# Efficient Transmit Antenna Subset Selection for Multiuser Space-Time Line Code Systems
## 3 TAS Selection Algorithms
### Proposed Incremental TAS Selection Algorithm  
In the MMSE-precoded MU-STLC systems, the received SNR can be used as the design criterion for the proposed incremental TAS selection optimization and readily derived from (8), (10), and (12) as  
\[\eta_{S}=\widehat{\beta}_{S}^{2}\frac{\sigma_{x}^{2}}{2\sigma_{z}^{2}} \tag{17}\]  
where  
\[\widehat{\beta}_{S}^{2}=\frac{1}{Tr\bigg{[}\Big{(}\mathbf{\hat{H}}_{S}\mathbf{ \hat{H}}_{S}^{H}\Big{)}^{-1}\bigg{]}} \tag{18}\]  
To significantly reduce the complexity of Algorithm 1, an SNR-based efficient incremental TAS selection algorithm can be developed by using the popular Woodbury formula [26]. The first-proposed TAS selection algorithm constructs the TAS by starting with an empty TAS and adding one antenna in each iteration. Assuming that \(k\) transmit antennas have been selected, the resulting selected subchannel is denoted as \(\mathbf{H}_{k}\in C^{2K\times k}\), where \(1\leq k<N_{S}\). Then, the channel matrix for the \((k+1)\)-th iteration process can be represented as  
\[\mathbf{H}_{k+1}=\left[ \begin{array}{cc}\mathbf{H}_{k}\mathbf{h}_{k+1} \end{array} \right] \tag{19}\]  
where \(\mathbf{h}_{k+1}\) denotes one of the unselected column vectors of \(\mathbf{H}\) after completing the \(k\)-th iteration. Using the selected channel submatrix \(\mathbf{H}_{k}\), the \((k+1)\)-th selected antenna can be determined by the following optimization. \[S_{k+1}=\underset{(k+1)\in R_{k}}{\text{arg min}}\,Tr\bigg{[}\big{(}\mathbf{ \hat{H}}_{k+1}\mathbf{\hat{H}}_{k+1}^{H}\big{)}^{-1}\bigg{]} \tag{20}\]  
where \(\mathbf{\hat{H}}_{k+1}=\left[\mathbf{H}_{k+1}\ \mathbf{Q}_{2K}^{T}\mathbf{H}_{k+1}^{ *}\right]\), \(S_{k+1}\) indicates the TAS determined after the \((k+1)\)-th selection procedure, and \(R_{k}\) denotes the TAS unselected after the \(k\)-th iteration. In each iteration, the computation of the matrix product \(\mathbf{\hat{H}}_{k+1}\mathbf{\hat{H}}_{k+1}^{H}\) and the matrix inversion in \(\left(\mathbf{\hat{H}}_{k+1}\mathbf{\hat{H}}_{k+1}^{H}\right)^{-1}\) requires an expensive computational load. To reduce the computational complexity further, we adopt the Woodbury formula [26] written as  
\[\left(\mathbf{A}+\mathbf{B}\mathbf{C}^{H}\right)^{-1}=\mathbf{A}^{-1}- \mathbf{A}^{-1}\mathbf{B}\Big{(}\mathbf{I}+\mathbf{C}^{H}\mathbf{A}^{-1} \mathbf{B}\Big{)}^{-1}\mathbf{C}^{H}\mathbf{A}^{-1} \tag{21}\]  
Then, \(\left(\mathbf{\hat{H}}_{k+1}\mathbf{\hat{H}}_{k+1}^{H}\right)^{-1}\) can be re-expressed as  
\[ \begin{array}{rl}\left(\mathbf{\hat{H}}_{k+1}\mathbf{\hat{H}}_{k+1}^{H} \right)^{-1}&=\left(\mathbf{\hat{H}}_{k}\mathbf{\hat{H}}_{k}^{H}+\mathbf{ \hat{h}}_{k+1}\mathbf{\hat{H}}_{k+1}^{H}\right)^{-1}\\ &=\left(\mathbf{\hat{H}}_{k}\mathbf{\hat{H}}_{k}^{H}\right)^{-1}-\left( \mathbf{\hat{H}}_{k}\mathbf{\hat{H}}_{k}^{H}\right)^{-1}\mathbf{\hat{h}}_{k+1} \bigg{(}\mathbf{I}_{2}+\mathbf{\hat{h}}_{k+1}^{H}\left(\mathbf{\hat{H}}_{k} \mathbf{\hat{H}}_{k}^{H}\right)^{-1}\mathbf{\hat{h}}_{k+1}\bigg{)}^{-1} \mathbf{\hat{h}}_{k+1}^{H}\left(\mathbf{\hat{H}}_{k}\mathbf{\hat{H}}_{k}^{H} \right)^{-1}\end{array}  \tag{22}\]  
where \(\mathbf{\hat{h}}_{k+1}=\left[\mathbf{h}_{k+1}\ \mathbf{Q}_{2K}^{T}\mathbf{h}_{k+1}^{ *}\right]\). Thus, by defining \(\mathbf{\hat{V}}_{k}=\left(\mathbf{\hat{H}}_{k}\mathbf{\hat{H}}_{k}^{H}\right) ^{-1}\), (20) can be written as  
\[S_{k+1}=\underset{(k+1)\in R_{k}}{\text{arg min}}\,Tr\bigg{[}\mathbf{\hat{V}}_{ k}-\mathbf{\hat{V}}_{k}\mathbf{\hat{h}}_{k+1}\Big{(}\mathbf{I}_{2}+\mathbf{\hat{h}}_{ k+1}^{H}\mathbf{\hat{V}}_{k}\mathbf{\hat{h}}_{k+1}\Big{)}^{-1}\mathbf{\hat{h}}_{k+1}^{H} \mathbf{\hat{V}}_{k}\bigg{]} \tag{23}\]  
Based on the above analysis, the procedure of the proposed SNR-based incremental TAS selection algorithm can be summarized in Algorithm 2. As an initial one of the matrix \(\mathbf{\hat{V}}_{k}\), \(\mathbf{\hat{V}}_{0}\) is employed with a \(2K\times 2K\) identity matrix of \(\mathbf{I}_{2K}\).