# A Novel 64 QAM-OFDM Optical Access System Based on Bit Reconstruction
## 2 Theory and Principle
As is shown in the bit reconstruction part of Figure 1, the pre-encoder, electric displacement, attenuator and electric adder are amalgamated to form the logic code circuit. The pre-encoder contains a logical NOT gate, logical XOR gate, and a delay circuit. We consider setting reference bits that are \(b(k-1)\) and \(c(k-1)\) at \((k-1)\) instance in the logic code circuits because of the presence of the delay unit. The value chosen can be logic 0 or logic 1. Different reference bit selections will also affect the output of the logic code circuit. \(C_{0}\) represents the initial value of the delay stage in the logic code circuit. It belongs to the reference bit matrix, as does \(B_{0}\). The reference level matrices \(B_{0}\) and \(C_{0}\) are all set to zero matrices. As the bit stream enters the pre-encoder module, the symbol matrix \(P\) refers to Equation (4):  
\[B(k)=\overline{P}\oplus B_{0} \tag{4}\]  
where \(P\) is the transmitted symbol matrix, \(B(k)\) is the pre-coded matrix and \(\oplus\) is the logic XOR gate. On the basis of the logic code circuit, when the signal passes through the level shifter (LS), the bit reconstruction matrix \(G_{1}\) is updated: \(B(k)\) is compared with the initialization bit reconstruction matrix \(G_{0}\); if \(B_{m,n}=0\), then \(G_{m,n}=-1\); if \(B_{m,n}=1\), then the elements in \(G_{m,n}\) do not change. \(G_{1}\) is continually updated:  
\[G_{2}=G_{1}+G_{0} \tag{5}\]  
In the logical code circuit, there is a 6 dB electric attenuator component to divide the amplitude by half. Therefore, the elements in \(M_{2n}\) rows of \(G_{2}\) are halved. After this, the row-column transformation of the bit reconstruction matrix is carried out, and every two rows are added as one channel to obtain a new matrix, \(G_{3}\). Here multilevel symbols are generated with amplitudes that range from 0 to 3 which can be referred to Equation (6). In particular, the above operations do not apply to the last two rows of the matrix. It should be noted that the number of bit streams output after the logic code circuit is reduced to half. In order to map the constellations, we fill some bits and update \(G_{3}\). We then set the filling rule: \(m\in 0,1\)  
\[ \begin{array}{llll}G_{3_{m,n}}=0,&G_{4_{m,n}}=0,&G_{4_{m+1,n}}=1,\\ G_{3_{m,n}}=1,&G_{4_{m,n}}=0,&G_{4_{m+1,n}}=0,\\ G_{3_{m,n}}=2,&G_{4_{m,n}}=1,&G_{4_{m+1,n}}=0,\\ G_{3_{m,n}}=3,&G_{4_{m,n}}=1,&G_{4_{m+1,n}}=1,\end{array}  \tag{6}\]  
The updated \(G_{4}\) is compared with the symbol matrix \(P\), and the elements in the same position of the \(P\) matrix are processed according to the value of \(G_{4}\) to obtain the bit reconstruction matrix \(P_{1}\):  
\[P_{1}= \begin{bmatrix}G_{4_{0,0}}&\cdots&G_{4_{0,n-1}}\\ \vdots&\ddots&\vdots\\ G_{4_{3,0}}&\cdots&G_{4_{3,n-1}}\\ P_{4,0}&\cdots&P_{4,n-1}\\ P_{5,0}&\cdots&P_{5,n-1}\end{bmatrix}  \tag{7}\]  
The partition mapping selects the constellation mapping of the reconstructed signal by the segmentation rule. As shown in the partition mapping part of Figure 1, in order to simplify the complexity of the mapping transformation, symmetry design is performed. The first four bits remain unchanged after a two-stage logic code circuit, i.e., each quadrant has a symmetrical position with the same four bits, while dividing the high- and low-power regions. It is well known that in the constellation diagram of a QAM signal, the constellation points at locations closer to the origin have lower power. As shown in the mapping rule in Figure 1, the yellow spheres represent the constellation points in the low-power region and the blue spheres represent the constellation points in the high-power region. The logic code circuit makes the bits values that enter into the partition mapping, i.e., the bit reconstruction matrix \(G_{3}\), flow mostly to the low-level amplitude \(0,1,2\). Filling bits are necessary to achieve the partition mapping. In this process, '\(00\)', '\(01\)' and '\(10\)' will be assigned to low-level amplitudes.