# Iterative QR-Based SFSIC Detection and Decoder Algorithm for a Reed-Muller Space-Time Turbo System
## 3 The Novel Iterative QR-Based SFSIC Detection and Decoder Algorithm
### RM-Turbo Iterative Decoder (Outer Decoder)
#### 3.2.1 Generating Extrinsic Information for the Vertical Decoder  
Once the soft bit stream coming from the inner decoder is de-multiplexed and decomposed to obtain the horizontal soft systematic bit stream and soft parity bit stream, the APP LLR of the extrinsic information for systematic bit \(d_{i_{n},m}\), i.e., \(u_{i_{n},m}\), of the horizontal decoder in the \(q\) outer iteration is defined as [4; 17]:  
\[L_{e}^{-}(i_{n},m,q)=L_{1}^{-}(i_{n},m,q)-L_{c}\cdot L^{-}(d_{i_{n},m},q)-L_{e ^{T}}^{\mid}(i_{n},m,q) \tag{17}\]  
where \(L_{c}\) is called the reliability value of the channel; \(L^{-}(d_{i_{n},m},q)\) denotes the received de-multiplexed soft output of the horizontal decoder, associated with the information bit \(u_{i_{n},m}\), estimated by the outer decoder; \(L_{e^{T}}^{\mid}(i_{n},m,q)\) is the transpose of the \([i_{n},m]\)th APP LLR extrinsic information \(L_{e}^{\mid}(i_{n},m,q)\) of the vertical decoder, which is equal to \(L_{a}^{-}(i_{n},m,q)\), the a priori LLR value for the \([i_{n},m]\)th information bit, at the first iteration; and \(L_{1}^{-}(i_{n},m,q)\) is the APP LLR associated with the information bit \(u_{i_{n},m}\) in the \(q\)-th iteration, expressed by:  
\[\begin{array}{ll}L_{1}^{-}(i_{n},m,q)&=\ln\frac{p\big{(}d_{i_{n},m}\!=\!+1 \big{|}L^{-}\big{(}d_{i_{n},m}\!=\!-1\big{)}\big{)}}{p\big{(}d_{i_{n},m}\!=\!- 1\big{|}L^{-}\big{(}d_{i_{n},m}\!=\!d\big{)}\big{)}}=\ln\frac{\sum_{(s^{\prime},s,l_{i_{n},k}\!=\!-1)}p\big{(}s^{\prime},s,\!L^{-}\big{(}d_{i_{n},m}\!=\!d \big{)}\big{)}}{\sum_{(s^{\prime},s,\!L_{i_{n},k}\!=\!-1)}p\big{(}s^{\prime}, \!L^{-}\big{(}d_{i_{n},k}\!=\!d\big{)}\big{)}}\\ &=\ln\frac{\sum_{(s^{\prime},s,l_{i_{n},k}\!=\!+1)}p\big{(}s^{\prime},\!L^{-} \big{(}d_{i_{n},j<k}\!\big{)}\big{)}p\big{(}s^{\prime},\!L^{-}\big{(}d_{i_{n}, k}\!=\!d\big{)}\big{)}p\big{(}s^{\prime},\!L^{-}\big{(}d_{i_{n},k}\!=\!d \big{)}\big{|}s^{\prime}\big{)}p\big{(}L^{-}\big{(}d_{i_{n},j>k}\!\big{)}\big{|} s\big{)}}{\sum_{(s^{\prime},s,\!L_{i_{n},k}\!=\!-1)}p\big{(}s^{\prime},\!L^{-} \big{(}d_{i_{n},j<k}\!\big{)}\big{)}p\big{(}s^{\prime},\!L^{-}\big{(}d_{i_{n}, k}\!=\!d\big{)}\big{|}s^{\prime}\big{)}p\big{(}L^{-}\big{(}d_{i_{n},j>k}\! \big{)}\big{|}s\big{)}}\\ &=\ln\frac{\sum_{(s^{\prime},d_{i_{n},k}\!=\!+1)}\alpha_{k-1}(s^{\prime})\cdot \lambda_{k}(s^{\prime},s)\cdot\omega_{k}(s)}{\sum_{(s^{\prime},s,\!\lambda_{i_ {n},k}\!=\!-1)}\alpha_{k-1}(s^{\prime})\cdot\lambda_{k}(s^{\prime},s)\cdot \omega_{k}(s)}\end{array} \tag{18}\]  
where \((s^{\prime},s,d_{i_{n}k}=+1)\) and \((s^{\prime},s,d_{i_{n},k}=-1)\) denote the set of the pair of states that connects the trellis state \(s^{\prime}\) at time \(k-1\) to the state \(s\) at time \(k\), respectively; and \(d_{i_{n},j<k}\) denotes the portion of the received sequence from bit \(0\) up to bit \(k-1\). Similarly, the received sequence from bit \(k\) up to bit \(m-1\) is  
Figure 2: Internal structure of the RM-turbo decoder.  
represented by \(d_{i_{n},j>k}\). Moreover, \(\alpha_{k}(s^{\prime})\), \(\omega_{k}(s)\), and \(\lambda_{k}(s^{\prime},s)\) are forward, backward, and branch metric recursions of the MAP decoding, obtained by:  
\[\begin{array}{l}\alpha_{k}(s)=\sum\limits_{s^{\prime}}\lambda_{k}(s^{\prime},s )\cdot\alpha_{k-1}(s^{\prime})\\ \omega_{k-1}(s)=\sum\limits_{s}\lambda_{k}(s^{\prime},s)\cdot\omega_{k}(s)\\ \lambda_{k}(s^{\prime},s)=P(s|s^{\prime})\cdot p\big{(}L^{-}(d_{i_{n},k^{ \prime}},q)\big{|}s^{\prime},s\big{)}=p(d_{i_{n},k^{\prime}}L^{-}\big{(}d_{i_{ n},k^{\prime}},q))\\ =\exp\Big{(}\frac{1}{2}d_{i_{n},k}(L_{c}\cdot L^{-}(d_{i_{n},k},q)+L_{a}^{-}(i_ {n},m,q))\Big{)}\\ =\exp\Big{(}L\Big{(}d_{i_{n},k};L^{-}(d_{i_{n},k},q)\cdot\frac{d_{n,k}}{2} \Big{)}\Big{)}\end{array} \tag{19}\]  
where \(L(d_{i_{n},k};L^{-}(d_{i_{n},k},q))=L_{c}\cdot L^{-}(d_{i_{n},k},q)+L_{a}^{-}( i_{n},m,q)\) for information bits and \(L(d_{i_{n},k};L^{-}(d_{i_{n},k},q))=L_{c}\cdot L^{-}(d_{i_{n},k},q)\) for parity bits.