# Distributed Joint Source-Channel Coding Using Quasi-Uniform Systematic Polar Codes
## 3 QSPC and Punctured QSPC
### Qspc  
A class of SPCs, called QSPCs, were introduced for the problem of JSCC with side information. For a \((MN,MK)\) QSPC (\(M=2^{m},N=2^{n},K\leq N\)), systematic bits \(\mathbf{x}_{\mathcal{B}}\) are first encoded by  
\[ \begin{split}\tilde{\mathbf{u}}_{\tilde{\mathcal{A}}}& =\left(\mathbf{x}_{\mathcal{B}}-\mathbf{u}_{\tilde{\mathcal{A}}^{c }}\tilde{G}_{MN}(\tilde{\mathcal{A}}^{c}\mathcal{B})\right)\left(\tilde{G}_{ MN}(\tilde{\mathcal{A}}\mathcal{B})\right)^{-1}\\ &=\mathbf{x}_{\mathcal{B}}\left(\tilde{G}_{MN}(\tilde{\mathcal{A} }\mathcal{B})\right)^{-1},\end{split}  \tag{7}\]  
where \(\tilde{\mathcal{A}}=\{i|MN-MK-1\leq i\leq MN\},\mathcal{B}=\mathcal{Q}(MN,MK)\), and \(\tilde{G}_{MN}(\tilde{\mathcal{A}}\mathcal{B})\) denotes the sub-matrix of \(\tilde{G}_{MN}\) with the rows indexed by \(\tilde{\mathcal{A}}\) and the columns indexed by \(\mathcal{B}\). Then the codeword is obtained by  
\[\mathbf{x}=\left(\tilde{\mathbf{u}}_{\tilde{\mathcal{A}}}\tilde{G}_{MN}( \tilde{\mathcal{A}})\right)\oplus\left(\tilde{\mathbf{u}}_{\tilde{\mathcal{A} }}\tilde{G}_{MN}(\tilde{\mathcal{A}}^{c})\right), \tag{8}\]  
where \(\tilde{G}_{MN}(\tilde{\mathcal{A}})\) and \(\tilde{G}_{MN}(\tilde{\mathcal{A}}^{c})\) denote the rows of \(\tilde{G}_{MN}\) indexed by \(\tilde{\mathcal{A}}\) and \(\tilde{\mathcal{A}}^{c}\), respectively. The generator matrix is given by  
\[\tilde{G}_{MN}=DG_{MN}=DB_{MN}\Bigg{[} \begin{array}{cc}1&0\\ 1&1\end{array} \Bigg{]}^{\otimes(m+n)}, \tag{9}\]  
where \(B_{MN}\) is a bit-reversal matrix, and \(D\) is an invertible bit-swap coding matrix (refer to Section IV.A of [11] for details). It can be seen that \(\mathbf{x}=\tilde{\mathbf{u}}\tilde{G}_{MN}=\mathbf{u}G_{MN}\) and \(\mathbf{u}=\tilde{\mathbf{u}}D\). For this code, the typical decoder of polar codes attempts to decode \(\mathbf{u}\), then the codeword \(\mathbf{x}\) is obtained by \(\mathbf{x}=\mathbf{u}G_{MN}\) or \(\mathbf{x}=\mathbf{u}D^{-1}\tilde{G}_{MN}\). The entropies/BERs (bit error rate) of \(\mathbf{u}\) are determined by underlying channels and the sources. With the aid of density evolution, we can locate those bits \(\mathbf{u}_{\mathcal{A}}\) which have low entropies/BERs. However, the reversibility of \(G_{\mathcal{A}\mathcal{B}}\) cannot be guaranteed, i.e., it is impossible to construct QSPCs via original polar coding. The bit-swap coding \(D\) is then proposed to change the information bits from \(\mathbf{u}_{\mathcal{A}}\) to \(\tilde{\mathbf{u}}_{\tilde{\mathcal{A}}}\) and modify the generator matrix \(\tilde{G}_{MN}=DG_{MN}\). This operation ensures that \(\tilde{G}_{\mathcal{A}\mathcal{B}}\) is always invertible. The block error rate (BLER) of \(\mathbf{x}\) under the SC decoder can be calculated by  
\[P_{\text{BLER}}=1-\prod_{i\in\tilde{\mathcal{A}}}\left[1-P_{\text{e}}(u_{i}) \right], \tag{10}\]  
where \(P_{\text{e}}(u_{i})\) is the BER of \(u_{i}\). Consider a source message \(\mathbf{v}=[v^{1},v^{2},...,v^{MK}]\) with side information \(\mathbf{w}=[w^{1},w^{2},...,w^{MK}]\). When \(\mathbf{v}\) is encoded by a \((MN,MK)\) QSPC for JSCC, only parity bits \(\mathbf{x}_{\mathcal{B}^{c}}\) are transmitted over the noisy channel \(W(y|x)\). Due to \(\mathcal{B}=\mathcal{Q}(MN,MK)\), the codeword structure can be depicted in Figure 3, and Theorem 1 can be obtained as follows. **Theorem 1** ([11]). : _For any \(0\leq\epsilon<\frac{1}{2}\),_  
\[ \begin{split}&\lim_{M\to\infty}\frac{1}{MN}\left|\{i:H(u_{i}| \boldsymbol{y}\mathbf{u}_{1:i-1})\in[0,\epsilon)\}\right|=\lim_{M\to\infty} \frac{1}{MN}|\{i:Z(u_{i}|\boldsymbol{y}\mathbf{u}_{1:i-1})\in[0,\epsilon)\}|\\ &=1-\frac{KH(v|w)+(N-K)H(x|y)}{N},\\ &\lim_{M\to\infty}\frac{1}{MN}|\{i:H(u_{i}|\boldsymbol{y}\mathbf{u }_{1:i-1})\in(1-\epsilon,1]\}|=\lim_{M\to\infty}\frac{1}{MN}|\{i:Z(u_{i}| \boldsymbol{y}\mathbf{u}_{1:i-1})\in(1-\epsilon,1]\}|\\ &=\frac{KH(v|w)+(N-K)H(x|y)}{N},\end{split}