# 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
### Iterative QR-Based SFSIC Detector (Inner Decoder)
#### 3.1.1 Forming a Reliability Metric Based on a Posterior Information
Similarly, if \(\boldsymbol{\psi}_{f,k:l}\) denotes the covariance matrix of \(\left[x_{k},\ldots,x_{l}\right]^{T}\), given a posterior LLRs, to obtain the \(q\)th diagonal element \(\mathbf{\Psi}_{i_{n}}(q,q)\), we should let \(\mathbf{L}_{q,1:M_{b}}^{(\cdot)}=\mathbf{L}_{q,1:M_{b}}^{f}\) and \(\left|x_{q}^{(\cdot)}\right|^{2}=\left|x_{q}^{f}\right|^{2}\). Thus, if the proposed algorithm operates \(l-1\) inner iterations, the symbol reliability metric \(\Lambda_{n}\) can be obtained according to the posterior LLR of encoded bits:  
\[\Lambda_{i_{n}}\triangleq\min_{1\leq i\leq 2^{M_{b}}}\left|L_{i_{n},i}^{f}\right| \tag{8}\]  
where \(L_{i_{n},i}^{f}\), composed of a prior LLR \(L_{i_{n},i}^{a1}\) and an extrinsic LLR \(L_{i_{n},i}^{e1}\), is the a posterior LLR of the \(i\)th encoded bit from the \(i_{n}\)th antenna, expressed by:  
\[ \begin{array}{ll}L_{i_{n},i}^{f}&=\ln\frac{P(d_{i_{n},i}=1\left|\hat{x}_{i_ {n}})}{P(d_{i_{n},i}=0\left|\hat{x}_{i_{n}})}\\ &=L_{i_{n},i}^{e1}+L_{i_{n},i}^{a1}\end{array}  \tag{9}\]  
When \(\Lambda_{i_{n}}>\tau\) (\(1\leq i_{n}\leq N_{t}\)), the detected symbol transmitted from the \(i_{n}\)th antenna is a reliable one such that it is saved in the reliable symbol segment \(\phi\) while the rest is put into the residual symbol segment \(\vartheta\). Obviously, \(\tau\) is the threshold whose value will affect the number of symbols whose interference is to be cancelled in the next inner iteration. However, the proper \(\tau\) is difficult to exploit. Based on the trade-off between symbols with a larger reliability metric, which should be enclosed in the reliable symbol segment, and relatively lower computational effort, we choose:  
\[\tau=\min(\left\{\Lambda_{1},\Lambda_{N_{t}}\right\})+\beta(\max(\left\{ \Lambda_{1},\Lambda_{N_{t}}\right\})-\min(\left\{\Lambda_{1},\Lambda_{N_{t}} \right\})) \tag{10}\]  
where \(\beta=0.85\) is the empirical value [13]. 1.2 Canceling Interference from Reliable Symbols and Forming the MMSE Estimate of Residual Symbols Iteratively  
To perform the \(l\)th inner iteration for detecting the symbols in segment \(\vartheta_{l}\), based on an a posteriori estimate \(\overline{x}_{i}^{f}\), we obtain the residual signal [13; 16] after canceling the interference resulting from the segment \(\phi_{l}\), i.e.,:  
\[\mathbf{y}^{l}=\mathbf{y}-\sum_{i\in\varphi_{l}}\mathbf{r}_{i}\overline{x}_{i }^{f} \tag{11}\]  
Thereafter, the MMSE estimate \(\hat{x}_{k}^{l}\) of the residual symbol segment \(\vartheta_{l}\) can be calculated as [13; 16]:  
\[\hat{x}_{k}^{l}=\mathbf{w}_{n}^{l}\mathbf{y}^{l}=\left( \begin{subarray}{c}l\\ n\end{subarray} \right)^{-1}\mathbf{r}_{n} \tag{12}\]  
\[\mathbf{Z}_{n}^{l}=\mathbf{r}_{n}\mathbf{r}_{n}^{H}+\sum_{k=1,k\
eq n}^{N_{t} }A_{k}\mathbf{r}_{k}\mathbf{r}_{k}^{H}+\sigma^{2}\mathbf{I}_{N_{t}} \tag{13}\]  
where \(A_{k}\) is the variance of an a priori estimate of \(x_{k}\) for \(k\in\vartheta_{l}\) and the variance of an a posteriori estimate for \(k\in\phi_{l}\), which can be acquired by Equations (6) and (7).