# Iterative QR-Based SFSIC Detection and Decoder Algorithm for a Reed-Muller Space-Time Turbo System
## 1 Introduction  
Multiple-input multiple-output (MIMO) systems achieve high data rate transmission and substantial gains in channel capacity over a rich-scattering environment [1; 2], which makes very high spectral efficiency available for the emerging wireless standard IEEE 802.11n/ac [3], etc. Moreover, if a powerful channel code, such as turbo codes [4], can be serially concatenated with this architecture, a kind of MIMO system, i.e., the turbo space-time system, is obtained in order to achieve near-capacity performance [5; 6]. Endowed with the turbo learning principle, iterative detection (inner decoder) and decoder (outer decoder) receivers for the turbo space-time system [5], detecting and decoding, by exchanging soft bits of information mutually, can approximately approach optimal performance with reduced complexity [5; 6]. Although the maximum a posteriori probability (MAP) MIMO detector could attain superior performance, it has a complexity that grows exponentially with the product of the number of transmit antennas and the modulation order, which, in reality, precludes it from implementation. Therefore, three major types of sub-optimal soft-input soft-output (SISO) MIMO detectors, i.e., the SISO sphere decoder (SD) [6; 7; 8], SISO interference cancellation (IC) detectors [9; 10; 11], and SISO decision-feedback (DF) detectors [12; 13], have attracted the most attention. Among them is SISO list sphere detection with a probabilistic radius tightening [7] and SISO sphere detection with perfect node enumeration [8], which, taking into account the characteristic of the extrinsic information estimates, can achieve the upper bound of the diversity [14; 15]. However, the smaller the radius, the more computationally efficient the method of the SISO SDs would be. Nevertheless, only a sphere of relatively large radius and list size would ensure SISO SDs to provide near-optimal detection, which results in an unobviously-reduced complexity comparable to that of the MAP detection algorithm. With recent advances in IC and DF, more and more novel achievements, such as SISO minimum-mean-square error (MMSE)-based successive interference cancelation receivers with a novel strategy [9]; an area-efficient MMSE with parallel interference cancellation (PIC) detector [10]; the MMSE-PIC-based SISO detector with Neumann series expansion to reduce the total complexity [11]; a novel low-complexity decision-feedback detection algorithm with constellation constraints [12]; a soft input, soft output, and soft feedback (SICF) detector [13]; etc., in this area have been applied to modify the strategy for mitigating error propagation (EP) for this kind of suboptimum detector. Previously, it was shown that the QR decomposition [16] of the channel matrix led to a simple successive detection algorithm with reduced complexity. In this way, we extend this idea to a SICF detector and propose an iterative QR-based soft feedback segment interference cancellation (QRSFSIC) detection and decoder algorithm. Furthermore, the originally-proposed turbo codes are generated by two parallel recursive systematic convolutional encoders separated by an interleaver and using an iterative SISO decoder [4]. Nevertheless, using Reed-Muller (RM) [4; 17], instead of convolutional codes as component codes, RM turbo codes, with a simple block interleaver, not only give very good performance for high code rates, but also avoid the problem of trellis termination. In this paper, it is serially concatenated with MIMO architecture to constitute a novel kind of MIMO system, named as the RM turbo space-time system. The proposed algorithm is composed of an iterative QRSFSIC detector (inner decoder) and a trellis-based MAP RM turbo decoder (outer decoder), which mitigates the multi-stream interference and obtains the desired information bit by exchanging information between inner and outer decoders. It, based on QR, not only embeds SFSIC, but also merges RM-turbo codes with minimal trellis representation into the QRSFSIC detector and decoder.