# Efficient Transmit Antenna Subset Selection for Multiuser Space-Time Line Code Systems
## 3 TAS Selection Algorithms
### Proposed Decremental TAS Selection Algorithm
\begin{array}{rl}J_{\overline{S}}&=J_{S^{\prime}}-J_{S}=2K\sigma_{z}^{2}Tr \big{(}\tilde{\mathbf{V}}_{S^{\prime}}-\tilde{\mathbf{V}}_{S}\big{)}\\ &=2K\sigma_{z}^{2}Tr\Big{(}\tilde{\mathbf{V}}_{S^{\prime}}-\tilde{\mathbf{H}}_ {S^{\prime}}^{H}-\tilde{\mathbf{H}}_{\overline{S}}\tilde{\mathbf{H}}_{ \overline{S}}^{H}\Big{)}^{-1}\Big{)}\\ &=-2K\sigma_{z}^{2}Tr\Big{(}\tilde{\mathbf{V}}_{S^{\prime}}\tilde{\mathbf{H}}_ {\overline{S}}\Big{(}\mathbf{1}_{2K}-\tilde{\mathbf{H}}_{\overline{S}}^{H} \tilde{\mathbf{V}}_{S^{\prime}}\tilde{\mathbf{H}}_{\overline{S}}\Big{)}^{-1} \tilde{\mathbf{H}}_{\overline{S}}^{H}\tilde{\mathbf{V}}_{S^{\prime}}\Big{)} \end{array}  \tag{40}\]  
where  
\[\tilde{\mathbf{V}}_{S^{\prime}}=\left(\tilde{\mathbf{H}}_{S^{\prime}}\tilde{ \mathbf{H}}_{S^{\prime}}^{H}+\frac{2K\sigma_{z}^{2}}{\sigma_{x}^{2}}\mathbf{ I}_{2K}\right)^{-1} \tag{41}\]  
\[\tilde{\mathbf{V}}_{\overline{S}}=\left(\tilde{\mathbf{H}}_{\overline{S}} \tilde{\mathbf{H}}_{\overline{S}}^{H}+\frac{2K\sigma_{z}^{2}}{\sigma_{x}^{2} }\mathbf{I}_{2K}\right)^{-1} \tag{42}\]  
\[\tilde{\mathbf{H}}_{S^{\prime}}\triangleq\left[\mathbf{H}_{S^{\prime}}\ \mathbf{Q}_{2K}^{T}\mathbf{H}_{S^{\prime}}^{*}\right] \tag{43}\]  
\[\tilde{\mathbf{H}}_{\overline{S}}\triangleq\left[\mathbf{H}_{\overline{S}}\ \mathbf{Q}_{2K}^{T}\mathbf{H}_{\overline{S}}^{*}\right] \tag{44}\]  
Since \(Tr\Big{(}\tilde{\mathbf{V}}_{S^{\prime}}\tilde{\mathbf{H}}_{\overline{S}} \Big{(}\mathbf{1}_{2K}-\tilde{\mathbf{H}}_{\overline{S}}^{H}\tilde{\mathbf{ V}}_{S^{\prime}}\tilde{\mathbf{H}}_{\overline{S}}\Big{)}^{-1}\tilde{ \mathbf{H}}_{\overline{S}}^{H}\tilde{\mathbf{V}}_{S^{\prime}}\Big{)}>0\), \(J_{\overline{S}}\) of (40) becomes negative. Thus, it can be concluded that when the transmit power constraint and the number \(N_{T}\) of transmit antennas are fixed, the MSE is monotonically decreasing with the number \(N_{S}\) of active transmit antennas in MMSE-precoded MU-STLC systems. Then, the optimal TAS selection algorithm based on the MSE criterion for the MMSE-precoded MU-STLC system can be formulated as [14]. \[ \begin{array}{rl}S_{opt}=\text{arg min}&S\in\ \{S_{n},n=1,2,\cdots,C\left(N_{T },N_{S}\right)\}&|J_{\overline{S}}|\\ &\overline{S}=S^{\prime}-S\end{array}  \tag{45}\]  
which is the same as that for the ZF-precoded MU-STLC system, except for \(\tilde{\mathbf{V}}_{S^{\prime}}\) given as (41). Therefore, for MSE-based efficient decremental TAS selection in MMSE-precoded MU-STLC systems, line 3 in Algorithm 4 is replaced with the following:  
\[\tilde{\mathbf{V}}_{N_{T}}=\bigg{(}\tilde{\mathbf{H}}_{S^{\prime}}\tilde{ \mathbf{H}}_{S^{\prime}}^{H}+\frac{2K\sigma_{z}^{2}}{\sigma_{x}^{2}}\mathbf{ I}_{2K}\bigg{)}^{-1} \tag{46}\]  
On the other hand, the received SINR penalty in MMSE-precoded MU-STLC systems can be defined as  
\[P_{MMSE,S}\triangleq\frac{\gamma_{S^{\prime}}}{\gamma_{S}} \tag{47}\]  
where the received SINRs for two TASs of \(S^{\prime}\) and \(S\) in MMSE-precoded MU-STLC systems are given as [14]. \[\gamma_{S^{\prime}}=\frac{\sigma_{x}^{2}}{\sigma_{z}^{2}Tr\big{(}\tilde{ \mathbf{V}}_{S^{\prime}}\big{)}}-1 \tag{48}\]  
\[\gamma_{S}=\frac{\sigma_{x}^{2}}{\sigma_{z}^{2}Tr\big{(}\tilde{\mathbf{V}}_{S} \big{)}}-1 \tag{49}\]The asymptotic received SINR penalty for large values of SNR in the MMSE- precoded MU-STLC systems can be obtained as  
\[