# Low-Complexity QRD-\(M\) with Path Eliminations in MIMO-OFDM Systems
## 3 Conventional QRD-\(M\)  
The conventional QRD-\(M\) starts from QR decomposition of the channel matrix \(\mathbf{H}\) as follows,  
\[\mathbf{H}=\mathbf{Q}\mathbf{R}, \tag{2}\]  
where \(\mathbf{Q}\) denotes \(N_{r}\times N_{r}\) unitary quadrature matrix and \(\mathbf{R}\) denotes \(N_{r}\times N_{t}\) upper triangular matrix. Using Equation (2), the MIMO received vector in Equation (1) is rewritten as follows,  
\[\mathbf{Y}=\mathbf{H}\mathbf{X}+\mathbf{W}=\mathbf{Q}\mathbf{R}\mathbf{X}+ \mathbf{W}, \tag{3}\]  
Multiplying \(\mathbf{Q}^{H}\) on both sides of Equation (3), the modified MIMO received vector \(\mathbf{Z}=\left[ \begin{array}{cccc}Z_{1}&Z_{2}&\cdots&Z_{N_{r}}\\ \end{array} \right]^{T}\) is as follows,  
\[\mathbf{Z} =\mathbf{Q}^{H}\mathbf{Y}=\mathbf{R}\mathbf{X}+\widetilde{ \mathbf{W}} \tag{4}\] \[=\left[ \begin{array}{cccc}R_{11}&R_{12}&\cdots&R_{1N_{t}}\\ 0&R_{22}&\cdots&R_{2N_{t}}\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&0&R_{N_{t}N_{t}}\\ 0&0&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&0&0\\ \end{array} \right]\left[ \begin{array}{c}X_{1}\\ X_{2}\\ \vdots\\ X_{N_{t}}\\ \end{array} \right]+\left[ \begin{array}{c}\widetilde{W}_{1}\\ \widetilde{W}_{2}\\ \vdots\\ \widetilde{W}_{N_{r}}\\ \end{array} \right],\]  
where \(\widetilde{\mathbf{W}}=\mathbf{Q}^{H}\mathbf{W}\) denotes a modified noise vector which has the same statistical properties as the existing noise vector \(\mathbf{W}\). In Equation (4), \(|S|\) times SEDs are calculated at the \(N_{t}\)-th layer (the \(N_{t}\)-th row of \(\mathbf{Z}\)) between \(Z_{N_{t}}\) and the \(k\)-th reference symbol \(C_{k}\); i.e., \(k=1,2,\cdots,|S|\), where \(|S|\) denotes constellation size. In all calculated SEDs at the \(N_{t}\)-th layer, \(M\) paths are selected as survival paths and the remaining \(|S|-M\) paths are discarded. These \(M\) survival paths are extended to the \((N_{t}-1)\)-th layer and operations at the \(N_{t}\)-th layer end. The SED \(E_{N_{t},k}\) between \(Z_{N_{t}}\) and \(C_{k}\) is as follows:  
\[E_{N_{t},k}=|Z_{N_{t}}-R_{N_{t}N_{t}}C_{k}|^{2}. \tag{5}\]  
At the \((N_{t}-1)\)-th layer, \(M|S|\) times SEDs are calculated. However, for accurate estimation of transmit symbols, the SED in Equation (5) is used to calculate the ASED. The ASED \(E_{N_{t}-1,k}^{d_{N_{t}}}\) between \(Z_{N_{t}-1}\) and \(C_{k}\) which considers the SED for the \(d_{N_{t}}\)-th survival path in Equation (5) is as follows:  
\[E_{N_{t}-1,k}^{d_{N_{t}}}=\left|Z_{N_{t}-1}-\left(R_{N_{t}-1N_{t}-1}C_{k}+R_{N_ {t}-1N_{t}}\hat{X}_{N_{t}}^{d_{N_{t}}}\right)\right|^{2}+E_{N_{t},d_{N_{t}}}, \tag{6}\]  
where \(\hat{X}_{N_{t}}^{d_{N_{t}}}\) is temporarily estimated symbol at the \(N_{t}\)-th layer corresponding to the \(d_{N_{t}}\)-th survival path. Like the \(N_{t}\)-th layer, \(M\) paths are selected as survival paths, and the remaining \(M|S|-M\) paths are discarded. The \(M\) survival paths are extended to the next layer and these operations continue until the first layer for accurate estimation of transmit symbols. At the \(m\)-th layer, \(M|S|\) times ASEDs are calculated. The ASED \(E_{m,k}^{d_{m+1}}\) between \(Z_{m}\) and \(C_{k}\) which considers the ASED for the \(d_{m+1}\)-th survival path is as follows:  
\[E_{m,k}^{d_{m+1}}=\left|Z_{m}-\left(R_{mm}C_{k}+\sum_{i=m+1}^{N_{t}}R_{mi}\hat{ X}_{i}^{d_{i}}\right)\right|^{2}+E_{m+1,d_{m+1}}^{d_{m+2}}. \tag{7}\]Finally, transmit symbols are estimated by selecting the path with the smallest ASED at the first layer and an algorithm of conventional QRD-\(M\) ends. Figure 2 shows the tree structure for the conventional QRD-\(M\) (\(M=4\)) using quadrature phase shift keying (QPSK) modulation in a \(4\times 4\) system. In Figure 2, bold lines are survival paths and bold circles are estimated transmit symbols. The conventional QRD-\(M\) has extremely high complexity in huge MIMO-OFDM systems because it calculates \(M|S|\) times SEDs at each layer. In Section 4, the authors try to reduce the complexity of the conventional QRD-\(M\) by eliminating unnecessary survival paths at all layers.