channel literature [13; 14; 15], both the shot noise and the thermal noise are modeled as AWGN. Thus, in the analysis, the noiseless received signal \(r(t)\) can be represented by \(Q[s(t)]\). To demodulate the received signal, we first convert the signal range back to \([-a,a]\) from \([0,n]\), and then multiply by \(e^{-2\pi if_{c}t}\) to separate the inphase and quadrature amplitudes into the real and imaginary parts. Therefore, the demodulated signal can be represented by  
\[\hat{r}(t)=(\frac{2a}{n}Q[s(t)]-a)e^{-2\pi if_{c}t}. \tag{3}\]  
For a noiseless channel, the sampled values of \(\hat{r}(t)\) at \(\frac{1}{2f_{c}}\) and \(\frac{1}{4f_{c}}\) generate the I and Q outputs, respectively, by forcing the double-frequency terms to be zero. Therefore, to quantify the I-Q offset, we derive the close-form expressions for \(\hat{r}(\frac{1}{2f_{c}})\) and \(\hat{r}(\frac{1}{4f_{c}})\). Figure 3: Signal distortion caused by multi-pixel design in Retro-VLC system. **Lemma 1**: \(Q[-s(t)]-\frac{n}{2}=-Q[s(t)]+\frac{n}{2}\)__  
The proof of Lemma 1 is provided in Appendix A. **Theorem 1**: _For a demodulated signal \(\hat{r}(t)=(\frac{2a}{n}Q[s(t)]-a)e^{-2\pi if\xi t}\), the \(I\) output is \(\hat{r}(\frac{1}{2f_{c}})=\frac{2a}{n}Q[\mathfrak{Re}\{u(t)\}]-a\), and the \(Q\) output is \(\hat{r}(\frac{1}{4f_{c}})=j(\frac{2a}{n}Q[\mathfrak{Im}\{u(t)\}]-a)\)._  
The proof of Theorem 1 is provided in Appendix B. For 4-QAM, if we denote the transmitted inphase and quadrature amplitudes both as \(\sqrt{E_{b}}\), according to Theorem 1, the I-Q outputs will be shifted to \(\frac{2a}{n}Q[\sqrt{E_{b}}]-a\). Therefore, an I-Q offset \(\Delta V\) can be defined as  
\[\Delta V=(\frac{2a}{n}Q[\sqrt{E_{b}}]-a)^{2}-E_{b}. \tag{4}\]  
The BER equation for 4-QAM without I-Q offset is \(\frac{1}{2}erfc\sqrt{\frac{E_{b}}{N_{0}}}\)[12]. \(E_{b}\) denotes the distance between the four I-Q points and the original point of the constellation diagram. Since \(\Delta V\) provides a further I-Q offset, it changes the numerator part of the BER equation to \(E_{b}+\Delta V\).