# Efficient Transmit Antenna Subset Selection for Multiuser Space-Time Line Code Systems
## 3 TAS Selection Algorithms
### Optimal Exhaustive Search-Based and SINR-Greedy-Based TAS Selection Algorithms  
It is easily shown that maximizing the received SINR of the MMSE-precoded MU-STLC systems is equivalent to maximizing the term \(\widehat{\beta}_{S}^{2}\). Hence, the TAS selection scheme for the MU-STLC systems can be expressed as  
\[S_{opt}=\operatorname*{arg\,max}_{S\in\{\,S_{n},n=1,2,\cdots,C(N_{T},N_{S})\,\}} \widehat{\beta}_{S}^{2} \tag{15}\]  
where \(S_{n}\) is the \(n\)-th enumeration of the set of all available TASs. Here, \(C(N_{T},N_{S})\) is the total number of combinations of selecting \(N_{S}\) transmit antennas out of \(N_{T}\) antennas. Then, the optimal TAS selection algorithm for the MMSE-precoded MU-STLC system is described as [14].  
\[S_{opt}=\operatorname*{arg\,min}_{S\in\,\{\,S_{n},n=1,2,\cdots,C(N_{T},N_{S})\,\} }Tr\big{[}\,\bar{\mathbf{V}}_{MMSE,S}\big{]} \tag{16}\]  
It should be pointed out that in the ZF precoding case, the previous works of [24; 25] use the minimization of \(Tr\Big{[}\big{(}\mathbf{H}_{S}\mathbf{H}_{S}^{H}\big{)}^{-1}\Big{]}\) for TAS selection in the PSM systems, whereas this work for the MU-STLC systems is based on the optimization of \(Tr\Big{[}\big{(}\bar{\mathbf{H}}_{S}\mathbf{\tilde{H}}_{S}^{H}\big{)}^{-1} \Big{]}\), where \(\bar{\mathbf{H}}_{S}\) is defined by (9). Thus, the TAS selection algorithms presented in [24; 25] are unsuitable for TAS selection in the MU-STLC systems. Due to the difference between two channel matrices of \(\mathbf{H}_{S}\in\mathbb{C}^{2K\times N_{S}}\) and \(\bar{\mathbf{H}}_{S}\in\mathbb{C}^{2K\times 2N_{S}}\), the Woodbury formula should be applied differently to the development process of the low-complexity algorithm, and thus, the succeeding efficient TAS selection algorithms proposed for the MU-STLC systems in this work are distinct from those in [24; 25]. Furthermore, note that most of the studies on antenna selection, including [16; 18; 23], are based on the channel \(\mathbf{H}_{S}\), not \(\bar{\mathbf{H}}_{S}\). That is why they are inappropriate for direct use in the MU-STLC systems.  
Obviously, the exhaustive search algorithm to solve (16) requires \(C(N_{T},N_{S})\) matrix inverse operations, whose computational complexity is tremendous, especially when the number of all possible TASs is large. Since the first effort in the MU-STLC systems is to reduce the complexity, an SINR-greedy-based TAS selection algorithm shown in Algorithm 1 has been proposed in [14].  
```
1:Inputs: \(\mathbf{H},\mathbf{Q}_{2K},N_{T},N_{S},K,\sigma_{x}^{2},\sigma_{z}^{2}\)
2:\(C_{n}=\left\|\,\mathbf{H}(:,n)\right\|_{F}\), \(n=1,2,\cdots,N_{T}\)
3:\([V,\,u]=\max\{\,C_{1},C_{2},\cdots,C_{N_{T}}\,\}\)
4:\(\mathbf{H}_{S}=\mathbf{H}(:,u)\)
5:\(\mathbf{H}_{0}=\mathbf{H}(:,1:(u-1)\,\,(u+1):\text{end})\)
6:for\(k=1,2,\cdots,N_{S}-1\)
7:for\(q=1,2,\cdots,(N_{T}-k)\)
8:\(\mathbf{H}_{temp}=\left[\mathbf{H}_{S}(:,1:k)\,\,\mathbf{H}_{k-1}(:,q)\right]\)
9:\(\mathbf{\bar{H}}_{temp}=\left[\mathbf{H}_{temp}\,\,\mathbf{Q}_{2K}^{T} \mathbf{H}_{temp}^{t}\right]\)
10:\(\lambda_{k,q}=Tr\Big{[}\,\bar{\mathbf{V}}_{k,q}\Big{]}\)
11:end
12:\(\hat{q}=\,\operatorname*{arg\,min}_{q}\lambda_{k,q}\)
13:\(\mathbf{H}_{S}(:,k+1)=\mathbf{H}_{k-1}(:,\hat{q})\)
14:\(\bar{\mathbf{H}}_{k}=\bar{\mathbf{H}}_{k-1}(:,[1:(\hat{q}-1)\,(\hat{q}+1): \text{end}])\)
15:end
```  
**Algorithm 1** SINR-greedy-based TAS selection algorithm.