# Securing Smart Grid Enabled Home Area Networks with Retro-Reflective Visible Light Communication
## 3 Numerical and Experimental Results  
### BER Performance  
To validate the BER (Equation (5)) derived in Section 2.1, we perform both numerical and experimental analysis for different numbers of pixels. Note that the total area of the reflecting surface of the pixelated shutter is fixed; as the number of pixels increases, the area of each pixel reduces. In BER experimental validation, to minimize the signal distortion caused by slow response of LCD shutter, we selected a low carrier frequency of 2 Hz.  
In Figure 6, theoretical results are compared to simulation results for 11, 18, and 32 pixels. The lower bound of BER is calculated based on the assumption of an infinite number of pixels (i.e., \(\Delta V=0\)). Different values of \(\frac{E_{b}}{N_{0}}\) are considered to evaluate BER at different distances between the VLC AP and the Retro-VLC tag. In the simulation, 50,000 QAM symbols are generated randomly for each test when BER \(>10^{-3}\), and the number of QAM symbols are increased to \(10^{6}\) when BER \(<10^{-3}\). The simulation was implemented in MATLAB R2020a. The simulator begins by generating a sampled time domain QAM signal at 100 samples per symbol. The samples are then quantized based on Equation (1). AWGN is superimposed on the quantized sample with a constant \(\frac{E_{b}}{N_{0}}\). Finally, demodulation is performed in line with the procedures described in Section 2.1. We can observe from Figure 6 that as the number of pixels increases, the BER performance asymptotically approaches the lower bound (blue curve). The simulation results match well with the theoretical results with a mean square error of 6.4678%.  
In Figure 7, the AP-tag distance is fixed and the theoretical, simulation ([[https://github.com/mathewsalas/pixelated_RVLC_simulation.git](https://github.com/mathewsalas/pixelated_RVLC_simulation.git)]([https://github.com/mathewsalas/pixelated_RVLC_simulation.git](https://github.com/mathewsalas/pixelated_RVLC_simulation.git)), accessed on 30 November 2022) and experimental BER results are plotted as the number of pixels increases. The asymptotic line is also provided by assuming an infinite number of pixels. For experiments, to emulate a lower number of pixels, individual pixels are paired into blocks. Since each pixel covers the same area of the retro-reflective surface, each block can be treated as a larger single pixel. The graphic shutter used in Figure 5 has a total of 36 pixels; using the pairing method discussed above, the number of pixels we evaluate in experiments is the divisors of 36. The theoretical, simulated, and experimental results match well, and all exhibit non-monotonic behavior.  
The sawtooth pattern of BER is the combinatorial result from the rounding operation and 4-QAM setting. First, the rounding operation in Equation (1) replaces the sample value by either a larger integer or a smaller integer depending on the number of pixels \(n\) and the signal amplitude \(a\). The larger or smaller integer replacement will result in an outward I-Q shift (\(\Delta V\) is positive) or an inward I-Q shift (\(\Delta V\) is negative), respectively. Using hard decision decoding in 4-QAM, an outward I-Q shift will improve the BER performance, which is not necessarily true as the QAM order increases. With the number of pixels increased, the sign of \(\Delta V\) will change multiple times, which is represented as the turning points of the BER curve.  
Figure 6: Theoretical results vs. simulation results for three different pixel numbers. The blue curve is plotted assuming an infinite number of pixels (i.e., \(\Delta V=0\)).  
Figure 7: Theoretical results (red curve) generated by Equation (5), simulation results (yellow circle) and experimental results (green cross) are presented. The blue line is plotted assuming an infinite number of pixels (i.e., \(\Delta V=0\)). \(\frac{E_{b}}{N_{0}}\) is set at 6.6 dB based on experimental measurement.