# Design and Analysis of Joint Source-Channel Code System with Fixed-Length Code
## 2 Joint Source-Channel Code System
### Design Example  
As an example, a Gaussian distribution source [26] is used. The discrete source symbols obey the occurrence probabilities that resulted from the 16-ary Lloyd-Max quantization of independent Gaussian-distributed source. The source entropy is \(H(S)\ =\ 3.76\ \text{bits/symbol}\). Set the crossover probability \(p_{c}\ =\ 10^{-4}\). For the codeword length \(L\ =\ 5,6,7,8\), the source codes are generated by applying Algorithm 1. The resulting codewords are detailed in Table 1, along with their corresponding minimum Hamming distances \(d_{H,\text{min}}\). For the proposed block codes, inverse EXIT curves of the SBSD are illustrated in Figure 3, and the EXIT curve of natural bit mapping [10] is also illustrated in the same figure.  
The natural bit mapping approach maps the source symbols to the binary representation of indices. The codewords have a length of \(L\ =\ 4\) and do not have any redundant bits. Even though this source code remainsthe nonuniform distribution redundancy of the source symbols, there is limited redundancy in the source codewords. As can be seen from Figure 3, the SBSD only provides a low amount of extrinsic information, even if perfect a priori information, i.e., \(I_{SD}^{a}\ =\ 1\), is provided. The early intersections with the EXIT curve of the decoders will lead to a poor convergence for iterative decoding. Thus, the source codes with low redundancy cannot benefit from iterative decoding.  
By increasing the codeword's redundancy, i.e., decreasing the source code rate, the SBSD can benefit more from a priori information provided by the channel decoder. This can be seen for our proposed code with length \(L\ =\ 5,6,7,8\). It is also shown that if the codeword's minimum Hamming distance satisfies \(d_{\min}\geq 2\), the EXIT curve of the SBSD reaches the \((1,1)\) point. In this case, it is possible to design an appropriate channel encoder that ensures reliable extrinsic information can be obtained by iterative decoding. In addition, the coding efficiency of FLCs may be reduced due to the high residential redundancy. In our joint system design, to keep the overall rate of the system constant, the code rate loss caused by source coding redundancy can be compensated by a channel code with code rate \(R_{C}>1\).  
The convergence of the iterative decoder relies on the amount of redundancy at both sides of the interleaver. If a source decoder has neither redundancy nor channel measures, the iterative decoding is almost useless and does not perform much better than separate decoding.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Length** & **Code Rate** & **Codewords in Decimal** & **Minimum** \\  & & & **Hamming Distance** \\ \hline \(L\ =\ 5\) & 0.75 & 24, 3, 6, 3, 18, 29, 15, 10, 9, 30, 5, 20, 27, 12, 0, 17 & 2 \\ \(L\ =\ 6\) & 0.63 & 6, 23, 51, 10, 30, 17, 43, 56, 39, 12, 3, 36, 29, 0, 45, 54 & 2 \\ \(L\ =\ 7\) & 0.54 & 41, 21, 2, 79, 86, 12, 125, 65, 100, 115, 88, 62, 27, 106, 39, 48 & 3 \\ \(L\ =\ 8\) & 0.47 & 120, 102, 85, 75, 51, 45, 30, 0, 135, 153, 170, 180, 204, 210, 225, 255 & 4 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of optimized source codewords with different code lengths.  
Figure 3: The EXIT curves of different source codes.