# Distributed Joint Source-Channel Coding Using Quasi-Uniform Systematic Polar Codes
## 2 System Model  
Consider the problem of transmitting distributed sources \(V_{1},V_{2},...,V_{s}\) over independent symmetric B-DMCs \(W_{1},W_{2},...,W_{s}\) to a common destination. The message from source \(V_{i},i=1,2,\ldots,s\), is denoted by a binary vector \(\mathbf{v}_{i}=[v_{i}^{1},v_{i}^{2},...,v_{i}^{K}]\) of length \(K\). We assume that \(\mathbf{v}^{j}=[v_{1}^{j},v_{2}^{j},...,v_{s}^{j}]\) for different \(j\in\{1,2,...,K\}\) are i.i.d. (identically and independently distributed) vectors drawn from \(\{0,1\}^{s}\) with a common probability distribution \(P_{V_{1},V_{2},...,V_{s}}\). From the Slepian-Wolf theorem, the compression rate \(\mathbf{R}=(R_{1},R_{2},...,R_{s})\) is achievable if  
\[\sum_{i\in\mathcal{T}}R_{i}\geq H(\mathbf{v}_{\mathcal{T}}^{j}|\mathbf{v}_{ \mathcal{T}^{\mathrm{c}}}^{j}) \tag{1}\]  
for any \(\mathcal{T}\subseteq\{1,2,...,s\}\), where \(H(\cdot)\) is entropy function, \(\mathcal{T}^{\mathrm{c}}=\{1,2,...,s\}\backslash\mathcal{T}\) is the complementary set of \(\mathcal{T}\), \(\mathbf{v}_{\mathcal{T}}^{j}\) and \(\mathbf{v}_{\mathcal{T}^{\mathrm{c}}}^{j}\) denote, respectively, the sub-vectors of \(\mathbf{v}^{j}\) indexed by \(\mathcal{T}\) and \(\mathcal{T}^{\mathrm{c}}\). This achievable compression region is also known as the Slepian-Wolf region \(\mathcal{R}_{SW}\). Let \(\tilde{R}_{i}=\widetilde{N}_{i}/K\) be the rate, measured in the number of channel-use per source-symbol, where \(\widetilde{N}_{i}\) is the number of channel uses of source \(V_{i}\) for transmitting \(\mathbf{v}_{i}\). Let \(C_{i}\) denote the channel capacity of \(W_{i}\). Since the source-channel separation theorem still holds for this problem [20], reliable transmission is possible if \(\tilde{\mathbf{R}}=(\tilde{R}_{1},\tilde{R}_{2},...,\tilde{R}_{s})\) falls into the achievable rate region \(\mathcal{R}_{\mathrm{rate}}\), which is defined as  
\[\mathcal{R}_{\mathrm{rate}}=\{\tilde{\mathbf{R}}|\tilde{R}_{i}\geq 0, \mathcal{R}_{SW}\cap\mathcal{R}_{c}(\tilde{\mathbf{R}})\
eq\emptyset\}, \tag{2}\]where \(\mathcal{R}_{c}(\tilde{\mathbf{R}})\) is the set of all points \((R_{1},R_{2},...,R_{s})\) satisfying  
\[0\leq R_{i}\leq\tilde{R}_{i}C_{i} \tag{3}\]  
for \(i=1,2,...,s\). Given \(\tilde{\mathbf{R}}_{1},\tilde{\mathbf{R}}_{2}\in\mathcal{R}_{\text{rate}}\) and \(0\leq\lambda\leq 1\), by the definition of convex sets, we can readily prove that \(\lambda\tilde{\mathbf{R}}_{1}+(1-\lambda)\tilde{\mathbf{R}}_{2}\in\mathcal{R }_{\text{rate}}\) as follows. There obviously exist \(\mathbf{R}_{1}^{*}\in\mathcal{R}_{\text{SW}}\cap\mathcal{R}_{c}(\tilde{ \mathbf{R}}_{1})\) and \(\mathbf{R}_{2}^{*}\in\mathcal{R}_{\text{SW}}\cap\mathcal{R}_{c}(\tilde{ \mathbf{R}}_{2})\). Due to the convexity of \(\mathcal{R}_{\text{SW}}\), \(\lambda\mathbf{R}_{1}^{*}+(1-\lambda)\mathbf{R}_{2}^{*}\in\mathcal{R}_{\text{ SW}}\). Besides, it can be observed that  
\[\lambda\mathbf{R}_{1}^{*}+(1-\lambda)\mathbf{R}_{2}^{*}\in\mathcal{R}_{c}( \lambda\tilde{\mathbf{R}}_{1}+(1-\lambda)\tilde{\mathbf{R}}_{2}). \tag{4}\]  
This indicates that a point at least exists in \(\mathcal{R}_{\text{SW}}\cap\mathcal{R}_{c}(\lambda\tilde{\mathbf{R}}_{1}+(1- \lambda)\tilde{\mathbf{R}}_{2})\), and \(\mathcal{R}_{\text{SW}}\cap\mathcal{R}_{c}(\lambda\tilde{\mathbf{R}}_{1}+(1- \lambda)\tilde{\mathbf{R}}_{2})\
eq\emptyset\). Therefore, \(\lambda\tilde{\mathbf{R}}_{1}+(1-\lambda)\tilde{\mathbf{R}}_{2}\in\mathcal{R }_{\text{rate}}\) and the achievable rate region of \(\mathcal{R}_{\text{rate}}\) is convex. To solve this transmission problem, one option is distributed SSCC (DSSCC) scheme as shown in Figure 1. The message from each source is first compressed into compressed bits with DSC. Then each source encodes its compressed bits into channel code bits with CCs and transmit these channel code bits over its channel. At the destination, the received signals are first decoded by channel decoders, and then decoded bits are jointly decompressed by a source joint decoder.