## 3 Proposed Solution
### Analyzing the Impact of MP-WFRFT Parameter Settings on BER Performance  
#### 3.2.1 Analyzing the Impact of MP-WFRFT Scale Vector Parameter Settings on BER Performance  
The direction of the \(\cos\gamma\) signal is characterized by a transform order of \(\alpha\) and a scale vector of \(V1=\begin{bmatrix}MV1&NV1\end{bmatrix}\), where \(MV1=\begin{bmatrix}m_{0}&m_{1}&m_{2}&m_{3}\end{bmatrix}\), and \(NV1=\begin{bmatrix}n_{0}&n_{1}&n_{2}&n_{3}\end{bmatrix}\). In the direction of the \(\sin\gamma e^{i\varphi}\) signal, the transform order is \(\beta\) with a scale vector of \(V2=\begin{bmatrix}MV2&NV2\end{bmatrix}\), \(MV2=\begin{bmatrix}m_{00}&m_{11}&m_{22}&m_{33}\end{bmatrix}\), where \(NV2=\begin{bmatrix}n_{00}&n_{11}&n_{22}&n_{33}\end{bmatrix}\). This analysis seeks to investigate the scanning properties of a single parameter within the scale vector in the \(\cos\gamma\) signal direction, assuming that the parameter in the \(\sin\gamma e^{i\varphi}\) signal direction is kept constant.  
When \(k\) in Equation (7) is zero in the case of \(MV1=\begin{bmatrix}m_{0}&m_{1}&m_{2}&m_{3}\end{bmatrix}\), the 0-th term of \(\omega_{l}\) is as follows:  
\[\omega_{l}(\alpha,V,k=0)=\exp\biggl{\{}\pm\frac{2\pi j}{T}[(Tm_{0}+1)\alpha \cdot Tm_{0}]\biggr{\}} \tag{12}\]  
From the above equation, it can be seen that the BER performance of the parameter \(m_{0}\) on the signal is affected by \(n_{0}\). In particular, when \(n_{0}\) is equal to zero, the variation in \(m_{0}\) has no enhancement on the anti-interception performance of satellite communication.  
When \(k\) in Equation (7) is greater than zero in the case of \(MV1=\begin{bmatrix}m_{0}&m_{1}&m_{2}&m_{3}\end{bmatrix}\), the other term of \(\omega_{l}\) is as follows:  
\[\omega_{l}(\alpha,V,k>0)=\exp\biggl{\{}\pm\frac{2\pi j}{T}[k(\alpha Tm_{k}+ \alpha-l)+(Tm_{k}+1)\alpha Tm_{k}]\biggr{\}} \tag{13}\]  
Based on the equation, the periods associated with \(m_{1}\), \(m_{2}\), and \(m_{3}\) are arranged in descending order as follows: \(P_{m1}\)\(>\)\(P_{m2}\)\(>\)\(P_{m3}\).  
In the case of \(NV1=\begin{bmatrix}n_{0}&n_{1}&n_{2}&n_{3}\end{bmatrix}\), the term of \(\omega_{l}\) is as follows:  
\[\omega_{l}(\alpha,V,k)=\exp\biggl{\{}\pm\frac{2\pi j}{T}[(Tm_{k}+1)\alpha Tm_{ k}+(Tm_{k}+1)\alpha k]-lk\biggr{\}} \tag{14}\]  
By analyzing the equation, it is evident that \(n_{k}\) is not directly influenced by \(k\). Consequently, the periods of the four parameters associated with \(n_{k}\) are essentially equivalent, which suggests a consistent level of security against eavesdropping for \(NV1\).  
Figure 6: Constellation diagrams of VPM.  
2.2 Analyzing the Impact of MP-WFRFT \(\cos\gamma\) and \(\sin\gamma e^{i\varphi}\) Direction Parameter Settings on BER Performance  
When signals in directions \(\cos\gamma\) and \(\sin\gamma e^{i\varphi}\) pass through the Gaussian white noise channel, the inverse MP-WFRFT is initially applied. Referring to VPM demodulation in Equation (4), it becomes apparent that the \(\sin\gamma e^{i\varphi}\) signal direction component serves as the numerator in the equation for determining the polarization phase descriptor \((\gamma,\varphi)\), while the \(\cos\gamma\) direction signal component functions as the denominator. A minor fluctuation in the \(\cos\gamma\) direction parameter can significantly alter the demodulated \((\gamma,\varphi)\). But the maximum likelihood decision criterion, a standard in variable polarization demodulation, may not fully counteract this susceptibility to error. However, as detailed in previous subsection, the \(\sin\gamma e^{i\varphi}\) direction signal component is subject to a periodic normal error, which is mitigated in two stages: firstly, by correcting the error in the \(\cos\gamma\) direction signal component, and subsequently, by the maximum likelihood decision criterion.