## 3 Proposed Solution
### Proposing the Fusion of MP-WFRFT and DM Algorithms Based on the VPAPM Scheme
 \tag{19}\]  
\[\alpha(dis_{0},t_{i})=\arccos\left(\frac{\tan dis_{0}}{\tan t_{i}}\right) \tag{20}\]  
\[dis_{0}=\frac{1}{2}\text{dis}_{\text{T}} \tag{21}\]  
\[ \begin{cases}dis_{11}=2\delta_{i},dis_{12}=\pi-2\delta_{i},&(M_{P}=4,J=2)\\ dis_{11}=\frac{\arccos[\cos(2\delta_{i})+\sin(2\delta_{i})\tan(2\delta_{i})]}{ \sqrt{1+\sec^{2}(A/2)\tan^{2}(2\delta_{i})}},dis_{12}=2\delta_{i},dis_{13}= \pi-2\delta_{i},&(M_{P}\geq 8,J=3)\end{cases}  \tag{22}\]  
where \(M_{P}=2^{R_{P}}\) is the polarization modulation order; \(dis_{0}\) represents half the spherical distance between adjacent constellation points, denoting the spherical distance between a polarization constellation point and its \(j\)-th decision region boundary point; \(J\) represents the number of decision region boundary points when employing the ML decision criterion; and \(A\) is the spherical angle:  
\[A=\angle P_{i}P_{H}P_{j}=2\arcsin\left[\frac{\sin(\text{dis}_{\text{T}}/2)}{ \sin(2\delta_{i})}\right] \tag{23}\]  
The polarization constellation point positions are dynamically selected by solving an optimization problem:  
\[\min_{\delta_{i},\phi_{i}}SER_{P}(\delta_{i},\phi_{i},\text{SNR}) \tag{24}\]  
The optimization objective is the minimization of \(SEP_{P}\), subject to the following constraints:  
1. The constellation points lie on the Poincare sphere, satisfying \(t_{i}=t(\delta_{i},\phi_{i})\) and \(\varphi_{i}=\varphi(\delta_{i},\phi_{i})\). 2. The constellation structure maintains central symmetry. It can be observed that adaptive optimization solely influences the integration boundaries and constellation point parameters pertinent to \(SEP_{P}\). The VPAPM scheme, by selecting appropriate transmit polarization states, aims to counteract channel polarization mode dispersion (PMD) or PDL. This ensures that the \(M_{P}\) polarization states arriving at the receiver preserve their ideal relative geometric configuration to the greatest extent possible, thereby minimizing the loss of signal power. 2. Analyzing the BER Performance of MP-WFRFT and DM fusion algorithm  
The receiver side is the inverse process of the transmitter side mentioned above. MP-WFRFT and DM are independent of each other because the former processes the baseband signals, while the latter deals with the RF signals. The BER of the MP-WFRFT and DM fusion algorithm consists of two main components: one is the inherent BER of the MP-WFRFT, and the other is the BER impact of the DM algorithm on the MP-WFRFT. Since the MP-WFRFT and DM algorithms are independent of each other, the total BER under Gaussian white noise channel conditions can be expressed by the following equation:  
\[BER=k\cdot BER_{1}\cdot BER_{2}+BER_{2} \tag{25}\]  
where \(BER_{1}\) represents the BER of the DM algorithm under Gaussian white noise channel conditions, while \(BER_{2}\) denotes the BER of the MP-WFRFT algorithm under the same channel conditions. The term \(k\) refers to the BER accumulation factor, which is typically an empirical parameter obtained by fitting to extensive simulation or experimental data. In practical signal processing or communication systems, an error in one module may disproportionately amplify or alter the impact of errors occurring in subsequent modules. The parameter \(k\) precisely attempts to capture this complex interaction effect. For example, if the output of the DM algorithm serves as a critical input or parameter for the MP-WFRFT algorithm, errors in DM may cause the nature of errors in MP-WFRFT, when they occur, to be more severe or their probability to be higher. In addition, \(k\) is a parameter that must be determined through empirical calibration. We calibrate \(k\) through a series of controlled-variable simulation experiments:  
1. Simulation 1 (Obtaining \(BER_{1}\)): Activate only the DM system (by setting the MP-WFRFT module to a passthrough mode). Simulate the BER for an eavesdropper at various offset angles (\(\theta_{e}\)), yielding \(BER_{1}(\theta_{e},\ SNR)\). 2.