# Waveform Design for Space-Time Coded MIMO Systems with High Secrecy Protection +
## 2 Related Works  
### Alamouti Coded PLS with AN  
The generic model for the PLS scheme, comprises a cooperative wireless network that consists of three nodes. Suppose that a legitimate transmitter node is referred to as Alice, which is the source node. The corresponding destination node is referred to as Bob, which is the legitimate receiver node. On the other hand, the third node, named Eve, is the passive eavesdropper node. Figure 1 shows a block diagram for the Alamouti-coded PLS with AN [10]. In this system, it is assumed that Alice is equipped with three antennae, while Bob and Eve are equipped with only one antenna, respectively. At the transmitter, the bit stream emitted by a binary source is divided into a block containing one bit to select one antenna among Ant 1 and Ant 2 for SM, and the other blocks containing information bits to transmit. The selected antenna will form the first virtual antenna for the Alamouti scheme, while Ant 3 in Figure 1 will form the second virtual antenna for the \(2\times 1\) Alamouti scheme. Therefore, the first virtual antenna is dynamically changed by time. During the first symbol period, \(s_{1}\) and \(s_{2}\) which are mapped by using \(m\) bits for each will be transmitted through two antennae, respectively. In addition, AN will be injected for secrecy purpose. During the second symbol period, the Alamouti encoded symbols, \(-s_{2}^{*}\) and \(s_{1}^{*}\), are transmitted along with ANs through two antennae, respectively. It is assumed that the channel coefficients are constant within each transmission block, i.e., two symbol periods, and they are changed independently from block to block, as in the conventional  
Figure 1: System model of the Alamouti-coded physical layer security (PLS) with artificial noise (AN). Alamouti scheme. The perfect channel state information (CSI) is assumed to be available at Bob and Eve, but it may not be available at Alice. The encoding matrix for the Alamouti code with AN, \(\mathbf{X}_{\text{AN}}\) can be expressed as:  
\[\mathbf{X}_{\text{AN}}=\left[ \begin{array}{cc}s_{1}+\beta_{11}v&s_{2}+\beta_{ 12}v\\ -\left(s_{2}+\beta_{21}v\right)^{*}&\left(s_{1}+\beta_{22}v\right)^{*}\end{array}  \right], \tag{1}\]  
where \(s_{1}\) and \(s_{2}\) are modulation symbols used for the Alamouti code during two symbol periods, and they are drawn from the \(M\)-ary quadrature amplitude modulation (QAM) constellation, where \(M=2^{m}\). In addition, \(v\) is the complex Gaussian AN with zero-mean and unit variance, and \(\beta_{ij}\)s are the coefficients of the \(v\) for the \(i\)-th timeslot at the \(j\)-th antenna. The coefficients \(\beta_{ij}\)s are designed to be nulled at the legitimate receiver. Therefore they should be computed by exploiting the CSI of the legitimate channel, which results in the following two possible sets of solutions for each symbol period, i.e., for the first time period:  
\[\beta_{11}=h_{2},\ \beta_{12}=-h_{1},\ \text{or}\ \beta_{11}=-h_{2},\ \beta_{12}=h_{1}, \tag{2}\]  
and for the second time period:  
\[\beta_{21}=h_{2}^{*},\ \beta_{22}=h_{1}^{*},\ \text{or}\ \beta_{21}=-h_{2}^{*},\ \beta_{22}=-h_{1}^{*}, \tag{3}\]  
where, \(h_{1}\) and \(h_{2}\) are the channel gains from the first and second virtual antennae at Alice to Bob, respectively. With the above solution, the received signal at Bob during two symbol periods, \(y_{\text{b1}}\) and \(y_{\text{b2}}\) can be represented by:  
\[y_{\text{b1}} = h_{1}s_{1}+h_{2}s_{2}+n_{\text{b1}}, \tag{4}\] \[y_{\text{b2}} = -h_{1}s_{2}^{*}+h_{2}s_{1}^{*}+n_{\text{b2}},\]  
where \(n_{\text{b1}}\) and \(n_{\text{b2}}\) are additive white Gaussian noise (AWGN) at Bob for the first and second symbol periods, respectively. Because the added ANs are perfectly canceled due to (2) and (3) at Bob, the detection can be successfully made as in the conventional \(2\times 1\) Alamouti scheme.