# Efficient MIMO Signal Predistortion for Secrecy-Enhancing
## 3 The Proposed Methods
### Secrecy Enhancement for Spatial Multiplexing  
We consider a wiretap SM MIMO system with \(N_{T}\) transmit antennae at [PERSON] as well as \(N_{R}\) receive antennae at [PERSON] and [PERSON], respectively. In this system, Alice transmits spatially multiplexed \(N_{T}\times 1\) complex signal vector \(\mathbf{z}=\begin{bmatrix}z_{1}&z_{2}&\ldots&z_{N_{T}}\end{bmatrix}^{T}\) via the \(N_{T}\) antennae for each time period. Then, \(\mathbf{z}\) is transmitted through the \(N_{R}\times N_{T}\) channel matrices of \(\mathbf{H}=\begin{bmatrix}\mathbf{h}_{1}&\mathbf{h}_{2}&\ldots&\mathbf{h}_{N_{ T}}\end{bmatrix}\) and \(\mathbf{G}=\begin{bmatrix}\mathbf{g}_{1}&\mathbf{g}_{2}&\ldots&\mathbf{g}_{N_{ T}}\end{bmatrix}\) at Bob and Eve, respectively, where \(\mathbf{h}_{k}\) and \(\mathbf{g}_{k}\) respectively denote the \(k\)-th column vectors of \(\mathbf{H}\) and \(\mathbf{G}\). We propose a new PLS scheme for this environment.  
By considering that the \(k\)-th signal component of \(\mathbf{z}\), \(z_{k}\) will be transmitted through \(\mathbf{h}_{k}\) to Bob, the proposed PLS scheme applies phase distortion to \(z_{k}\) by using the sum of the channel phases of \(\mathbf{h}_{k}\). For this purpose, the phase distorting vector, \(\mathbf{q}\), is first found as follows:  
\[\mathbf{q}=\begin{bmatrix}\theta_{1}&\theta_{2}&\ldots&\theta_{N_{T}}\end{bmatrix} ^{T}, \tag{18}\]  
where  
\[\theta_{k}=e^{\left(-j\sum_{i=1}^{N_{R}}\angle{h_{k}}\right)}, \tag{19}\]  
and \(h_{k}\) and \(\angle{h_{k}}\) are the complex channel gain from the \(k\)-th transmit antenna to the \(i\)-th receive antenna and its phase, respectively.  
Next, the pre-distorted signal vector at Alice, \(\mathbf{z}^{\prime}\), can be found as follows:  
\[\begin{split}\mathbf{z}^{\prime}&=\text{diag}(\mathbf{z}) \mathbf{q}\\ &=\begin{bmatrix}z_{1}&0&\ldots&0\\ 0&z_{2}&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&z_{N_{T}}\end{bmatrix}\begin{bmatrix}\theta_{1}\\ \theta_{2}\\ \vdots\\ \theta_{N_{T}}\end{bmatrix}\\ &=\begin{bmatrix}\theta_{1}z_{1}&\theta_{2}z_{2}&\ldots&\theta_{N_{T}}z_{N_{ T}}\end{bmatrix}^{T}.\end{split} \tag{20}\]  
Then, the received signal at Bob can be represented by:  
\[\begin{split}\mathbf{y_{b}}&=\mathbf{H}\mathbf{z}^{\prime}+ \mathbf{n_{b}}\\ &=\begin{bmatrix}\mathbf{h}_{1}&\mathbf{h}_{2}&\ldots&\mathbf{h}_{N_{T}} \end{bmatrix}\begin{bmatrix}\theta_{1}z_{1}\\ \theta_{2}z_{2}\\ \vdots\\ \theta_{N_{T}}z_{N_{T}}\end{bmatrix}+\mathbf{n_{b}}\\ &=\begin{bmatrix}\theta_{1}z_{1}\mathbf{h}_{1}+\ldots+\theta_{N_{T}}z_{N_{ T}}\mathbf{h}_{N_{T}}\end{bmatrix}+\mathbf{n_{b}}\\ &=\mathbf{H}^{\prime}\mathbf{z}+\mathbf{n_{b}},\end{split} \tag{21}\]where the resulting channel matrix can be represented by:  
\[\mathbf{H^{\prime}}=\big{[}\theta_{1}\mathbf{h}_{1}\quad\theta_{2}\mathbf{h}_{2} \quad\ldots\quad\theta_{N_{T}}\mathbf{h}_{N_{T}}\big{]}. \tag{22}\]  
Therefore, the signal detection can be achieved by using various well known detection methods, as follows:  
\[\mathbf{\hat{z}_{b}}=\mathbf{F}\mathbf{y}_{b}, \tag{23}\]  
where \(\mathbf{F}\) is a detection matrix, e.g., \(\mathbf{F}=(\mathbf{H^{\prime}}^{H}\mathbf{H^{\prime}})^{-1}\mathbf{H^{\prime }}^{H}\) for zero forcing (ZF) detection and \(\mathbf{F}=(\mathbf{H^{\prime}}^{H}\mathbf{H^{\prime}}+\sigma^{2}\mathbf{I})^ {-1}\mathbf{H^{\prime}}^{H}\) for minimum mean square error (MMSE) detection, where \(\sigma^{2}\) is the noise variance and \(\mathbf{I}\) is the \(N_{T}\times N_{T}\) identity matrix. On the other hand, the received signal at [PERSON] is represented by:  
\[\mathbf{y_{e}} =\mathbf{G}\mathbf{z^{\prime}}+\mathbf{n_{e}} \tag{24}\] \[=\big{[}\theta_{1}z_{1}\mathbf{g}_{1}+\ldots+\theta_{N_{T}}z_{N_ {T}}\mathbf{g}_{N_{T}}\big{]}+\mathbf{n_{e}}.\]  
As in the previous case for STBC schemes, \(\mathbf{z^{\prime}}\) contains \(\mathbf{H}\)-dependent distortion, and as a result, [PERSON] cannot retrieve any information on the signal.