# Efficient Transmit Antenna Subset Selection for Multiuser Space-Time Line Code Systems
## 2 System Model of MU-STLC with TAS Selection  
We consider a downlink \(N_{T}-\)by\(-2K\) MU-STLC system with \(N_{T}\) transmit antennas and \(K\) users. Each user has two receive antennas for STLC [9; 14; 15]. Thus, the total number of receive antennas is \(2K\). The transmitter is equipped with only \(N_{S}(K\leq N_{S}\leq N_{T})\) RF transmission units. Thus, it is assumed that \(N_{S}\) transmit antennas are selected from \(N_{T}\) antennas. Let \(x_{k,t}\) be the \(t\)-th transmitted symbol of the \(k\)-th user, with \(E[x_{k,t}x_{k,t}^{*}]=\sigma_{x}^{2}\). Then, the MU-STLC signal matrix is defined as  
\[\mathbf{U}=[\mathbf{u}_{1}\ \mathbf{u}_{2}]=\mathbf{W}_{S}\mathbf{X}\in C^{N_{S} \times 2} \tag{1}\]  
where \(\mathbf{u}_{t}=[u_{1,t}\,u_{2,t}\ \cdots\ u_{N_{S},t}]^{T}\in C^{N_{S}\times 1}\), \(t=1,2\), and \(\mathbf{X}=[\mathbf{X}_{1}\,\mathbf{X}_{2}\ \cdots\,\mathbf{X}_{K}]^{T}\in C^{2K\times 2}\) with  
\[\mathbf{X}_{k}=\left[ \begin{array}{cc}x_{k,1}&x_{k,2}\\ -x_{k,2}^{*}&x_{k,1}^{*}\end{array} \right]\in C^{2\times 2},\ k=1,2,\cdots,K \tag{2}\]  
and \(\mathbf{W}_{S}\in C^{N_{S}\times 2K}\) is the MU-STLC precoding matrix for all users such that \(\|\mathbf{W}_{S}\|_{F}^{2}=1\). The received signals with TAS selection are then represented as  
\[[\ \mathbf{r}_{1}\ \mathbf{r}_{2}]=\mathbf{H}_{S}\mathbf{U}+\mathbf{Z}\in C^{2K \times 2} \tag{3}\]  
where \(\mathbf{r}_{t}=[\ \mathbf{r}_{1,t}\,\mathbf{r}_{2,t}\ \cdots\ \mathbf{r}_{K,t}]^{T}\in C^{2K\times 1}\). Here, \(\mathbf{r}_{k,t}=[\ \mathbf{r}_{k,t}^{1}\,\mathbf{r}_{k,t}^{2}]^{T}\in C^{2\times 1}\) is the received signal vector where \(\mathbf{r}_{k,t}^{n}\) is the received signal at the \(n\)-th receive antenna of the \(k\)-th user at time \(t\). \(\mathbf{H}_{S}\in C^{2K\times N_{S}}\) denotes the channel submatrix obtained by selecting \(N_{S}\) columns from the full channel matrix \(\mathbf{H}=[\mathbf{h}_{1}\,\mathbf{h}_{2}\ \cdots\,\mathbf{h}_{N_{T}}]\in C^{2K\times N_{T}}\). Here, \(\mathbf{h}_{m}\), \(m=1,2,\cdots,N_{T}\), is a channel vector between the \(m\)-th transmit antenna and all users, which is static for \(t=1\) and \(t=2\), and whose elements are independent and identically distributed (i.i.d.) circularly symmetric complex Gaussian random variables with zero mean and unit variance. \(\mathbf{Z}=[\mathbf{Z}_{1}\,\mathbf{Z}_{2}\ \cdots\,\mathbf{Z}_{K}]^{T}\in C^{2K \times 2}\) with \(\mathbf{Z}_{k}=[\mathbf{z}_{1}\,\mathbf{z}_{2}]\in C^{2\times 2}\) and \(\mathbf{z}_{t}=[z_{1,t}\,z_{2,t}]^{T}\in C^{2\times 1}\)\(t=1,2\), is an independent and identically distributed (i.i.d.) additive white Gaussian noise (AWGN) matrix whose elements are the zero-mean circular complex white Gaussian noise component of variance of \(\sigma_{z}^{2}\). For the MU-STLC decoding, the received signal matrix of (3) is re-expressed in a linear form as [14]  
\[\mathbf{r}\triangleq\left[\mathbf{r}_{1}^{T}\,\mathbf{r}_{2}^{H}\right]^{T}= \left[ \begin{array}{c}\mathbf{H}_{5}\mathbf{W}_{S}\\ \left(\mathbf{H}_{5}\mathbf{W}_{S}\right)^{*}\mathbf{Q}_{2K}\end{array} \right] \mathbf{x}+\mathbf{z}\in C^{4K\times 1} \tag{4}\]  
where  
\[\mathbf{x}\triangleq\left[x_{1,1}\,x_{1,2}\,\cdots\,x_{k,1}\,x_{k,2}\,\cdots\, x_{K,1}\,x_{K,2}\right]^{T} \tag{5}\]  
\[\mathbf{Q}_{2K}=blkdiag\{\,\mathbf{Q}_{2},\cdots,\mathbf{Q}_{2}\}\in R^{2K \times 2K} \tag{6}\]  
\[\mathbf{Q}_{2}=\left[ \begin{array}{cc}0&-1\\ 1&0\end{array} \right] \tag{7}\]  
and \(\mathbf{z}\in C^{4K\times 1}\) is the AWGN vector with \(E[\mathbf{z}\mathbf{z}^{H}]=\sigma_{z}^{2}\mathbf{I}_{4K}\).