# Efficient MIMO Signal Predistortion for Secrecy-Enhancing
## 2 Related Works  
### Adding AN to STBC MIMO Schemes for Secrecy Enhancing  
We consider a multi-input-single-output (MISO) wiretap channel, which includes a transmitter [PERSON] with \(N_{T}\) transmit antennae, a legitimate receiver [PERSON], and a passive eavesdropper [PERSON]. Let \(\mathbf{h}=\begin{bmatrix}h_{1}&h_{2}&\ldots&h_{N_{T}}\end{bmatrix}^{T}\) and \(\mathbf{g}=\begin{bmatrix}g_{1}&g_{2}&\ldots&g_{N_{T}}\end{bmatrix}^{T}\) respectively denote the channel state information (CSI) vectors of [PERSON] and [PERSON], where \(h_{k}\) and \(g_{k}\) represent the complex channel gains for the \(k\)-th transmit antenna from [PERSON] to [PERSON] and [PERSON], respectively. The perfect CSI is assumed to be available at both [PERSON] and [PERSON], but [PERSON] only shares the CSI with [PERSON]. It is also assumed that the channel states of \(h_{k}\) and \(g_{k}\) are independent of each other. For the purpose of employing AN [7], the \(2\times 1\) Alamouti scheme is adopted with the following encoding matrix, \(\mathbf{Z}_{A}\)[13],\[\mathbf{Z}_{\mathrm{A}}=\begin{bmatrix}z_{1}&z_{2}\\ -{z_{2}}^{*}&{z_{1}}^{*}\end{bmatrix}, \tag{1}\]  
where \(z_{1}\) and \(z_{2}\) are the signals transmitted during the two symbol periods. Prior to transmission, [PERSON] adds the following AN, \(\mathbf{W}\), to \(\mathbf{Z}_{\mathrm{A}}\):  
\[\mathbf{W}=\begin{bmatrix}\alpha_{11}\
u&\alpha_{12}\
u\\ -(\alpha_{21}\
u)^{*}&(\alpha_{22}\
u)^{*}\end{bmatrix}, \tag{2}\]  
where \(\
u\) is the complex Gaussian AN with zero-mean and unit variance and \(\alpha_{ik}\)s are the coefficients of the \(\
u\) for the \(i\)-th time slot at the \(k\)-th antenna, which are designed to be nulled at Bob. Therefore, one of the possible coefficients sets can be \(\alpha_{11}=h_{2}\), \(\alpha_{12}=-h_{1}\), \(\alpha_{21}={h_{2}}^{*}\), and \(\alpha_{22}={h_{1}}^{*}\)[7].  
Then, [PERSON] transmits \((\mathbf{Z}_{\mathrm{A}}+\mathbf{W})\) and the received signal vectors at [PERSON] and [PERSON] during the two symbol periods, \(\mathbf{y}_{\mathrm{b}}\) and \(\mathbf{y}_{\mathrm{e}}\) can, respectively, be expressed by:  
\[\mathbf{y}_{\mathrm{b}} =(\mathbf{Z}_{\mathrm{A}}+\mathbf{W})\mathbf{h}+\mathbf{n}_{ \mathrm{b}}, \tag{3}\] \[=\mathbf{Z}_{\mathrm{A}}\mathbf{h}+\mathbf{n}_{\mathrm{b}},\] \[\mathbf{y}_{\mathrm{e}} =(\mathbf{Z}_{\mathrm{A}}+\mathbf{W})\mathbf{g}+\mathbf{n}_{ \mathrm{e}},\]  
where \(\mathbf{n}_{\mathrm{b}}\) and \(\mathbf{n}_{\mathrm{e}}\) are additive white Gaussian noise (AWGN) vectors at [PERSON] and [PERSON], respectively. Since the AN is designed to be nulled out at [PERSON], [PERSON] received \(\mathbf{y}_{\mathrm{b}}\) as if [PERSON] transmitted without any AN. Meanwhile, since the channel gains \(h_{k}\) and \(g_{k}\) are independent of each other, [PERSON] cannot retrieve any signal due to uncancelled \(\mathbf{W}\).  
Because this method requires additional energy for AN, a new method was proposed that does not require any additional energy for secrecy protection. The idea of this proposed method is to utilize the quasi-orthogonal STBC (QO-STBC) scheme with linear detection capability, called linear decoding QO-STBC (LD-QO-STBC) [15], and to re-allocate the signal power across the transmission time and antennae [9].