# Experimental Characterization of RGB LED Transceiver in Low-Complexity LED-to-LED Link
## 3 Results and Analysis
### BER Improvement with Matched Filter  
Now that the photodetector noise has been determined, the bit error rate (BER) and the SNR of the system can be deduced to investigate the performance of the system. The BER can be measured using two methods:  
The error bits are deduced from the generated PRBS20 bits to calculate the BER:  
\[BER\ =\ \frac{error\ bits}{total\ number\ of\ bits} \tag{4}\]  
In order to verify the measured BER results, the BER is also calculated via the quality factor (Q-factor) obtained from the captured eye diagram [24]:  
\[Q\ =\ \frac{\left|\mu_{1}-\mu_{0}\right|}{\sigma_{1}+\sigma_{0}} \tag{5}\]  
where \(\mu_{1}\) and \(\mu_{0}\) are the mean voltages for the '1'-bit voltage rail and '0'-bit voltage rail and \(\sigma_{1}\) and \(\sigma_{0}\) are the standard deviation values of the '1' and '0' rails due to the AWGN respectively. Hence, the BER can be calculated using the following relation [24]:  
\[BER(V_{th})=\frac{1}{2}\Bigg{[}erfc\Bigg{(}\frac{\left|\mu_{1}-V_{th}\right|}{ \sigma_{1}}\Bigg{)}\ +\ erfc\Bigg{(}\frac{\left|V_{th}\ -\ \mu_{0}\right|}{\sigma_{0}}\Bigg{)}\Bigg{]} \tag{6}\]  
Figure 4: Bode plot of TIA receiver circuit. where \(V_{th}\) is the threshold voltage which determines where the decision is made. In this scenario it is taken as the centre rail between the '0' and '1'. The histograms for 0 and 10 cm at 10 kbps as well as the eye diagrams and Q-factor's parameters are depicted in Figure 5a,b, respectively. For the back-to-back case in Figure 5a, the two rails of the signal are distinct and far away from each other, allowing a right decision to be taken with \(V_{th}\) at the center. This agrees with the wide eye opening. On the contrary, this distinction vanishes in the signal constellation at 10 cm distance as seen in Figure 5b, where the normal distribution functions of the '0' and '1' rails are overlapping. In agreement therewith, the eye is closed, which corresponds to a higher probability of error. The comparison between the measured BER results employing Equation (4) and the calculated BER from Equation (6) for all data rates shows less than 9% deviation between the two methods as seen in Figure 6a. The deviation is calculated by subtracting the measured from the calculated BER and dividing by the measured (true) value. Hence, both approaches have similar responses which validates the practical approach. Figure 6b shows the average BER at varying distances. As expected, the BER rises with longer transmission distance and higher data rate. Error-free data transmission can be achieved at a maximum distance of 1 cm at 200 kbps, 1.5 cm at 100 kbps and 2 cm at 50 kbps while it reaches 3 cm at 10 kbps. Figure 5: Eye diagram (inset) and signal constellation in (a) back-to-back configuration and (b) at 10 cm distance. Figure 6: (a) Comparison between the measured and calculated BER; (b) BER and SNR at different distances and data rates. The BER stays below the acceptable \(10^{-3}\) range [17] up to 7 cm for 10 and 50 kbps, up to 4 cm for 100 kbps and up to 3 cm for 200 kbps. This low sensitivity is because an RGB LED has a much smaller active area for light emission and detection per sub-LED compared to a regular single-colour LED. This is apparent after placing both LEDs under a microscope and comparing their active areas when dimly lit. The active area of the single-colour LED is 1.99 times larger than that of the sub-LED of the RGB LED, which means only 50% of light is captured compared to single-colour LED. Figure 7 compares the active areas (circled in red) of the single-colour red LED in (a) with that of the red sub-LED in a typical off-the-shelf RGB LED in (b) under the same scale. Moreover, it can be concluded that the small data rates achieved by this small-area LED photodetector can be interpolated to match the higher data rates achieved in [PERSON]'s research [17].