## 3 Proposed Solution
### Proposing the Fusion of MP-WFRFT and DM Algorithms Based on the VPAPM Scheme  
In this subsection of the study, the DM technique with MP-WFRFT is explored in depth, and BER performance is evaluated for two modulation strategies respectively: PAPM and VPAPM schemes. 1. The Principles of PAPM and VPAPM Schemes  
The PAPM modulation scheme is presented in Figure 7 and is in a position to both multiply the information transmission rate and improve the anti-interception performance through the MP-WFRFT and DM algorithms. Furthermore, as can be seen in Figure 8, the VPAPM modulation scheme is capable of adjusting the polarization pattern of the modulated signal adaptively, employing polarization orthogonality to suppress interference. Although satellite channels are susceptible to polarization-dependent loss (PDL) effects, the adoption of pre-compensation techniques can effectively eliminate this impairment. Consequently, under such conditions, VPM is primarily subjected to additive white gaussian noise (AWGN), which endows VPM with considerable advantages in satellite communication systems. Given that the VPAPM scheme employs a hybrid approach combining QPSK modulation and VPM, the derivation of its overall BER necessitates separate consideration of the BER performance attributable to each of these constituent modulation techniques. The BER for VPAPM is given by the following [22]:  
\[SER=\frac{SER_{P}\times R_{P}+SER_{Q}\times R_{Q}}{R_{P}+R_{Q}} \tag{15}\]  
where \(SER_{P}\) is the symbol error rate (SER) attributable to VPM modulation, and \(SER_{Q}\) is the SER attributable to QPSK modulation. \(R_{P}\) and \(R_{Q}\) represent the number of symbols transmitted per unit time for PM and QPSK, respectively. First, calculate the theoretical SER for QPSK:  
\[SER_{Q}=2Q\Big{(}\sqrt{SNR}\Big{)}-\Big{[}Q\Big{(}\sqrt{SNR}\Big{)}\Big{]}^{2} \tag{16}\]  
where \(M_{Q}=2^{R_{Q}}\) is the modulation order, and SNR is the signal-to-noise ratio (SNR) at the receiver. \(Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-t^{2}/2}dt\) stands for the Gaussian Q-function. Under the influence of an AWGN channel, the polarization constellation points of the received signal will exhibit a displacement relative to their ideal transmitted positions. The probability density function (PDF) characterizing the locations of these noise-perturbed received constellation points can be expressed as follows [24]:  
\[f(t_{i},\varphi_{i})=\frac{1}{4\pi}\sin t_{i}\bigg{(}e^{-SNR\frac{1-\cos t_{i}}{ 2}}\bigg{)}\bigg{[}1+SNR\frac{1+\cos t_{i}}{2}\bigg{]} \tag{17}\]  
Consider that in the \(M_{P}\)-order three-dimensional constellation diagram, each constellation point exhibits central symmetry with respect to the horizontal polarization point \(P_{H}\), and assume that each symbol occurs with equal probability. Due to the adoption of the VPAPM scheme, it is necessary to introduce adaptive constellation point position parameters for polarization modulation. The SER formula is given by  
\[SER_{P}=\frac{1}{M_{P}}\underset{i=1}{\overset{M_{P}}{\sum}}SER_{P}^{i}(\delta _{i}(SNR),\phi_{i}(SNR)) \tag{18}\]  
where \(\delta_{i}\) and \(\phi_{i}\) are functions of SNR, dynamically selected by an optimization algorithm. The decision region is defined by the adaptive parameter \(\delta_{i}\):  
Figure 8: Block diagram of directional variable polarization modulation transmitter based on MP-WFRFT. Figure 7: Block diagram of directional polarization modulation transmitter based on MP-WFRFT. \[SEP_{P}^{i}= \begin{cases}\int_{\pi-di_{0}}^{\pi}\int_{0}^{2\pi}f(t_{i},\varphi_{i} )d\varphi_{i}dt_{i}+\int_{dis_{0}}^{\pi-di_{0}}\int_{0}^{\alpha(dis_{0},t_{i})}f (t_{i},\varphi_{i})d\varphi_{i}dt_{i},&(M_{P}=2),\\ \sum\limits_{i=1}^{I}2\left[\int_{dis_{1j}}^{\pi}\int_{0}^{\alpha(dis_{0},dis_{ 1j})}f(t_{i},\varphi_{i})d\varphi_{i}dt_{i}+\int_{dis_{0}}^{dis_{1j}}\int_{0}^{ \alpha(dis_{0},t_{i})}f(t_{i},\varphi_{i})d\varphi_{i}dt_{i}\right],&(M_{P}>2) \end{cases}