## 2 System Model
### PAPM Modulation and Demodulation Model  
PAPM represents an advanced and efficient strategy for wireless transmission. As depicted in Figure 3, the overall input information bitstream is denoted as \(I\), and the encoded information bitstream is \(I_{x}\), which is first processed by a serial-to-parallel (S/P) converter.  
Figure 1: Poincare polarization sphere for Stokes parameter coordinate mapping.  
Figure 2: Constellation point weight mapping.  
The S/P converter splits the stream into two sub-streams: \(I_{p}\) and \(I_{q}\). The sub-stream \(I_{q}\) is used for PM to determine the polarization state, while the sub-stream \(I_{p}\) is used for amplitude-phase modulation (APM), typically quadrature phase shift keying (QPSK), to modulate the signal's amplitude and phase. Subsequently, the PM signal is split into two identical components. Each component is then multiplied by a distinct weighting factor, tailored to optimize the characteristics of the signal for transmission. This dual-stream approach allows for the simultaneous transmission of information through both the polarization and the conventional amplitude-phase dimensions of the electromagnetic wave. Following this, the signals undergo conversion from digital to analog format using a digital-to-analog converter (DAC). The final step involves up-conversion (UC), which shifts the signals to the desired frequency band to transmit respectively through horizontal polarization antenna (HPA) and vertical polarization antenna (VPA). The specific characteristics of the signals in the vertical and horizontal orientations are as follows:  
\[\left\{\begin{array}{l}E_{H}=\sum\limits_{k=1}^{K}\cos(\gamma_{k})A_{k}e^{i2 \pi f_{k}t+\phi_{0}(k)}\\ E_{V}=\sum\limits_{k=1}^{K}\sin(\gamma_{k})A_{k}e^{i2\pi f_{k}t+\phi_{0}(k)+ \varphi_{k}}\end{array}\right. \tag{3}\]  
where \(A_{k}\) denotes the amplitude value of the carrier \(f_{k}\) in the \(k\) symbol period, and \(\phi_{0}(k)\) is the initial phase value of the \(f_{k}\) carrier in the \(k\) symbol period.  
For PAPM demodulation at the receiver side, polarization matching method is used for demodulation. Figure 4 shows the block diagram of the demodulation scheme for PAPM. Firstly, the received signals are first down converted (DC), and then analog-to-digital conversion (ADC) is performed on the signals. Subsequently, the polarized phase descriptors of the two orthogonal signals are extracted, and polarization demodulation uses the maximum likelihood (ML) method. For APM demodulation, the polarization information is removed by using the polarization state matching method. The phase descriptor of the \(k\) symbol period is as follows:  
\[\left\{\begin{array}{l}\gamma^{\prime}_{kR}=\text{arctan}\Big{(}\frac{|y_{kR }|}{|y_{kR}|}\Big{)}\\ \varphi^{\prime}_{kR}=\Psi(y_{kV})-\Psi(y_{kH})\end{array}\right. \tag{4}\]  
Figure 3: Block diagram of PAPM transmitter.  
The polarization matching method to obtain APM modulation information is shown below:  
\[y^{APM}_{kR}=\begin{bmatrix}\cos\gamma^{\prime}_{kR}\\ \sin\gamma^{\prime}_{kR}e^{j\theta^{\prime}_{kR}}\end{bmatrix}^{H}y^{\prime}_{k }=\begin{bmatrix}\cos\gamma^{\prime}_{kR}\\ \sin\gamma^{\prime}_{kR}e^{j\theta^{\prime}_{kR}}\end{bmatrix}^{H}\cdot \begin{bmatrix}\cos\gamma_{k}\\ \sin\gamma_{k}e^{j\theta_{k}}\end{bmatrix}A_{k}e^{j(\omega_{k}t+\theta_{k})} \tag{5}\]  
where \(y^{\prime}_{k}\) is the received signal, and the first term on the right-hand side of the equation is the polarized phase descriptor of the PM demodulated by the maximum likelihood criterion.