# Linear Approximation Signal Detection Scheme in MIMO-OFDM Systems
## 4 Proposed Detection Scheme
### QRD-M Stage  
The QRD-\(M\) is applied to the two MIMO-OFDM systems in Equations (18) and (20). For low complexity QRD-\(M\), the value of \(M\) is fixed with respect to the used modulation order, like Figure 2. Figure 2 shows the method for candidate selections in the 16-quadrature amplitude modulation (QAM) constellation with respect to three cases. The candidates are both center point (black colored circle) and neighbor points (dash-lined circles). The center point is a symbol, which is estimated by the DFE stage, as shown in Section 4.2. Also, the neighbor points are symbols that are within one point from the center point. The case (**a**) is that the center point is in the middle of the constellation and the case (**b**) is that the center point is in the edge of the constellation. Finally, the case (**c**) is that the center point is the nearest point from the edge point. The value of \(M\) is different according to the estimated center point, and it is fixed as 5, 4, 6 for cases (**a-c**), respectively. The neighbor points may have high probability for the original transmit symbol if the DFE stage fails to estimate the accurate transmit symbols. From the QRD-\(M\), all of the transmit symbols are estimated again and all processes of the proposed detection scheme are completed. With a slight increase of the complexity, the estimated symbols are more reliable than the estimated symbols in the DFE stage. Table 1 represents the number of complex multiplications for the proposed detection scheme in \(N\times N\) MIMO-OFDM system. Also, it is assumed that one multiplication between two complex numbers requires four real multiplications. Finally, Figure 3 shows the flow chart for the proposed detection scheme. It is composed of three stages which are explained in Sections 4.1-4.3.  
\begin{table}
\begin{tabular}{c c} \hline
**Detection Scheme** & **The Number of Complex Multiplications** \\ \hline Conventional ZF & \(8N^{3}+4N^{2}\) \\ \hline Conventional DFE & \(12N^{3}+4N^{2}+4\sum\limits_{k=2}^{N}(k+1)+4\) \\ \hline Proposed scheme (Partial ZF stage) & \(\sum\limits_{n=2}^{N/2}(12n^{2}+4n)N\)\((n=2,4,8,\cdots,N/2)\) \\ \hline Proposed scheme (DFE stage) & \(64N\) \\ \hline Proposed scheme (QRD-\(M\) ZF stage) & \((4M^{2}+8M)N\) \\ \hline Group Iterative Linear ZF in [7] & \(14N^{3}+8N^{2}\) \\ \hline Hybrid ZF and QR Receiver in [8] & \(15N^{3}+6N^{2}+4\sum\limits_{k=2}^{N/2}(k+1)+4\) \\ \hline Hybrid dynamic QRD-\(M\) and ZF in [16] & \(12N^{3}+4N^{2}(1+\lceil N/3\rceil)+\) \\ \(4N\Big{(}\lceil N/3\rceil^{2}+\lceil N/3\rceil\Big{)}8L+4 LM\sum\limits_{k=2}^{N -\lceil N/3\rceil+1}(l+1)\) \\  & where \(\lceil N/3\rceil\) denotes the integer number which is larger than \(N/3\) or equal to \(N/3\). \\ \hline \end{tabular}
\end{table}
Table 1: The number of complex multiplications ZF: zero-forcing; DFE: decision feedback equalizer; QRD-M: QR decomposition-\(M\) algorithm).  
Figure 3: The flow chart of the proposed detection scheme (ZF: zero-forcing: DFE: decision feedback equalizer).  
Figure 2: The method for candidate selections in the 16-QAM constellation (**a**): center point is in the middle of the constellation; (**b**): center point is in the edge of the constellation; (**c**): center point is the nearest point from the edge point (QAM: quadrature amplitude modulation).