# Hybrid Window Decoding for Joint Source Channel Anytime Coding System
## 3 Hybrid Window Decoding for JSCAC System  
### Hybrid Window Decoding Structure  
In this section, we first describe the proposed JHWD algorithm for JSCAC systems in detail, then we provide the improved DE analysis method for the proposed JHWD algorithm. The proposed hybrid window decoding structure is shown in Figure 2, which is composed of two main parts, i.e., the initial expanding window decoding phase and the sliding window decoding phase. Each part is numerically marked. In order to simplify the description, the source and channel couple lengths are set the same as \(\gamma^{sc}=\gamma^{cc}=\gamma\). In the example shown in Figure 2, \(\gamma\) is set as 5. We explain the details of the proposed JHWD scheme as follows. _Application scenarios_: The proposed JHWD scheme is designed for the source streaming with short block length, and with a chain length \(L\) far longer than the coupling length \(\gamma\), i.e., \(L\gg\gamma\). Accordingly, as shown in Figure 2, there is not any special decoding process  
Figure 2: Hybrid window decoding structure of the proposed JHWD algorithm. for blocks at the end of the data chain. Moreover, this work focuses on the decoding performances for the source streaming with short block lengths in low-latency and high-reliability communications. _Selection of window size_: In the JSCAC system, the coupling length \(\gamma\) is supposed to be finite, and the encoding structure of any block is directly related to its \(\gamma-1\) coupled blocks. However, all blocks in the data chain are causally coupled and connected one by one. Therefore, the decoding performance of any block is still related to the backward-connected blocks beyond the coupling length. However, the mutual influence will gradually weaken as distance increases. In order to trade off the error correction performance and the decoding complexity, we adopt the sliding decoding window size as \(3\gamma-2\) block lengths, covering the coupling range where the object block is located and the two nearest adjacent coupling ranges with the greatest influences. These are called the first, the second and the third coupling interval, respectively. _Emendatory decoding strategies_: Combined with the encoding structure of JSCAC system, some emendatory decoding strategies are adopted in the proposed JHWD scheme. (1) Update the source _a posteriori_ log-likelihood ratios (LLRs) at each time slot \(t\) after decoding as follows  
\[\mathbf{L}_{t}^{sc}=(\mathbf{L}_{t-1}^{sc}+\mathbf{L}_{t}^{sc})/2. \tag{5}\]  
The reason for updating the source LLRs instead of the channel LLRs is that the source sequence is not actually transmitted through the channel; rather, it is represented by its compressed syndrome. It is found that decoding the object block with the previously well-decoded source information can effectively improve the error correction performance. (2) In order to alleviate the error propagation caused by sliding window decoding [23], a kind of syndrome-check based detection, shown in (6), is carried out before the local expanding window decoding, where \(\mathbf{\hat{S}}_{k}^{sc}\) and \(\mathbf{\hat{U}}_{k}^{cc}\) are the source and channel decoding results of \(k\)-th joint BP, and \(\mathbf{\hat{H}}_{k}^{sc}\) is the involved source parity-check matrix. If the detection result is false, a local expanding window decoding is adaptively carried out within the decoding window until the detection result is true. This syndrome-check-based detection is implemented once, prior to the matrix multiplication in the joint BP decoding stage of the local expanding window decoding. However, it controls the adaptive execution of the local expanding window decoding, and thus can greatly reduce the decoding iterations and message updates. \[S_{check}= \begin{cases}1&\text{if}\quad\mathbf{\hat{S}}_{k}^{sc}\cdot\mathbf{ \hat{H}}_{k}^{sc}=\mathbf{\hat{U}}_{k}^{cc},\\ 0&else.\end{cases}  \tag{6}\]  
_Decoding structure descriptions_: The details of the JHWD algorithm are provided in Algorithm 1.