# Secure Image Signal Transmission Scheme Using Poly-Polarization Filtering and Orthogonal Matrix
## 3 Signal Model and Problem Description  
The \(k\)-th transmit image signal can be represented, as in [4]:  
\[\mathbf{s}_{k}=\left[ \begin{array}{c}s_{\mathrm{Hk}}\\ s_{\mathrm{Vk}}\end{array} \right]\!\!=\!\underbrace{\left[ \begin{array}{c} \cos\gamma_{k}\\ \sin\gamma_{k}e^{i\eta_{k}}\end{array} \right]}_{\mathrm{PS}}\underbrace{A_{k }e^{i\eta_{k}}}_{\mathrm{AMsignal}} \tag{1}\]  
where \(\mathbf{P}_{k}\!\!=\!\!\left[ \begin{array}{c}\cos\gamma_{k}\\ \sin\gamma_{k}e^{i\eta_{k}}\end{array} \right]\) is the PS and \(\gamma_{k}\in\left[0,\frac{\pi}{2}\right]\) is the polarized angle. \(\eta_{k}\in\left[0,2\pi\right]\) denotes the auxiliary polarized angle. \(A_{k}e^{i\varphi_{k}}\) denotes the \(k\)-th amplitude-phase modulation (AM) symbol, where \(A_{k}\) and \(\varphi_{k}\) denote the amplitude and phase, respectively. For receivers, the \(k\)-th received image signal was denoted, as follows:  
\[\mathbf{y}_{k}=\left[ \begin{array}{c}y_{\mathrm{Hk}}\\ y_{\mathrm{Vk}}\end{array} \right]=\mathbf{s}_{k}+\mathbf{n}_{k} \tag{2}\]  
where \(\mathbf{n}_{k}=\left[ \begin{array}{c}n_{\mathrm{Hk}}\\ n_{\mathrm{Vk}}\end{array} \right]\) is the noise vector with the probability density function (PDF) as \(\mathcal{CN}\!\left(0,\sigma^{2}\mathbf{I}_{2}\right)\). If the receivers have the PSs, then the AM signal could be obtained through polarization matching, as follows:  
\[\mathbf{s}_{\mathrm{Rk}}=\mathbf{P}_{k}^{\mathrm{H}}\mathbf{y}_{k}=\mathbf{s }_{k}+\mathbf{P}_{k}^{\mathrm{H}}\mathbf{n}_{k} \tag{3}\]  
Next, the maximum likelihood estimate (MLE) was used to demodulate the AM image signal. If receivers did not know the PSs, the blind demodulation method was applied, as follows:  
1. Based on the \(k\)-th received image signal, the PS \(\mathrm{P}_{\mathrm{Rk}}\) was obtained, as follows: \[ \begin{array}{l}\gamma_{\mathrm{Rk}}\!\!=\arctan\!\left(\frac{\mathrm{abs} (y_{\mathrm{Vk}})}{\mathrm{abs}(y_{\mathrm{Hk}})}\right)\\ \eta_{\mathrm{Rk}}=\Xi\!\left(y_{\mathrm{Vk}}\right)-\Xi\!\left(y_{\mathrm{Hk }}\right)\end{array} \] (4)
2. With \(\mathrm{P}_{\mathrm{Rk}}\), the polarization matching was carried out according to the \(k\)-th received signal, such as in Equation (3), as follows: \[ \begin{array}{l}\mathrm{P}_{\mathrm{Rk}}{}^{\mathrm{H}}\mathbf{s}_{k}\\ =\mathrm{P}_{\mathrm{Rk}}{}^{\mathrm{H}}\!\left(\mathrm{P}_{k}A_{k}e^{i\varphi _{k}}+\mathbf{n}_{k}\right)\\ \approx A_{k}e^{i\varphi_{k}}+\mathrm{P}_{\mathrm{Rk}}{}^{\mathrm{H}}\mathbf{n} _{k}\end{array} \] (5)  
In this manner, the information carried by AM signals could also be recovered by MLE. Thus, if the signals were broadcast without protection, it was possible for [PERSON] to overhear on the information. However, we considered a \(2\times 2\) polarimetric multi-tap fading channel, where the transfer function for each channel estimation was written according to [21]:  
\[ \begin{array}{l}\mathbf{H}\!\!=\!\!\left[\begin{array}{cc}h_{\mathrm{HH}} &h_{\mathrm{VH}}\\ h_{\mathrm{HV}}&h_{\mathrm{VV}}\end{array}\right]\\ =\left[\begin{array}{cc}\sum\limits_{i=1}^{W_{\mathrm{Hk}}}h_{i}^{\mathrm{HH }}e^{2\pi f\tau_{i}^{\mathrm{Hk}}}&\sum\limits_{i=1}^{W_{\mathrm{V}}}h_{i}^{ \mathrm{VH}}e^{2\pi f\tau_{i}^{\mathrm{V}}}\\ \sum\limits_{i=1}^{W_{\mathrm{Hk}}}h_{i}^{\mathrm{HV}}e^{2\pi f\tau_{i}^{ \mathrm{Hk}}}&\sum\limits_{i=1}^{W_{\mathrm{V}}}h_{i}^{\mathrm{VV}}e^{2\pi f \tau_{i}^{\mathrm{V}}}\end{array}\right]\end{array}  \tag{6}\]where \(W_{\rm H}\) and \(W_{\rm V}\) are the number of the multipaths associated with horizontal (H) polarization and vertical (V) polarization, respectively; \(\tau_{i}^{\rm H}\) (\(\tau_{i}^{\rm V}\)) denotes the propagation delay associated with the \(i\)-th propagation path for the H (V)-polarized component; \(f\) denotes the frequency; and \(h_{i}^{\rm XY}\) denotes the \(i\)-th complex channel gain between the X-polarized transmission antenna and the Y-polarized receiver antenna. For other types of channels, the analysis method was the same.