# 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  
At first, in terms of QR decomposition of the channel matrix \(\mathbf{H}=\mathbf{Q}\mathbf{R}\), where \(\mathbf{Q}\) is a unitary matrix, and \(\mathbf{R}\) is the upper triangular, multiplying the received signal \(\mathbf{Y}\) with \(\mathbf{Q}^{\mathbf{H}}\), the algorithm yields the sufficient statistic [13; 16] for soft feedback successive interference cancellation of \(x_{i_{n}}\):  
\[ \begin{array}{ll}\mathbf{y}_{i_{n}}&=\mathbf{Q}^{H}\left(\mathbf{Y}-\mathbf{ H}\left[\begin{array}{c}\overline{\mathbf{x}}_{i_{n-1}}^{a}\\ 0\\ \overline{\mathbf{x}}_{i_{n+1}}^{a}\end{array}\right]\right)=\mathbf{y}-\mathbf{ R}\left[\begin{array}{c}\overline{\mathbf{x}}_{i_{n-1}}^{a}\\ 0\\ \overline{\mathbf{x}}_{i_{n+1}}^{a}\end{array}\right]\\ &=\mathbf{r}_{i_{n}}x_{i_{n}}+\left[\mathbf{r}_{i_{1}i_{n-1}}\mathbf{r}_{i_{n+ 1}:N_{t}}\right]\left[\begin{array}{c}\mathbf{x}_{i_{n-1}}-\overline{\mathbf{ x}}_{i_{n-1}}^{a}\\ \mathbf{x}_{i_{n+1}}-\overline{\mathbf{x}}_{i_{n+1}}^{a}\end{array}\right]+ \mathbf{n}\end{array}  \tag{3}\]  
where the first term on the right of the third equal sign comprises the desired symbol that needs to be detected and the rest of the terms represent the interference that is not completely cancelled plus the noise. \(\overline{\mathbf{x}}_{i_{n}}^{a}\) are the soft estimates of \(\mathbf{x}_{i_{n}}\) derived from the a priori LLRs, written as:  
\[ \begin{array}{ll}\overline{x}_{i_{n}}^{a}&=E[x_{i_{n}}|\mathbf{L}_{i_{n},i: \mathbf{M}_{b}}^{a1}]\\ &=\sum\limits_{\theta\in\Theta}\theta\prod\limits_{m=1}^{\Omega}\frac{1}{2} \left(1+(2d_{i_{n},m}-1)\text{tanh}\bigg{(}\frac{L_{i_{n},m}^{a1}}{2}\bigg{)} \right)\qquad 1\leq i_{n}\leq N_{t}\end{array}  \tag{4}\]  
where \(\mathbf{L}_{i_{n},1:\mathbf{M}_{b}}^{a}=\left[L_{i_{n},1}^{a},\ldots,L_{i_{n}, \mathbf{M}_{b}}^{a}\right]^{T}\), with the entry \(L_{i_{n},m}^{a}=\ln P(d_{i_{n},m}=1)-\ln P(d_{i_{n},m}=0)\), which indicates the prior the set of all constellation points, each of which corresponds to an encoded bit vector \(d_{i_{n},1},\ldots,d_{i_{n},M_{b}}\) according to a certain mapping. Then, the MMSE estimate \(\hat{x}_{i_{n}}\) is denoted by [16; 13]:  
\[ \begin{array}{ll}\hat{x}_{i_{n}}&=(\mathbf{w}_{i_{n}})^{H}\mathbf{y}_{i_{n}}= \left(\left(\mathbf{R}\mathbf{\Psi}_{i_{n}}\mathbf{R}^{H}+\sigma^{2}\mathbf{I} _{N_{r}}\right)^{-1}\mathbf{r}_{i_{n}}\right)^{H}\mathbf{y}_{i_{n}}\\ &=\left(\left(\mathbf{R}\left[\begin{array}{ccc}\boldsymbol{\psi}_{a,i_{1}: i_{n-1}}&0&0\\ 0&1&0\\ 0&0&\boldsymbol{\psi}_{a,i_{n+1}:N_{t}}\end{array}\right]\mathbf{R}^{H}+\sigma^ {2}\mathbf{I}_{N_{t}}\right)^{-1}\mathbf{r}_{i_{n}}\right)^{H}\mathbf{y}_{i_{n}} \end{array}  \tag{5}\]  
where \(\mathbf{w}_{i_{n}}\) is a weighted coefficient vector of the MMSE estimate. \(\boldsymbol{\psi}_{a,k:l}\) indicates the covariance matrix of \(\left[x_{k},\ldots,x_{l}\right]^{T}\), given a prior LLRs. Due to the interleaver, symbols from each transmit antenna would be regarded as being independent. Thus, each non-diagonal element of \(\boldsymbol{\psi}_{a,k:l}\) is zero. The \(q\)th diagonal element \(\mathbf{\Psi}_{i_{n}}(q,q)\) of \(\mathbf{\Psi}_{i_{n}}\) can be represented as:  
\[\mathbf{\Psi}_{i_{n}}(q,q)=E\Big{[}\left|x_{q}\right|^{2}\left|\mathbf{L}_{q,1 :M_{b}}^{(\cdot)}\right]-\left|\mathbf{\bar{x}}_{q}^{(\cdot)}\right|^{2} \tag{6}\]  
\[E\Big{[}\left|x_{q}\right|^{2}\left|\mathbf{L}_{q,1:M_{b}}^{(\cdot)}\right]= \sum_{\theta\in\Theta}\left|\theta\right|^{2}\prod_{m=1}^{M}\frac{1}{2}\left( 1+(2d_{q,m}-1)\text{tanh}\left(\frac{L_{q,m}^{(\cdot)}}{2}\right)\right) \tag{7}\]  
where \(\mathbf{L}_{q,1:M_{b}}^{(\cdot)}=\mathbf{L}_{q,1:M_{b}}^{a1}\) and \(\left|x_{q}^{(\cdot)}\right|^{2}=\left|x_{q}^{a1}\right|^{2}\).