# Design of Indoor Visible Light Communication PAM4 System
## 2 Theoretical Basis  
Most VLC systems are designed as intensity modulation/direct detection (IM/DD) systems. OOK is the easiest modulation method to implement in an IM/DD system, but the transmission efficiency of OOK is too low [14], while PAM4 can transmit 2 bits of information per waveform, effectively improving the communication rate and modulation efficiency, which can meet the requirements of high transmission efficiency. Under the same transmission rate, the bandwidth utilization of PAM4 is higher than that of OOK, PPM, and DPPM. Table 1 shows the bandwidth requirements of four modulation methods [15], where R\({}_{\text{b}}\) refers to the transmission rate, and M refers to the modulation order. As can be seen from Table 1, in the case of the same transmission rate, PPM requires the largest bandwidth, followed by DPPM, and PAM requires the smallest bandwidth. Therefore, the bandwidth utilization rate of PAM is higher than that of the other three methods, with higher spectral efficiency, and the data transmission rate is the highest under the same bandwidth requirements. As the order M of PAM increases, the bandwidth utilization rate and data transmission rate will also increase, but the anti-noise performance of the system will gradually deteriorate, and the required hardware will also increase its complexity [16; 17; 18]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Modulation** & **Bandwidth Requirements** \\ \hline OOK & R\({}_{\text{b}}\) \\ PAM & R\({}_{\text{b}}\)/log2(M) \\ PPM & M R\({}_{\text{b}}\)/log2(M) \\ DPPM & (M+1) R\({}_{\text{b}}\)/log2(M) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The bandwidth requirements for different modulation methods [19]. PAM4 uses periodic pulses as the carrier, and the symbol information of the source controls the amplitude of the pulse. The waveform diagram of the PAM4 signal is shown in Figure 1, and four different amplitudes control different symbol information. For the PAM4 signal, its waveform is expressed as [19]:  
\[s_{m}(t)=A_{m}p(t),\ 0\leq t\leq T,\ m=1,2,3,4 \tag{1}\]  
where \(T\) is the pulse duration, which is called the symbol interval, \(t\) is the time, \(p(t)\) is the rectangular pulse with duration \(T\), and \(A_{m}\) is the amplitude of the \(m\)-th waveform, which is expressed as:  
\[A_{m}=(2m-5)d,\ m=1,2,3,4 \tag{2}\]  
where \(d\) is half the Euclidean distance between adjacent levels. \(p(t)\) is defined as:  
\[p(t)=\sqrt{1/T} \tag{3}\]  
Therefore, the energy of the pulse \(p(t)\) is normalized to \(1\). From Equation (1), it can be seen that the energy of the PAM4 signal is [14]:  
\[\varepsilon_{m}=\int_{-\infty}^{+\infty}A_{m}^{2}p^{2}(t)dt=A_{m}^{2} \varepsilon_{p} \tag{4}\]  
where \(\varepsilon_{p}\) is the energy of the rectangular pulse \(p(t)\). From this, the average energy of the PAM4 signal can be obtained as:  
\[\varepsilon_{avg}=\frac{\varepsilon_{p}}{4}\sum_{m=1}^{4}A_{m}^{2}=\frac{2 \varepsilon_{p}}{4}[1^{2}+3^{2}]=5d^{2}\varepsilon_{p}=5d^{2} \tag{5}\]  
Since the PAM4 signal only modulates the amplitude, the constellation map of the PAM4 signal only has values on the in-phase component, and the signal points are all located on the in-phase component. Therefore, the PAM4 map belongs to the one-dimensional real constellation map, and the Euclidean distance \(d_{\min}\) of any pair of adjacent signal points is [14]:  
\[d_{\min}=\sqrt{\parallel s_{m}-s_{n}\parallel^{2}}=[A_{m}-A_{n}]\sqrt{ \varepsilon_{p}}=2\sqrt{\varepsilon_{p}}=2\sqrt{\frac{1}{5}\varepsilon_{avg}} =2d \tag{6}\]  
where \(\varepsilon_{avg}\) is the average energy of the PAM4 signal. \(S_{n}\) and \(S_{m}\) represent the energy values of any two adjacent signal points. It can be seen from Equation (6) that the Euclidean distance obtained is exactly the same as the expression of Equation (2). Since the amplitude of the PAM4 signal cannot be negative in VLC, and the amplitude of the PAM4 signal has no fixed value, it can be adjusted appropriately to meet the demand, so the amplitude after PAM4 encoding is set to \(0,1,2,\) and \(3\).