# Iterative QR-Based SFSIC Detection and Decoder Algorithm for a Reed-Muller Space-Time Turbo System
## 2 Reed-Muller Space-Time Turbo System Model  
Figure 1 presents the configuration of a Reed-Muller space-time turbo system with a receiver based on a QR-based SFSIC detection concatenated with an iterative trellis-based MAP RM turbo decoder. In this system, there are \(N_{t}\) transmit antennas and \(N_{r}\) receive antennas. Let \(M_{b}\) be the number of symbols in the complex constellation \(\mathbf{C}\). The user's information bits \(\mathbf{u}\) are first encoded into the Reed-Muller turbo code stream by the outer encoder, which is then bit-interleaved using an offline-designed pseudo-random interleaver referred to as \(\Pi\). Followed by de-multiplexing, the high-speed RM turbo code stream transfers into low-speed \(N_{t}\) parallel sub-vectors. In the end, each sub-vector is independently mapped into the transmitted vector constellation symbol \(\mathbf{x}\) whose entries \([x_{1},\cdots,x_{m},\cdots,x_{N_{t}}]^{T}\) are chosen from \(\mathbf{C}\), where \(x_{m}\) is a symbol transmitted out of the \(m\)th transmit antenna at the \(i\)th interval. The received \(N_{r}\times 1\) vector can be represented by:  
\[\mathbf{Y}=\mathbf{H}\mathbf{x}+\mathbf{\eta}=\sum_{k=1}^{N_{t}}\mathbf{h}_{k}x_{k }+\mathbf{\eta} \tag{1}\]  
where \(\mathbf{H}\) denotes the \(N_{r}\times N_{t}\) channel matrix, the elements of which are independent and identically distributed (i.i.d.) with the zero-mean unit-variance complex Gaussian distribution. \(\mathbf{\eta}\) is a \(N_{r}\times 1\) complex white Gaussian noise with zero mean and a covariance of \(2\sigma^{2}\mathbf{I}\), where \(\mathbf{I}\) is an \(N_{r}\times N_{r}\) identity matrix. In order to obtain the most likely information vector, we apply the statistical features among all components of the received signals. By doing this, the multi-stream interference from other transmit antennas is mitigated, and different types of detectors are also acquired. The MAP algorithm selects the most likely encoded bit with respect to all combinations of transmitted signals by maximizing the log-likelihood ratio (LLR) [4,18] as given by:  
\[\Lambda[d_{k,m}]=\ln\frac{P(d_{k,m}=+1\big{|}\mathbf{y},\mathbf{H})}{P(d_{k,m} =-1\big{|}\mathbf{y},\mathbf{H})}=\ln\frac{\sum\limits_{\mathbf{x}\in\mathbf{ X}(d_{k,m}=-1)}p(\mathbf{x},\mathbf{y},\mathbf{H})}{\sum\limits_{\mathbf{x}\in \mathbf{X}(d_{k,m}=-1)}p(\mathbf{x},\mathbf{y},\mathbf{H})} \tag{2}\]  
where \(\mathbf{X}(d_{k,m}=\pm 1)\) is the set of all possible RM turbo-coded bit vectors, the \(m\)th bit position of the constellation symbol for the \(k\)th transmit antenna, which is fixed with \(d_{k,m}=\pm 1\). The Log MAP algorithm must calculate \(2^{N_{t}\Gamma}\), where \(\Gamma\) is the code constraint length sequence corresponding to a sequence of modulated symbols to be transmitted during the \(i\)th interval, the metric being expressed in Equation (2), and the vector that yields the largest metric being selected. It is a considerably complicated NP-complete (nondeterministic polynomial time-complete) problem when \(N_{t}\Gamma\) is large. Although the bit error rate (BER) of it is considerably low, its computational complexity is unacceptable in reality. To reduce the computational complexity, an alternative approach is adopted. By employing the sphere decoder technique, the list sphere decoder [6,7] and other improved search vector versions [8] that are composed of the constellation points are limited to a sphere of radius C centered  
Figure 1: The configuration of a Reed–[PERSON] space-time turbo system with a receiver based on a QR-based SFSIC detector and decoder. at the received signal. These approaches aim to implement suboptimum detection, with its complexity depending on the size of the candidate vector set to be searched for detections per information bit. However, it is still a rather complicated NP-complete problem when the size of the candidate vector set is large. Furthermore, the SIOF detectors are proposed to avoid EP and reduce the complexity.