# Constant Transmission Efficiency Dimming Control Scheme for VLC Systems
## 4 Simulation Results
### Error Performance  
Error performance is another essential index for VLC systems. It is represented by curves of the bit error rate (BER). The curves describe the relationship between the signal-to-noise ratio (SNR) and BER. The SNR can be expressed as [21]:  
\[\text{SNR}=10\lg\frac{1}{R_{C}\sigma^{2}}, \tag{12}\]  
where \(R_{C}\) is the code rate. In this paper, we compare the block coding dimming control scheme and the proposed scheme. From the discussion before and the contents in the last subsection, we know that the length of the coded binary sequence \(n\) of the proposed ECWCS scheme is not unique when \(k\) is fixed. As shown in Equation (5), for the proposed scheme \(n\geqslant 2^{k}\). While for the block coding scheme, \(n=k^{2}\). Therefore, we fix the value of \(n\), \(k\), and \(\gamma\) to compare the error performance of the two schemes for fairness. From Figure 3, we can conclude that when the value of \(n\), \(k\), and \(\gamma\) are the same, the error performance of the ECWCS scheme is better than that of the block coding scheme. That is because the minimum Hamming distance between the codewords of the ECWCS scheme is larger than that of the block coding scheme. We can also see that for the proposed ECWCS scheme, the error performance with \(k=2,n=4\) outperformed \(k=3\), \(n=9\) under the same dimming factor. For clarity, we demonstrate the adaptability of this system in terms of provided illuminance. First, we introduce the relationship between the dimming factor and illuminance. We know that the dimming factor is the ratio of the average power to peak power, and the range of dimming factor is \(0<\gamma<1\). The value of the dimming factor reflects the illuminance of the LED. With the increase of the dimming factor, the illuminance of the LED increases gradually. Then we demonstrate the relationship between BER and illuminance. We can assume a set \(\mathcal{D}\) consists of the Euclidean distance between any two distinct received signals. The minimum Euclidean distance (MED) of the received signals can be defined as the minimum of the elements in set \(\mathcal{D}\). Therefore, the MED of the received signals can be expressed as:  
\[d_{R}=\min\sqrt{\|y^{a}-y^{b}\|^{2}}, \tag{13}\]  
where \(y^{a}\) and \(y^{b}\) are two distinct received signals. Figure 3: Error performance of the block coding scheme and the ECWCS scheme. Similarly, the MED of the transmitted signals is:  
\[d_{T}=\min\sqrt{\|x^{a}-x^{b}\|^{2}}, \tag{14}\]  
where \(x^{a}\) and \(x^{b}\) are two distinct transmitted signals. We know that the received signal can be expressed as Equation (1) in Section 2. We substitute Equation (1) into Equation (13) and the MED of the received signals can be also expressed as  
\[d_{R}=\min\sqrt{\|(\mu hx^{a}+n)-(\mu hx^{b}+n)\|^{2}}=\min\sqrt{\mu h\|x^{a}-x ^{b}\|^{2}}=d_{T}\sqrt{\mu h}. \tag{15}\]  
There is a point that should be noted: In terms of sequences, the received signal is the received sequence **r** and the transmitted signal is the coded binary sequence **c**. The distance between the signals means the distance between the sequences, and the numbers in the sequence represent the intensity level of the optical signal. From the contents in paper [22], we know that the BER curve can represent the error performance, and the error performance is determined by the MED of the received signals when the SNR is fixed. Therefore, in the proposed system model of this paper, the error performance is determined by the MED of the transmitted signals when the SNR is fixed. The way we explain the BER metric according to minimum Euclidean distance is not only for OOK modulation but also for pulse amplitude modulation. In this paper, we provide OOK modulation as an example to introduce the ECWCS scheme. For the proposed ECWCS scheme, the MED of the transmitted signals is constant when the dimming factor varies. Therefore, the BER curves are the same with different dimming factors. However, when the dimming factor varies, the average power of the transmitted signals changes.