# Semantic Communication on Digital Wireless Communication Systems
## 4 Optimization for the Bit-Conversion JSCC Scheme  
### Minimum Manhattan Distance Constellation Mapping Scheme  
Based on the simulation in Section 3.3, it can be observed that different integer-to-bit conversion coding methods will impact the received data directly. The key factor of this phenomenon is the numerical difference in the integers mapped onto adjacent constellation points. In existing wireless systems, error correction is based on bit-level and channel coding. Therefore, the bits mapped onto adjacent constellation points are based on Gray-code mapping, meaning that adjacent constellation points differ by only one bit. However, in JSCC, the channel coding is removed, and the received symbols will be directly converted to the mapped integers and then sent to the JSCC decoder. Thus, the principle of the JSCC constellation mapping scheme design should be changed to the numerical difference in integers mapped onto adjacent constellation points instead of the difference in bits. According to this JSCC constellation mapping scheme design principle, the minimum Manhattan distance constellation mapping scheme of m-QAM modulation is proposed. And the algorithm to generate the binary encoding table of this constellation mapping scheme to be compatible with the current system, called Manhattan distance binary coding, is described. For regular m-QAM modulation, we design a minimum Manhattan distance constellation mapper. This mapper maps the integer number pair to the constellation point according to Formula (4). \[\mathrm{M}(\mathrm{p},\mathrm{q},\mathrm{K}_{\mathrm{mod}},\mathrm{Q}_{\mathrm{ m}})=-\mathrm{K}_{\mathrm{mod}}\times\{(2\mathrm{p}-\mathrm{Q}_{\mathrm{m}}+1)+ \mathrm{j}(2\mathrm{q}-\mathrm{Q}_{\mathrm{m}}+1)\} \tag{4}\]  
Here, \(\mathrm{Q}_{\mathrm{m}}\) is the modulation order and \(\mathrm{K}_{\mathrm{mod}}\) is the normalization parameter of the adopted system for different m-QAM. Function \(\mathrm{M}(\mathrm{p},\mathrm{q},\mathrm{K}_{\mathrm{mod}},\mathrm{Q}_{\mathrm{ m}})\) takes \(\mathrm{p},\mathrm{q}\), the non-negative integer in the range \([0,\mathrm{Q}_{\mathrm{m}}+1]\), as the input and produces complex-valued modulation symbols as the output. For example, in the case of 5G-64 QAM,5G-16 QAM modulation, the integer constellation mapping scheme generated by this formula is shown in Figure 13.  
Then, for compatibility with the adopted system, binary coding should be introduced. We design an algorithm for Manhattan distance binary coding generation, as Algorithm 1 illustrates. Figure 13: Minimum Manhattan distance integer mapping schemes. (**a**) 64 QAM integer mapping schemes; (**b**) 16 QAM integer mapping schemes. The input of this algorithm, besides the m-QAM integer constellation mapper, is the modulation order \(Q_{\text{m}}\) and the responding m-QAM bit constellation mapping scheme d(i) of the adopted system. The modulation mapper \(\text{d}(\text{i})\) takes binary digits as the input and produces complex-valued modulation symbols as the output. For example, in the case of 5G-64 QAM modulation, hextuplets of bits, \(\text{b}(\text{6i})\), \(\text{b}(\text{6i}+1)\), \(\text{b}(\text{6i}+2)\), \(\text{b}(\text{6i}+3)\), \(\text{b}(\text{6i}+4)\), \(\text{b}(\text{6i}+5)\), are mapped to complex-valued modulation symbols d(i), according to Formula (5) [27]. Here, b(x) is the Gary code and x is the non-negative integer in the range \(\left[0,Q_{\text{m}}^{2}-1\right]\). The constellation mapping scheme generated by Formula (5) is shown in Figure 14a. Then, follow the data process in the algorithm, determine the \(K_{\text{mod}}\) according to \(Q_{\text{m}}\), and map the integer number pairs \((\text{p},\text{q})\) with the bits at the same constellation points; following this, the binary coding table can be generated. For example, in the case of 5G  
Figure 14: (a) 5G 64 QAM bit constellation mapping scheme; (b) the 64 QAM Manhattan distance binary coding table adapted to 5G.