# Secure Image Signal Transmission Scheme Using Poly-Polarization Filtering and Orthogonal Matrix
## 4 The Principle of the PPF-OM Scheme  
### Signal Processing on the Transmitter Side  
In the PPF-OM scheme, the information sequence was divided into two parts, \(\mathbb{I}_{p}\) and \(\mathbb{I}_{q}\), which were translated into AM symbols with the variable \(L\), as shown in Figure 2.  
We assumed the length of a signal block was \(u\), and the symbols were divided into \(\frac{L}{u}\) groups. Here, we used the first-part signal as an example and chose one group as an example (for the other groups, the analysis was the same). The symbol vector is written as  
\[\bf X=\left[ \begin{array}{cccc}x_{1}&x_{2}&\cdots&x_{u}\end{array} \right] \tag{9}\]  
Then, \(\bf X\) was reshaped into a matrix with the size \(\frac{l}{2}\times\frac{2u}{l}\), as follows:  
\[\bf\bar{X}=\left[ \begin{array}{c}\bf x_{1}\\ \bf x_{2}\\ \vdots\\ \bf x_{\frac{l}{2}}\end{array} \right]=\left[ \begin{array}{cccc}x_{1}&x_{2}& \cdots&x_{\frac{2u}{l}}\\ x_{\frac{2u}{l}+1}&x_{\frac{2u}{l}+2}&\cdots&x_{\frac{2u}{l}\times 2}\\ \vdots&\vdots&\ddots&\vdots\\ x_{\frac{2u}{l}\times\left(\frac{l}{2}-1\right)+1}&x_{\frac{2u}{l}\times\left( \frac{l}{2}-1\right)+2}&\cdots&x_{u}\end{array} \right] \tag{10}\]  
In the same way, the second-part signal \(\bf Y\) was reshaped into \(\bf\bar{Y}\) with the size \(\frac{l}{2}\times\frac{2u}{l}\), which had the same expression as Equation (10). For the first matrix, each row was multiplied with a PS from the PS matrix \(\mathbf{P}\), with the same index, to obtain the following:  
\[\mathbf{\tilde{X}}=\left[ \begin{array}{c}\mathbf{s}_{1}\\ \mathbf{s}_{2}\\ \vdots\\ \mathbf{s}_{\frac{1}{2}}\end{array} \right]=\left[ \begin{array}{c}\mathbf{P}_ {1}\mathbf{x}_{1}\\ \mathbf{P}_{2}\mathbf{x}_{2}\\ \vdots\\ \mathbf{P}_{\frac{1}{2}}\mathbf{x}_{\frac{1}{2}}\end{array} \right]=\left[ \begin{array}{c}\cos\gamma_{1}\mathbf{x}_{1}\\ \sin\gamma_{1}e^{i\eta_{1}}\mathbf{x}_{1}\\ \cos\gamma_{2}\mathbf{x}_{2}\\ \sin\gamma_{2}e^{i\eta_{2}}\mathbf{x}_{2}\\ \vdots\\ \cos\gamma_{\frac{1}{2}}\mathbf{x}_{\frac{1}{2}}\\ \sin\gamma_{\frac{1}{2}}e^{i\eta_{1}}\mathbf{x}_{\frac{1}{2}}\end{array} \right] \tag{11}\]  
For the second matrix, each row was multiplied with a PS from the PS matrix \(\mathbf{P}^{\perp}\) and with the same index, where the corresponding PSs of \(\mathbf{P}\) and \(\mathbf{P}^{\perp}\) were orthogonal to each other, e.g., \(\mathbf{P}_{1}^{i\mathbf{P}}\mathbf{P}_{1}^{\perp}=0\). Thus, we obtained \(\mathbf{\tilde{Y}}\). Afterwards, we assumed \(\mathbf{F}=\mathbf{\tilde{X}}+\mathbf{\tilde{Y}}\), and the odd and even rows of \(\mathbf{F}\) were denoted by \(\mathbf{F}^{\mathrm{o}}\) and \(\mathbf{F}^{\mathrm{e}}\), respectively, as follows:  
\[\mathbf{F}^{\mathrm{o}} =\left[ \begin{array}{c}\cos\gamma_{1}\mathbf{x}_{1}+\cos\gamma _{1+}\frac{1}{2}\mathbf{y}_{1}\\ \cos\gamma_{2}\mathbf{x}_{2}+\cos\gamma_{2+}\frac{1}{2}\mathbf{y}_{2}\\ \vdots\\ \cos\gamma_{\frac{1}{2}}\mathbf{x}_{\frac{1}{2}}+\cos\gamma_{1}\mathbf{y}_{ \frac{1}{2}}\\ \end{array} \right] \tag{12}\] \[\mathbf{F}^{\mathrm{e}} =\left[ \begin{array}{c}\sin\gamma_{1}e^{i\eta_{1}}\mathbf{x}_{1 }+\sin\gamma_{1+}\frac{1}{2}e^{i\eta_{1+1}/2}\mathbf{y}_{1}\\ \sin\gamma_{2}e^{i\eta_{2}}\mathbf{x}_{2}+\sin\gamma_{2+}\frac{1}{2}e^{i\eta_{ 2+1}/2}\mathbf{y}_{2}\\ \vdots\\ \sin\gamma_{\frac{1}{2}}e^{i\eta_{1}}\mathbf{x}_{\frac{1}{2}}+\sin\gamma_{1}e^ {i\eta_{1}}\mathbf{y}_{\frac{1}{2}}\end{array} \right] \tag{13}\]  
Then, a size \(M\times M\) (\(M>l\)) orthogonal matrix \(\mathbf{\Gamma}\) was created with column vectors that were orthogonal to each other. The first \(\frac{l}{2}\) columns of \(\mathbf{\Gamma}\) were denoted by \(\mathbf{\Gamma}^{\mathrm{o}}\), and the  
Figure 2: Diagram of the signal processing on the transmitter side. other \(\frac{l}{2}\) columns of \(\mathbf{\Gamma}\) were denoted by \(\mathbf{\Gamma}^{\mathrm{e}}\). Then, each column vector of \(\left[ \begin{array}{cc}\mathbf{\Gamma}^{\mathrm{o}}&\mathbf{\Gamma}^{ \mathrm{e}}\end{array} \right]\) was multiplied by each row of \(\left[