# Secure Image Signal Transmission Scheme Using Poly-Polarization Filtering and Orthogonal Matrix
## 4 The Principle of the PPF-OM Scheme
### Signal Processing on the Receiver Side
As in Equation (12), both \(\mathbf{F}_{\text{R}}^{\text{o}}\) and \(\mathbf{F}_{\text{R}}^{\text{e}}\) resulted from the addition of two signals, which would be separated for demodulation. Here, the corresponding rows of \(\mathbf{F}_{\text{R}}^{\text{o}}\) and \(\mathbf{F}_{\text{R}}^{\text{e}}\) were transformed into pairs for polarization filtering. The \(k\)-th pair was taken as an example, which could be written, as follows:  
\[ \begin{split}&\left[\begin{array}{c}\mathbf{F}_{\text{R}}^{ \text{o}}(k,\cdot)\\ \mathbf{F}_{\text{R}}^{\text{e}}(k,\cdot)\end{array}\right]\\ &=\underbrace{\left[\begin{array}{c}\cos\gamma_{k}e^{j\eta_{k}} \\ \sin\gamma_{k}e^{j\eta_{k}}\end{array}\right]}_{\mathbf{P}_{k}}\mathbf{x}_{k}+ \underbrace{\left[\begin{array}{c}\cos\gamma_{k+l/2}\\ \sin\gamma_{k+l/2}e^{j\eta_{k+l/2}}\end{array}\right]}_{\mathbf{P}_{k}^{\text{ e}}}\mathbf{y}_{k}+\mathbf{n}_{k}\end{split}  \tag{19}\]  
We assumed two PSs were \(\mathbf{P}_{k}\) and \(\mathbf{P}_{k}^{\perp}\); then, the polarization filtering matrices were calculated, as in [20]:  
\[ \begin{split}\mathbf{Q}_{x}&=\!\mathbf{P}_{k}\Big{(}(\mathbf{P }_{k})^{\text{H}}\mathbf{P}_{k}^{\dagger}\mathbf{P}_{k}\Big{)}^{-1}(\mathbf{P} _{k})^{\text{H}}\mathbf{P}_{k}^{\dagger}\\ \mathbf{Q}_{y}&=\!\mathbf{P}_{k}^{\perp}\Big{(}(\mathbf{P}_{k}^{ \perp})^{\text{H}}\mathbf{P}_{k}^{\perp\dagger}\mathbf{P}_{k}^{\perp}\Big{)}^ {-1}\big{(}\mathbf{P}_{k}^{\perp}\big{)}^{\text{H}}\mathbf{P}_{k}^{\perp\dagger} \end{split}  \tag{20}\]  
where \(\mathbf{I}\) denotes the unit matrix and  
\[ \begin{split}\mathbf{P}_{k}^{\dagger}&=\!\mathbf{I} \!-\!\mathbf{P}_{k}^{\perp}\Big{(}(\mathbf{P}_{k}^{\perp})^{\text{H}}\mathbf{P} _{k}^{\perp}\Big{)}^{-1}\big{(}\mathbf{P}_{k}^{\perp}\big{)}^{\text{H}}\\ \mathbf{P}_{k}^{\perp\dagger}&=\!\mathbf{I}\!-\!\mathbf{P}_{k} \Big{(}(\mathbf{P}_{k})^{\text{H}}\mathbf{P}_{k}\Big{)}^{-1}(\mathbf{P}_{k})^{ \text{H}}\end{split} \]  
Based on Equation (20), it was determined that  
\[ \begin{split}\mathbf{Q}_{x}\mathbf{P}_{k}&=\mathbf{P} _{k},\mathbf{Q}_{x}\mathbf{P}_{k}^{\perp}=\mathbf{0}\\ \mathbf{Q}_{y}\mathbf{P}_{k}&=\mathbf{0},\mathbf{Q}_{y} \mathbf{P}_{k}^{\perp}=\mathbf{P}_{k}^{\perp}\end{split}  \tag{21}\]Based on Equations (20) and (21), it was determined that  
\[ \begin{split}\mathbf{Q}_{x}\bigg{[}\begin{array}{c}\mathbf{F}_{ \mathbf{R}}^{\mathrm{o}}(k,:)\\ \mathbf{F}_{\mathbf{R}}^{\mathrm{e}}(k,:)\end{array}\bigg{]}=\underbrace{ \left[\begin{array}{c}\cos\gamma_{k}\\ \sin\gamma_{k}e^{i\eta_{k}}\end{array}\right]}_{\mathbf{F}_{k}}\mathbf{x}_{k}+ \mathbf{Q}_{x}\mathbf{n}_{k}\\ \mathbf{Q}_{y}\bigg{[}\begin{array}{c}\mathbf{F}_{\mathbf{R}}^{\mathrm{o}}(k,: )\\ \mathbf{F}_{\mathbf{R}}^{\mathrm{e}}(k,:)\end{array}\bigg{]}=\underbrace{ \left[\begin{array}{c}\cos\gamma_{k+}\
icefrac{{1}}{{2}}\\ \sin\gamma_{k+}\
icefrac{{1}}{{2}}\end{array}\right]_{\mathbf{F}_{k}^{\perp}} \
icefrac{{1}}{{2}}}_{\mathbf{P}_{k}^{\perp}}\mathbf{y}_{k}+\mathbf{Q}_{y} \mathbf{n}_{k}\end{split}  \tag{22}\]  
According to Equation (22), the noise power after being processed by polarization filtering matrices was calculated, as follows:  
\[\tilde{\sigma}^{2}=trace\Big{(}\mathbf{Q}_{x}(\mathbf{Q}_{x})^{H}\sigma^{2} \Big{)}\!=\!\frac{\sigma^{2}}{\sin^{2}\!\xi}\geq\sigma^{2} \tag{23}\]  
where \(\xi\) is the principal angle between subspace \(\mathbf{P}_{k}\) and \(\mathbf{P}_{k}^{\perp}\) on Poincare sphere [26]. As \(\mathbf{P}_{k}\) and \(\mathbf{P}_{k}^{\perp}\) were both orthogonal to each other, \(\xi\)=\(\frac{\pi}{2}\) and \(\tilde{\sigma}^{2}=\sigma^{2}\), the noise power was not amplified. Based on Equation (22), two PSs were used to conduct the polarization matching to recover two signals, as follows:  
\[