# Distributed Joint Source-Channel Coding Using Quasi-Uniform Systematic Polar Codes
## 5 Simulation Results  
In the previous section, we have proposed a DJSCC scheme and shown its asymptotic optimality: In this section, we present the simulation results to verify the performance of the proposed scheme under finite code-length. In the simulation, two sources \(V_{1}\), \(V_{2}\) are considered and \(\Pr\{v_{2}^{j}|v_{1}^{j}\}\) is modeled as binary symmetric channel (BSC) with a crossover probability \(q=0.11\), and \(\Pr\{v_{1}^{j}=0\}=0.5\). The transmission channels \(W_{1}\), \(W_{2}\) are BI-AWGN channels with the same signal-to-noise ratio (SNR). The \((64\cdot 32,64\cdot 10)\) QSPC and the \((64\cdot 32,64\cdot 10,64\cdot 11)\) punctured QSPC are constructed with density evolution. The QSPC and the punctured QSPC are respectively used to encode \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) for DJSCC. For this DJSCC scheme, the rates are \(\tilde{\mathbf{R}}=(\tilde{R}_{1},\tilde{R}_{2})=(2.2,1.1)\) in channel-use/source-symbol. The Shannon limits for both two sources are approximately -0.4 dB, which are respectively obtained by \(MK\cdot H(v_{1}^{j})\leq M(N-K)\cdot C(\text{SNR})\) and \(MK\cdot H(v_{2}^{j}|v_{1}^{j})\leq M(N-K-P)\cdot C(\text{SNR})\), where \(C(\text{SNR})\) is the capacity of BI-AWGN channels. As a reference, we consider a DSSCC scheme where both DSC and CC are designed using polar codes of length \(N_{s}=N_{c}=1024\). The source compression rate (in bit/source-symbol) of DSC \(R_{si}\), and the channel code rate (in bit/channel-use) of CC \(R_{ci}\) for source \(V_{i},i=1,2\) are illustrated in Table 1. The rate of this DSSCC scheme can be calculated by  
\[\tilde{R}_{i}=\frac{R_{si}}{R_{ci}},\qquad i=1,2 \tag{19}\]  
in channel-use/source-symbol. In the source layer, each \(N_{s}\) bits of source message are divided into a block for DSC and the polar code based DSC is designed as follows. After polar transform \(\mathbf{u}=[v_{1}^{1}\oplus v_{2}^{1},v_{1}^{2}\oplus v_{2}^{2},...,v_{1}^{N_ {s}}\oplus v_{2}^{N_{s}}]G_{N}\), the BERs \(P_{\text{e}}(u_{i})\) of \(\mathbf{u}\) are calculated with density evolution. As shown in Figure 6, both \(\mathbf{u}_{1}=\mathbf{v}_{1}G_{N}\) and \(\mathbf{u}_{2}=\mathbf{v}_{2}G_{N}\) are sorted in a descend order according to \(P_{\text{e}}(u_{i})\), and the bits represented by black boxes are compressed bits that will be fed into the buffer for CC. At the receiver, the decoder firstly tries to recover \(\mathbf{u}_{1}\oplus\mathbf{u}_{2}\) with LLRs \(\log\frac{1-q}{q}\), then the bits represented by gray boxes are reconstructed with some modulo-2 additions. Finally, the sources are obtained by \(\hat{\mathbf{v}}_{1}=\hat{\mathbf{u}}_{1}G_{N}\) and \(\hat{\mathbf{v}}_{2}=\hat{\mathbf{u}}_{2}G_{N}\). In the channel layer, each \(N_{c}R_{ci}\) compressed bits in the buffer are divided into a block for \((N_{c},N_{c}R_{ci})\) CC based on polar codes. \begin{table}
\begin{tabular}{l c c c c c} \hline  & \(R_{s1}\) & \(R_{s2}\) & \(R_{c1}\) & \(R_{c2}\) & \(\tilde{\mathbf{R}}\) \\ \hline DSSCC 1 & 1024/1024 & 650/1024 & 465/1024 & 591/1024 & \((2.2,1.1)\) \\ DSSCC 2 & 900/1024 & 900/1024 & 409/1024 & 818/1024 & \((2.2,1.1)\) \\ DSSCC 3 & 1024/1024 & 900/1024 & 465/1024 & 818/1024 & \((2.2,1.1)\) \\ \hline \end{tabular}
\end{table} 
Table 1: Distributed separate source-channel coding (DSSCC) schemes with different \(R_{si},R_{ci}\). Besides, we consider the DJSCC based on classic SPCs as a reference. For source \(V_{1}\), the \((64\cdot 32,64\cdot 10)\) SPC is constructed via Gaussian approximation where the underlying channel is BI-AWGN channels with the equivalent capacity \(C_{equ1}=\frac{K}{N}\cdot(1-H(v_{1}^{j}))+\frac{N-K}{N}C_{1}=\frac{N-K}{N}C_{1}\). For source \(V_{2}\), the \((64\cdot 32,64\cdot 10)\) SPC is constructed via Gaussian approximation where the underlying channel is BI-AWGN channels with equivalent capacity \(C_{equ2}=\frac{K}{N}\cdot(1-H(v_{2}^{j}|v_{1}^{j}))+\frac{N-K-P}{N}C_{2}=\frac{ K}{N}+\frac{N-K-P}{N}C_{2}\).