# Adaptive Bit-Labeling Design for Probabilistic Shaping Based on Residual Source Redundancy
## 2 System Model  
The structure of the JSCCM system over AWGN channels is shown in Figure 1. A non-uniform memoryless binary (\"0\" and \"1\") source is considered in this paper, where the probability of \"1\" is represented as \(p\) (\(p\
eq 0.5\)). The source entropy is therefore given by  
\[H(p)=-p\log_{2}p-(1-p)\log_{2}(1-p)<1. \tag{1}\]  
Let us consider source protograph LDPC codes with base matrix \(\mathbf{B}_{s}\) and channel protograph LDPC codes with base matrix \(\mathbf{B}_{c}\). Then, the progressive edge growth (PEG) algorithm is employed to generate the corresponding low-density matrices \(\mathbf{H}_{s}\) and \(\mathbf{H}_{c}\) by the copy-and-permute operation [17].  
The encoding process of the JSCCM system comprises two steps as follows. Firstly, the source bit sequence \(\mathbf{s}\) is compressed by using the source code as  
\[\mathbf{b}^{T}=\mathbf{H}_{s}\mathbf{s}^{T}, \tag{2}\]  
Figure 1: The structure of a JSCCM system.  
where \(\mathbf{b}\) is the compressed bit sequence. Then, the compressed bit sequence is protected by the channel code. For systematic binary channel encoding, a systematic generator matrix can be constructed from the parity-check matrix \(\mathbf{H}_{c}\) and is represented by  
\[\mathbf{G}_{c}=[\mathbf{P}\mid\mathbf{I}], \tag{3}\]  
where \(\mathbf{I}\) is the identity matrix. The systematic channel codeword \(\mathbf{x}\) is thus obtained as  
\[\mathbf{x}=\mathbf{b}\mathbf{G}_{c}=[\mathbf{c}\mid\mathbf{b}], \tag{4}\]  
where \(\mathbf{c}=\mathbf{b}\mathbf{P}\) is the parity bit sequence.  
Assume a quadrature amplitude (QAM) alphabet with \(2^{m}\) signal points, where \(m\) is the modulation order. By setting the channel code rate to be \((m-1)/m\), the channel codeword \(\mathbf{x}\) can be composed of a parity bit sequence \(\mathbf{c}\) of length \(N\) and a compressed bit sequence \(\mathbf{b}\) of length \(N\times(m-1)\), where \(N\) represents the lifting factor. Thus, the sequence \(\mathbf{x}\) is organized as a \(N\times m\) binary matrix by the interleaver as  
\[\mathbf{M}_{\mathbf{x}}^{(1)}=\begin{bmatrix}c_{1}&b_{1}^{1}&b_{1}^{2}&\cdots &b_{1}^{m-1}\\ c_{2}&b_{2}^{1}&b_{2}^{2}&\cdots&b_{2}^{m-1}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ c_{N}&\underbrace{b_{N}^{1}}&\underbrace{b_{N}^{2}}&\cdots&\underbrace{b_{N} ^{m-1}}\\ b^{0}&b^{1}&b^{2}&\cdots&b^{m-1}\end{bmatrix}, \tag{5}\]  
where every column forms a bit level \(b^{i}\) for \(0\leq i\leq m-1\).  
In (5), the parity bits in \(\mathbf{c}\) are all transferred to the bit level \(b^{0}\) and the compressed bits in block \(b^{i}\), \(1\leq i\leq m-1\) are transferred to the bit level \(b^{i}\). Subsequently, the QAM symbol sequence \(\mathbf{X}=(X_{1},X_{2},\ldots,X_{N})\) for transmission is obtained by row-wise mapping according to the mapping rule.  
With the constellation scaling \(\alpha>0\), the AWGN channel is described by the input-output relationship  
\[\mathbf{Y}=\alpha\mathbf{X}+\mathbf{Z}, \tag{6}\]  
where \(\mathbf{X}\) is the modulated symbol sequence, \(\mathbf{Y}\) is the received symbol sequence, and \(\mathbf{Z}\) is the AWGN sequence.  
The overall transmission rate of the JSCCM system is defined in \"source bits/channel symbol\" as  
\[R=\frac{mR_{c}}{R_{s}}=\frac{m-1}{R_{s}}, \tag{7}\]  
where \(R_{s}\) and \(R_{c}\) represent the source coding rate and the channel coding rate, respectively.  
At the receiver, after demodulation and de-interleaving, the joint source and channel decoder is applied to reconstruct the source bit sequence.