# Low-Complexity QRD-\(M\) with Path Eliminations in MIMO-OFDM Systems
## 5 Simulation Results  
This section shows the simulation results for the error performance and the complexity of the proposed QRD-\(M\). As simulation parameters, the size of FFT was 128 and the size of CP was 32. Finally, all transmit symbols go through a Rayleigh fading channel with seven multi-paths. In Figure 5, bit error rate (BER) performances for the conventional and the proposed QRD-\(M\) using 16-QAM modulation are shown in \(2\times 2\) and \(4\times 4\) systems. Additionally, in Figure 6, BER performances for the conventional and the proposed QRD-\(M\) using 64-QAM modulation are shown in \(4\times 4\) and \(8\times 8\) systems. In Figures 5 and 6, the BER performance for the ML is also shown to present the optimal error performance of the proposed QRD-\(M\). The BER performances for the proposed QRD-\(M\) is the same as conventional QRD-\(M\) and ML regardless of the used modulation scheme. Thus, the proposed QRD-\(M\) successfully eliminates unnecessary survival paths from the \(N_{t}\)-th layer to the first layer due to the properly calculated thresholds at all layers. In Figures 7-9, complexities for the conventional and the proposed QRD-\(M\) using 16-QAM modulation are shown by calculating average number of metric operations in \(2\times 2\), \(4\times 4\), and \(8\times 8\) systems, respectively. In Figure 7, the complexities for the adaptive QRD-\(M\) in [10] and LR-aided QRD-\(M\) in [12] are also shown for comparison. The average number of metric operations for the conventional QRD-\(M\) is shown with \(M=1,4,8,16\), and all results are not changed with respect to the signal-to-noise ratio (SNR) because the number of survival paths at all layers is \(M\). However, the average number of metric operations of the proposed QRD-\(M\) is lower than the conventional QRD-\(M\) (\(M=4\)), and is approximated to the conventional QRD-\(M\) (\(M=1\)) with respect to increased SNR because the difference of ASEDs between necessary survival paths and unnecessary survival paths is large. Additionally, the average number of metric operations of the proposed QRD-\(M\) is lower than the adaptive QRD-\(M\) and LR-aided QRD-\(M\) due to more path eliminations by adaptively calculated threshold at each layer. \begin{table}
\begin{tabular}{c c} \hline \hline
**Detection Scheme** & **The Number of Complex Multiplications** \\ \hline Conventional QRD-\(M\) & \(12N_{t}^{3}+4N_{t}^{2}+8L+4 LM\sum\limits_{l=2}^{N_{t}}(l+1)\) \\ \hline Proposed QRD-\(M\) & \(12N_{t}^{3}+4N_{t}^{2}+8L+4L\sum\limits_{l=2}^{N_{t}}M_{l}(l+1)\) \\ \hline Extended DFE & \(12N_{t}^{3}+4N_{t}^{2}+4+L+L\sum\limits_{l=2}^{N_{t}}(4l+1)+4 LN_{t}^{2}+ \mathrm{CLLL}_{N_{t}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The number of complex multiplications. DFE: decision feedback equalizer. Figure 5: The bit error rate (BER) performances for the conventional and the proposed QRD-\(M\) using 16-QAM modulation in \(2\times 2\) and \(4\times 4\) systems. ML: maximum likelihood. Figure 6: The BER performances for the conventional and the proposed QRD-\(M\) using 64-QAM modulation in \(4\times 4\) and \(8\times 8\) systems. Figure 8: Average number of metric operations for the conventional and the proposed QRD-\(M\) using 16-QAM modulation in a \(4\times 4\) system. Figure 7: Average number of metric operations for the conventional and the proposed QRD-\(M\) using 16-QAM modulation in a \(2\times 2\) system. In Figures 8 and 9, complexities for the conventional and the proposed QRD-\(M\) using 16-QAM modulation are shown in \(4\times 4\) and \(8\times 8\) systems. Specifically, modified tree structure QRD-\(M\) is also shown in Figure 9 to provide visible results of complexity improvement for the proposed QRD-\(M\). In Figures 8 and 9, the average number of metric operations for the proposed QRD-\(M\) is approximated to the conventional QRD-\(M\)\((M=1)\) with respect to increased SNR.