# Efficient Transmit Antenna Subset Selection for Multiuser Space-Time Line Code Systems
## 3 TAS Selection Algorithms
### Proposed Decremental TAS Selection Algorithm  
In the ZF-precoded MU-STLC systems, the received SNR of (17) can be also employed as a design criterion for the proposed decremental TAS selection optimization. In order to see how TAS selection affects the SNR, let \(S\) and \(S^{\prime}\) be two TASs in the ZF-precoded MU-STLC systems, where \(S\subset S^{\prime}\subseteq\{\,1,\,2,\,\cdots,N_{T}\}\). Let \(\overline{S}=S^{\prime}-S\) and \(\mathbf{H}_{S^{\prime}}=\left[\,\mathbf{H}_{S}\,\mathbf{H}_{\overline{S}}\right]\). Then, the expression of \(\mathbf{\hat{H}}_{S^{\prime}}\) can be written as  
\[\mathbf{\hat{H}}_{S^{\prime}}=\left[\mathbf{H}_{S^{\prime}}\,\,\mathbf{Q}_{2K} ^{T}\mathbf{H}_{S^{\prime}}^{*}\right]=\left[\mathbf{H}_{S}\,\,\mathbf{H}_{ \overline{S}}\,\,\mathbf{Q}_{2K}^{T}\mathbf{H}_{\overline{S}}^{*}\right] \tag{24}\]  
Thus, it is easily shown that \(\mathbf{\hat{H}}_{S^{\prime}}\mathbf{\hat{H}}_{S^{\prime}}^{H}=\mathbf{\hat{ H}}_{S}\mathbf{\hat{H}}_{S}^{H}+\mathbf{\hat{H}}_{\overline{S}}\mathbf{\hat{H}}_{S}^{H}\), where \(\mathbf{\hat{H}}_{\overline{S}}\triangleq\left[\mathbf{H}_{\overline{S}}\, \,\mathbf{Q}_{2K}^{T}\mathbf{H}_{\overline{S}}^{*}\right]\in C^{2K\times 2N_{S}}\). Now we have  
\[\left(\mathbf{\hat{H}}_{S}\mathbf{\hat{H}}_{S}^{H}\right)^{-1}=\left(\mathbf{ \hat{H}}_{S^{\prime}}\mathbf{\hat{H}}_{S^{\prime}}^{H}-\mathbf{\hat{H}}_{ \overline{S}}\mathbf{\hat{H}}_{\overline{S}}^{H}\right)^{-1} \tag{25}\]  
Then, according to the Woodbury formula [26] written as  
\[\left(\mathbf{A}-\mathbf{B}\mathbf{C}^{H}\right)^{-1}=\mathbf{A}^{-1}+\mathbf{ A}^{-1}\mathbf{B}\left(\mathbf{I}-\mathbf{C}^{H}\mathbf{A}^{-1}\mathbf{B}\right)^{-1} \mathbf{C}^{H}\mathbf{A}^{-1} \tag{26}\]  
and the inverse of \(\mathbf{\hat{H}}_{S}\mathbf{\hat{H}}_{S}^{H}\) can be calculated as  
\[\left(\mathbf{\hat{H}}_{S}\mathbf{\hat{H}}_{S}^{H}\right)^{-1}=\left(\mathbf{ \hat{H}}_{S^{\prime}}\mathbf{\hat{H}}_{S^{\prime}}^{H}\right)^{-1}+\left( \mathbf{\hat{H}}_{S^{\prime}}\mathbf{\hat{H}}_{S^{\prime}}^{H}\right)^{-1} \mathbf{\hat{H}}_{\overline{S}}\left(\mathbf{I}-\mathbf{\hat{H}}_{S}^{H}( \mathbf{\hat{H}}_{S}\mathbf{\hat{H}}_{S^{\prime}}^{H})^{-1}\mathbf{\hat{H}}_{ \overline{S}}\right)^{-1}\mathbf{\hat{H}}_{S}^{H}(\mathbf{\hat{H}}_{S^{\prime}} \mathbf{\hat{H}}_{S}^{H})^{-1} \tag{27}\]  
Using (27), \(\widehat{\beta}_{S}^{2}\) of (18) can be rewritten as  
\[\widehat{\beta}_{S}^{2}=\frac{1}{Tr\left[\left(\mathbf{\hat{H}}_{S^{\prime}} \mathbf{\hat{H}}_{S^{\prime}}^{H}\right)^{-1}\right]+Tr\left(\mathbf{\hat{V}} _{\overline{S}}\right)} \tag{28}\]  
where  
\[\mathbf{\hat{V}}_{\overline{S}}=\mathbf{\hat{V}}_{ZF,S^{\prime}}\mathbf{\hat{H}}_ {\overline{S}}\left(\mathbf{I}-\mathbf{\hat{H}}_{\overline{S}}^{H}\mathbf{ \hat{V}}_{ZF,S^{\prime}}\mathbf{\hat{H}}_{\overline{S}}\right)^{-1}\mathbf{\hat{H} }_{\overline{S}}^{H}\mathbf{\hat{V}}_{ZF,S^{\prime}} \tag{29}\]  
then the corresponding remained channel submatrix is denoted by \(\mathbf{H}_{N_{T}-k}\in C^{2K\times(N_{T}-k)}\), where \(1\leq k\leq(N_{T}-N_{S})\), and can be expressed as  
\[\mathbf{H}_{N_{T}-k}=\big{[}\,\mathbf{H}_{N_{T}-k-1}\mathbf{h}_{k}\big{]} \tag{34}\]  
where \(\mathbf{h}_{k}\) is the column vector of \(\mathbf{H}_{N_{T}-k}\) corresponding to the \((k+1)\)-th deleted antenna.