# Efficient MIMO Signal Predistortion for Secrecy-Enhancing
## 3 The Proposed Methods  
### Secrecy Enhancement for STBC Schemes  
Motivated by the fact that a full-rate O-STBC scheme can be achieved for real-numbered signal transmission through the use of more than two transmit antennae [12], we propose an efficient PLS scheme for STBC. In the proposed scheme, we first form a non-orthogonal full-rate encoding matrix for complex-numbered signals by inserting complex signals into \(\mathbf{X}_{N_{T}}\). Then, the proposed scheme transforms the encoding matrix using a phase distortion matrix which represents the null space of the phase of the channel gain. During the transmission, the channel phase will be nulled out, and eventually only the amplitude component of the channel gain will arrive at the receiver. Therefore, we can have a full orthogonal real-numbered channel matrix at the receiver. Even though the encoding matrix with complex numbered signals is not orthogonal, the proposed scheme can achieve performance equivalent to that of the scheme with full-rate and full-diversity. It can also achieve high secrecy protection. Here, we detail the operational principle of the proposed scheme. At the transmitter, [PERSON] first generates a complex signal vector \(\mathbf{z}=\left[z_{1},\dots,z_{p}\right]^{T}\) for \(p\) time periods and constructs a non-orthogonal encoding matrix \(\mathbf{Z}_{N_{T}}\) by inserting complex signals into \(\mathbf{X}_{N_{T}}\). For example, the following \(\mathbf{Z}_{4}\) can be found for a system with four transmit antennae. \[\mathbf{Z}_{4}= \begin{bmatrix}z_{1}&z_{2}&z_{3}&z_{4}\\ -z_{2}&z_{1}&-z_{4}&z_{3}\\ -z_{3}&z_{4}&z_{1}&-z_{2}\\ -z_{4}&-z_{3}&z_{2}&z_{1}\end{bmatrix} . \tag{10}\]Note that \(\mathbf{Z}_{N_{T}}\) keeps full-rate but is not orthogonal. Next, the transmitting signals are predistorted so as to null out the phase of the channel gain. To this end, we form a phase-distorting vector, \(\mathbf{p}\), as follows:  
\[\mathbf{p}= \begin{bmatrix}e^{-j\angle h_{1}}&e^{-j\angle h_{2}}&\ldots&e^{-j \angle h_{N_{T}}}\end{bmatrix} ^{T}, \tag{11}\]  
where \(\angle h_{k}\) represents the phase of \(h_{k}\). The next step is to determine the encoding matrix with the predistortion capability, \(\mathbf{Z}_{N_{T}}^{\prime}\), as follows:  
\[ \begin{split}&\mathbf{Z}_{N_{T}}^{\prime}=\mathbf{Z}_{N_{T}} \text{diag}(\mathbf{p})\\ &=\begin{bmatrix}\mathbf{z}_{1}&\mathbf{z}_{2}&\ldots&\mathbf{z}_{N_{T}} \end{bmatrix}\begin{bmatrix}e^{-j\angle h_{1}}&0&\ldots&0\\ 0&e^{-j\angle h_{2}}&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&e^{-j\angle h_{N_{T}}}\end{bmatrix}\\ &=\begin{bmatrix}e^{-j\angle h_{1}}\mathbf{z}_{1}&e^{-j\angle h_{2}}\mathbf{z }_{2}&\ldots&e^{-j\angle h_{N_{T}}}\mathbf{z}_{N_{T}}\end{bmatrix},\end{split}  \tag{12}\]  
where \(\mathbf{z}_{k}\) is the \(k\)-th column vector of \(\mathbf{Z}_{N_{T}}\). Note that multiplying \(e^{-j\angle h_{k}}\) to \(\mathbf{z}_{k}\) will lead to nulling of the phase of the complex channel gain for \(\mathbf{z}_{k}\) at Bob. Finally, the received signal vector at Bob for \(p\) time periods can be represented by:  
\[ \begin{split}&\mathbf{y}_{\text{b}}=\mathbf{Z}_{N_{T}}^{\prime} \mathbf{h}+\mathbf{n}_{\text{b}}\\ &=\begin{bmatrix}e^{-j\angle h_{1}}\mathbf{z}_{1}&e^{-j\angle h_{2}} \mathbf{z}_{2}&\ldots&e^{-j\angle h_{T}}\mathbf{z}_{N_{T}}\end{bmatrix} \begin{bmatrix}h_{1}\\ h_{2}\\ \vdots\\ h_{N_{T}}\end{bmatrix}+\mathbf{n}_{\text{b}}\\ &=\begin{bmatrix}||h_{1}||\mathbf{z}_{1}+||h_{2}||\mathbf{z}_{2}+\ldots+||h_{ N_{T}}||\mathbf{z}_{N_{T}}\end{bmatrix}+\mathbf{n}_{\text{b}},\end{split}  \tag{13}\]  
because \(h_{k}=||h_{k}||e^{j\angle h_{k}}\) and \(||h_{k}||e^{j\angle h_{k}}e^{-j\angle h_{k}}\mathbf{z}_{k}=||h_{k}||\mathbf{z }_{k}\). Further, \(\mathbf{y}_{\text{b}}\) can be equivalently expressed by using a channel matrix as follows:  
\[\mathbf{y}_{\text{b}}=\mathbf{H}_{\text{R},N_{T}}\mathbf{z}+\mathbf{n}_{\text {b}}, \tag{14}\]  
where the channel matrix \(\mathbf{H}_{\text{R},N_{T}}\) results in a real-numbered orthogonal matrix.