# A Nonbinary LDPC-Coded Probabilistic Shaping Scheme for a Rayleigh Fading Channel
## 2 Methods  
The proposed NB LDPC-coded PS scheme for Rayleigh fading channel is shown in Figure 1. In the transmitter, a length of \(k\) uniform data bits \(\mathbf{u}_{bin}^{A}\) is used for generating the nonuniform \(n\) amplitude symbols according to the unequal probability distribution \(P_{A}\), which is generated by distribution matcher (DM). For Gray-labeled QAM constellations, such as 16 QAM, 64 QAM and 256 QAM constellations, we consider using two-dimensional PS. Hereby, we use two \(\sqrt{M}\)-ASK constellations for one M-QAM constellation. Let \(\chi_{A}(\cdot)\) denote the amplitude labeling function, where \(\chi_{A}:\tilde{A}\rightarrow\{0,1\}^{m-1}\), \(m=\frac{\sqrt{M}}{2}\). Assume the binary bits sequence of one 4-ASK symbol is \(b_{1}b_{2}\), where \(b_{1}\) denotes the sign bit, and \(b_{2}\) represents the amplitude bits. The symbols \(-\) and \(+\) correspond to \(b_{1}=0\) and \(b_{1}=1\), respectively. The amplitudes 3 and 1 correspond to \(b_{2}=0\) and \(b_{2}=1\), respectively. Assume the binary bits sequence of one 8-ASK symbol is \(b_{1}b_{2}b_{3}\), where \(b_{1}\) denotes the sign bit, and \(b_{2}b_{3}\) represents the amplitude bits. The symbols \(-\) and \(+\) correspond to \(b_{1}=0\) and \(b_{1}=1\), respectively. The amplitudes 7,5,3 and 1 correspond to \(b_{2}b_{3}=00\), \(b_{2}b_{3}=01\), \(b_{2}b_{3}=11\) and \(b_{2}b_{3}=10\), respectively. Assume the binary bits sequence of one 16-ASK symbol is \(b_{1}b_{2}b_{3}b_{4}\), where \(b_{1}\) denotes the sign bit, and \(b_{2}b_{3}b_{4}\) represents the amplitude bits. The symbols \(-\) and \(+\) correspond to \(b_{1}=0\) and \(b_{1}=1\), respectively. The amplitudes 15, 13, 11, 9, 7, 5, 3 and 1 correspond to \(b_{2}b_{3}b_{4}=000\), \(b_{2}b_{3}b_{4}=001\), \(b_{2}b_{3}b_{4}=010\), \(b_{2}b_{3}b_{4}=100\), \(b_{2}b_{3}b_{4}=101\), \(b_{2}b_{3}b_{4}=111\) and \(b_{2}b_{3}b_{4}=110\), respectively. For 4-ASK constellations,  
\[\chi_{\tilde{A}}(3)=0,\chi_{\tilde{A}}(1)=1. \tag{1}\]  
For 8-ASK constellations,  
\[\chi_{\tilde{A}}(7)=00,\chi_{\tilde{A}}(5)=01,\chi_{\tilde{A}}(3)=11,\chi_{ \tilde{A}}(1)=10. \tag{2}\]  
For 16-ASK constellations,  
\[\chi_{\tilde{A}}(15)=000,\chi_{\tilde{A}}(13)=001,\chi_{\tilde{A}}(11)=011, \chi_{\tilde{A}}(9)=010, \tag{3}\]  
\[\chi_{\tilde{A}}(7)=100,\chi_{\tilde{A}}(5)=101,\chi_{\tilde{A}}(3)=111,\chi_{ \tilde{A}}(1)=110.\]The binary data bits \(\mathbf{a_{bin}}\) are converted into \(\mathbf{u^{A}}\) by series-to-parallel (S/P) conversion, which is one part of the \(GF(2^{p})\) LDPC information symbols. We assume that the \(GF(2^{p})\) LDPC code rate is \(r\), and the length of information symbols and parity symbols are \(k_{c}\) and \(n_{c}-k_{c}\), respectively. The length of the additional data source bits \(\mathbf{u_{bin}^{S}}\) is \(nmr-(m-1)n\). The total code rate is \(R=\frac{k+mmr-(m-1)n}{mn}\). \(\mathbf{u_{bin}^{S}}\) are converted into \(\mathbf{u^{S}}\) by S/P conversion, which is the other part of the \(GF(2^{p})\) LDPC information symbols. The \(GF(2^{p})\) LDPC codes use systematic encoding with the generator matrix \(\mathbf{G}\), which can be obtained by the parity-check matrix \(\mathbf{H}\), and the parity-check matrix \(\mathbf{H}\) is constructed by the progressive edge-growth algorithm [14]. After passing through NB LDPC encoding, the uniformly distributed parity symbols \(c\) can be transferred into data bits \(\mathbf{u_{p1}}\) by parallel-to-series (P/S) conversion. The additional data source bits \(\mathbf{u_{bin}^{S}}\) are assumed as \(\mathbf{u_{p2}}\), and the connected data sequences \(\{\mathbf{u_{p1}},\mathbf{u_{p2}}\}\) are used for generating modulated sign parts, while \(\mathbf{a_{bin}}\) are used for generating modulated amplitude parts. For one \(\sqrt{M}\)-ASK constellation symbol, the length of the modulated sign part is one bit, while the length of the modulated sign part is \(\frac{\sqrt{M}}{2}-1\) bits. The signal \(X=[X_{I},X_{Q}]^{T}\) is obtained by the rotation of the traditional signal \(S=[S_{I},S_{Q}]^{T}\), which can be expressed as  
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