# Soft Iterative Decoding Algorithms for Rateless Codes in Satellite Systems
## 3 Efficient Soft Iterative Decoding Algorithms for HARQ with Rateless Codes in Satellite Systems
### HARQ with Rateless LDPC Codes
#### 3.2.2 Soft Iterative Decoding Algorithms  
At the receiver, joint iterative decoding with soft information can be performed by adopting the modified Tanner graph approach [12]. Figure 3 shows the Tanner graph of the rateless LDPC code, and two component codes, i.e., the LT and LDPC codes are connected by sharing the same bit nodes in the Tanner graph. This node sharing in the Tanner graph can lead to a parallel decoding process at each component code; a similar approach was attempted earlier in the orthogonal frequency division multiplexing (OFDM) system [13]. Referring to the outer LDPC code part of the Tanner graph in Figure 3, nodes \(c_{1}\) to \(c_{K}\) are included in the systematic part, i.e., original information, while nodes \(c_{K+1}\) to \(c_{k}\) are included in the parity part. We note that \(k\) is the length of the systematic part of the inner LT code, and it is the same as the length of the outer LDPC codeword of \(N\). Therefore, there is direct connection with \(c_{K+1}\) to \(c_{k}\) to the check nodes \(P_{1}\) to \(P_{N-K}\) of the outer LDPC code. The nodes \(c_{k+1}\) to \(c_{n}\) denote the parity part of the inner LT code, and are used to form check nodes \(p_{1}\) to \(p_{n-k}\). As shown in the figure, the inner and outer codes in the Tanner graph are represented by check and bit nodes; the same decoding algorithm can be used. In addition, bit nodes of both codes are shared, so there will be automatic interaction between two decoders and this enables the parallel processing of two iterative decoders. Figure 2: Hybrid automatic repeat request (HARQ) transmissions with rateless low-density parity check (LDPC) codes in satellite systems. In order to feed input to the decoder, the demodulated soft information, \(r_{i}\), for the \(i\)th transmitted signal, \(c_{i}\), given the \(i\)th received signal, \(\gamma_{i}\), is estimated by using a soft demapping method [14; 15], which is an approximation of the following equation [8]. \[r_{i}=\log\left(\frac{\Pr(c_{i}=1|\gamma_{i})}{\Pr(c_{i}=0|\gamma_{i})}\right). \tag{4}\]  
Then, the joint iterative soft decoding is started by initializing the a priori information as the soft information on the bit nodes, \(\lambda_{i}^{\left(\text{R}\right)}\), and it is used for both LT and LDPC codes due to the bit node sharing property in the Tanner graph. After initializing \(\lambda_{i}^{\left(\text{R}\right)}=r_{i}\), soft information exchanges are iteratively performed from bit to check nodes and vice versa at both codes. At the initial iteration, soft information from bit node \(i\) to check node \(t\) of the LT code, \(\mu_{i,t}\), is initialized with \(\lambda_{i}^{\left(\text{R}\right)}\) as follows:  
\[\mu_{i,t}=\lambda_{i}^{\left(\text{R}\right)},\ t\in R_{i}, \tag{5}\]  
where \(R_{i}\) is the set of indices for the check nodes connecting to bit node \(i\). Afterwards, soft information between check node \(t\) and bit node \(i\), \(\omega_{t,i}\) and \(\mu_{i,t}\) are alternately estimated for soft information exchange across the nodes, as follows:  
\[\omega_{t,i}=2\tan^{-1}\left(\prod_{s\in C_{i},s\
eq i}\tanh\frac{\mu_{s,t}}{ 2}\right), \tag{6}\]  
\[\mu_{i,t}=\lambda_{i}^{\left(\text{R}\right)}+\sum_{y\in R_{i},y\
eq t}(\omega _{y,i}), \tag{7}\]  
Figure 3: Tanner graph of the rateless LDPC code. where \(C_{t}\) is the set of indices for the bit nodes connecting to check node \(t\).