## 4 System Architecture and Performance Analysis
### Performance Analysis
#### 4.3.2 Performance Analysis
More precisely, the dimming level is the ratio of the blur kernel size and the LED area on the image. From Figure 10, when the blur kernel area is within the LED area of an image, the blur effect for the central points of LEDs will occur in each LED illumination area. Thus, in this case, the impact of dimming level on the decoding performance is trivial, as illustrated in Figure 12 with a blur kernel size of \(40\times 40\). On the other hand, when the blur kernel size exceeds the area of one LED on the image, the blur effect to the central point of each LED is also mixed with a part of other nearby LED illumination area. Thus, the impact of the dimming level could be significant. Figure 13 shows the decoding performance of our NN-based decoder for DS8-PSK, which are evaluated based on the SER value with a blur kernel size of \(60\times 60\). The similarity feature that can be realized from both cases is that there always the significant separation of the dimming 1 and dimming 7 lines from other dimming lines. To be more detailed, these two dimming cases always yield the worst SER performance. The reason is that the two cases have one LED ON (or OFF in dimming 7/8) and seven other LEDs have the opposite state. Consequently, the probability of wrongly decoding in these cases happens when the unique LED and one among other seven similar LEDs are detected with false states. This error probability can be expressed as follows:  
\[\text{P}_{\text{e,decoding}}=7\text{P}(\text{s}_{\text{ON}}|_{\text{detected =OFF}},\text{s}_{\text{OFF}}|_{\text{detected=ON}})=7\text{P}_{\text{e}}^{2} \tag{11}\]  
Figure 12: The comparison of symbol error rate (SER) performance between an NN-based decoder and a matched filter on different dimming cases with a blur kernel size of \(40\times 40\). With the dimming from 2/8 to 6/8, the performance lines are approximately similar. Moreover, the theoretical error decoding probability of each dimming case could be easily calculated as follows:  
\[\text{Dimming 2/8 or 6/8}:\ \text{P}_{\text{e,decoding}}=2\text{p}_{\text{e}}^{2}+5 \text{p}_{\text{e}}^{4}; \tag{12}\]  
\[\text{Dimming 3/8 or 5/8}:\ \text{P}_{\text{e,decoding}}=2\text{p}_{\text{e}}^{2}+2 \text{p}_{\text{e}}^{4}+3\text{p}_{\text{e}}^{6}; \tag{13}\]  
\[\text{Dimming 4/8}:\ \text{P}_{\text{e,decoding}}=2\text{p}_{\text{e}}^{2}+2 \text{p}_{\text{e}}^{4}+2\text{p}_{\text{e}}^{6}+\text{p}_{\text{e}}^{8}; \tag{14}\]  
Notably, it can also be assumed that the portions of error probability with an exponent equal or greater than four could be insignificant, in which case the error probability of all dimming cases from 2/8 to 6/8 could be approximately the same (\(\approx 2\text{p}_{\text{e}}^{2}\)), provided that the bit error probability (p\({}_{\text{e}}\)) is equal for all LEDs. With the blur kernel size within the LED area as in Figure 10, \(\text{p}_{\text{e}}\) could be the same for all eight LEDs within that group. However, when the blur kernel size exceeds an LED area as in Figure 11, the \(\text{p}_{\text{e}}\) in each LED will now be different, also depending on the LED array arrangement. Figures 12 and 13 also show that compared with the decoding technique using a matched filter, our proposed decoding method of using the neural network-based decoder for the DS8-PSK scheme performed excellently well and enhanced the SER performance of all of the dimming cases considered.