# Low-Complexity Hybrid Optical OFDM with High Spectrum Efficiency for Dimming Compatible VLC System
## 2 Brief Review of HACO-OFDM  
To enhance the spectrum efficiency, HACO-OFDM scheme amalgamates the signal of ACO-OFDM with PAM-DMT for achieving an ambitious transmission rate [12]. In order to guarantee real-valued outputs, Hermitian symmetry is invoked in each branch. As demonstrated in Figure 1a, the mapped \(M\)-ary quadrature amplitude modulation (\(M\)-QAM) symbols \(A_{i}\) in ACO-OFDM branch occupy the odd-indexed subcarriers, which is expressed as:  
\[\mathbf{X}=\left[0,A_{1},0,A_{2},\cdots,A_{N/4},0,A_{N/4}^{*},\cdots,A_{2}^{*},0,A_{1}^{*}\right], \tag{1}\]  
where the total number of subcarriers is denoted by \(N\) and \(A_{i}^{*}\) is invoked to present the Hermitian symmetry of symbol \(A_{i}\). After performing the \(N\)-point IFFT of \(\mathbf{X}\), a bipolar time-domain signal \(x_{ACO,n}\) is obtained. In order to acquire the positive signal appropriating for VLC propagation, the bipolar signal of \(x_{ACO,n}\) has to be directly clipped at zero as:  
\[\left\lfloor x_{ACO,n}\right\rfloor_{c}=\left\{\begin{array}{ll}0,&x_{ACO,n }<0\\ x_{ACO,n},&x_{ACO,n}\geq 0\end{array}\right., \tag{2}\]where the resultant clipping distortion is imposed on the even-indexed subcarriers, leaving the odd-indexed subcarriers carrying ACO-OFDM symbols uncontaminated [25]. For the PAM-DMT link, as depicted in Figure 1a, the mapped \(L\)-ary pulse amplitude modulation (\(L\)-PAM) symbols \(B_{i}\) are assigned to the the even-indexed subcarriers, with only the imaginary component being modulated. It can be expressed as:  
\[\mathbf{Y}=\left[0,0,jB_{1},0,jB_{2},\cdots,jB_{N/4-1},0,0,0,jB_{N/4-1}^{*},0, \cdots,jB_{2}^{*},0,jB_{1}^{*},0\right], \tag{3}\]  
where we have \(j=\sqrt{-1}\). A bipolar signal of \(y_{PAM,n}\) can be then obtained, upon performing the \(N\)-point IFFT operation of \(\mathbf{Y}\). For the sake of producing the non-negative result, \(y_{PAM,n}\) is clipped at zero-level, which is given by:  
\[\left\lfloor y_{PAM,n}\right\rfloor_{c}=\left\{\begin{array}{ll}0,&y_{PAM,n }<0\\ y_{PAM,n},&y_{PAM,n}\geq 0\end{array}\right.. \tag{4}\]  
Consequently, in HACO-OFDM scheme, the clipped signal \(\left\lfloor x_{ACO,n}\right\rfloor_{c}\) and \(\left\lfloor y_{PAM,n}\right\rfloor_{c}\) is capable of being simultaneously transmitted, achieving a beneficially high spectrum efficiency. As the detrimental clipping noise generated by ACO-OFDM contaminates the subcarriers carrying PAM-DMT signal, the receiver of HACO-OFDM has to recover the ACO-OFDM symbols first and then retrieve the clipping distortion imposed on the even-indexed subcarriers, so that the detection of PAM-DMT signal can be successively implemented.  
For achieving dimming control, directly combining the HACO-OFDM signal with PWM is deemed to be inefficient. Due to the non-negativity of the clipped unipolar signal \(\left\lfloor x_{ACO,n}\right\rfloor_{c}\) and \(\left\lfloor y_{PAM,n}\right\rfloor_{c}\), the hybrid signal may exceed the dynamic range of LEDs with a large probability [26]. To overcome, a negative-valued NHACO-OFDM signal is invoked to superpose to the high brightness level of PWM signal to avoid non-linear distortion, while the HACO-OFDM with positive-polarity is invoked for the low-brightness scenario [22]. In this regime, the PWM signal has to be detected first by a receiver in order to recover the dissimilar clipping distortion generated by NHACO-OFDM  
Figure 1: Frequency-domain demonstration including both the real component and the imaginary component of the proposed DCHO-OFDM signal, with \(N=16\) subcarriers. (**a**) presents the symbol allocation for ACO-OFDM and for PAM-DMT branch, respectively; (**b**) demonstrates the hybrid unclipped ACO-OFDM and the down/upper-clipped PAM-DMT signals; (**c**) indicates the combined signal after performing time-varying biasing scheme; (**d**) depicts the hybrid signal further incorporating the PWM signal.  
and HACO-OFDM, respectively. Afterwards, the transmitted information can be then extracted by successively demodulating the ACO-OFDM and PAM-DMT branch.