# 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  
For Bob, signal demodulation procedures are shown in Figure 3.  
The received signal vectors were written, as follows:  
\[ \begin{array}{l}\mathbf{\bar{T}}_{\mathrm{RH}}{=}h_{\mathrm{HH}}\mathbf{ \bar{T}}_{\mathrm{H}}+h_{\mathrm{VH}}\mathbf{\bar{T}}_{\mathrm{V}}+\mathbf{n}_ {\mathrm{H}}\\ \mathbf{\bar{T}}_{\mathrm{RV}}{=}h_{\mathrm{HV}}\mathbf{\bar{T}}_{\mathrm{H}} +h_{\mathrm{VV}}\mathbf{\bar{T}}_{\mathrm{V}}+\mathbf{n}_{\mathrm{V}}\end{array}  \tag{14}\]  
where \(\mathbf{n}_{\mathrm{H}}\) and \(\mathbf{n}_{\mathrm{V}}\) are noise vectors. To simplify the formula analysis, the noise were ignored temporarily. \(\mathbf{\bar{T}}_{\mathrm{RH}}\) and \(\mathbf{\bar{T}}_{\mathrm{RV}}\) should first be reshaped into the \(M\times\frac{2\mu}{l}\) matrix, which was written, as follows:  
\[ \begin{array}{l}\mathbf{T}_{\mathrm{RH}}{=}h_{\mathrm{HH}}\mathbf{T}_{ \mathrm{H}}+h_{\mathrm{VH}}\mathbf{T}_{\mathrm{V}}\\ \mathbf{T}_{\mathrm{RV}}{=}h_{\mathrm{HV}}\mathbf{T}_{\mathrm{H}}+h_{\mathrm{ VV}}\mathbf{T}_{\mathrm{V}}\end{array}  \tag{15}\]  
According to Equation (15), interferences \(h_{\mathrm{VH}}\mathbf{T}_{\mathrm{V}}\) and \(h_{\mathrm{VV}}\mathbf{T}_{\mathrm{V}}\) were found in the received signals. To remove the interferences, the orthogonal matrix \(\left[ \begin{array}{cc}\mathbf{\Gamma}^{\mathrm{o}}&\mathbf{\Gamma}^{ \mathrm{e}}\end{array} \right]\) was used to process the received signals:  
\[\mathbf{F}_{\mathrm{R}}^{\mathrm{o}}=\left[ \begin{array}{c}\mathbf{\Gamma} ^{\mathrm{o}}(:,1)^{\mathrm{H}}\mathbf{T}_{\mathrm{RH}}\\ \mathbf{\Gamma}^{\mathrm{o}}(:,2)^{\mathrm{H}}\mathbf{T}_{\mathrm{RH}}\\ \vdots\\ \mathbf{\Gamma}^{\mathrm{o}}(:,\frac{l}{2})^{\mathrm{H}}\mathbf{T}_{\mathrm{RH}} \end{array} \right],\mathbf{F}_{\mathrm{R}}^{\mathrm{e}}=\left[ \begin{array}{c }\mathbf{\Gamma}^{\mathrm{e}}(:,1)^{\mathrm{H}}\mathbf{T}_{\mathrm{RV}}\\ \mathbf{\Gamma}^{\mathrm{e}}(:,2)^{\mathrm{H}}\mathbf{T}_{\mathrm{RV}}\\ \vdots\\ \mathbf{\Gamma}^{\mathrm{e}}(:,\frac{l}{2})^{\mathrm{H}}\mathbf{T}_{\mathrm{RV}} \end{array} \right] \tag{16}\]  
Figure 3: Signal processing on the receiver side. where the superscript H denotes the conjugate inversion. Then, we used the \(k\)-th row of \(\mathbf{F}_{\text{R}}^{\text{o}}\) as an example, according to the orthogonality of the vectors of \(\mathbf{\Gamma}\), and we obtained the following:  
\[ \begin{split}&\mathbf{F}_{\text{R}}^{\text{o}}(k,\cdot)=\mathbf{ \Gamma}^{\text{o}}(\cdot,k)^{\text{H}}(h_{\text{HH}}\mathbf{T}_{\text{H}}+h_{ \text{VH}}\mathbf{T}_{\text{V}})\\ &=h_{\text{HH}}\mathbf{\Gamma}^{\text{o}}(\cdot,k)^{\text{H}} \sum\limits_{a=1}^{l}\mathbf{\Gamma}^{\text{o}}(\cdot,a)\bigg{(}\cos\gamma_{a} \mathbf{x}_{a}+\cos\gamma_{a+l/2}\mathbf{y}_{a}\bigg{)}+\\ & h_{\text{VH}}\mathbf{\Gamma}^{\text{o}}(\cdot,k)^{\text{H}} \sum\limits_{a=1}^{l}\mathbf{\Gamma}^{\text{e}}(\cdot,a)\left(\begin{array}{ c}\sin\gamma_{a}e^{j\eta_{a}}\mathbf{x}_{a}\\ +\sin\gamma_{a+l/2}e^{j\eta_{a+l/2}}\mathbf{y}_{a}\end{array}\right)\\ &=h_{\text{HH}}\mathbf{\Gamma}^{\text{o}}(\cdot,k)^{\text{H}}\mathbf{\Gamma}^{ \text{o}}(\cdot,k)\bigg{(}\cos\gamma_{k}\mathbf{x}_{k}+\cos\gamma_{k+l/2} \mathbf{y}_{k}\bigg{)}\\ &=h_{\text{HH}}\bigg{(}\cos\gamma_{k}\mathbf{x}_{k}+\cos\gamma_{k+l/2} \mathbf{y}_{k}\bigg{)}=h_{\text{HH}}\mathbf{F}^{\text{o}}(k,\cdot)\end{split}  \tag{17}\]  
According to the interoperability of the wireless channels, both [PERSON] and [PERSON] obtained the channel state information (CSI); thus, the impact of the CSI on \(\mathbf{F}_{\text{R}}^{\text{o}}(k,\cdot)\) was eliminated. In addition, as column vectors of \(\left[ \begin{array}{cc}\mathbf{\Gamma}^{\text{o}}&\mathbf{\Gamma}^{\text{e}} \end{array} \right]\) were not the same, according to Equation (17), each row vector could be separately recovered. Thus, we obtained the following:  
\[\mathbf{F}_{\text{R}}^{\text{o}}=\mathbf{F}^{\text{o}},\mathbf{F}_{\text{R}}^{ \text{e}}=\mathbf{F}^{\text{e}} \tag{18}\]  
In this manner, interferences for both polarization signals were eliminated.