# Iterative QR-Based SFSIC Detection and Decoder Algorithm for a Reed-Muller Space-Time Turbo System
## 4 Simulation Results and Discussion  
In this section, a simulation investigation which, in terms of a time-domain Monte Carlo method, was designed in MATLAB (Matrix Laboratory) R2010a under the Windows 7 Ultimate 64-bit operating system, using a computer with a 3.4-GHz Intel(R) Core(TM) i7-2600K CPU, is carried out to show the performance of the proposed scheme in this context. We use three iterative detection schemes, i.e., an iterative QR-based soft feedback segment interference cancellation detector and decoder (IQRSFSICDD), an iterative soft input-soft output and soft feedback detector and decoder (ISIOFDD), and an iterative minimum mean square error parallel interference cancellation detector and decoder (IMMSEPICDD). The simulation results are described in two aspects, i.e., the performance of ISISOICSPDD over a number of iterations between the detector and the decoder loop (the major iteration) and the bit error rate (BER), which, counted by comparisons between the estimated bits and the transmitted bits, is measured as a function of signal-to-noise ratio SNR (signal-to-noise ratio), which is followed by a discussion of the diversity order of the proposed scheme. Let us consider that, in a Rayleigh fading channel, there is an RM space-time turbo system, shared by \(N_{t}\) transmitters and \(N_{r}\) receivers. The length of the frame is 1000 symbols and the transmitted bits for each SNR is larger than 10/BER, which is consistent with the requirements of the Monte Carlo method. The transmitters employ the 4-ary quadrature amplitude modulation (4-QAM) with Gray mapping. An overall code rate R = 121/240, or R = 676/988 outer channel RM turbo code with the same RM (2, 4), or an RM (3, 5) [17; 18] code in each dimension is used. In Figure 1, the pseudo-random interleaver (\(\mathbb{T}\)) is used not only to de-correlate the fading channel and maximize the diversity order of the system, but also to eliminate the correlation in the sequence of the RM turbo coded bits, which is crucial for the proposed algorithm. The number of iterations of the RM turbo decoder (outer iteration) is limited to 5. Figure 3 depicts the BERs of the IQRSFSICDD versus SNR values over a range of iterations, with the transmitters using an R = 121/240 outer channel RM turbo code and the 4-QAM for \(N_{t}=N_{r}=3\), and an R = 676/988 outer channel RM turbo code accompanied by the 4-QAM for \(N_{t}=N_{r}=4\), respectively. For the graphic clarification, not all of the simulation results are shown and the number of the iteration being one means that there is no soft information exchanging between the iterative IQRSFSIC detector and minimal trellis-based MAP RM turbo decoder. From Figure 3, we can obtain the following observation:  
Originally, in the relatively lower input SNR interval, the BER of the proposed algorithm is almost not favorable to the numbers of detector and decoder iterations being executed. Gradually, with the increase in the input SNR, the BERs between successive iterations could be distinguished. Then, at a certain input SNR or, more accurately, an input SNR threshold, the BER curves go into a waterfall region with their characteristic sharp drop, where the proposed algorithm would converge to zero BER within a finite number of iterations. Figure 3: The BERs of the IQRSFSICDD versus SNR values over a range of iterations, with the transmitters using the 4-QAM and (**a**) the R = 121/240 outer channel RM turbo code for \(N_{t}=N_{r}=3\) (**left**); and (**b**) the R = 676/988 outer channel RM turbo code for \(N_{t}=N_{r}=4\) (**right**). Figure 3 has also shown that though the greater number of iterations would lead to the lower bit error rate, the gradient of the BER with respect to the number of iterations for the proposed algorithm does not obviously decrease when the iteration size is increased from 4 to 5. Henceforth, we fix the iterations to 5 as a fundamental trade-off between the BER and computational complexity of the proposed algorithm in all our following simulations.