# Visible-Light CDMA Communications Using Inverted Spread Sequences
## 3 MUI Cancellation Schemes with Inverted MPSC
### Modified Liu's Scheme with Inverted MPSC  
If we use [PERSON]'s scheme with an inverted MPSC in a fashion analogous to that of the other two schemes, it is not possible to decode transmitted data correctly. This is because the numbers of chip locations where two correlations are calculated at the decoder are different. Therefore, we propose a modified Liu's scheme that can decode data correctly even if an inverted MPSC is used. In our modified Liu's scheme, the \(k\)th user uses a spread sequence \(\mathbf{w}^{\prime}_{k,1}\) of length \(q^{2}+q+1\) for \(k=0,\ 1,\ \cdots,\ N_{\text{max}}-1\), where \(N_{\text{max}}=q^{2}\). The first \(q^{2}+q\) chips of \(\mathbf{w}^{\prime}_{k,1}\) are the inverted sequence of \(\mathbf{w}_{k,1}\) for the conventional Liu's scheme, and the last chip of \(\mathbf{w}^{\prime}_{k,1}\) is always a mark. An example of the codeword assignment for the modified Liu's scheme is shown in Table 4. By using these spread sequences, the numbers of chip locations where two correlations are calculated at the decoder are made to be the same value. The \(k\)th user's encoder of this scheme is also illustrated by Figure 2a, although \(\mathbf{w}_{k,1}\) is replaced with \(\mathbf{w}^{\prime}_{k,1}\) and \(\mathbf{w}_{k,0}\) is replaced with an all-one sequence with a length of \(q^{2}+q+1\). The first \(q^{2}+q\) chips of the multiplexed sequence in this scheme form the same sequence as that obtained by subtracting the multiplexed sequence of the conventional Liu's scheme from the constant-level sequence whose level is the maximum light intensity of \(N_{\text{L}}\) light sources. In addition, the last chip of the multiplexed sequence is always equal to the maximum light intensity of \(N_{\text{L}}\) light sources. The \(k\)th user's decoder is illustrated by Figure 2b. At the decoder, Correlator 1 computes \(\Gamma^{\prime}_{1}\), which is the correlation of the first \(q^{2}\) chips between the received sequence \(\mathbf{r}\) and the non-inverted sequence \(\mathbf{w}_{k,1}\). Correlator 0 computes \(\Gamma^{\prime}_{0}\), which is the correlation of the last \(q+1\) chips between \(\mathbf{r}^{\prime}\) and the inverted sequence \(\mathbf{w}^{\prime}_{k,1}\). For example, \(\Gamma^{\prime}_{1}\) and \(\Gamma^{\prime}_{0}\) at the fourth user's decoder are represented as follows:  
\[\Gamma^{\prime}_{1}=\Gamma\Big{(}\mathbf{\dot{r}^{\prime}},(1000\ 010\ 0010\ 001\ 000\ 0) \Big{)}, \tag{14}\]  
\[\Gamma^{\prime}_{0}=\Gamma\big{(}\mathbf{r}^{\prime},(0000\ 0000\ 000\ 000\ 1011\ 1) \big{)}, \tag{15}\]  
where the spread sequence of the fourth user sending \(I=1\ \mathbf{is}\ \mathbf{w}^{\prime}_{4,1}=(0111\ 101\ 1101\ 1110\ 1011\ 1)\). Then, the decoder calculates the difference \(\Gamma^{\prime}_{0}-\Gamma^{\prime}_{1}\). If the difference is greater than or equal to the threshold, the decision circuit outputs \(\hat{I}=1\). Otherwise, it outputs \(\hat{I}=0\). The ideal threshold is \(q/2\) when we suppose that the light intensity of each user's positive chip is 1. For an ideal link, \(\Gamma^{\prime}_{1}\) and \(\Gamma^{\prime}_{0}\) are \(qN_{\text{L}}-\Gamma_{1}\) and \(qN_{\text{L}}-\Gamma_{0}\), respectively, where \(\Gamma_{1}\), \(\Gamma_{0}\), \(\Gamma^{\prime}_{1}\), and \(\Gamma^{\prime}_{0}\) are given by Equations (8), (9), (14) and (15), respectively. Therefore, the difference \(\Gamma^{\prime}_{0}-\Gamma^{\prime}_{1}\) is equal to \(\Gamma_{1}-\Gamma_{0}\) in the conventional [PERSON]'s scheme. This means that the modified Liu's scheme with an inverted MPSC can cancel MUI and decode data correctly, as is the case with the conventional Liu's scheme. The decision distance is also \(q\). In Table 3, the three MUI cancellation schemes using inverted MPSCs are also compared. As for the EWO scheme and [PERSON]'s scheme, spread sequence length \(n\), the maximum number of users \(N_{\text{max}}\), and the decision distance \(d\) are not different between the non-inverted and inverted forms.