# Efficient Transmit Antenna Subset Selection for Multiuser Space-Time Line Code Systems
## 3 TAS Selection Algorithms
### Proposed Decremental TAS Selection Algorithm
Then, based on (33), the proposed ZF-based efficient decremental TAS selection problem can be rewritten as  
\[S_{k}=\underset{S_{k}}{\arg\min}\,Tr\bigg{[}\hat{\mathbf{V}}_{N_{T}-k}\tilde{ \mathbf{H}}_{k}\Big{(}\mathbf{I}_{2}-\tilde{\mathbf{H}}_{k}^{H}\tilde{\mathbf{ V}}_{N_{T}-k}\tilde{\mathbf{H}}_{k}\Big{)}^{-1}\tilde{\mathbf{H}}_{k}^{H}\tilde{ \mathbf{V}}_{N_{T}-k}\bigg{]} \tag{35}\]  
where \(S_{k}\) denotes the antenna subset remained at the \(k\)-th decremental step and  
\[\tilde{\mathbf{V}}_{N_{T}-k}=\Big{(}\mathbf{H}_{N_{T}-k}\tilde{\mathbf{H}}_{ N_{T}-k}^{H}\Big{)}^{-1} \tag{36}\]  
\[\tilde{\mathbf{H}}_{N_{T}-k}\triangleq\Big{[}\mathbf{H}_{N_{T}-k}\ \mathbf{Q}_{2K}^{T}\mathbf{H}_{N_{T}-k}^{*}\Big{]}=\Big{[}\mathbf{H}_{N_{T}-k- 1}\,\mathbf{h}_{k}\ \ \mathbf{Q}_{2K}^{T}\mathbf{H}_{N_{T}-k-1}^{*}\ \ \mathbf{Q}_{2K}^{T}\mathbf{h}_{k}^{*}\Big{]} \tag{37}\]  
\[\left(\mathbf{\hat{H}}_{S}\mathbf{\hat{H}}_{S}^{H}\right)^{-1}=\left(\mathbf{ \hat{H}}_{S^{\prime}}\mathbf{\hat{H}}_{S^{\prime}}^{H}-\mathbf{\hat{H}}_{ \overline{S}}\mathbf{\hat{H}}_{\overline{S}}^{H}\right)^{-1} \tag{25}\]0  
In Algorithm 4, the computation of \(\tilde{\mathbf{V}}_{N_{T}}\) using matrix inverse operation is carried out only once in the beginning, and then the matrix \(\tilde{\mathbf{V}}_{N_{T}-k+2}\) is updated using \(\tilde{\mathbf{V}}_{N_{T}-k+1}\) obtained at the previous iteration. This procedure is different from Algorithm 3 and thus makes a contribution to have lower complexity. ```
1:Inputs: \(\mathbf{H},\mathbf{Q}_{2K},N_{T},N_{S}\)
2:\(\mathbf{H}_{Q}=\mathbf{Q}_{2K}^{T}\mathbf{H}^{*}\)
3:\(\mathbf{H}_{S^{\prime}}=\big{[}\mathbf{H}\ \mathbf{H}_{Q}\big{]}\)
4:\(\tilde{\mathbf{V}}_{N_{T}}=\Big{(}\ \tilde{\mathbf{H}}_{S^{\prime}}\tilde{ \mathbf{H}}_{S^{\prime}}^{H}\Big{)}^{-1}\)
5:for\(k=1,2,\cdots,N_{T}-N_{S}\)
6:for\(q=1,2,\cdots,(N_{T}-k+1)\)
7:\(\tilde{\mathbf{H}}_{k,q}=\Big{[}\mathbf{H}(;q)\ \mathbf{H}_{Q}(;q)\Big{]}\)
8:\(\mathbf{\Theta}_{k,q}=\tilde{\mathbf{V}}_{N_{T}-k+1}\tilde{\mathbf{H}}_{k,q}\)
9:\(\lambda_{k,q}=Tr\Big{[}\tilde{\Theta}_{k,q}\Big{(}\ \mathbf{I}_{2}-\tilde{\mathbf{H}}_{k,q}^{H}\tilde{\Theta}_{k,q}\Big{)}^{-1} \tilde{\Theta}_{k,q}^{H}\Big{]}\)
10:end
11:\(\tilde{q}=\underset{q}{\arg\min}\,\lambda_{k,q}\)
12:\(\tilde{\mathbf{V}}_{N_{T}-k+2}=\tilde{\mathbf{V}}_{N_{T}-k+1}+\tilde{\Theta}_ {k,q}\Big{(}\ \mathbf{I}_{2}-\tilde{\mathbf{H}}_{k,q}^{H}\tilde{\Theta}_{k,q}\Big{)}^{-1} \tilde{\Theta}_{k,q}^{H}\)
13:\(\mathbf{H}=\mathbf{H}(;[1:(\tilde{q}-1)(\tilde{q}+1):\text{end}])\)
14:\(\mathbf{H}_{Q}=\mathbf{H}_{Q}(;[1:(\tilde{q}-1)(\tilde{q}+1):\text{end}])\)
15:end
16:\(\mathbf{H}_{S}=\mathbf{H}\)
17:Output:\(\mathbf{H}_{S}\)
```  
**Algorithm 4** Proposed ZF-based efficient decremental TAS selection algorithm. In the MMSE-precoded MU-STLC systems, the mean square error (MSE) criterion can be adopted for the proposed efficient decremental TAS selection. For a channel matrix \(\mathbf{H}_{S}\), the MSE derived in [14] for MMSE-precoded MU-STLC systems is given as  
\[\left(\mathbf{\hat{H}}_{S}\mathbf{\hat{H}}_{S}^{H}\right)^{-1}=\left(\mathbf{ \hat{H}}_{S^{\prime}}\mathbf{\hat{H}}_{S^{\prime}}^{H}-\mathbf{\hat{H}}_{ \overline{S}}\mathbf{\hat{H}}_{\overline{S}}^{H}\right)^{-1} \tag{25}\]1  
Similar to the ZF-precoded MU-STLC systems, we want to see how TAS selection affects the MSE. By denoting \(S\) and \(S^{\prime}\) by two TASs to be satisfied with \(\{1,\,2,\,\cdots,N_{T}\}\) and \(\overline{S}=S^{\prime}-S\), the MSE difference between two sets of \(S^{\prime}\) and \(S\) in MMSE-precoded MU-STLC systems can be written as  
\[