# Low-Complexity QRD-\(M\) with Path Eliminations in MIMO-OFDM Systems
## 4 Proposed Low-Complexity QRD-\(M\)
It is assumed that the number of existing survival paths is \(M_{N_{t}}\) after the eliminations of unnecessary survival paths at the \(N_{t}\)-th layer. \(|S|M_{N_{t}}\) paths are extended to the next layer to select \(M\) paths at the \((N_{t}-1)\)-th layer. The only difference between the \(N_{t}\)-th layer and the \((N_{t}-1)\)-th layer is that ASED is used at the \((N_{t}-1)\)-th layer, unlike the \(N_{t}\)-th layer. To eliminate unnecessary paths at the \((N_{t}-1)\)-th layer, the \((N_{t}-1)\)-th threshold \(\eta_{N_{t}-1}\) should be calculated like \(\eta_{N_{t}}\). The way to calculate \(\eta_{N_{t}-1}\) is similar to the calculation of \(\eta_{N_{t}}\). The \(l\)-th threshold \(\eta_{l}\) is calculated on the \(l\)-th operation loop by selecting the smallest element in \(\left[\mathbf{E}\right]_{l}\), like in Equation (10). Then, each element in ASEDs vector \(\left[\mathbf{E}_{l}\right]_{l}\) is compared with \(\eta_{l}\) and the element larger than \(\eta_{l}\) is eliminated. It is assumed that the number of existing survival paths is \(M_{l}\) after the elimination of the unnecessary survival paths at the \(l\)-th layer. The \(|S|M_{l}\) paths are extended to the next layer to select \(M\) paths at the \((l-1)\)-th layer, and these operations continue until the first layer for lower complexity. Finally, MIMO transmit vector \(\mathbf{\hat{X}}_{\text{P}}=\left[ \begin{array}{cc}\hat{X}_{\text{P},1}& \hat{X}_{\text{P},2}&\cdots&\hat{X}_{\text{P},N_{t}}\end{array} \right]^{T}\) is estimated at the first layer by selecting the path which has the smallest ASED and an algorithm of proposed QRD-\(M\) ends. In Figure 2(a), the calculation of threshold is illustrated as follows. The value of \(T\) is 2 because the used modulation scheme is QPSK \((T=\log_{2}4=2)\). At the second layer, SEDs in Equation (8) are calculated as 10, 7, 16, 9. In four SEDs, two small values (7 and 9) are selected. Then, at the first layer, ASEDs in Equation (10) are calculated by using SEDs for 7 and 9. The final ASEDs are 16, 21, 14, and 17 from SED for 7 and 18, 13, 20, and 16 from SED for 9. Among eight ASEDs, the smallest value is 13, and it is selected as threshold for path eliminations at the second layer. Note that the smallest value is not 14, which is extended from SED for 7. So, the used modulation scheme must be considered for low complexity. In Figure 2(b), path eliminations are illustrated as follows. At the second layer, the SED for 16 is eliminated because it is larger than the threshold. Then, at the first layer, ASEDs in Equation (10) are calculated for SEDs 10, 7, and 9. Note that ASEDs in Equation (10) are calculated for SEDs 10, 7, 16, and 9 in the conventional QRD-\(M\). The final ASEDs are 19, 18, 12, and 16 for SED 10 and 16, 21, 14, and 17 for SED 7 and 18, 13, 20, and 16 for SED 9. Among twelve ASEDs, the smallest value is 12 and all transmit symbols are estimated. Figure 4 shows the flow chart of the proposed QRD-\(M\). It is composed of two stages for the calculation of the threshold and path eliminations. The detailed steps of the proposed QRD-\(M\) algorithm are presented in Table 1.  
Figure 3: The tree structure of the proposed QRD-\(M\) (\(M=4\)) using QPSK modulation in 2 \(\times\) 2 system. (**a**)The calculation of threshold; (**b**) The path elimination.