# Secure Image Signal Transmission Scheme Using Poly-Polarization Filtering and Orthogonal Matrix
## 5 Security Performance Evaluation
In this condition, the PF matrices of [PERSON] were written, as follows: \[ \begin{array}{c}\mathbf{Q}_{\text{E}x}{=}\mathbf{Q}_{x}+\mathbf{Q}_{x}^{ \text{e}}\\ \mathbf{Q}_{\text{E}y}{=}\mathbf{Q}_{y}+\mathbf{Q}_{y}^{\text{e}}\end{array} \] (27) where \(\mathbf{Q}_{x}^{\text{e}}\) and \(\mathbf{Q}_{y}^{\text{e}}\) denote the errors. Here, we assumed \(\left[ \begin{array}{cc}\mathbf{\Gamma}_{\text{ER}}^{\text{o}}&\mathbf{ \Gamma}_{\text{ER}}^{\text{e}}\end{array} \right]\)\(=\)\(\left[ \begin{array}{cc}\mathbf{\Gamma}^{\text{o}}&\mathbf{\Gamma}^{\text{e}}\end{array} \right]\), and the signal separation of the \(k\)-th row signal vector was taken as an example: \[ \begin{array}{c}\mathbf{\tilde{x}}_{\text{ER}k}{=}\mathbf{Q}_{\text{E}x} \left[\begin{array}{c}\mathbf{F}_{\text{ER}}^{\text{o}}(k\cdot)\\ \mathbf{F}_{\text{ER}}^{\text{e}}(k\cdot)\end{array}\right]=\left[\begin{array} []{c}\cos\gamma_{k}\\ \sin\gamma_{k}e^{\
icefrac{{\
icefrac{{\
icefrac{{\pi_{k}}}}{{2}}}}{{\
icefrac{{ \pi_{k}}}{{2}}}}}}\end{array}\right]\mathbf{x}_{k}\\ +\mathbf{J}_{\text{E}x}+\mathbf{Q}_{\text{E}x}\mathbf{n}_{\text{E}k}\end{array} \] (28) where \(\mathbf{J}_{\text{E}x}{=}\) \[\mathbf{Q}_{x}^{\text{e}}\left(\underbrace{\left[ \begin{array}{c}\cos \gamma_{k}\\ \sin\gamma_{k}e^{\
icefrac{{\
icefrac{{\pi_{k}}}{{2}}}}{{\
icefrac{{\pi_{k}}}{{2}}}}} \end{array} \right]}_{\mathbf{P}_{k}}\mathbf{x}_{k}+\underbrace{\left[ \begin{array} []{c}\cos\gamma_{k+\
icefrac{{\pi_{k}}}{{2}}}\\ \sin\gamma_{k+\
icefrac{{\pi_{k}}}{{2}}}e^{\
icefrac{{\
icefrac{{\pi_{k+1}}}{{2}}}} }\end{array} \right]}_{\mathbf{P}_{k}}\mathbf{y}_{k}\right)\]We found \(\mathbf{J}_{\text{Ex}}\
eq 0\), and after polarization matching, Equation (28) was written, as follows:  
\[ \begin{split}&\mathbf{x}_{\text{ER}k}{=}(\mathbf{P}_{k})^{\text{H}} \bar{\mathbf{x}}_{\text{ER}k}{=}\Bigg{[}\begin{array}{c}\mathbf{F}_{\text{ ER}}^{\text{o}}(k,:)\\ \mathbf{F}_{\text{ER}}^{\text{e}}(k,:)\end{array}\Bigg{]}=\\ &(\mathbf{P}_{k})^{\text{H}}\underbrace{\left[\begin{array}{c}\cos\gamma_{k }\\ \sin\gamma_{k}e^{\beta_{k}}\end{array}\right]}_{\mathbf{P}_{k}}\mathbf{x}_{k}+( \mathbf{P}_{k})^{\text{H}}(\mathbf{J}_{\text{Ex}}+\mathbf{Q}_{\text{Ex}} \mathbf{n}_{\text{E}k})\\ =&\mathbf{x}_{k}+\underbrace{(\mathbf{P}_{k})^{\text{H}}\mathbf{J}_{\text{Ex}} }_{\Omega_{\text{Ex}}}+(\mathbf{P}_{k})^{\text{H}}\mathbf{Q}_{\text{Ex}} \mathbf{n}_{\text{E}k}\end{split}  \tag{29}\]  
In the same manner, we found the following:  
\[\mathbf{y}_{\text{ER}k}{=}\mathbf{y}_{k}+\underbrace{\left(\mathbf{P}_{k}^{ \perp}\right)^{\text{H}}\mathbf{J}_{\text{Ey}}}_{\Omega_{\text{Ey}}}+\left( \mathbf{P}_{k}^{\perp}\right)^{\text{H}}\mathbf{Q}_{\text{Ey}}\mathbf{n}_{ \text{E}k} \tag{30}\]  
where \(\mathbf{J}_{\text{Ey}}{=}\mathbf{Q}_{y}^{\text{e}}(\mathbf{P}_{k}\mathbf{x}_ {k}+\mathbf{P}_{k}^{\perp}\mathbf{y}_{k})\). According to Equations (29) and (30), we found that there were interferences in recovered signals by [PERSON] that would degrade BER performance. Thus, for [PERSON] and [PERSON], the signal-to-noise ratio (SNR) was calculated as:  
\[ \begin{split}&\bar{\xi}_{\text{Bx},j}{=}\frac{l}{2u}\frac{\| \mathbf{x}_{j}\|_{2}}{\sigma},\xi_{\text{By},j}{=}\frac{l}{2u}\frac{\|\mathbf{ y}_{j}\|_{2}}{\sigma^{2}}\\ &\bar{\xi}_{\text{Ex},j}{=}\frac{\|\mathbf{x}_{j}\|_{2}}{\| \Omega_{\text{E}}\|_{2}+\frac{2u}{\sigma}^{2}},\xi_{\text{Ey},j}{=}\frac{\| \mathbf{y}_{j}\|_{2}}{\|\Omega_{\text{Ey}}\|_{2}+\frac{2u}{\sigma}^{2}}\end{split}  \tag{31}\]  
Then, the average secrecy rate with the proposed scheme could be calculated, as follows:  
\[ \begin{split} C_{\text{ave}}{=}\frac{\text{E}}{i}\Bigg{[}10\text {log}\Big{(}\frac{(1+\xi_{\text{Bx},j})\big{(}1+\xi_{\text{By},j}\big{)}}{(1+ \xi_{\text{Ex},j})\big{(}1+\xi_{\text{Ex},j})}\Big{)}\Bigg{]}\end{split}