# Hybrid Adaptive Bias OFDM-Based IM/DD Visible Light Communication System
## 2 System Model
### Transmitter  
The block diagram structure of the transmitter of the proposed HABO-OFDM system is presented in Figure 1.  
In the proposed scheme, for the \((2m+1)\)-th \((m=0,1,\cdot\cdot\cdot,N/8-1)\) subcarriers, the transmitted signal is mapped in line with the Hermitian symmetry [11]. Thus, the frequency-domain signal of this component is expressed as  
\[\mathbf{X}_{1}=[0,Q_{1},0,Q_{2},0,\cdot\cdot\cdot,Q_{N/4},0,Q_{N/4}^{*},0,\cdot \cdot\cdot,Q_{2}^{*},0,Q_{1}^{*}], \tag{1}\]  
where \(N\) is the number of subcarriers, and \(Q_{i}\) is the quadrature amplitude modulation (QAM) symbol. For the \((4m+2)\)-th \((m=0,1,\cdot\cdot\cdot,N/8-1)\) subcarriers, the modulation of the transmitted signal is QAM. Hence, the frequency-domain signal is given as follows:  
\[\mathbf{X}_{2}=[0,0,Q_{N/4+1},0,\cdot\cdot\cdot,Q_{3N/8},0,0,0,Q_{3N/8}^{*}, \cdot\cdot\cdot,0,Q_{N/4+1}^{*},0], \tag{2}\]  
where \(Q_{i}\) denotes the QAM symbol. Furthermore, the modulated transmission signals in \(\mathbf{X}_{1}\) and \(\mathbf{X}_{2}\) are allocated on different subcarriers. Then, the frequency-domain signal of the superimposition of two different component is given by  
\[\mathbf{X}=[0,Q_{1},Q_{N/4+1},Q_{2},0,\cdot\cdot\cdot,Q_{3N/8},Q_{N/4},0,Q_{N/ 4}^{*},Q_{3N/8}^{*},\cdot\cdot\cdot,0,Q_{N/4+1}^{*},0], \tag{3}\]  
where \(Q_{i}\) is the QAM symbol. The remaining component in HABO-OFDM only modulates the imaginary parts of the \(4m\)-th \((m=0,1,\cdot\cdot\cdot,N/8-1)\) subcarriers, which can be presented as  
\[\mathbf{Y}=j[0,0,0,0,P_{1},0,\cdot\cdot\cdot,P_{N/8-1},0,0,0,0,0,0,P_{N/8-1}^ {*},0,\cdot\cdot\cdot,0,P_{1}^{*},0,0,0], \tag{4}\]  
where \(Y_{4k}=jP_{k}\) and \(P_{k}\) represents the real PAM symbol, \(k=1,\cdot\cdot\cdot,N/2-1\). Additionally, the time-domain signal \(y_{n}\) holds anti-symmetry and periodic property [18],  
\[ \begin{cases}y_{0}=y_{N/4}=y_{N/2}=y_{3N/4}=0,\\ y_{n}=y_{n+N/2}=-y_{N/2-n}=-y_{N-n},\ \ \ n=1,2,\cdot\cdot\cdot,N/4-1\end{cases} . \tag{5}\]  
Therefore, the combined time-domain signal \(w_{n}\) of the proposed scheme, is as follows:  
\[w_{n}=x_{n}+\lfloor y_{n}\rfloor,\ \ \ \ n=0,1,\cdot\cdot\cdot,N-1, \tag{6}\]  
where \(x_{n}\) denotes the time-domain signal of \(\mathbf{X}\), and \(\lfloor y_{n}\rfloor_{c}=\max\{y_{n},0\}\) presents the clipped value of \(y_{n}\). Thus, \(x_{n}\) does not interfere with the \(4m\)-th subcarriers of the PAM component due to \(x_{n}\) is unclipped. Moreover, the interference generated by \(\lfloor y_{n}\rfloor\) only falls on the real parts of the \(4m\)-th subcarriers, which implies that the \((2m+1)\)-th and the \((4m+2)\)-th subcarriers with the imaginary parts of the \(4m\)-th subcarriers are not interfered with the clipping operation of \(y_{n}\). The combination of QAM and PAM achieves the advantage of high spectral efficiency. However, the hybrid optical OFDM signal of \(w_{n}\) is a bipolar signal, and it is necessary to ensure the non-negativity of the signal in the optical OFDM  
Figure 1: Block diagram structure of the transmitter of the proposed HABO-OFDM system. scheme [19]. In order to guarantee the non-negativity of \(w_{n}\), we design the adaptive bias according to the characteristics of the superimposed signal. Furthermore, the properties which are satisfied with the adaptive bias \(s_{n}\) can be expressed as  
\[ \begin{split}& s_{n}=s_{n+N/4}=s_{n+N/2}=s_{n+3N/4}=-\min\{w_{n},w _{n+N/4},w_{n+N/2},w_{n+3N/4}\},\\ & n=0,N/8,\end{split}  \tag{7}\]  
\[ \begin{split}& s_{n}=s_{N/4-n}=s_{N/4+n}=s_{N/2-n}=s_{N/2+n}=s_{3N /4-n}=s_{3N/4+n}=s_{N-n}\\ &\quad=-\min\{w_{n},w_{N/4-n},w_{N/4+n},w_{N/2-n},w_{N/2+n},w_{3N /4-n},w_{3N/4+n},w_{N-n}\},\\ & n=1,\cdots,N/8-1\end{split} . \tag{8}\]  
According to (7), it is obvious that, when at least one of the four signals, \(w_{n}\), \(w_{n+N/4}\), \(w_{n+N/2}\), \(w_{n+3N/4}\), for \(n=0,N/8\) is negative, the inverse polarity of the minimum of \(\{w_{n},w_{n+N/4},w_{n+N/2},w_{n+3N/4}\}\) can be selected to produce the adaptive bias.