# Low-Complexity Hybrid Optical OFDM with High Spectrum Efficiency for Dimming Compatible VLC System
## 3 Proposed Dimming Compatible Hybrid O-OFDM
### Dcho-Ofdm
: _If a time-domain signal \(z_{n}\) has the characteristic as:_  
\[\left\{ \begin{array}{ll}z_{n}=z_{\frac{N}{2}+n},&n=0,\frac{N}{4}\\ z_{n}=z_{\frac{N}{2}-n}=z_{\frac{N}{2}+n}=z_{N-n},&n=1,\cdots,\frac{N}{4}-1 \end{array} \right., \tag{9}\]  
_the interference generated by \(z_{n}\) imposes only on the real-part of the even-indexed subcarriers._  
Proof. : Please see Appendix B. For the hybrid signal \(s_{n}^{\text{on}}\), it is expected to be negative for further incorporating the PWM signal during ON state. In the light of Proposition 1, a bias of \(-\max\left\{s_{n}^{\text{on}},s_{N/2+n}^{\text{on}}\right\}\) can be added to sample \(s_{n}^{\text{on}}\) and \(s_{N/2+n}^{\text{on}}\), for \(n=0,N/4\), so that to achieve the negative-valued samples. Uniformly, a bias of \(-\max\left\{s_{n}^{\text{on}},s_{N/2-n}^{\text{on}},s_{N/2+n}^{\text{on}},s_{ N-n}^{\text{on}}\right\}\) is applied for sample \(s_{n}^{\text{on}},s_{N/2-n}^{\text{on}},s_{N/2+n}^{\text{on}}\), and \(s_{N-n}^{\text{on}}\) to acquire the non-positive values, when we have \(n=1,2,\cdots,N/4-1\). Therefore, the introduced time-varying bias \(z_{n}^{\text{on}}\) for signal \(s_{n}^{\text{on}}\) is given by:(10)  
After performing the time-varying biasing, the bipolar signal \(s_{n}^{\text{on}}\) is able to be converted to the unipolar signal as:  
\[s_{\text{conv},n}^{\text{on}}=x_{ACO,n}+\left[y_{PAM,n}\right]_{c}+z_{n}^{\text{ on}},\quad n=0,1,\cdots,N-1, \tag{11}\]  
where all the \(N\) samples experience non-positive values, which is demonstrated in Figure 2c. On the contrary, a positive-valued signal is desired during OFF state transmission. Therefore, for the bipolar signal \(s_{n}^{\text{off}}\), the time-varying bias \(z_{n}^{\text{off}}\) is introduced as:  
Figure 2: A demonstration of the superposed signal in DCHO-OFDM. (**a**,**b**) depict the bipolar signal \(s_{n}^{\text{on}}\) and \(s_{n}^{\text{off}}\), where the ACO-OFDM signal is directly combined with upper/down-clipped PAM-DMT; (**c**,**d**) present the unipolar signal of \(s_{\text{conv},n}^{\text{on}}\) and \(s_{\text{conv},n}^{\text{off}}\), aided by the introduced time-varying biasing scheme. \[\left\{ \begin{array}{l}z_{n}^{\text{off}}=z_{\frac{N}{2}+n}^{\text{off}}=-\min \left\{s_{n}^{\text{off}},s_{\frac{N}{2}+n}^{\text{off}}\right\},\qquad\quad n=0,\frac{N}{4}\\ z_{n}^{\text{off}}=z_{\frac{N}{2}-n}^{\text{off}}=z_{\frac{N}{2}+n}^{\text{ off}}=z_{N-n}^{\text{off}}\\ \qquad=-\min\left\{s_{n}^{\text{off}},s_{\frac{N}{2}-n}^{\text{off}},s_{\frac{ N}{2}+n}^{\text{off}},s_{N-n}^{\text{off}}\right\},\quad\quad n=1,\cdots,\frac{N}{4}-1 \end{array} \right.. \tag{12}\]  
Hence, the converted signal is given by:  
\[s_{\text{conv},n}^{\text{off}}=x_{ACO,n}+\lfloor y_{PAM,n}\rfloor_{c}+z_{n}^{ \text{off}},\quad n=0,1,\cdots N-1, \tag{13}\]  
resulting in a positive-valued signal as shown in Figure 2d, which is appropriated for OFF state transmission. Assisted by the time-varying biasing scheme, the acquired unipolar signals relying on the Equations (11) and (13) can be readily amalgamated with PWM signal. To this end, the proposed DCHO-OFDM signal can be expressed as:  
\[ \begin{split} v_{n}&=\left\{\begin{array}{ll}s_{ \text{conv},n}^{\text{on}}+I_{h},&\text{ON state}\\ s_{\text{conv},n}^{\text{off}}+I_{l},&\text{OFF state}\end{array}\right.\\ &=s_{\text{conv},n}+d_{PAM,n},\quad n=0,1,\cdots,N-1.\end{split}  \tag{14}\]  
Note that the signal of \(s_{\text{conv},n}\) consists of the ON state transmitted signal, \(s_{\text{conv},n}^{\text{on}}\), and the OFF state transmitted signal, \(s_{\text{conv},n}^{\text{off}}\), while \(d_{PAM,n}\) is the sample of PWM signal \(d_{PAM}(t)\). It is observed from Equation (14) that the PWM dimming control and the OFDM transmission can be well integrated in the conceived DCHO-OFDM. Specifically, Figure 3 depicts the block diagram of the proposed DCHO-OFDM. Furthermore, according to Equation (14), the average amplitude of signal \(v_{n}\) is given by:  
\[